ENERGY HARVESTING FROM LOCALIZED DYNAMIC
TRANSITIONS IN POST-BUCKLED ELASTIC BEAMS UNDER
QUASI-STATIC LOADING
By
Wassim Borchani

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Civil Engineering - Doctor of Philosophy
2015

ABSTRACT
ENERGY HARVESTING FROM LOCALIZED DYNAMIC TRANSITIONS IN POSTBUCKLED ELASTIC BEAMS UNDER QUASI-STATIC LOADING
By
Wassim Borchani
The deployability of structural health monitoring self-powered sensors relies on their
capability to harvest energy from signals being monitored. Many of the signals required to assess
the structure condition are quasi-static events which limits the levels of power that can be extracted.
Several vibration-based techniques have been proposed to increase the transferred level of power
and broaden the harvester operating bandwidth. However, these techniques require vibration input
excitations at frequencies higher than dominant structural response frequencies which makes them
inefficient and not suitable for ambient quasi-static excitations.
This research proposes a novel sensing and energy harvesting technique at low frequencies
using mechanical energy concentrators and triggers. These mechanisms consist of axially-loaded
bilaterally-constrained beams with attached piezoelectric energy harvesters. When the quasi-static
axial load reaches a certain mechanical threshold, a sudden snap-through mode-switching occurs.
These transitions excite the attached piezoelectric scavengers with high-rate input accelerations,
generating then electric power. The main objectives are to understand and model the post-buckling
behavior of bilaterally-constrained beams, control it by tailoring geometry and material properties
of the buckled elements or stacking them into system assemblies, and finally characterize the
energy harvesting and sensing capability of the system under quasi-static excitations.
The fundamental principle relies on the following concept. Under axial load, a straight
slender beam buckles in the first buckling mode. The increased transverse deformations from a
buckled shape lead to contact interaction with the lateral boundaries. The contact interaction

generates transverse forces that induce the development of higher order buckling configurations.
Transitions between the buckled configurations occur not only during loading, but also unloading.
In this work, the post-buckling response of the bilaterally constrained beam subjected to
axial loading is investigated experimentally, numerically, and theoretically. The capability of the
system to generate electric energy under quasi-static excitation is also assessed experimentally.
The post-buckling behavior is reproducible under cyclic loadings and independent of the input
loading frequency. The static and dynamic response of the beam is theoretically studied using an
energy method. The model adequately predicts the beam geometry at every loading stage,
including the flattening behavior just before the snap buckling transitions, the mode transition
events and the released kinetic energy as well as accelerations of the beam during transitions. The
buckling transitions generate high kinetic energy and acceleration spikes. However, the location
of the maximum acceleration differs from one transition to another.
Tuning the parameters of the system affects dramatically the accelerations generated during
snap-through transitions. However, it does not affect the number and spacing between these events.
To achieve better control of the system, multiple slender beams with different geometric and
material properties are stacked in parallel configurations. The system allows then to control the
spacing between energy bursts and reduce the energy leakage in electronic circuits.
As an application example, the mechanical energy concentrators and triggers were
integrated with a piezo-floating gate events sensor. This allowed for harvesting and recording of
bursts and impulses of released energy at very low frequencies. The system can be calibrated to
determine the number of times the magnitude of the input signal exceeded a mechanical threshold.
The mechanism allows for frequency up-conversion from the low input frequency (in the order of
mHz) to the natural frequency of the piezoelectric scavenger.

Copyright by
WASSIM BORCHANI
2015

To my parents, for their devotion and sacrifice,
To my fiancée, for her love and support,
To my siblings, for their help and encouragement,
To my friends, with whom I spent so many precious moments.
I dedicate my work,
Symbol of gratitude and love.

v

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to Dr. Nizar Lajnef and Dr. Rigoberto Burgueño
for their excellent guidance, valuable knowledge, and precious comments and advices that helped
me carry out my research.
I would like to extend my sincere appreciation to the rest of my committee members,
Professor Brian Feeny and Professor Venkatesh Kodur, for their availability, guidance and
valuable comments.
Special thanks should go to Dr. Shantanu Chakrabartty for his contribution on the sensor
work and Dr. Yi Sun for his help with the finite element modeling.
Finally, I would like to thank my family and friends for their support and encouragement.

vi

TABLE OF CONTENTS
LIST OF TABLES ....................................................................................................................... ix
LIST OF FIGURES ...................................................................................................................... x
CHAPTER 1: INTRODUCTION ................................................................................................ 1
1.1. Motivation and vision .................................................................................................... 1
1.2. Literature review ............................................................................................................ 5
1.2.1. Buckling and post-buckling of elastic structures ...................................................... 5
1.2.2. Limitation of existing energy harvesting techniques at very low frequency
excitations ................................................................................................................................ 8
1.3. Research hypothesis and objectives ............................................................................ 11
1.3.1. Hypothesis ............................................................................................................... 11
1.3.2. Objectives ................................................................................................................ 11
1.4. Outline ........................................................................................................................... 12
CHAPTER 2: BACKGROUND ................................................................................................ 14
2.1. Overview ....................................................................................................................... 14
2.2. Broadband energy harvesting using bistable buckled elements .............................. 14
2.3. Buckling analysis .......................................................................................................... 19
2.4. Post-buckling response of a bilaterally constrained beam ....................................... 21
2.4.1. Pre-contact .............................................................................................................. 22
2.4.2. Point contact ........................................................................................................... 23
2.4.3. Line contact ............................................................................................................. 24
2.4.4. Mode transition ....................................................................................................... 24
2.4.5. Multiple buckles ...................................................................................................... 26
2.5. Contact between an elastica and its boundary involving friction ............................ 28
2.6. Summary ....................................................................................................................... 30
CHAPTER 3: EXPERIMENTAL AND NUMERICAL STUDIES OF THE POSTBUCKLING RESPONSE OF BILATERALLY CONSTRAINED BEAMS UNDER AXIAL
LOADING ................................................................................................................................... 32
3.1. Overview ....................................................................................................................... 32
3.2. Post-buckling response of one bilaterally constrained beam ................................... 33
3.2.1. Experimental study.................................................................................................. 33
3.2.2. Numerical study ...................................................................................................... 40
3.3. Post-buckling response of multiple bilaterally constrained beams in parallel
configuration ............................................................................................................................ 56
3.4. Summary ....................................................................................................................... 64
CHAPTER 4: THEORETICAL STUDY OF THE POST-BUCKLING RESPONSE OF A
BILATERALLY CONSTRAINED BEAM UNDER AXIAL LOADING ............................. 66
4.1. Overview ....................................................................................................................... 66
4.2. Strain-displacement Relationship ............................................................................... 67
vii

4.3. Buckling analysis .......................................................................................................... 69
4.4. Derivation of kinetic, potential and dissipation energies .......................................... 71
4.4.1. Normalized kinetic energy ...................................................................................... 73
4.4.2. Normalized potential energy ................................................................................... 74
4.4.3. Normalized energy dissipated due to friction ......................................................... 77
4.5. Frictionless static model............................................................................................... 82
4.5.1. Force-control formulation ...................................................................................... 84
4.5.2. Displacement-control formulation .......................................................................... 87
4.6. Dynamic analysis .......................................................................................................... 88
4.7. Summary ..................................................................................................................... 100
CHAPTER 5: ENERGY HARVESTING AND SENSING BASED ON QUASI-STATIC
MECHANICAL ENERGY CONCENTRATORS AND TRIGGERS ................................. 102
5.1. Introduction ................................................................................................................ 102
5.2. Principle of operation................................................................................................. 104
5.3. Mechanical energy concentrators and triggers ....................................................... 106
5.4. Self-powered piezo-floating-gate sensing and data logging .................................... 111
5.5. Summary ..................................................................................................................... 119
CHAPTER 6: CONCLUSIONS .............................................................................................. 121
6.1. Research contributions .............................................................................................. 121
6.1.1. Post-buckling response of bilaterally constrained beams using an energy method 121
6.1.2. Dynamic response and transitions in multiple bifurcation points mechanical
energy concentrators ........................................................................................................... 121
6.1.3. Energy transfer at quasi-static excitations using mechanical energy
concentrators ....................................................................................................................... 121
6.1.4. Self-powered sensing at micro Hertz .................................................................... 122
6.2. Conducted work ......................................................................................................... 122
6.3. Future research .......................................................................................................... 125
6.3.1. Optimization of the piezoelectric energy scavenger ............................................. 125
6.3.2. Optimization of mechanism’s parameters for different strain ranges .................. 125
6.3.3. Optimization of the algorithm to include friction effects ...................................... 126
APPENDIX ................................................................................................................................ 127
REFERENCES .......................................................................................................................... 132

viii

LIST OF TABLES

Table 2-1: Effect of the device on the output power. ................................................................... 18
Table 3-1: Mode transitions mean points and standard deviations. .............................................. 39
Table 3-2: Comparison of simulation post-buckling results with experimental data. .................. 43
Table 3-3: Modified Parameter at each simulated case. ............................................................... 59
Table 4-1: Value and location of the maximum displacement at transitions. ............................... 97
Table 5-1: Properties of the piezoelectric vibrator. .................................................................... 107
Table A-1: Cases in parametric studies. Case 1 corresponds to the experimentally evaluated
system. Parameter values in bold text indicate changes from the values in Case 1. .................. 128

ix

LIST OF FIGURES

Figure 1-1: Schematic illustrating the integration and relevance of the presented concept for
enhanced energy harvesting capabilities from quasi-static deformations in civil infrastructure. ... 4
Figure 2-1: (a) Schematic view of proposed energy-harvesting device. (b) Photograph of
device and experimental setup (Jung and Yun, 2010). ................................................................. 15
Figure 2-2: Resonance test results of the fabricated device (Jung and Yun 2010). ...................... 16
Figure 2-3: Frequency response for constant input power (Jung and Yun 2010). ........................ 17
Figure 2-4: Power output generated by a single cantilever beam (Jung and Yun 2010). ............. 19
Figure 2-5: The normalized square root axial load, ζ, vs the normalized end shortening, η
(Chai 1998). .................................................................................................................................. 27
Figure 2-6: Equilibrium of a beam element subjected to lateral pressure with friction. .............. 28
Figure 3-1: Schematic and diagrams of the test setup. ................................................................. 34
Figure 3-2: Views of the loaded beam showing the obtained multiple stable post-buckled
configurations. .............................................................................................................................. 35
Figure 3-3: Force-displacement response for 10-cycle compression test. .................................... 36
Figure 3-4: Post-buckling behavior for displacement control cyclic loading tests at 0.16 Hz,
0.05 Hz, 0.025 Hz and 0.006 Hz. .................................................................................................. 37
Figure 3-5: Transition points distribution. .................................................................................... 38
Figure 3-6: Comparison between the post-buckling response of the system with rigid and
flexible walls. ................................................................................................................................ 40
Figure 3-7: Comparison of experimental and simulated force-displacement responses of the
bilaterally constrained beam. ........................................................................................................ 42
Figure 3-8: Buckled shapes from the finite element analysis of the experimentally evaluated
system. The color contours represent transverse deformations with the yellow (lightest color)
defining nodal lines. ...................................................................................................................... 44
Figure 3-9: Trace of axial force vs. transverse mid-height displacement during a loading
cycle indicating the post-buckling branch switching events......................................................... 45

x

Figure 3-10: Acceleration generated from the multiple mode-jumping response of the axially
loaded bilaterally constrained beam at mid-height of the beam during one loading cycle: (a)
experiment, (b) finite element analysis. The labels Φ3 to Φ7 indicate the mode transition
events. ........................................................................................................................................... 46
Figure 3-11: Effect of (a) Young’s modulus, (b) walls gap and (c) beam’s thickness on axial
forces and displacements at snap-through transitions. .................................................................. 48
Figure 3-12: Results from the numerical parametric study showing the effect of (a) beam
thickness, (b) elastic modulus, (c) wall gap, (d) beam length, and (e) boundary conditions on
the maximum acceleration generated by snap-through behavior. ................................................ 50
Figure 3-13: Effect of beam thickness and length on the maximum acceleration. ....................... 54
Figure 3-14: Effect of gap and length on the maximum acceleration........................................... 55
Figure 3-15: Effect of Young’s modulus and length on the maximum acceleration. ................... 55
Figure 3-16: Test setup with three parallel constrained beams. .................................................... 56
Figure 3-17: Force-displacement response of three similar parallel bilaterally constrained
beams. ........................................................................................................................................... 58
Figure 3-18: Force-displacement response of three parallel bilaterally constrained beams: two
of them are similar and the third is thinner. .................................................................................. 58
Figure 3-19: Loading branch of the force-displacement response and accelerations generated
during transitions for case 1 in Table 3-4 (same geometry and material properties for all beams).
....................................................................................................................................................... 60
Figure 3-20: Loading branch of the force-displacement response and accelerations generated
during transitions for case 2 in Table 3-4 (variation of Young’s moduli). ................................... 61
Figure 3-21: Loading branch of the force-displacement response and accelerations generated
during transitions for case 3 in Table 3-4 (variation of beams thicknesses). ................................ 62
Figure 3-22: Loading branch of the force-displacement response and accelerations generated
during transitions for case 4 in Table 3-4 (variation of gaps between the lateral walls). ............. 63
Figure 4-1: Geometry of the beam buckled in the first mode. ...................................................... 67
Figure 4-2: Deformation of beam element.................................................................................... 67
Figure 4-3: Variation of the axial force in the beam under different loading forces and
corresponding buckling shapes. .................................................................................................... 79

xi

Figure 4-4: Deflected shape for increasing axial load. ................................................................. 84
Figure 4-5: Theoretical and experimental Force-displacement response under force-control
loading........................................................................................................................................... 86
Figure 4-6: Comparison between the presented static model and Chai’s model (Chai 1998). ..... 86
Figure 4-7: Theoretical and experimental force-displacement response under displacementcontrol loading. ............................................................................................................................. 88
Figure 4-8: Variation of axial loading force, 𝑝̂ . ............................................................................ 89
Figure 4-9: Dynamic force-displacement response under force-controlled cyclic loading. ......... 90
Figure 4-10: (a) Displacement response, (b) kinetic energy and (c) accelerations during one
loading cycle. ................................................................................................................................ 91
Figure 4-11: Deflected shapes before and after each mode transition. ......................................... 93
Figure 4-12: Relative displacement, 𝑊𝑅 = 𝑊𝑓 − 𝑊𝑖, at mode transitions where 𝑊𝑖 and 𝑊𝑓
represent, respectively, the deflected shapes before and after transitions. ................................... 95
Figure 4-13: Non-dimensional accelerations at the locations (a) X = 0.21, (b) X = 0.37, (c)
X = 0.5 and (d) X = 0.69 during a loading cycle of 10 s. ............................................................. 98
Figure 5-1: (a) System architecture of a generic piezoelectricity driven self-powered sensor; and
(b) adverse effect of electronic leakage on energy scavenging for quasi-static excitation. ........ 103
Figure 5-2: (a) System architecture of the proposed self-powered sensor comprising of
mechanical energy concentrators; and (b) reduction of electronic leakage during the impulsive
energy release of the energy concentrators. ................................................................................ 103
Figure 5-3: Principle of operation of the proposed self-powered sensor: (a)-(c) different buckling
states of a bilaterally constrained elastic beam; (d) energy diagram illustrating how the released
potential energy induces hot-electrons in a floating-gate transistor channel. ............................. 105
Figure 5-4: Piezoelectric output voltage (output frequency = 6.6667Hz) generated for a full load
cycle at 0.05 Hz........................................................................................................................... 108
Figure 5-5: Harvested energy for different piezoelectric elements and input loading frequencies.
..................................................................................................................................................... 109
Figure 5-6: Frequency up-conversion principle for alternating loading between the third and the
seventh modes at 0.11 Hz. The fifth mode was used as the initial equilibrium position............ 110

xii

Figure 5-7: Schematic of a linear p-IHEI based data logging circuit: (a) when operating in the
data logging mode; and (b) when operating in a data read-out mode (Lajnef et al. 2014b). ...... 113
Figure 5-8: Measured output range and the linearity for linear injection circuit (Sarkar et al.
2013). .......................................................................................................................................... 114
Figure 5-9: Micrograph of a fabricated prototype integrating an array of linear floating-gate
injectors along with digital command-control and programming circuits (Lajnef et al., 2014b).
..................................................................................................................................................... 114
Figure 5-10: Measured output piezoelectric voltage generated for loading cycles at 0.1 Hz,
showing the system’s ability for energy generation under quasi-static loading. ........................ 116
Figure 5-11: Recorded variations of the voltage at the analog floating gate for input cyclic
loading events at 0.1Hz. .............................................................................................................. 116
Figure 5-12: Recorded variations of the voltage at the analog floating gate for input cyclic
loading events at 0.5 Hz. ............................................................................................................. 117
Figure 5-13: Illustration of the energy conversion principle under cyclic loading at 0.1 Hz for the
bilaterally constrained beam configuration. ................................................................................ 118
Figure 5-14: Recorded variations of the voltage at the analog floating gate for input cyclic
loading events at 0.1 Hz. ............................................................................................................. 119

xiii

CHAPTER 1: INTRODUCTION

The presented work in this thesis builds on the novel idea of exploiting snap-through
transitions in post-buckled elastic elements to harvest energy at very low frequencies. The original
concept is a result of collaboration between Dr. Nizar Lajnef and Dr. Rigoberto Burgueño. The
developed theoretical model, detailed in Chapter 4, constitute the major contribution of this thesis.
The sensor related work presented in Chapter 5 was developed in collaboration with Dr.
Chakrabartty.

1.1.

Motivation and vision
Development of smart structures capable of monitoring critical events and alerting users

has been a quest for decades. Different sensors have been developed to monitor the events that a
structure can be subjected to, such as vibrations, displacements, pressure, temperature, etc.
Monitoring systems, composed of a network of wireless sensors capable to communicate between
themselves and with a central computer, can detect the change in the structure response, due to
crack initiation or failure of one or multiple elements, and alert engineers for maintenance.
However wireless monitoring systems require a continuous source of power that a battery cannot
provide due to its limited lifetime. Therefore, self-powered sensors capable of harvesting energy
from the signal being sensed are needed.
Although many potential sources of energy are available such as solar, thermal gradient,
radio frequency, vibrations and strain energies, piezoelectric harvesters are the most promising for
deployment in structures, given the size limitations and the possibility of being embedded within
the construction material. One of the major disadvantages of the electro-mechanical energy
1

scavengers available for civil, mechanical and biomechanical applications is their narrow-band
frequency response, which is concentrated around high frequencies. For instance a vibration-based
scavengers with an overall volume limited to less than 5 cm3 will exhibit a resonant frequency in
the range 50-300 Hz (Najafi et al. 2011), while most civil structures have a fundamental vibration
mode at frequencies less than 5 Hz. Likewise, changes in physical processes like temperature and
pressure variations between day and night induce a stress/strain response in structures that occur
at frequencies lower than 1 mHz. In biomedical engineering, changes in in-vivo strain levels during
bone-healing and spinal fusion processes span from a few days to up to a few months. In all these
processes monitoring the evolution of quasi-static strain is important and could provide significant
benefits. For instance, the history of mechanical loading inside different structures could be used
to predict the life expectancy of the structure. A more challenging prospect is to design batteryless sensors that can self-power by harvesting energy directly from these quasi-static processes.
The mismatch between the input excitation frequency and the natural frequency of the
energy harvester significantly limits the levels of extractable power. Several techniques have been
proposed to increase the transferred power level, including improvements through the optimization
of the piezoelectric material, altering of the electrode patterns and system configuration, use of
matching networks, and tuning of the device’s resonant frequency. However, energy harvesting
from loads, deformations, or motions within the quasi-static frequency range is beyond the reach
of current vibration-based energy harvesting methods. The developed energy harvesters to date are
still inefficient and not suitable for low frequency vibration sources such as human walking motion
and deformations in large civil structures (Green et al. 2013).
This research presents a new technique for sensing and energy harvesting at very low
frequencies (less than 1 Hz) using mechanical energy concentrators and triggers. These systems
2

exploit the sudden snap-through behavior between the multiple buckling modes of an axiallyloaded bilaterally-constrained beam. As the axial loading increases, the strain energy that is stored
in the buckled element is released as kinetic energy during mode transitions. The approach relies
on the high-rate motions that are generated during the transitions between the system’s multiple
potential energy wells. The input is not a vibration motion but a slow quasi-static deformation.
The vision for the proposed energy harvesting concept is illustrated in Figure 1-1. The presented
concept is shown packaged into a device for embedment or attachment to structural components
in a form-factor similar to vibrating wire strain gages. Under quasi-static structural deformations
(e.g., due to service deformations, temperature fluctuations, or slow degradation in material
properties), the device capsule is compressed (or tensioned), inducing variations in axial
deformations to the enclosed beam element. The axial deformations imposed on the internal
element leads to multiple bifurcation points in its post-buckling response due to the presence of
bilateral constraining walls. Energy harvesting oscillators, attached to the axially compressed
element, are excited by the sudden snap-through transitions in the beam. This conversion, from
low-rate axial deformations to dynamic high-rate lateral accelerations allows the harvesting of
energy from a global quasi-static input.
A piezoelectricity driven self-powered sensor contains a piezoelectric transducer that
powers a minimal set of electronic modules by harvesting energy from ambient strain variations.
The sensor electronics comprises of a rectification module to extract energy from the transducer,
a triggering module that detects events of interest and a data-logging module that records events
either on a non-volatile memory or using remote data transmission. When the piezoelectric
transducer is excited quasi-statically the load voltage generated by the transducer is approximately
constant. Unfortunately, the majority of the charge generated by the transducer is lost as leakage

3

through the electronics (for example through diode leakage) and the residual energy is insufficient
to drive the rest of the sensor modules (triggering and data-logging modules).

Integration in Structural
Elements and Systems

Packaging for Embedding
or Attachment in a Structure

Multi-stable
Buckling
Element
Energy Harvesting
Oscillator

Energy Harvesting
Device

Figure 1-1: Schematic illustrating the integration and relevance of the presented concept for
enhanced energy harvesting capabilities from quasi-static deformations in civil infrastructure.
This work proposes the use of mechanical energy concentrators and triggers as a front-end
to circumvent the challenge of quasi-static self-powering. The role of the mechanical energy
concentrator is to harvest potential energy over a long time-span and release the stored energy as
bursts and impulses. The impulsive nature of the energy-release not only results in piezoelectric
output voltage levels that are significantly higher than the triggering but also reduces the effect of
energy leakage. Management of the extracted energy for use or storage is an important component
for implementation of the concept.

4

1.2.

Literature review
Buckling and post-buckling theories of elastic structures have been studied extensively for

over 80 years (Euler, 1744; Hutchinson and Koiter, 1970 and Burgess, 1971). Most efforts have
been focused on preventing global and local instability. However, recent attention has been paid
to the potential of using post-buckling behavior for deployable and active structures. Buckled
structures have been used to create bistable and multistable mechanisms that have proven to be
efficient in many applications such as MEMS and energy harvesting devices (Fang and Wickert,
1994; Qiu et al., 2004; Jung and Yun, 2010; Masana and Daqaq, 2011; Lajnef et al., 2014b).
1.2.1. Buckling and post-buckling of elastic structures
Buckling and post-buckling of an unrestrained beam under compressive axial stress was
studied by Fang and Wickert (1994). Accounting for beam imperfections and nonlinear
dependence of the transverse deflection and the compressive load, it was shown that the beam
deflects only in the first mode and exhibits a continuous increase in the transverse deflection
amplitude during the post-buckling response. Therefore the beam does not exhibit snap-through
behavior during its post-buckling responses.
Bistability of a bow-shaped curved beam subjected to transverse force acting at its midspan
without axial residual stress was theoretically investigated by Qiu et al. (2004) using energy
methods. It was shown that the beam snaps under the applied actuation but does not stay in the
snapped shape if the force is released. Park and Hah (2008) analyzed pre-shaped buckled-beam
actuators subjected to a transverse distributed force taking into consideration the existence of an
axial residual stress. It was shown that the stability of the second equilibrium position depends on
the ratio between the beam’s thickness and initial transverse curvature at midspan as well as

5

residual stresses in the beam. Results also indicate that transitions between the two equilibrium
configurations do not follow the same equilibrium path.
Post-buckling behavior of buckled beams subjected to both axial and concentrated
transverse loading has also been theoretically studied for central and non-central actuations
(Vangbo, 1998; Cazottes et al., 2009). Equilibrium configurations were determined by
minimization of the system’s energy taking into consideration bending and compression. It was
shown that the transverse force that should be applied to snap the beam into the second stable state
depends on the pre-compression force. In addition, the position of the actuation was shown to
significantly affect the system’s behavior.
In an effort to generate higher buckling modes beyond the bistable configurations, several
studies have been conducted on the post-buckling response of unilaterally constrained beams
subjected to axial compressive forces. Stein and Wriggers (1984) used Timoshenko beam theory
with large deformations and rotations to numerically investigate the problem. It was shown that
the elastica buckles until it touches the lateral constraint. The system then regains stiffness and is
able to carry higher loading. Chateau and Nguyen (1991) demonstrated that a contact zone
develops under increased loading leading to localized buckling in that area and thus an upper
buckling mode. Increasing the input loading will result in multiple contact regions with distributed
reaction forces (Essenburg, 1975). Soong and Choi (1986) studied the contact problem between a
beam and its boundary. The contact involves friction. The normal pressures as well as tangential
forces at the contact zone were determined. It was shown that the axial compression force inside
the beam is not constant due to friction. Line contacts were simplified into multiple discrete points
under the condition that none of the contact points has a curvature that cuts into the local boundary.
Naghdi and Rubin (1989) showed that accounting for normal extensional deformation describes
6

better the contact problem and correctly predicts the conditions under which the beam loses
contacts.
Methods based on geometric assumptions have been the most common way to model the
observed behavior for this class of problems. However, this approach significantly overestimates
the required force-displacement conditions for the higher modes transitions. Numerical
simulations are typically used to overcome the mismatch with measured experimental results. In
the work developed by two research groups (Domokos et al., 1997; Holmes et al., 1999; Chai,
1998), the post-buckling behavior of an inextensible beam confined between two frictionless rigid
side-walls and subjected to axial displacement was investigated using analytical-computational
methods. Large deformations and geometric elliptic integrations were used. It was shown that a
combination of line and point contacts can exist and that the deflected shape can be asymmetric.
Therefore friction has to be considered in the study to more accurately describe equilibrium states
and critical loads and displacements (Domokos et al., 1997; Holmes et al., 1999). Transitions to a
higher buckling mode were assumed to be always bounded between a lower limit, corresponding
to a single flattened contact zone, and an upper limit, corresponding to multiple flattened contact
segments of equal lengths (Chai, 1998). The lateral reactions exerted by the constraints were
modeled as a set of discrete point forces acting at the ends of the free standing segments of the
beam. The case of a bilaterally constrained thin plate subjected to height reduction was also
investigated by Roman and Pocheau (2002) through a combined experimental, theoretical, and
numerical study. Experimental measurements of the contact zone length were fed back into the
model to calibrate for the plate reactions. It was shown that even though friction is an important
parameter, the assumed frictionless symmetric Euler’s model combined with the added corrections
can represent the considered behavior. Based on the collected literature for this class of problems,

7

it was determined that a theoretical model that is based on the physics of the device is relevant,
thus the interest in this work for energy methods.
1.2.2. Limitation of existing energy harvesting techniques at very low frequency
excitations
Energy harvesting from loads, deformations, or motions within the quasi-static frequency
range is beyond the reach of current vibration-based energy harvesting methods. Although many
research works have focused on vibration energy harvesting (Elfrink et al.. 2008; Elvin et al., 2006;
Finkel et al., 2009; Park et al., 2008; Rahimi et al., 2003; Sodano et al., 2004), the developed
energy harvesters to date are still inefficient and not suitable for low frequency vibration sources
such as human walking motion and deformations in large civil structures (Green et al., 2013). The
major limitation in vibration energy harvesting revolves around the fact that the energy harvester
should be excited at, or near, its resonance frequency in order to generate the most energy. Thus
the vibrator should be tuned to match the most distinct frequency of the vibration source. In
general, the excitation frequency varies over a large spectrum, which considerably reduces the
efficiency of the harvester. Different methods to broaden the generator’s operating bandwidth have
been explored. Twiefel and Westermann (2013) present a survey on broadband techniques for
vibration energy harvesting and classify the techniques as advanced electronic networks, linear
generators and nonlinear generators. The latter two methods deal with mechanical systems and are
the closest related to the work presented here. Thus, their features, advantages and disadvantages
are summarized next.
Linear generators, characterized by the harvester’s specific natural frequencies, are well
suited for narrowband excitations. However, they are less efficient if the excitation frequency is
distributed over a wide spectrum or if the frequency is very low. One way to increase their
8

operation bandwidth is by using an array of piezoelectric generators (e.g., cantilevered beams)
with different tuned resonant frequencies (Liu et al., 2008). Another approach is to couple bimorph
beams with elastically connected masses and tune their response such that they have resonant
frequencies close to each other, and a wider band can be achieved by connecting more beams
(Erturk et al., 2009; Yang and Yang, 2009). These methods can be effective and simple. Yet the
drawbacks are that the setups can become voluminous for a wide frequency band and that mode
coupling can reduce the overall response.
A number of research studies have been focused on exploiting response nonlinearities of
vibration harvesters to improve energy generation and extend bandwidth. Proposed techniques
include vibroimpact, monostable (Duffing) and bistable/multistable systems. Vibroimpact systems
use a mechanical stopper to limit the amplitude of the harvester’s vibrations such that if the
excitation frequency increases the system response remains on a plateau defined by the stopper’s
positions (Soliman et al, 2008; Moss et al, 2010; Blystad and Halvorsen, 2011). The technique
provides a wider bandwidth around the harvester’s resonant frequencies but it is not practical since
a slowly increasing frequency sweep is required to reach a wide plateau.
Nonlinear monostable harvesters exhibit a wider bandwidth and higher efficiency in a nonstationary vibratory environment than linear generators. The harvester’s behavior is governed by
Duffing equations where cubic stiffness nonlinearities can be incorporated into the harvester’s
design using nonlinear magnetic levitation (Mann and Sims, 2009) or by axially loading the
harvester in the pre-buckling state (Daqdaq, 2010). The potential energy function of these
harvesters exhibits monostable characteristics that can broaden the frequency bandwidth.
However, the nonlinearity in monostable harvesters can reduce the mean output power under

9

colored random excitations (Daqdaq, 2010; Barton et al., 2010), and it has been shown that it has
no effect on the output under white noise excitation (Langley, 2014).
Another class of nonlinear harvesters is bistable and multistable generators, which are
characterized by two or more potential energy wells defining stable equilibrium positions.
Depending on the excitation level and frequency these devices can exhibit either a periodic or a
chaotic inter-well vibration. This behavior increases the range of frequency at which energy can
be harvested (Tang et al., 2010; Zhu et al., 2010). Bistability can be introduced into energy
harvesters using different techniques such as magnetic repulsion (Tang et al., 2010; Stanton et al.,
2009), magnetic attraction (Erturk et al, 2009; Galchev et al., 2011) and mechanical buckling (Jung
and Yun, 2010; Sneller et al., 2011). Multistable systems using multiple attracting positions to the
oscillating magnet have also been investigated and shown to improve efficiency for low frequency
input (Wickenheiser and Garcia, 2010). Bistable/multistable generators exhibit good performance
in terms of power conversion and frequency broadband; but for very low-frequency applications
they should be designed accurately in order to be efficient.
Green et al. (2013) numerically investigated the effectiveness of current nonlinear energy
harvesting solutions for real ambient low frequency vibration sources. It was concluded that the
efficiency of nonlinear harvesters is sensitive to the nature and type of the excitation and that the
development of new devices capable of harvesting energy at very low frequency is needed. The
dominant frequencies of human motions and large civil structure vibrations are less than 10 Hz
(Najafi et al., 2011). Furthermore, under service loading, civil structures typically exhibit small
amplitude low-rate deformations and significantly small vibration motions. Hence, the vibration
energy harvesting devices mentioned above are inefficient and not suitable for this kind of
application.
10

1.3.

Research hypothesis and objectives
1.3.1. Hypothesis
The main hypothesis behind this research is that energy can be harvested under quasi-static

excitations using sudden transitions between equilibrium positions of bilaterally constrained
beams. Instabilities of buckled elements can be exploited to transform the low-amplitude and lowrate service and ambient deformations into an amplified input to the piezoelectric transducer. Postbuckling response of elastic elements can be modeled and controlled for enhanced power
management.
1.3.2. Objectives
The main objective of this research is to develop a sensing and energy harvesting technique
that is efficient at very low frequency and quasi-static excitations. The technique uses the snapthrough behavior between the multiple buckling modes of a bilaterally constrained axially loaded
beam to convert the low-amplitude and low-rate excitations into an amplified input to the
piezoelectric scavenger. This work aims to model and control the post-buckling response of
bilaterally constrained beams to maximize the levels of the extractable energy by optimizing
system’s parameters. To provide better understanding of the behavior and control it, theoretical
modeling, finite element simulations and experiments were carried out.
A theoretical model of the post-buckling behavior of bilaterally constrained beams was
developed based on an energy approach. The method minimizes the energy stored in the system
to determine the deflected shapes for different load values as well as the transition states. The
model detects the loads at which mode transitions occur. It also computes velocities, accelerations
and energies during transitions.

11

Finite element simulations were performed in order to study the effect of the system’s
parameters on the post-buckling response of the bilaterally constrained beam. The parameters’
effect on the maximum accelerations during transitions was investigated to maximize the levels of
the extractable energy. The finite element method was also used to investigate the effect of
combining multiple slender elastic beams, in parallel configurations, on the number and magnitude
of mode branch switching events during the post-buckling response of the system.
To validate the theoretical and numerical models, an experimental apparatus was built and
the post-buckling response of the beam was experimentally investigated. Piezoelectric scavengers
with different natural frequencies were used to convert the high-rate motion of the beam, at
transitions, into electric power. The effect of the excitation frequency as well as the natural
frequency of the piezoelectric transducer on the levels of the extractable energy was
experimentally investigated. The sensing ability of the developed system to low-frequency and
quasi-static events was experimentally investigated by combining the mechanical energy
concentrators and triggers with piezo-floating-gate sensors.

1.4.

Outline
This dissertation is organized as follows: Chapter 2 presents a background review on

energy harvesting techniques using bistable mechanisms, buckling eigenvalue analysis and the
study of post-buckling behavior of bilaterally constrained beams. Chapter 3 provides experimental
and numerical investigations of the post-buckling response of one and multiple parallel bilaterally
constrained beams. This chapter assesses the repeatability and the reproducibility of the behavior
and its dependency to the input frequency and the parameters of the system such as thickness,
length and Young’s modulus of the beam and the gap between the lateral rigid walls. Chapter 4

12

presents a theoretical study of the post-buckling behavior of the system under consideration. The
study is based on an energy method. Static and dynamic models are developed to determine the
response of the beam as well as energy balance and conversion during buckling mode transitions.
Chapter 5 provides a novel technique for energy harvesting and self-powered sensing at very low
frequency environments. The technique consists in using mechanical energy concentrators
(composed of axially-loaded bilaterally-constrained beams) to convert a quasi-static strain input
into a high-rate acceleration that transfers the input into a piezoelectric scavenger. The technique
combines the physics of mechanical buckling in bilaterally constrained elastic beams with the
physics of piezoelectricity driven hot-electron injection, thus creating a novel sensing mechanism
able to self-power at milli- and micro-Hertz frequencies.

13

CHAPTER 2: BACKGROUND

2.1.

Overview
This chapter presents some experimental and theoretical background related to buckling

and post-buckling of elastic structures and energy harvesting using bistability of buckled elements.
The theoretical background presented in this chapter was used in the development of the theoretical
model established in the current work. The chapter is divided into 4 separate sections. Section 2.2
presents a broadband energy harvesting technique that exploits bistability of buckled elements to
up-convert low input frequencies into the energy harvester’s resonant frequency. Section 2.3
presents a buckling eigenvalue analysis developed by Nayfeh and Emam (2004) to determine the
buckling critical loads and modes. Section 2.4 summarizes a theoretical model on the postbuckling behavior of a bilaterally constrained beam under axial loads developed by Chai (1998).
The results of this model will be compared to our theoretical model’s in the following chapters.
Section 2.5 presents a study of the contact between an elastica and its boundary involving friction.
The formulation was used to investigate the effect of friction on the post-buckling behavior of the
bilaterally constrained beam under study.

2.2.

Broadband energy harvesting using bistable buckled elements
Jung and Yun (2010) presented an energy-harvesting device that employs both mechanical

frequency up-conversion and bistable behavior of buckled bridges for wide-bandwidth operations
at an ambient vibration frequency. The proposed device enables energy harvesting at slow and
time-varying mechanical vibrations. The schematic of the proposed energy harvester is shown in

14

Figure 2-1 (a), and a photograph of the experimental setup of the fabricated device is shown in
Figure 2-1 (b). The device consists of two pre-buckled slender bridges and cantilever beams
attached to the bridges. A proof mass is mounted on top of the bridge at the center to provide
sufficient inertial force at a given acceleration. A piezoelectric polymer polyvinylidene fluoride
(PVDF) layer is attached on top of each cantilever beam to convert the induced strain into electrical
signal. A small mass is mounted at the end of each beam in order to maximize the swing.
When the threshold acceleration is exceeded, the buckled bridge snaps through to the other
equilibrium state. The transition between the two equilibrium positions generates an impulse-like
excitation of the cantilever beams that vibrate freely at their resonant frequency independently of
the input frequency.

Figure 2-1: (a) Schematic view of proposed energy-harvesting device. (b) Photograph of device
and experimental setup (Jung and Yun, 2010).

15

Figure 2-1 (cont’d)

Figure 2-2: Resonance test results of the fabricated device (Jung and Yun 2010).

16

Figure 2-2 displays the frequency up-conversion characteristic of the device. The red line
indicates the applied acceleration and the blue line represents the generated output voltage. Low
frequency input vibration is converted into free and high-frequency output vibration. The input
frequency is 12 Hz and the output frequency is 143 Hz. Transitions between the equilibrium
positions occur at the positive and negative peaks of the input accelerations.
The proposed device exhibits a wide operational-frequency range. The snap-through
motion of the buckled bridge structure occurs at various vibration input frequencies. However, the
input acceleration should be high enough to obtain a snap-through motion. Figure 2-3 displays the
frequency response determined experimentally by subjecting the device to harmonic excitations
with different frequencies. The (acceleration2/frequency) ratio was kept constant for all
frequencies in order to apply same input power for all experiments. The frequency for maximum
power output was approximately 30 Hz, which was much lower than the resonant frequency of the
cantilever.

Figure 2-3: Frequency response for constant input power (Jung and Yun 2010).

17

Table 2-1 summarizes the experimental results for the device and a simple cantilever beam
without the buckled bridge structure. The maximum output power generated by a single cantilever
was 131 µW for the device at 30 Hz excitation. The average power was measured as 2.9 µW,
which is significantly larger than the output power of 0.57 µW generated by a simple cantilever
beam without buckled bridges. Output power amplification is due to the increase in the deflection
magnitude of the cantilever beams caused by the snap-through accelerations of the bridge in
addition to the base acceleration. The power generated by one of the device’s cantilever beams is
shown in Figure 2-4.
Table 2-1: Effect of the device on the output power.
Cantilever with

Simple cantilever without

buckled bridge

buckled bridge

Cantilever size (mm3)

8.0 × 4.0 × 0.2

8.0 × 4.0 × 0.2

Resonant frequency (Hz)

143

124

Maximum voltage (V)

27.1 (at 30 Hz)

4.13 (at 124 Hz)

Maximum power (µW)

131 (at 30 Hz)

2.37 (at 124 Hz)

Average power (µW)

2.9 (at 30 Hz)

0.57 (at 124 Hz)

Parameter

18

Figure 2-4: Power output generated by a single cantilever beam (Jung and Yun 2010).

2.3.

Buckling analysis
The buckling problem of a fixed-fixed beam subjected to axial load was studied by Nayfeh

and Emam (2004). The governing equation of a fixed-fixed straight beam subjected to axial load
P accounting for midplane stretching is given by
1
1
2
𝜓 𝑖𝑣 + 𝑃𝜓′′ − 𝜓′′ ∫ 𝜓 ′ 𝑑𝑥 = 0
2
0

(2-1)

𝜓 = 0 and 𝜓 ′ = 0 at 𝑥 = 0 and 𝑥 = 1

(2-2)

where 𝜓(𝑥) denotes the static buckled configuration associated with the load P and the prime
denotes differentiation with respect to x. Since the integral in Equation (2-1) is a constant for a
given 𝜓(𝑥), the buckling equation can be reduced into

19

𝜓 𝑖𝑣 + 𝜆2 𝜓′′ = 0
1

(2-3)

2

where 𝜆2 = 𝑃 − 1/2 ∫0 𝜓 ′ 𝑑𝑥 is a constant that represents the critical buckling load.
Equation (2-3) is a fourth order ordinary differential equation with constant coefficients. Its general
solution is given by
𝜓(𝑥) = 𝑐1 + 𝑐2 𝑥 + 𝑐3 cos(𝜆𝑥) + 𝑐4 sin(𝜆𝑥)

(2-4)

where the ci are constants. Applying the boundary conditions (2-2) yields the following four
algebraic equations
𝑐1 + 𝑐3 = 0

(2-5)

𝑐2 + 𝜆𝑐4 = 0

(2-6)

𝑐1 + 𝑐2 + 𝑐3 cos 𝜆 + 𝑐4 sin λ = 0

(2-7)

𝑐2 − 𝜆𝑐3 sin 𝜆 + 𝑐4 𝜆 cos 𝜆 = 0

(2-8)

This system of equations represents an eigenvalue problem for λ. Computing the
determinant of the coefficient matrix and equating it to zero, the characteristic equation for 𝜆 is
given by
2 − 2 cos 𝜆 − 𝜆 sin 𝜆 = 0

(2-9)

Manipulating Equation (2-9) using trigonometric identities, the characteristic equation is
simplified into

20

𝜆
𝜆 𝜆
𝜆
sin (sin − cos ) = 0
2
2 2
2

(2-10)

Solving the characteristic Equation (2-10) results in two kinds of buckling modes:
symmetric and antisymmetric. Symmetric modes are extracted by solving sin 𝜆/2 = 0 . They can
be expressed by
𝜓(𝑥) = 1 − cos 𝜆 𝑥

(2-11)

where 𝜆 = 2𝑚𝜋, 𝑚 = 1,2, … and antisymmetric modes are extracted by solving tan 𝜆/2 = 𝜆/2.
Their expression is given by
2
𝜓(𝑥) = 1 − 2𝑥 − cos 𝜆 𝑥 + sin 𝜆 𝑥
𝜆

(2-12)

where 𝜆 = 2.86𝜋, 4.92𝜋, 6.94𝜋, 8.95𝜋, …

2.4.

Post-buckling response of a bilaterally constrained beam
The post-buckling response of a straight prismatic bilaterally constrained beam has been

theoretically studied by Chai (1998) using small and large deformation assumptions. The beam is
placed between two flat rigid walls separated by a distance h0 such that it lies along the surface of
one of the walls. Both of its ends are clamped. The elastica has a length L0, thickness t, width b
and Young’s modulus E. The net gap between the beam and the lateral constraints is denoted h
and is defined as ℎ = ℎ0 − 𝑡. Under axial end shortening Δ, the beam buckles. The fourth-order
linearized differential equation for an Euler beam under axial compression is given by
𝑦 ′′′′ + 𝑘 2 𝑦 ′′ = 0,

21

𝑘2 =

𝑃
𝐸𝐼

(2-13)

where a prime denotes differentiation with respect to x, y is the transverse deflection, P is the
compression force and I is the moment of inertia of the beam.
2.4.1. Pre-contact
After buckling the beam separates from the adjacent constraint. The mid-span rise, A,
increases with Δ until touching the opposite wall. Subjected to the following boundary conditions
𝑦(0) = 𝑦 ′ (0) = 𝑦(𝐿0 ) = 𝑦 ′ (𝐿0 ) = 0,

(2-14)

the solution to Equation (2-13) is given by
𝑦(𝑥) =

𝐴
(1 − cos 𝑘𝑥)
2

(2-15)

Note that the first critical buckling load for the Euler buckling problem of fixed-fixed beams is
defined as

𝑃𝑐𝑟 =

4𝜋 2 𝐸𝐼
𝐿20

(2-16)

A normalized square-root axial force parameter 𝜁 is introduced as follows

𝜁≡√

𝑃
𝑘𝐿0
=
𝑃𝑐𝑟
2𝜋

(2-17)

In the Pre-contact configuration the post-buckling load is defined by 𝜁 = 1. Therefore the
deflected shape of Equation (2-15) becomes
𝑦(𝑥) =

𝐴
2𝜋𝑥
(1 − cos
)
2
𝐿0

The total end shortening is given by
22

(2-18)

𝐿0 /2
𝑃
(𝑦 ′ )2 𝑑𝑥
Δ=
𝐿 +∫
𝐸𝑏𝑡 0
0

(2-19)

Using Equations (2-13) and (2-16)-(2-19) the midspan rise A can be expressed as
𝐴=

2𝑡
√3

√𝜂 − 1,

1 ≤ 𝜂 ≤ 1 + 0.75ℎ̃2

(2-20)

3𝐿0 Δ
(πt)2

(2-21)

where
ℎ̃ ≡

ℎ
𝑡

and 𝜂 ≡

2.4.2. Point contact
Subjected to an increased axial displacement the beam touches the opposite wall at its
midspan. The boundary conditions for the free standing segment are
𝑦(0) = 𝑦 ′ (0) = 𝑦 ′ (𝐿0 /2) = 0,

𝑦(𝐿0 /2) = ℎ

(2-22)

Applying these boundary conditions to Equation (2-13) and using Equations (2-19) and (2-21),
transverse displacement and normalized end shortening are expressed, respectively, as
𝜋𝜁𝑥̅
𝜋𝜁𝑥̅
𝜋𝜁𝑥̅
𝜋𝜁
𝑦 sin ( 𝑛 ) − 𝑛 + (1 − cos ( 𝑛 )) tan (2𝑛)
=
𝜋𝜁
𝜋𝜁
ℎ
2 (tan (2𝑛) − 2𝑛)

(𝜁 2 (2 +
𝜂 = 𝜁 2 + 0.75ℎ̃2

6𝑛𝜁
𝜋𝜁
) − 𝜋 tan (2𝑛))
𝜋𝜁
cos 2 (2𝑛)
𝜋𝜁
𝜋𝜁 2
(tan (2𝑛) − 2𝑛)

(2-23)

1

where n denotes the number of buckles in the beam and 𝑥̅ ≡ 𝑥/(𝐿0 /2).

23

(2-24)

2.4.3. Line contact
It was assumed that a line contact develops over three segments situated at the ends and
the midspan of the beam. Taking into account the vanishing of the bending moment along the
contact zones, the independent boundary conditions for the free standing segment of length H are
𝑦(0) = 𝑦 ′ (0) = 𝑦 ′′ (0) = 𝑦 ′ (𝐻) = 0,

𝑦(𝐻) = ℎ

(2-25)

The solution to Equations (2-13) and (2-25) is
𝑦 2𝜋𝑥̅ − sin(2𝜋𝑥̅ )
=
,
ℎ
2𝜋

0 ≤ 𝑥̅ ≡ 𝑥/𝐻 ≤ 1

(2-26)

with 𝐻/𝐿0 = 1/𝜁.
Using Equations (2-19) and (2-26) the relationship between the normalized axial load, 𝜁, and end
shortening, 𝜂, is given by
9𝑛ℎ̃2
𝜂=𝜁 +
𝜁
2𝜋 2
2

(2-27)

2.4.4. Mode transition
Further increase of the axial end shortening, Δ, results in increasing the contact zones
lengths until the longer flat segment buckles. Let a and b denote the lengths of the flat segments at
the ends of the beam and c denote the length at its midspan. From geometric compatibility
𝑎 + 𝑏 + 𝑐 + 2𝐻 = 𝐿0
Substituting 𝐻/𝐿0 by 1/𝜁 in Equation (2-28) yields

24

(2-28)

𝑎+𝑏+𝑐
2
=1−
𝐿0
𝜁

(2-29)

The load at which mode transition occurs was assumed to be bounded between a lower
limit corresponding to a single contact zone and an upper limit corresponding to equi-length
contact zones. The lower load limit is determined by eliminating the lengths a and b in Equation
(2-29) and using Equation (2-17) as
𝑘𝑐
=𝜁−2
2𝜋

(2-30)

The upper limit corresponds to the case where 𝑎 = 𝑏 = 𝑐. Therefore Equation (2-29) is reduced
into
𝑘𝑐
𝜁−2
=
2𝜋
3

(2-31)

Guided by experimental observations, and due to the absence of friction, it was assumed that the
most probable configuration corresponds to the symmetric configuration with 2𝑎 = 2𝑏 = 𝑐. In
this case Equation (2-29) reduces to
𝑘𝑐
𝜁−2
=
2𝜋
2

(2-32)

The flat segment of length c can be considered as Euler beam with fixed ends. Therefore, similar
to Equation (2-17), the buckling critical load is such that 𝑘𝑐/2𝜋 = 1. From Equations (2-30) - (232), the transition to the second mode is bounded in the range 3 ≤ 𝜁 ≤ 5. The symmetric buckling
occurs at 𝜁 = 4.

25

2.4.5. Multiple buckles
When the beam has multiple buckles, the buckling configuration can be either a point or a
line contact. The length and the end-to-end shortening of each free standing segment of the beam,
denoted, respectively, by H and Δ𝐻 , are given by
𝐻=

𝐿0
Δ
and Δ𝐻 =
2𝑛
2𝑛

(2-33)

where n denotes the total number of buckles in the buckled configuration. The deflection of each
segment and the total end shortening for the point contact configuration are given, respectively, by
Equations (2-23) and (2-24). The transition from a point to a line contact takes place at 𝜁 = 2𝑛.
The results for the symmetric line contact configuration are given by Equations (2-26) and (2-27).
Loads at transitions to higher buckling modes are bounded by 1 + 2𝑛 ≤ 𝜁 ≤ 1 + 4𝑛. Symmetric
buckling transitions occur at 𝜁 = 4𝑛. In this case, Equation (2-28) is reduced to
2𝑛(𝑐 + 𝐻) = 𝐿0

(2-34)

The relationship between the contact zone and the load is determined by multiplying both
side of Equation (2-34) by 𝑘/2𝜋 and is given by
2𝑛

𝑐
2𝑛
=1− .
𝐿0
𝜁

26

(2-35)

Normalized square root of axial force,

Normalized end shortening,
Figure 2-5: The normalized square root axial load, ζ, vs the normalized end shortening, η (Chai
1998).
Figure 2-5 shows the variation of the normalized load, 𝜁, with respect to the normalized
end shortening, 𝜂. The results shown in the figure were constructed for ℎ̃ = 3. The dashed and the
solid lines correspond to the point contact and the line contact configurations, respectively.
Following buckling, 𝜁 remains fixed until the beam touches the opposite constraint. The point
contact holds up to 𝜁 = 2. Then a line contact develops and increases until the beam snaps into
two buckles at 𝜁 = 4. During transition the force drops and the end shortening remains fixed. The
next mode transition (i.e. to n = 3) occurs at 𝜁 = 8, and so forth. The response of the beam depends

27

on the loading direction. If the beam is unloaded rather than loaded, it returns to a lower buckling
mode through a different path.

2.5.

Contact between an elastica and its boundary involving friction

Figure 2-6: Equilibrium of a beam element subjected to lateral pressure with friction.
Soong and Choi (1986) presented a theoretical formulation of the friction force generated
by the dry contact between the buckled beam and the lateral rigid walls. The beam is modeled
using Euler Bernoulli beam theory and Coulomb model is used for friction. Figure 2-6 displays
the force equilibrium of a buckled beam element subjected to normal pressure and tangential
̂ is the bending moment, N
̂ is the axial compressive force, V
̂ is the transverse
friction force where M
shear force, F̂n is the normal pressure force, μk is the coefficient of friction, 𝑑𝑠 is the arc length of
the element, and 𝑅 is the local radius of curvature for the deformed beam.

28

The beam under consideration is homogeneous, uniform and elastic with constant bending
stiffness EI. The Euler–Bernoulli equation describes the dependence of the local radius of
curvature, bending moment and transverse shear to the beam's deflection 𝜔
̂ as
1 𝑑2𝜔
̂
=
𝑅 𝑑𝑠 2

(2-36 a)

̂ = 𝐸𝐼
𝑀

𝑑2 𝜔
̂
𝑑𝑠 2

(2-36 b)

𝑉̂ = 𝐸𝐼

𝑑3𝜔
̂
𝑑𝑠 3

(2-36 c)

From the equilibrium of tangential and normal forces the normal and tangential pressure at the
contact zone are expressed as follows
𝐹̂𝑘 =

̂ 𝑑𝑉̂
𝑁
+
𝑅 𝑑𝑠

𝜇𝑘 𝐹̂𝑘 =

̂
𝑉̂ 𝑑𝑁
−
𝑅 𝑑𝑠

(2-37)

(2-38)

Substituting Equations (2-36) into (2-37) and (2-38) yields
̂
𝐹̂𝑘 = 𝑁

𝑑2𝜔
̂
𝑑4 𝜔
̂
+
𝐸𝐼
2
𝑑𝑠
𝑑𝑠 4

𝜇𝑘 𝐹̂𝑘 = 𝐸𝐼

̂
𝑑3 𝜔
̂ 𝑑2𝜔
̂ 𝑑𝑁
−
𝑑𝑠 3 𝑑𝑠 2
𝑑𝑠

(2-39)

(2-40)

Combining Equations (2-39) and (2-40), the variation of the axial compression force is governed
by the following ordinary differential equation

29

̂
𝑑𝑁
𝑑2𝜔
̂
𝑑3𝜔
̂ 𝑑2 𝜔
̂
𝑑4 𝜔
̂
+ 𝜇𝑘 𝑁 2 = 𝐸𝐼 ( 3
− 𝜇𝑘 4 )
2
𝑑𝑠
𝑑𝑠
𝑑𝑠 𝑑𝑠
𝑑𝑠

(2-41)

̂ to Equation (2-41) can be expressed as
The solution 𝑁
𝑠

𝑠

𝜉

̂ = 𝑒 − ∫𝑠0 𝑓(𝜏) 𝑑𝜏 (𝑁
̂𝑠=𝑠 + ∫ 𝑔(𝜉) 𝑒 ∫𝜉0 𝑓(𝜏) 𝑑𝜏 𝑑𝜉 )
𝑁
0

(2-42)

𝑠0

where

𝑓(𝑠) = 𝜇𝑘

𝑑2 𝜔
̂
𝑑𝑠 2

(2-43)

𝑑3 𝜔
̂ 𝑑2𝜔
̂
𝑑4𝜔
̂
𝑔(𝑠) = 𝐸𝐼 ( 3
−
𝜇
)
𝑘
2
𝑑𝑠 𝑑𝑠
𝑑𝑠 4

(2-44)

Substituting Equation (2-42) into (2-40), the tangential friction force is expressed by
𝑠

𝑠

𝜉

− ∫ 𝑓(𝜏) 𝑑𝜏
̂𝑠=𝑠 + ∫ 𝑔(𝜉) 𝑒 ∫𝜉0 𝑓(𝜏) 𝑑𝜏 𝑑𝜉 )
𝜇𝑘 𝐹̂𝑛 = 𝜇𝑘 [𝑒 𝑠0
(𝑁
0
𝑠0

2.6.

𝑑2𝜔
̂
𝑑4 𝜔
̂
+
𝐸𝐼
]
𝑑𝑠 2
𝑑𝑠 4

(2-45)

Summary
Energy harvesting using snap-through buckling for mechanical frequency up-conversion

has been proposed by Jung and Yun (2010). The proposed device consists of buckled slender
bridges with a proof mass to provide inertial excitation force and cantilever beams to harvest free
vibration energy. Snap-through transitions between the equilibrium positions of the buckled
bridges allows for frequency up-conversion. It was shown that the device harvests power over a
wide frequency range. However, the input excitation has to be a vibration with accelerations high
enough to activate snap-through motion.

30

The eigenvalue buckling analysis of a uniform fixed-fixed straight beam subjected to axial
loading results in two kinds of solutions: symmetric and antisymmetric modes. Therefore, for the
post-buckling analysis, if the contact between the beam and the lateral constraints involves friction,
all the modes have to be included in the analysis. However, antisymmetric modes can be neglected
if the contact is frictionless and the load distribution is symmetric.
The post-buckling behavior of a bilaterally constrained beam under axial loading was
investigated by Chai (1998). The contact was assumed frictionless and the transitions to a higher
buckling mode were assumed to be always bounded between a lower limit, corresponding to a
single flattened contact zone, and an upper limit, corresponding to multiple flattened contact
segments of equal lengths. The force-displacement response of the beam was investigated
assuming that the buckling configuration is always symmetric.
A study of the contact problem between a beam and its boundary was carried out by Soong
and Choi (1986). The contact formulation involves friction. The normal pressure as well as
tangential force at the contact zones were determined. It was shown that, due to friction, the axial
compression force inside the beam is not constant.

31

CHAPTER 3: EXPERIMENTAL AND NUMERICAL STUDIES
OF THE POST-BUCKLING RESPONSE OF BILATERALLY
CONSTRAINED BEAMS UNDER AXIAL LOADING 1

3.1.

Overview
This chapter presents experimental and numerical studies of the post-buckling response of

one and multiple parallel bilaterally-constrained beams. Section 3.1 deals with the response of one
beam. An experimental apparatus was built and tested to provide better understanding of the postbuckling behavior of the beam and to assess its repeatability under cyclic loading events with
different input frequencies. A finite element (FE) model was developed to extract information on
the transverse accelerations generated during mode jumping and investigate the effect of the
system’s parameters on the beam’s response. In order to control the number and the spacing
between the snap-through events, multiple slender beams can be stacked, in parallel or series
configurations, into a system. Section 3.3 explores the post-buckling response of multiple
bilaterally constrained beams combined in a parallel configuration. The experimental setup and
the FE model developed for one beam were extended to accommodate three parallel equi-length
beams with fixed end supports. The effect of the system’s parameters on the post-buckling
response of the beams and the generated accelerations was studied. The impact of changing the
thicknesses of the beams is investigated numerically and experimentally. However, other

1

The presented results in this chapter were published in Lajnef et al. (2012 and 2014a) and

Borchani et al. (2013)
32

parameters, such as Young’s modulus and gaps between the walls, are only investigated
numerically.

3.2.

Post-buckling response of one bilaterally constrained beam
The multistable post-buckling response of an axially loaded elastic element can be obtained

and controlled by providing appropriate lateral constraints to the buckling element so as to allow
multiple stable equilibrium configurations. In such conditions, the increased transverse
deformations from a buckled shape under axial compressive demands lead to contact interaction
with discrete or continuous boundaries. The contact interaction generates secondary restraining
forces that induce the development of a higher order buckling configuration. Increased transverse
deformations of the second post-buckled configuration lead to new transverse forces from the
lateral constraints, thus inducing a third post-buckled stable configuration, and so forth.
3.2.1. Experimental study
The snap buckling, or mode jumping, characteristics of the post-buckling response of a
bilaterally constrained beam was experimentally investigated. A polycarbonate beam with fixed
end supports was placed between rigid continuous bilateral plexiglass frame as shown in Figure
3-1. The optically clear plexiglass acrylic frame was chosen to be 1 inch thick and was designed
to have a gap, h0, of 4 mm for transverse deformations. The walls were constrained at the bottom
and top to avoid opening of the wall gap during testing. The beam was placed inside the gap,
adjacent to one of the constraining walls. The net gap, h, for traverse deformations of the beam is
the difference between the wall gap, h0, and the beam thickness, t. The geometry and material
properties of the used polycarbonate beam are detailed in Table 3-1.The testing procedure
consisted in applying a gradually increasing compressive load under displacement control to the

33

top of the beam using a mechanical test system (MTS model Flextest 40 with series 370 load unit)
as shown in Figure 3-1.

Loading

Loading

Constraining
Walls

t
h0
h

Constraining
Walls

PZT Energy
Harvester
L

Strip
Column

Specimen

L/2

Cantilever
Bimorph

Figure 3-1: Schematic and diagrams of the test setup.
34

(a) Mode 1 -Φ1

(b) Mode 3 - Φ3

(c) Mode 5 - Φ5

(d) Mode 7 - Φ7

Figure 3-2: Views of the loaded beam showing the obtained multiple stable post-buckled
configurations.
The applied axial load induces multistable post-buckling configurations, as shown by the
experimental mode shapes in Figure 3-2. The post-buckling branch switching effect can be seen
by the discontinuities (load drops) in the force-deformation response (applied force versus the top
vertical displacement) plotted in Figure 3-3. The figure labels (Φ3, Φ5, etc.) indicate the buckling
mode shape to which the beam transitions after the snap-through event. Each of the mode
transitions represents the switch between a stable equilibrium branch to another that reduces the
total potential energy in the system. The stable branches are associated with an eigenvector or
buckling mode, and a branch switch constitutes a global unstable transition or a snap-through
event.

35

3000
2500

Axial Force (N)

2000

Φ7

Loading
Φ5

1500
1000

Φ3

500

Φ5

Φ3
0
-500
0

Unloading

Φ1
1

2
3
4
Axial Displacement (mm)

5

Figure 3-3: Force-displacement response for 10-cycle compression test.

To evaluate the repeatability of the post-buckling behavior and to assess its cyclic response,
the bilaterally constrained beam setup was subjected to a series of displacement-controlled
unidirectional compression tests. For all runs, the beam was compressed past its 7th buckling mode
then unloaded until a zero stress state was reached. The force-displacement responses for 10
different loading cycles are displayed in Figure 3-3. The total duration of the loading/unloading
cycles was set at 20 seconds, which represents a displacement rate of 0.5 mm/s. It can be seen that
all of the response curves coincide, with small deviations observed at the higher transition points
for modes 5 and 7.
The dependency of the beam’s post-buckling response to the frequency of the input axial
forces (loading rate) was also evaluated by a second set of displacement-controlled cyclic loading

36

tests. Tests were conducted at 0.16, 0.05, 0.025 and 0.006 Hz (i.e., 6, 20, 40 and 150 second cycle
periods). Ten-cycle tests were performed for each input frequency. Figure 3-4 displays the mean
force-displacement response of the beam for each frequency. It can be noticed that the input
frequency had little influence on the beam post-buckling response. The post-buckling response
curves are superposed and the transition points are very close. The distribution of all the transition
points is presented in Figure 3-5. The transition points for each individual mode are within a
relatively small range of displacement and force values. Table 3-2 summarizes the mean and
standard deviation values for the force and displacement at each mode transition. The displacement
and force standard deviations at the transition points were lower than 3% and 5%, respectively.
Therefore, it can be concluded that the beam post-buckling response is reproducible and
independent of the load rate under the considered test conditions.

3000

Axial Force (N)

2500
2000

Loading
1500
1000
500

Unloading

0
-500
0

1

2

3

4

5

Axial Displacement (mm)
Figure 3-4: Post-buckling behavior for displacement control cyclic loading tests at 0.16 Hz, 0.05
Hz, 0.025 Hz and 0.006 Hz.
37

Mode 7 To Mode 5 Transition Points
770

500

760

490

750

Force (N)

Force (N)

Mode 1 To Mode 3 Transition Points
510

480
470

740
730

460

720

450

710

440
0.85

0.9

0.95
1
1.05
Displacement (mm)

700
1.75

1.1

1.85

Mode 5 To Mode 3 Transition Points

Mode 3 to Mode 5 Transition Points
1200

500

1180

480

Force (N)

Force (N)

1.8
Displacement (mm)

1160
1140

460
440

1120
2.1

2.2
2.3
Displacement (mm)

420
1

2.4

1.25

195
190

Force (N)

1800

Force (N)

1.1
1.15
1.2
Displacement (mm)

Mode 3 To Mode 1 Transition Points

Mode 5 To Mode 7 Transition Points
1850

1750
1700

185
180
175
170

1650
1600
3

1.05

3.2
3.4
Displacement (mm)

165
0.4

3.6

0.45
0.5
Displacement (mm)

Figure 3-5: Transition points distribution.

38

0.55

Table 3-1: Mode transitions mean points and standard deviations.

Mode

Mean points

Standard deviations

Transition

Displacement (mm), Force (N)

Displacement (mm), Force (N)

1 to 3

0.97, 477.4

0.02, 13.3

3 to 5

2.23 , 1145.3

0.02, 20.4

5 to 7

3.33 , 1716.7

0.10, 59.8

7 to 5

1.79 , 730.8

0.01, 16.0

5 to 3

1.11 , 455.2

0.01 , 24.8

3 to 1

0.46 , 177.1

0.01 , 8.4

The effect of the interaction between the lateral constraints and the beam was
experimentally investigated. The lateral rigid walls were replaced by two soft silicone blocks and
the net gap between them remains the same. As the axial load increases, the pressure that the
column applies on the walls increases forcing them to open the gap wider. The overall stiffness of
the system drops as the loading increases. Figure 3-6 displays a comparison between the postbuckling response of the system with rigid and flexible walls. These results show that the shape of
the force-displacement response curve as well as the dynamic response of the energy concentrators
can be controlled.

39

3000

Axial Force (N)

2500

Rigid Walls
Flexible Walls

2000
1500
1000
500
0
0

1

2

3

4

5

Axial Displacement (mm)
Figure 3-6: Comparison between the post-buckling response of the system with rigid and flexible
walls.

3.2.2. Numerical study
Small and large-deformation analytical models for predicting the force-deformation
response of bilaterally constrained beams, including the identification of mode transition events,
have been proposed by Chai (1998). However, the models neglect the friction between the wall
and the beam and assume that the buckled configurations are symmetric. The models can
adequately capture the system stiffness but the mode transitions are incrementally overestimated
and the small deformation model becomes increasingly inaccurate as the number of buckled waves
increases. Therefore, a finite element (FE) approach was chosen to provide further insight into the

40

beam’s post-buckling behavior and extract information on the transverse acceleration generated
during mode jumping, which determine the input to the piezoelectric energy harvester.
The numerical study was conducted using the general purpose finite element program
ABAQUS. Large-strain nonlinear dynamic analyses were conducted using four-node
isoparametric shell elements to model the beam and rigid no-penetration contact behavior with
friction was defined for the interaction between the beam and the rigid side walls. Consideration
of friction is important because this force creates a pseudo-softening effect on the load-deformation
response since it acts against the direction of loading. Because shell elements have no physical
thickness, the model of the test setup was such that the distance between the rigid walls was the
net gap h = h0 – t. To accommodate the seeding of initial imperfections for the large deformation
analysis and to avoid interaction with the no-penetration rigid wall boundary, the location of the
beam in the model was offset from the near wall by 5% of the net gap. Thus, the beam in the FE
model had a gap of 0.05h to one side and 0.95h on the other. The element size was approximately
4 mm square and was chosen based on a sensitivity study. The friction coefficient between the
wall and the beam surfaces was 0.2, chosen by calibration of the model with data from the noted
test setup. Geometric imperfection was introduced to the beam by the superposition of buckling
eigen modes. An implicit dynamic analysis procedure was used whereby a displacement amplitude
history was prescribed to the top edge of the beam with a rate of 0.5 mm/sec.

41

3000

Axial Force (N)

2500

Experiment
FE Model

2000

Loading
Loading
1500

Unloading

1000

Unloading
500

0
0

1

2

3

4

5

End Shortening (mm)
Figure 3-7: Comparison of experimental and simulated force-displacement responses of the
bilaterally constrained beam.
The numerically simulated force-displacement response from the FE model is shown in
Figure 3-7 together with the experimental data. It can be seen that the FE solution adequately
resembled the experimental response by capturing the buckling mode transitions (both in
force/deformation location and in the amount of energy loss, interpreted by the magnitude of the
load drop), the loading and unloading stiffness, and the hysteretic nature of the response (induced
by the friction between the beam and the walls). A comparison of the simulation and experimental
post-buckling results is provided in Table 3-3. The finite element solution is considered acceptable
given that parameters such as the actual values for the beam imperfection, friction coefficient and
boundary conditions are unknown.

42

Table 3-2: Comparison of simulation post-buckling results with experimental data.
Experimental
Mode

Disp.

Transition (mm)

FEA

Force

Disp.

Force

(N)

(mm) % Diff.

(N)

% Diff.

1 to 3

0.97

477.4

0.734

-25

397.8

-17

3 to 5

2.23

1145.3

2.785

25

1551.4

36

5 to 7

3.33

1716.7

4.346

30

2436.9

42

7 to 5

1.79

730.8

1.702

-5

767.5

5

5 to 3

1.11

455.2

0.934

-16

416.7

-9

3 to 1

0.46

177.1

0.404

-13

177.5

0

The buckled shapes obtained from the FE analysis are shown in Figure 3-8 with superposed
contour maps for the transverse deformations. It can be seen that the modes correspond well with
the experimentally observed buckling shapes shown in Figure 3-2. The asymmetric nature of the
buckling shapes from the FE analysis, which follows from the friction between the beam and the
wall, can be discerned by the nodal lines (lightest contour color). The post-buckling branch
switching events of the beam during loading and unloading can also be seen in the trace of axial
force vs. transverse mid-height displacement captured from the finite element solution for the
experimentally evaluated system and shown in Figure 3-9. The horizontal lines in the trace
represent the instantaneous transverse movements during branch switching or snap-through
between post-buckled configurations. The labels note the buckled mode at the end of the unstable

43

branch and the annotations L and U denote if they correspond to the loading or unloading path,
respectively.

Φ1

Φ3

Φ5

Φ7

Figure 3-8: Buckled shapes from the finite element analysis of the experimentally evaluated
system. The color contours represent transverse deformations with the yellow (lightest color)
defining nodal lines.

44

3000
Loading
Unloading

Axial Force (N)

2500

7,L

2000

1500
5,L

7,U

1000

500

5,U

3,U
1,U

3,L

1,L

0

0

0.5

1

1.5

Transverse Displacement at Mid Height (mm)
Figure 3-9: Trace of axial force vs. transverse mid-height displacement during a loading cycle
indicating the post-buckling branch switching events.
Of more interest is the acceleration created by the impact of the beam on the rigid side
walls as evidenced by the step-function-type response in Figure 3-10 where each vertical line is
the result of a mode jump. Experimental measurements are shown in Figure 3-10 (a) while the
values from the FE simulation are shown in Figure 3-10 (b). The acceleration created during the
impact of the beam with the side wall creates an input to the energy harvester analogous to a step
function. This effect dominates the input to the energy harvester and allows for the efficient
frequency conversion of the quasi-static external input to the dynamic input for the harvester.

45

1500

Acceleration (g)

7

1000

500

5

3
1

0
3

5

-500
0

5

10

15

20

Time (sec)
(a)

Acceleration (g)

2000

3
3

5

7

5

1000
7

1

0

-1000

-2000
0

10

20

30

40

50

Time (sec)
(b)
Figure 3-10: Acceleration generated from the multiple mode-jumping response of the axially
loaded bilaterally constrained beam at mid-height of the beam during one loading cycle: (a)
experiment, (b) finite element analysis. The labels Φ3 to Φ7 indicate the mode transition events.

46

The results from Figure 3-10 show that the side acceleration from the FE model for mode
7 during loading is only about 6% higher than the experimentally measured value. Discrepancies
with the other mode jumps are attributed to the fact that the damping from the beam and the side
wall in the experiment are not considered in the FE model.
As previously noted, the snap-through response between the equilibrium positions induces
accelerations in the transverse direction of the beam upon its impact with the side walls. It has
been stated that these accelerations define the input to the piezoelectric energy harvester. The
ability of the FE modeling approach to capture the post-buckling response of the bilaterally
constrained beam provides confidence that it can be used to evaluate the effect of system
parameters (e.g., beam length, stiffness, thickness, etc.). Their effect on the energy harvesting
capabilities is thus related to their effect on the generated accelerations during snap-through. A
parametric study was thus performed based on the calibrated finite element model to study the
effect of system parameters on the generated lateral acceleration from mode transitions. The
considered parameters are the boundary conditions, position of the beam, elastic modulus, E, beam
length, L, beam thickness, t, and gap between the constraining walls, h0.
Table A.1 summarizes the cases considered in the study, where Case 1 is the base case
previously presented in the experimental study. The remaining cases varied one or two parameters
at a time as shown in bold text in table A.1. The beam width, b, and the coefficient of friction were,
respectively, 30 mm and 0.2 for all cases.

47

6000
E=4.5 GPa
E=2.25GPa

Axial Force (N)

5000

Ø7
4000
Ø5

3000

Ø7

2000
1000

Ø5

Ø3
Ø3
1

0
0

2
3
4
Axial Displacement (mm)

5

(a) Effect of Young’s modulus.

3000
h0=4mm

Axial Force (N)

2500

Ø7

h0=4.83mm
Ø7

2000
Ø5

1500

Ø5

1000
500
0
0

Ø3
Ø3
1

2
3
4
Axial Displacement (mm)

5

(b) Effect of the gap between the walls.
Figure 3-11: Effect of (a) Young’s modulus, (b) walls gap and (c) beam’s thickness on axial
forces and displacements at snap-through transitions.
48

Figure 3-11 (cont’d)

3000
t=2.34mm
t=1.56mm
t=1.17mm
t=0.78mm

Axial Force (N)

2500
2000

Ø7

Ø5

1500
1000
500

Ø3

0
Ø3
-500
0

Ø5
1

Ø7

2
3
4
Axial Displacement (mm)

5

(c) Effect of beam’s thickness.
Axial displacements and forces at which the transitions occur are determined by the
aforementioned parameters. Their dependency to the system parameters was numerically
investigated. Figure 3-11 displays the effect of Young’s modulus, thickness and gap between the
lateral constraints on the loading branch of the post-buckling response of the beam. This figure
shows that by increasing the young’s modulus the forces increase, but the axial displacements at
transitions are invariant. In contrast, increasing the gap between the walls increases the axial
displacements but the forces at transitions are the same. It also shows that the variation of the beam
thickness affects both forces and displacements at mode transitions. However, the spacing between
the events is proportional.

49

Variation of the Acceleration (%)

100
80

Position= side
Position= center

60
40
20
0
0.5

1

1.5
2
Thickness (mm)

2.5

Variation of the Acceleration (%)

(a)

800
Position= side
Position= center
600

400

200

0
2

3

4
5
6
Youngs Modulus (GPa)

7

(b)
Figure 3-12: Results from the numerical parametric study showing the effect of (a) beam
thickness, (b) elastic modulus, (c) wall gap, (d) beam length, and (e) boundary conditions on the
maximum acceleration generated by snap-through behavior.
50

Variation of the Acceleration (%)

Figure 3-12 (cont’d)

200

Position= side
Position= center

150

100

50
4

5

6
Gap (mm)

8

7

Variation of the Acceleration (%)

(c)

700
Position= side
Position= center

600
500
400
300
200
100
0
100

150
200
Length (mm)
(d)
51

250

Figure 3-12 (cont’d)

Variation of Acceleration (%)

100
80

Top End
Bottom End
Both Ends

60
40
20
0
Free

Fixed
Boundary Conditions
(e)

The maximum generated accelerations along the beam's height from the snap-through
events for each of the first thirty five cases are compared in Figure 3-12 for each of the evaluated
parameters. Each of the five plots corresponds to one of the parameters in the study and the result
from Case 1 is presented in all plots. For all cases the data points correspond to the snap-through
transition to the 7th buckling mode. Thus, the total vertical end shortening was varied to ensure
that the 7th mode transition was achieved in all cases. Effect of the boundary conditions on the
snap-through accelerations was investigated by gradually releasing the end rotations. Partial
restraints were simulated by considering rotation springs at each end.

52

The results show that acceleration from snap-through will increase with: (a) an increase in
beam thickness, (b) an increase in the beam elastic modulus, (c) an increase in the wall gap and
(d) a decrease in the beam length. The noted trends are considered to be reasonable because the
beam is stiffer if its thickness or Young’s modulus increases or if its length is reduced. Also, an
increase in the gap between the walls allows higher rotations of the beam in its buckled shape,
which results in higher potential energy due to bending. Therefore the beam is able to store higher
strain energy that is then released during the snap-through transitions. Results also show that the
generated accelerations during transitions increase by placing the beam adjacent to one of the
lateral walls (figures 3-12 (a) - (d)) and by adopting fixed-end configuration (Figure 3-12 (e)).
Therefore for the remaining analyses the beam was placed adjacent to one of the lateral walls with
fixed-end supports.
The results shown indicate the possibilities for optimizing the system performance and
support the feasibility of down-scaling the system into sizes adequate for a device that can be
embedded inside a structure as shown in figure 1-1 for the concept integration. For that purpose, a
second set of simulations was carried out to investigate the combined effect of reducing the length
and varying one of the other parameters.
Figure 3-13 displays the variation of the maximum acceleration with respect to the beam’s
length for different thickness values. The gap between the walls and the young’s modulus remain
constant. It should be noted that for case 16 (L = 76.2 mm and t = 2.34 mm) and case 45 (L = 76.2
mm and t = 1.56 mm) the beam fails to reach the seventh buckling mode. It was shown in Figure
3-12 (a) that, for the original length (L = 250 mm), the thicker is the beam, the higher are the snapthrough accelerations. However, the results shown in Figure 3-13 indicate that for shorter beams

53

(L = 76.2 mm) the thickness should be reduced in order to reach the seventh mode and generate
higher accelerations.

Variation of the Acceleration (%)

3500
t=2.34 mm
t=1.56 mm
t=1.17 mm
t=0.78 mm

3000
2500
2000
1500
1000
500
0
50

100

150
Length (mm)

200

250

Figure 3-13: Effect of beam thickness and length on the maximum acceleration.
The combined effect of the gap between the walls and the length of the beam is presented
in Figure 3-14. The cases 56 and 59 (G = 7.32 mm, L = 125 mm and L = 76.2 mm) fail to reach
the seventh buckling mode. Therefore for reduced lengths the gap should be increased with an
upper bound representing the limit after which the beam is locked in a lower buckling mode.
The net gap and the thickness are related. Therefore results shown in figures 3-13 and 314 are equivalent since a decrease in the thickness is equivalent to an increase in the gap. Hence,
to design shorter beams, the ratio t/h is an important parameter to consider.

54

Variation of the Acceleration (%)

6000
G=4 mm
G=4.83 mm
G=5.66 mm
G=7.32 mm

5000
4000
3000
2000
1000
0
50

100

150
Length (mm)

200

250

Figure 3-14: Effect of gap and length on the maximum acceleration.

Variation of the Acceleration (%)

5000
E=2.25 GPa
E=3.375 GPa
E=4.5GPa
E=6.75 GPa

4000

3000

2000

1000

0
50

100

150
Length (mm)

200

250

Figure 3-15: Effect of Young’s modulus and length on the maximum acceleration.
55

Figure 3-15 displays the effect of the young’s modulus and the length on the maximum
acceleration generated during transitions. Similar to the gap, increasing the young’s modulus
amplify the levels of the accelerations. However, higher modes risk to be blocked for shorter beams
(L = 76.2 mm, E = 6.75 GPa). Therefore the young’s modulus should be bounded.

3.3.

Post-buckling response of multiple bilaterally constrained beams in parallel

configuration
Although tuning the parameters of the system affects dramatically the accelerations
generated in snap-through transitions, the number and the spacing between these events was shown
non-sensitive to the studied parameters. In order to tailor the number and spacing of the mode
branch switching during the post-buckling response, the experimental setup described above for
one bilaterally constrained beam was extended to accommodate for three parallel beams with fixed
end supports. The constraining walls were kept separated by a 4 mm gap and three equi-length
beams were placed inside the gap, adjacent to one of the constraining walls as shown in Figure 316.

Figure 3-16: Test setup with three parallel constrained beams.

56

Two sets of experiments were conducted to demonstrate the system’s ability to increase
the number and force drop magnitude in the buckling mode transitions without exceeding the
seventh mode in each beam. In the first experiment three similar beams with the same geometry
and material properties as presented in Table 3-1 were used. The beams were axially loaded in
compression under displacement control to a total end shortening of 5 mm and then unloaded. The
top extremities of the beams exhibit the same loading displacement. In the second experiment one
of the beams was substituted with a thinner polycarbonate beam (1.47 mm thickness) and the same
experiment procedure was applied. Only one parameter changed between the two experiments,
which is the thickness of one beam. The net gap between the walls was kept constant.
Figure 3-17 and Figure 3-18, successively, display the resulting force-displacement
response of the system in the first and second experiment. Comparing the two plots it can be
concluded that by changing one parameter, which in this case was the thickness of one beam, the
number and force drop magnitude of the transitions change. Furthermore the location of the
transitions and the overall stiffness of the system are affected by any modification in the major
parameters. The finite element model described above was modified to investigate the effect of the
beams thicknesses, young’s modulus and the gap between the rigid walls on the number, positions
and drop force magnitude of mode branch switchings during the post-buckling response, and to
extract information on the transverse acceleration generated during mode jumping as well, which
define the input to the piezoelectric energy harvester.

57

6000

Axial Force (N)

5000
4000
3000
2000
1000
0
0

1

2
3
4
Axial Displacement (mm)

5

Figure 3-17: Force-displacement response of three similar parallel bilaterally constrained beams.
4000
3500

Axial Force (N)

3000
2500
2000
1500
1000
500
0
0

1

2
3
4
Axial Displacement (mm)

5

Figure 3-18: Force-displacement response of three parallel bilaterally constrained beams: two of
them are similar and the third is thinner.
58

Table 3-3: Modified Parameter at each simulated case.
Young's modulus (MPa)

Beams thickness (mm)

Gap (mm)

Case
beam1

beam2

beam3

beam1

beam2

beam3

beam1

beam2

beam3

1

2250

2250

2250

2.34

2.34

2.34

4

4

4

2

2250

3375

4500

2.34

2.34

2.34

4

4

4

3

2250

2250

2250

2.34

1.47

0.78

4

4

4

4

2250

2250

2250

2.34

2.34

2.34

4

5

6

The overall force-displacement response of the three parallel constrained beam system is
sensitive to the beams' geometry, material properties and gap between the constraining walls. To
evaluate the impact of each parameter, four cases were numerically simulated. The first case
consists of three beams with the same geometry and material properties as presented in Table 3-1
and the other cases change one of the pre-mentioned parameters at a time. Table 3-4 highlights the
modified property of the beams at each case. In all cases the top edges of the beams were subjected
to a total end shortening of 5 mm. The loading branches of the force displacement responses as
well as transverse accelerations in the beams were extracted from the FE analyses database and
are displayed in Figure 3-19 to Figure 3-22. Each acceleration spike is resulted from one or
multiple beams movement or transitions between modes. The force drop magnitude and the
acceleration levels can reflect the level of the released energy when the system transitions from
one position to another. The number and spacing between the mode switching events can be tuned
by modifying one or multiple parameters. Although the number and the locations of the transitions
are very sensitive to the investigated parameters, the levels of the maximum acceleration are of the
59

same order. Hence the system transitions can be controlled such that the snap-through happens at

Axial
Axial
Force
Force
(KN)
(KN)

certain predetermined axial forces or displacements without sacrificing acceleration levels.

10
10

5
5

0
00
0

4
3
2
1
3 (mm)4
2
1Axial Displacement
Axial Displacement (mm)

5
5

4
3
2
1
3 (mm)4
2
1Axial Displacement
Axial Displacement (mm)

5
5

Accelerations
Accelerations
(g)(g)

4

x 10
2 x 104
2
1
1
0
0
-1
-1
-2
-2
-3
-30
0

Figure 3-19: Loading branch of the force-displacement response and accelerations generated
during transitions for Case 1 in Table 3-4 (same geometry and material properties for all beams).
60

Axial Force
Axial
Force (KN)
(KN)

15

10
15
5
10
0
50

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

4

Accelerations
Accelerations
(g)
(g)

x 10
0
40

2
4
x 10
4
0
2
-2
0
-4
0
-2
-4
0

Figure 3-20: Loading branch of the force-displacement response and accelerations generated
during transitions for Case 2 in Table 3-4 (variation of Young’s moduli).
61

Axial Force
Axial
Force (KN)
(KN)

5
4

5
3
4
2
3
1
2
0
0
1

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

4

Accelerations
Accelerations
(g)
(g)

x 10
0
20

4
1
x 10
2
0
1

-1
0
-2
-10

-2
0

Figure 3-21: Loading branch of the force-displacement response and accelerations generated
during transitions for Case 3 in Table 3-4 (variation of beams thicknesses).
62

Axial Force
Axial (KN)
Force (KN)

8
6
8
4
6
2
4
0
20

Accelerations
Accelerations
(g)
(g)

0
4
0x 10
4

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

1
2
3
4
Axial Displacement (mm)

5

4

4
2

x 10

2
0
0
-2
0
-2
0

Figure 3-22: Loading branch of the force-displacement response and accelerations generated
during transitions for Case 4 in Table 3-4 (variation of gaps between the lateral walls).

63

3.4.

Summary
The post-buckling response of the axially loaded bilaterally constrained beams was

investigated experimentally and numerically to provide better understanding of its behavior. It was
shown that the beam can achieve higher buckling modes by providing the appropriate lateral
constraints. Furthermore, the transitions between modes are sudden and generate high-rate
transverse accelerations.
Experimental results show that the post-buckling behavior is reproducible under cyclic
loadings and independent of the input loading frequency. An FE model that accounts for the
friction between the beam and the lateral walls was developed to extract information on the
transverse accelerations and to investigate the effect of system parameters on the accelerations as
well as axial forces and displacements at mode transitions. It was shown that placing the beam
adjacent to one of the lateral walls with fixed-end supports generates higher accelerations.
Furthermore, the accelerations amplify by increasing the thickness, Young’s modulus or gap
between the walls, or by decreasing the length of the beam. However, the combination between
length and one of the other parameters may have opposite effect. Results show that for short beams
the thickness, young’s modulus and gap should be upper-bounded in order to allow for the
development of higher modes and hence the generation of high accelerations.
Although tuning the parameters of the system affects dramatically the accelerations
generated in snap-through transitions, the number and the spacing between these events was shown
to be insensitive to the studied parameters. In order to control the number and the spacing between
mode transitions, multiple slender beams were stacked in parallel configurations. Experiments and
numerical studies were conducted to investigate the global response of the system and the
accelerations generated from each beam. It was shown that the global force-displacement response
64

is sensitive to the beams thicknesses, Young’s moduli and gaps between the walls. Any
modification of these parameters affects the locations of the transitions and the overall stiffness of
the system. More important is that the number and the spacing between these events can be
controlled by tuning the aforementioned parameters without sacrificing the levels of generated
accelerations.

65

CHAPTER 4: THEORETICAL STUDY OF THE POSTBUCKLING RESPONSE OF A BILATERALLY CONSTRAINED
BEAM UNDER AXIAL LOADING 2

4.1.

Overview

This chapter presents a theoretical study of the post-buckling snap-through behavior of a bilaterally
constrained beam subjected to axial loading using energy methods. In Section 4.2 the nonlinear
relationship between strain and displacement is determined. Section 4.3 provides an eigenvalue
buckling analysis that accounts for longitudinal deformations to determine the buckling modes.
Using the Galerkin discretization method, the transverse deflection is expressed as a linear
combination of buckling modes multiplied by weight coefficients that define the contribution of
each mode to the deflection. The normalized kinetic, potential and friction-induced energies are
expressed in terms of the unknown weight coefficients in Section 4.4. Section 4.5 presents a
frictionless static analysis of the beam’s response based on minimization of the potential energy
under the confinement constraints. The potential energy is composed of bending and compression
energies and the work of the external load. Both displacement- and force- control formulations are
presented. The dynamic snap-through transitions under axial force loading are investigated in
Section 4.6. The released kinetic energy as well as accelerations of the beam during buckling mode
switching are determined by minimizing the total energy which is the sum of kinetic and potential
energies.

2

Results presented in this chapter were published in Borchani et al. (2014a and 2014b).
66

4.2. Strain-displacement Relationship
The problem under consideration consists of a straight beam subjected to increasing axial
load, p̂. The loaded end is allowed to slide axially and the other end is fixed as shown in Figure 41. The beam has a length L, a uniform cross section area A, a moment of inertia I and a modulus
of elasticity E. The beam is confined between two rigid surfaces spaced by a distance h. û(x) and
w
̂ (x) denote the axial displacement and transverse deflection, respectively.

h

(x)

x=0

x=L/2

x=L

Figure 4-1: Geometry of the beam buckled in the first mode.

y

B*
θ
A*

A

x

+d
B

x
dx

+d

Figure 4-2: Deformation of beam element.
The nonlinear strain-displacement relationship is obtained by considering a differential
element AB located at a distance x from the origin as shown in Figure 4-2. The element length in

67

the undeformed beam is dx. The beam under study is slender and therefore it can be modeled using
nonlinear Euler-Bernoulli beam theory.
The length of the element in the deformed configuration can be related to the longitudinal and
transversal displacements by:
𝑑𝑠 = √(𝑑𝑥 + 𝑑𝑢̂)2 + 𝑑𝑤
̂ 2.

(4-1)

The total elongation of the differential element is given by:
𝑒 = 𝑑𝑠 − 𝑑𝑥
= √(𝑑𝑥 + 𝑑𝑢̂)2 + 𝑑𝑤
̂ 2 − 𝑑𝑥.

(4-2)

The resulting strain 𝑒/𝑑𝑥 is then expressed as:
𝜀 = √(1 + 𝑢̂′)2 + 𝑤
̂ ′2 − 1
= √1 + 2𝑢̂′ + 𝑢̂′2 + 𝑤
̂ ′2 − 1

(4-3)

where the over primes denote the derivative with respect to 𝑥. Using a binomial series expansion
for the quadratic terms and neglecting higher order terms in the displacement gradient, the resulting
strain can be expressed as:
𝑤
̂′2
𝜀 = 𝑢̂ +
.
2
′

(4-4)

By integrating the strain over the beam’s span, the variation in length due to the buckled
configuration can be expressed as:

68

𝐿

𝑤
̂′2
𝛥 = 𝛥𝑐 + ∫
𝑑𝑥.
0 2

(4-5)

L

The first term, Δc = û(L) − û(0), is due to axial compression and the second term, ∫0 w
̂ ′2 /2 dx,
is induced by the midplane rotation due to buckling. The midplane stretching induces an axial
compressive force that can be expressed by:
𝑆=

𝐸𝐴
𝛥
𝐿 𝑐

1 𝐿 ′2
= EA (𝜀 −
∫ 𝑤
̂ 𝑑𝑥 ).
2𝐿 0

4.3.

(4-6)

Buckling analysis
The governing equations that account for longitudinal deformations in a fixed-fixed

straight beam subjected to axial load 𝑝̂ are:

𝐸𝐼

𝑑4 𝑤
̂(𝑥)
𝐸𝐴 𝐿 ′ 2
𝑑2𝑤
̂(𝑥)
+
(𝑝̂
−
∫
𝑤
̂
𝑑𝑥
)
=0
4
𝑑𝑥
2𝐿 0
𝑑𝑥 2

𝑤
̂(0) = 𝑤
̂(𝐿) = 0,

𝑑𝑤
̂(𝑥)
𝑑𝑤
̂(𝑥)
|
=
|
= 0.
𝑑𝑥 𝑥=0
𝑑𝑥 𝑥=𝐿

(4-7)

For convenience, the following non-dimensional variables are used:

𝑋=

𝑥
,
𝐿

𝑊(𝑋) =

𝑤
̂(𝑋 𝐿)
,
ℎ

𝑁2 =

𝑝̂ 𝐿2 𝐴ℎ2 1 𝑑𝑊 2
−
∫ (
) 𝑑𝑋.
𝐸𝐼
2𝐼 0 𝑑𝑋

(4-8)

Substituting Equation (4-8) into (4-7), the non-dimensional buckling equation is then expressed
as:

69

𝑑4 𝑊(𝑋)
𝑑 2 𝑊(𝑋)
2
+𝑁
=0
𝑑𝑋 4
𝑑𝑋 2

𝑊(0) = 𝑊(1) = 0,

𝑑𝑊(𝑋)
𝑑𝑊(𝑋)
|
=
|
= 0.
𝑑𝑋 𝑋=0
𝑑𝑋 𝑋=1

(4-9)

The general solution of Equation (4-9) is given by Equation (2-4). The buckling analysis
was presented in details in Section 2.3. The symmetric and antisymmetric mode shapes are
expressed, respectively, by:
𝑊𝑗 (𝑋) = 1 − 𝑐𝑜𝑠(𝑁𝑗 𝑋)
𝑁𝑗 = (𝑗 + 1)𝜋

𝑊𝑗 (𝑋) = 1 − 2𝑋 − 𝑐𝑜𝑠(𝑁𝑗 𝑋) +

} 𝑗 = 1, 3, 5, …

2𝑠𝑖𝑛(𝑁𝑗 𝑋)
𝑁𝑗
} 𝑗 = 2, 4, 6, … .

(4-10)

(4-11)

𝑁𝑗 = 2.86𝜋, 4.92𝜋, 6.94𝜋, 8.95𝜋, , …
The Buckling modes given in Equations (4-10) and (4-11) form an orthogonal basis. Hence mode
superposition method can be used to express the beam deflection as a linear combination of
buckling modes, written as:
∞

𝑊(𝑋) = ∑ 𝐴𝑗 𝑊𝑗 (𝑋),

(4-12)

𝑗=1

where 𝐴𝑗 are weight coefficients that determine the contribution of each mode to the transverse
deflection.

70

4.4.

Derivation of kinetic, potential and dissipation energies
Every stable equilibrium configuration corresponds to an energy minimum. If the loading

is static with no dissipation of energy, the equilibrium corresponds to a potential energy well. As
the axial force increases the equilibrium at a certain configuration becomes unstable, thus the beam
snaps into a different stable configuration with lower energy. The total energy is the sum of the
potential energy, kinetic energy and dissipated energy. Since the axial loading is quasi-static,
̂ , can be
kinetic energy in the axial direction can be neglected. Therefore the kinetic energy, K
expressed by:
2

1 𝐿
𝜕𝜔
̂(𝑥, 𝑡̂)
̂ (𝑡̂) = ∫ 𝑚(𝑥) (
𝐾
) 𝑑𝑥,
2 0
𝜕𝑡̂

(4-13)

where 𝑚(𝑥) is a constant representing the mass per unit length of the undeformed beam and 𝑡̂ is
the time variable. The total potential energy is the sum of the bending and axial compression strain
energies stored in the deformed elastic element and the potential energy due to external force.
In general, if a force is applied to a prismatic beam in a gradual manner, i.e. the magnitude
of the force increases from 0 to 𝑝, and the bar stretches by Δ, when the material behaves in a linearelastic manner 𝐹 = 𝑝𝑥/𝛥 then the work of the external force is given by:
𝛥

𝑢𝑒 = ∫

0

𝑝
1
𝑥 𝑑𝑥 = 𝑝 𝛥.
𝛥
2

(4-14)

Since the force is gradually increasing, the potential energies ub ,uc and up due to bending,
compression and external applied force, respectively, are given by:

71

2

𝐿
1
𝑑2𝑤
̂(𝑥)
𝑢𝑏 = 𝐸𝐼 ∫ (
) 𝑑𝑥
2
𝑑𝑥 2
0

(4-15)

1
𝑢𝑐 = 𝑆𝛥𝑐
2

(4-16)

1
𝑢𝑝 = − 𝑝̂ 𝛥.
2

(4-17)

The following non-dimensional variables are introduced:

𝑑=

𝛥𝐿
,
ℎ2

𝑈(𝑋) =

𝑢𝑏 𝐿3
𝑉𝑏 =
,
𝐸𝐼ℎ2

𝑢̂(𝑋𝐿)𝐿
,
ℎ2

𝑄=

𝑡
,
ℎ

𝑢𝑝 𝐿3
𝑉𝑃 =
,
𝐸𝐼ℎ2

𝑢𝑐 𝐿3
𝑉𝑐 =
,
𝐸𝐼ℎ2

𝛿=

𝛥𝑐 𝐿
,
ℎ2

̂ 𝐿3
𝐾
𝐾=
,
𝐸𝐼ℎ2

𝑃=

𝑝̂ 𝐿2
𝐸𝐼

𝐸𝐼
𝑡 = 𝑡̂√ 4
𝑚𝐿

(4-18)

The normalized transverse displacement is expressed in Equation (4-12) as a linear
combination of all the buckling modes expressed in Equations (4-10) and (4-11). Substituting their
expressions and differentiating the normalized transverse displacement up to four times gives
∞

𝑊(𝑋) =

∑

𝐴𝑗 (1 − 𝑐𝑜𝑠(𝑁𝑗 𝑋))

𝑗=1,3,5,…
∞

+

∑

𝐴𝑗 (1 − 2𝑋 − 𝑐𝑜𝑠(𝑁𝑗 𝑋) +

𝑗=2,4,6,…
∞

𝑊

′ (𝑋)

=

∑

(4-19 a)

∞

𝐴𝑗 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋) +

𝑗=1,3,5,…

∑

∑
𝑗=1,3,5,…

𝐴𝑗 (−2 + 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋) + 2 𝑐𝑜𝑠(𝑁𝑗 𝑋))

(4-19 b)

𝐴𝑗 (𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋) − 2𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋))

(4-19 c)

𝑗=2,4,6,…

∞

𝑊 ′′ (𝑋) =

2 𝑠𝑖𝑛(𝑁𝑗 𝑋)
)
𝑁𝑗

∞

𝐴𝑗 𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋) +

∑
𝑗=2,4,6,…

72

∞

𝑊

′′′ (𝑋)

=−

𝐴𝑗 𝑁𝑗3 𝑠𝑖𝑛(𝑁𝑗 𝑋)

∑
𝑗=1,3,5,…

∞

−

∑

𝐴𝑗 (𝑁𝑗3 𝑠𝑖𝑛(𝑁𝑗 𝑋) + 2𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋))

(4-19 d)

𝑗=2,4,6,…
∞

𝑊 (4) (𝑋) = −

𝐴𝑗 𝑁𝑗4 𝑐𝑜𝑠(𝑁𝑗 𝑋)

∑
𝑗=1,3,5,…
∞

−

∑

𝐴𝑗 (𝑁𝑗4 𝑐𝑜𝑠(𝑁𝑗 𝑋) − 2𝑁𝑗3 𝑠𝑖𝑛(𝑁𝑗 𝑋))

(4-19 e)

𝑗=2,4,6,…

4.4.1. Normalized kinetic energy
Substituting the non-dimensional variables expressed in Equation (4-18) into the expression
of the kinetic energy (Equation (4-13)), the normalized kinetic energy can be expressed by
1

1
𝜕𝑊(𝑋, 𝑡) 2
𝐾 = ∫(
) 𝑑𝑋
2
𝜕𝑡

(4-20)

0

Time and space coordinates can be separated using Galerkin discretization method. The transverse
displacement can be expressed as a linear combination of admissible functions multiplied by
unknown temporal coordinate:
∞

𝑊(𝑋, 𝑡) = ∑ 𝐴𝑖 (𝑡)𝑊𝑖 (𝑋)

(4-21)

𝑖=1

where 𝐴𝑖 (𝑡) are the generalized temporal coordinate and 𝑊𝑖 (𝑋) are the buckling mode shapes
which constitute a set of orthonormal admissible functions. Substituting the buckling mode shape
expressions in Equations (4-10) and (4-11) into Equation (4-20), the normalized kinetic energy is
then expressed by:

73

2

∞

1 1
𝜕𝐴𝑖 (𝑡)
𝐾 = ∫ (∑
𝑊𝑖 (𝑋)) 𝑑𝑋
2 0
𝜕𝑡
𝑖=1
∞

∞

2

1
3 𝜕𝐴𝑖 (𝑡)
= ( ∑
(
) +2 ∑
2
2
𝜕𝑡
𝑖=1,3,5,…

∞

+

∞

∑

𝑖=1,3,5,… 𝑗=3,5,…>𝑖

𝜕𝐴𝑖 (𝑡) 𝜕𝐴𝑗 (𝑡)
𝜕𝑡
𝜕𝑡

2

𝜕𝐴𝑖 (𝑡)
1
2
(
(
)
( )
(
)
∑ (
)
3 2 Ni 6 + 5 Ni + 96 Ni cos Ni + 12 Ni cos 2 Ni
𝜕𝑡
12 Ni

𝑖=2,4,6,…

+ 24 (−4 + N2i ) sin(Ni ) + 3 (−4 + N2i ) sin(2 Ni ))
+2

∞

∞

∑

∑

𝜕𝐴𝑖 (𝑡) 𝜕𝐴𝑗 (𝑡)
1
(3 N2i Nj cos(Nj ) (4 Ni (N2i
𝜕𝑡
𝜕𝑡 3 N3 (N − N )N3 (N + N )
i
j
i
j
i
𝑖=2,4,6,… 𝑗=4,6,…>𝑖
j

− N2j ) + (4 + N2i ) N2j sin(Ni )) − 3Ni N2j cos(Ni ) (−4 N2i Nj + 4 N3j + N2i (4 + N2j ) sin(Nj ))
+ (N2i − N2j ) (3 N2j sin(Ni) ((−4 + N2i ) Nj − 2N2i sin(Nj ))
+ N3i (N3j + 3 (−4 + N2j ) sin(Nj))))

+2

∞

∞

∑

∑

𝑖=1,3,5,… 𝑗=2,4,6,…

𝜕𝐴𝑖 (𝑡) 𝜕𝐴𝑗 (𝑡)
1
2
(2 N4i − 2 N2i Nj + 2 N4j
2
2
𝜕𝑡
𝜕𝑡 Ni (Ni − Nj ) Nj (Ni + Nj )

2

+ N3i Nj sin(Ni ) − Ni N4j sin(Ni) + N2i cos(Nj ) (−2 N2i + 2 N2j + Ni N2j sin(Ni ))
3

− N4i Nj sin(Nj ) + N2i Nj sin(Nj ) − 2 N3i Nj sin(Ni ) sin(Nj )

− N2j cos(Ni ) (−2 N2i + 2 N2j + 2N2i cos(Nj ) + N2i Nj sin(Nj )))).

(4-22)

4.4.2. Normalized potential energy
The normalized midplane stretching and potential energies are expressed by substituting
Equations (4-8) and (4-18) into Equations (4-5) and (4-15)-(4-17) as:
74

2

𝐿 ′
𝐿
𝑤
𝑑 = 2 [𝛥𝑐 + ∫
𝑑𝑥]
ℎ
0 2

1 1 𝑑𝑊 2
=𝛿+ ∫ (
) 𝑑𝑋,
2 0 𝑑𝑋

(4-23)

2

1 1 𝑑 2 𝑊(𝑋)
𝑉𝑏 = ∫ (
) 𝑑𝑋,
2 0
𝑑𝑋 2

(4-24)

1 𝐿3 𝐸𝐼𝑃 ℎ2 𝑑
𝑉𝑃 = −
2 𝐸𝐼ℎ2 𝐿2 𝐿
=−

𝑃𝑑
,
2

(4-25)

1 𝐿3
𝐸𝐴 ℎ2 𝛿 ℎ2 𝛿
𝑉𝑐 = (
)(
)
2 𝐸𝐼ℎ2
𝐿 𝐿
𝐿
=

6 2
𝛿 .
𝑄2

(4-26)

Substituting Equation (4-19) into Equations (4-23) and (4-24), the normalized end shortening and
bending strain energy are expressed, respectively, as

1 1 𝑑𝑊 2
𝑑= 𝛿+ ∫ (
) 𝑑𝑋
2 0 𝑑𝑋
∞

𝑃𝑄 2 1 1
=
+ ∫ ( ∑
12
2 0

𝐴𝑗 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋)

𝑗=1,3,5,…
2

∞

+

∑

𝐴𝑗 (−2 + 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋) + 2 𝑐𝑜𝑠(𝑁𝑗 𝑋))) 𝑑𝑋

𝑗=2,4,6,…

75

∞

𝑃𝑄 2 1
=
+
12
2

∑
𝑗=1,3,5,…

𝑁𝑗2 2
𝐴 +
2 𝑗

[

∞

∑
𝑗=2,4,6,…

𝐴𝑗2
(6𝑁𝑗 + 𝑁𝑗3 − 2𝑁𝑗 cos (2𝑁𝑗 ) − 16 sin(𝑁𝑗 )
2𝑁𝑗

∞

+ cos(𝑁𝑗 ) sin(𝑁𝑗 ) (8𝑁𝑗 −

(𝑁𝑗2

− 4))) + 2 ∑

∞

∑

−2

𝑖=2,4,6,… 𝑗=4,6,8,…>𝑖

(
−

1
𝑁𝑗
𝑁
4𝑁𝑖 𝑁𝑗 (𝑁𝑖 − 𝑁𝑗 )(𝑁𝑖 + 𝑁𝑗 ) (−2 + 𝑁𝑖 cot ( 2𝑖 )) (−2 + 𝑁𝑗 cot ( 2 ))
𝑁𝑗 2
𝑁𝑖 2
× (csc ( ) csc ( ) (−2 + 2 cos(𝑁𝑖 ) + 𝑁𝑖 sin(𝑁𝑖 ))
2
2
× (−2 + 2 cos(𝑁𝑗 ) + 𝑁𝑗 sin(𝑁𝑗 )) (−𝑁𝑖 𝑁𝑗 cos(𝑁𝑗 ) (2 (𝑁𝑖2 − 𝑁𝑗2 )

+ 𝑁𝑖 (4 + 𝑁𝑗2 ) sin(𝑁𝑖 ) ) + 𝑁𝑖 𝑁𝑗 cos(𝑁𝑖 ) (−2𝑁𝑖2 + 2𝑁𝑗2 + 2(𝑁𝑖2 − 𝑁𝑗2 ) cos(𝑁𝑗 )

+ (4 + 𝑁𝑖2 )𝑁𝑗 sin(𝑁𝑗 ) + 4(𝑁𝑖2 − 𝑁𝑗2 ) (𝑁𝑗 sin(𝑁𝑖 ) + 𝑁𝑖 (−𝑁𝑗 + sin(𝑁𝑗 ))))) 𝐴𝑖 𝐴𝑗
)
∞

+2 ∑

∞

∑

𝑖=1,3,5,… 𝑗=2,4,6,…

2𝑁𝑖2 − 2𝑁𝑖2 cos(𝑁𝑗 ) − 𝑁𝑖2 𝑁𝑗 sin(𝑁𝑗 )
𝐴𝑖 𝐴𝑗 ,
𝑁𝑖2 − 𝑁𝑗2
]

76

(4-27)

∞

1 1
𝑉𝑏 = ∫ ( ∑ 𝐴𝑗 𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋)
2 0
𝑗=1,3,...

2

∞

+ ∑ 𝐴𝑗 (𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋) − 2𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋))) 𝑑𝑋
𝑗=2,4,…
∞

1
= [ ∑
2

𝑗=1,3,5,…

∞

+

∑
𝑖=2,4,6,…
∞

+2 ∑

𝑁𝑗4 2
𝐴
2 𝑗

𝑁𝑗
(2𝑁𝑖 (2 + 𝑁𝑖2 ) + 4𝑁𝑖 𝑐𝑜𝑠(2𝑁𝑖 ) + (−4 + 𝑁𝑖2 ) 𝑠𝑖𝑛(2𝑁𝑖 ))𝐴2𝑖
4
∞

∑

𝑁𝑖2 𝑁𝑗 (2𝑁𝑗 − 𝑁𝑗 (2 𝑐𝑜𝑠(𝑁𝑗 ) + 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 )))
𝑁𝑖2 − 𝑁𝑗2

𝑖=1,3,5,… 𝑗=2,4,6,…
∞

+2 ∑

∞

∑

𝐴𝑖 𝐴𝑗

𝑁𝑖 𝑁𝑗
(𝑁𝑗 (4 + 𝑁𝑖2 ) 𝑐𝑜𝑠(𝑁𝑗 ) 𝑠𝑖𝑛(𝑁𝑖 )
− 𝑁𝑗2

𝑁2
𝑖=2,4,6,… 𝑗=4,6,8,…>𝑖 𝑖

− 𝑁𝑖 (4 + 𝑁𝑗2 ) 𝑐𝑜𝑠(𝑁𝑖 ) 𝑠𝑖𝑛(𝑁𝑗 ) − 2(𝑁𝑖 − 𝑁𝑗 )(𝑁𝑖 + 𝑁𝑗 ) 𝑠𝑖𝑛(𝑁𝑖 ) 𝑠𝑖𝑛(𝑁𝑗 )) 𝐴𝑖 𝐴𝑗 ].

(4-28)

4.4.3. Normalized energy dissipated due to friction
The dry friction between the beam and the walls is modeled using Coulomb model. The
friction force is considered distributed over the contact surface. The energy dissipated due to
friction depends on the developed normal pressure as well as contact length and displacement of
the contact points. Dissipation of energy occurs only when the beam is in contact with the lateral
walls (i.e. 𝜔
̂(𝑥) = 0 or 𝜔
̂(𝑥) = ℎ). Hence, the dissipated energy can be expressed as:

77

𝐿

𝑤𝑓 = ∫ 𝜇𝑘 𝐹̂𝑛 (𝑥) 𝑢̂(𝑥) (𝐻(−𝜔
̂(𝑥)) + 𝐻(𝜔
̂(𝑥) − ℎ)) 𝑑𝑥,

(4-29)

0

where 𝐻(𝑥) is the Heaviside function and 𝜇𝑘 is the kinetic coefficient of friction. The normalized
expression of the dissipated energy is expressed as:
1

𝑊𝑓 = ∫ 𝜇𝑘 𝐹𝑛 (𝑋) 𝑈(𝑋) (𝐻(−𝑊(𝑋)) + 𝐻(𝑊(𝑋) − 1)) 𝑑𝑋

(4-30)

0

where
𝑤𝑓 𝐿3
𝑊𝑓 =
𝐸𝐼ℎ2

and

𝐹̂𝑛 (𝑋𝐿) 𝐿3
𝐹𝑛 (𝑋) =
.
𝐸𝐼

(4-31)

Due to friction, the axial compression force inside the beam was shown to fluctuate in
̂ s=s is the compression
Section 2.5. Its expression is given by Equation (2-42) where s0 = 0 and N
0
force applied at the end which is equal to p̂. Figure 4-3 displays the variation of the axial force
inside the beam under different axial forces and corresponding buckling modes. In all cases the
force variation is less than 1% of the applied load. The relative force variation was computed
̂−N
̂ s=s )/N
̂ s=s . Therefore, for simplification of the computation, the axial
by 𝑅𝑉% = 100 × (N
0
0
̂ , will be considered constant and equal to the applied force, p̂.
force inside the beam, N
The tangential force at contact zones is given by Equation (2-45). Assuming that the axial
compression force inside the beam is constant and using non-dimensional variables expressed in
Equations (4-8), (4-18) and (4-31), the friction induced tangential force can be simplified into

𝜇𝑘 𝐹𝑛 (𝑋) =

ℎ
𝑑 4 𝑊(𝑋)
𝑑2 𝑊(𝑋)
𝜇𝑘 [
+
𝑃
].
𝐿
𝑑𝑋 4
𝑑𝑋 2

78

(4-32)

Relative Variation ( % )

Relative Variation ( % )

0.4
0.2
0
-0.2
-0.4
0

0.5
Normalized Coordinate

0.5

0

-0.5
0

1

1

(b) 3rd buckling mode, 𝑝̂ =1000 N
Relative Variation ( % )

Relative Variation ( % )

(a) 1st buckling mode, 𝑝̂ =300 N

0.5
Normalized Coordinate

1
0.5

1

0.5

0

0

-0.5

-0.5
-1
0

0.5
Normalized Coordinate

-1
0

1

(c) 5th buckling mode, 𝑝̂ =2000 N

0.5
Normalized Coordinate

1

(d) 7th buckling mode, 𝑝̂ =2700 N

Figure 4-3: Variation of the axial force in the beam under different loading forces and
corresponding buckling shapes.
Substituting Equations (4-19) in Equation (4-32) the tangential force is expressed in terms of the
unknown coefficients, 𝐴𝑗 , as:
∞

ℎ
𝜇𝑘 𝐹𝑛 (𝑋) = 𝜇𝑘 ( ∑ (𝑃 − 𝑁𝑗2 ) 𝐴𝑗 𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋)
𝐿
𝑗=1,3,5,…
∞

+

∑ (𝑃 − 𝑁𝑗2 ) 𝐴𝑗 (𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋) − 2𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋))).
𝑗=2,4,6,…

79

(4-33)

On the other hand, from Equation (4-4) the displacement gradient over the beam is
expressed as
2

1 𝑑𝜔
̂(𝑥)
𝑢̂ (𝑥) = 𝜀 − (
)
2
𝑑𝑥

(4-34)

′

Integrating Equation (4-34) between 0 and the 𝑥 coordinate yields the axial displacement field as
2

𝑥

𝑝̂
1 𝑑𝜔
̂(𝑥)
𝑢̂(𝑥) = ∫ ( − (
) ) 𝑑𝑥
𝐸𝐴 2
𝑑𝑥
0

(4-35)

Using the non-dimensional variables expressed in Equations (4-8) and (4-18), the normalized axial
displacement field is expressed by:

𝑋

2

∞

𝜕𝑊𝑗 (𝑋)
𝑃 2
1
𝑈(𝑋) =
𝑄 𝑋 − ∫ (∑ 𝐴𝑗
) 𝑑𝑥
12
2 0
𝜕𝑋
𝑗=1

∞

𝑃 2
1 𝑋
=
𝑄 𝑋− ∫ ( ∑
12
2 0

𝐴𝑗 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋)

𝑗=1,3,5,…
2

∞

+

∑

𝐴𝑗 (−2 + 𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋) + 2 𝑐𝑜𝑠(𝑁𝑗 𝑋))) 𝑑𝑥

𝑗=2,4,6,…

80

∞

𝑃 2
1
=
𝑄 𝑋−
12
2

∞

∑
𝑗=1,3,5,…

𝑁𝑗
(2 𝑁𝑗 𝑋 − sin(2 𝑁𝑗 𝑋)) 𝐴𝑗2 + ∑ (−3 + 4 cos(𝑁𝑖 𝑋)
4
𝑖=2,4,6,…

[
−

−2Ni (12 + Ni2 ) 𝑋 + 4 Ni cos(2Ni 𝑋) + 32 sin(Ni 𝑋) + (−4 + Ni2 ) sin(2Ni 𝑋)
) 𝐴2𝑖
4Ni
∞

+2 ∑

∞

∑

Ni N𝑗 (Nj cos(Nj X) sin(Ni X) − Ni cos(Ni 𝑋) sin(Nj 𝑋))
Ni2 − Nj2

𝑖=1,3,5,… 𝑗=3,5,7,…>𝑖
∞

+2 ∑

∞

∑

N2
𝑖=1,3,5,… 𝑗=2,4,6,… i

𝐴𝑖 𝐴𝑗

1
(− cos(Ni 𝑋) (−2Ni2 + 2Nj2 + 2Ni2 cos(Nj 𝑋)
− Nj2

+ Ni2 Nj sin(Nj 𝑋)) + Nj (2Nj + Ni Nj cos(Nj 𝑋) sin(Ni 𝑋)
∞

∞

− 2Ni sin(N𝑖 𝑋) sin(Nj 𝑋))) 𝐴𝑖 𝐴𝑗 + 2 ∑

∑

−2

𝑖=2,4,6,… 𝑗=4,6,8,…>𝑖

(
−

1
Nj 𝑋
Ni 𝑋
4Ni (Ni − Nj )Nj (Ni + Nj ) (−2 + Ni cot ( 2
)) (−2 + Nj cot ( 2 ))

Ni 𝑋 2
(csc (
)
2

Nj 𝑋 2
× csc (
) (−2 + 2 cos(𝑁𝑖 𝑋) + Ni sin(𝑁𝑖 𝑋))(−2 + 2 cos(𝑁𝑗 𝑋)
2
+ Nj sin(𝑁𝑗 𝑋)) (−Ni Nj cos(Nj 𝑋) (2(Ni2 − Nj2 ) + Ni (4 + Nj2 ) sin(𝑁𝑖 𝑋))
+ Ni Nj cos(Ni 𝑋)(−2Ni2 + 2Nj2 + 2(Ni2 − Nj2 ) cos(𝑁𝑗 𝑋) + (4 + Ni2 )Nj sin(𝑁𝑗 𝑋))

+ 4(Ni2 − Nj2 ) (Nj sin(Ni 𝑋) + Ni (−Nj 𝑋 + sin (Nj 𝑋))))) 𝐴𝑖 𝐴𝑗 .
)

81

]

(4-36)

Similarly to the compression force inside the beam, the axial displacement field can be
reduced to the first linear term, P Q2 X/12, and the displacements due to mid-plane rotation can
be neglected. Therefore the energy dissipated due to friction is expressed as
∞

ℎ 𝑃𝑄 2 1
𝑊𝑓 = 𝜇𝑘
∫ 𝑋 ( ∑ (𝑃 − 𝑁𝑗2 ) 𝐴𝑗 𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋) +
𝐿 12 0
𝑗=1,3,5,…

∞

∑ (𝑃 − 𝑁𝑗2 ) 𝐴𝑗
𝑗=2,4,6,…

× (𝑁𝑗2 𝑐𝑜𝑠(𝑁𝑗 𝑋) − 2𝑁𝑗 𝑠𝑖𝑛(𝑁𝑗 𝑋))) (𝐻(−𝑊 (𝑋)) + 𝐻 (𝑊 (𝑋) − 1)) 𝑑𝑋

(4-37)

Due to the existence of the Heaviside function that ensures that the dissipation of energy
occurs only at the contact zones, the integral in Equation (4-37) is computed numerically to
approximate the energy 𝑊𝑓 .

4.5.

Frictionless static model
In this study, the beam is placed adjacent to one of the lateral walls. Hence, it deflects only

in the direction of the other constraint when it buckles in the first mode. Moreover, in this section,
the contact between the beam and the lateral constraints is considered frictionless. Therefore, the
transverse deflection can be expressed as a linear combination of only symmetric modes as
∞

𝑊(𝑋) = ∑ 𝐴𝑗 (1 − 𝑐𝑜𝑠(𝑁𝑗 𝑋))

(4-38)

𝑗=1

where Nj = 2jπ.
The normalized end shortening, work of external force, bending strain energy and
compression strain energy are reduced, respectively, to

82

∞

1
𝑑 = 𝛿 + ∑ A2j Nj2
4

(4-39)

j=1

∞

𝑃
1
𝑉𝑃 = − [𝛿 + ∑ A2j Nj2 ]
2
4

(4-40)

j=1

∞

1
𝑉𝑏 = ∑ A2j Nj4
4

(4-41)

j=1

𝑉𝑐 =

6 2
𝛿
𝑄2

(4-42)

In order to determine the deflected shape of the beam under axial loading, the potential
energy is minimized with respect to Aj. Lateral constraints allow the beam to undergo higher
modes, rather than buckle in the first mode only. The presence of the lateral walls can be modeled
mathematically as a constraint imposed on the transverse deflection. Therefore, the deflection is
always bounded by the distance between the walls. The constrained minimization problem of the
potential energy can be expressed as
𝑀𝑖𝑛 Π(𝐴𝑗 )
{
0 ≤ 𝑊(𝑋) ≤ 1

(4-43)

where Π = Vb + 𝑉𝑐 + 𝑉𝑝 is the normalized total potential energy.
The problem is then solved for the coefficients Aj that will simultaneously generate an
accurate representation of the deformations at each loading state, determine the balance of
energies, and the exact mode transition points. Figure 4-4 displays the normalized deflected shapes
of the beam determined by the presented model under increasing axial load up to the fifth mode.
The dimensions and the material properties of the beam used for all the theoretical results are
summarized in Table 3-1. Under increasing load the beam deflects until it touches the opposite
constraint at a single point (𝑃 = 89), it flattens (𝑃 = 390) and then snaps into the third mode (𝑃 =
83

410). Flattening of the contact zones can be also seen before the transition to the fifth mode (𝑃
=1020).

P = 89

Normalized Deflection

1
0
0
1

P = 390

1

0
0
1

P = 410

1

0
0
1

P = 1000

1

0
0
1

P = 1020

1

0
0

1
Normalized Coordinate

Figure 4-4: Deflected shape for increasing axial load.
4.5.1. Force-control formulation
In this section the beam shown in Figure 4-1 is subjected to a gradually-increased axial
force p̂. Axial shortening Δc due to the axial load can be expressed as
𝛥𝑐 =

𝑝̂ 𝐿
𝐸𝐴

Substituting Equation (4-18) into Equation (4-44) yields

84

(4-44)

𝛿=

𝑃 2
𝑄
12

(4-45)

Inserting Equation (4-45) into Equations (4-39) - (4-42), the non-dimensional midplane stretching
and potential energies are expressed as
∞

𝑃 2 1
𝑑=
𝑄 + ∑ A2j Nj2
12
4

(4-46)

j=1

∞

𝑃2
𝑃
𝑉𝑃 = − 𝑄 2 − ∑ A2j Nj2
24
8

(4-47)

j=1

∞

1
𝑉𝑏 = ∑ A2j Nj4
4

(4-48)

𝑃2 2
𝑉𝑐 =
𝑄
24

(4-49)

j=1

Therefore, the total potential energy can be expressed in terms of P and Aj as
∞

∞

1
𝑃
Π = ∑ A2j Nj4 − ∑ A2j Nj2
4
8
j=1

(4-50)

j=1

Figure 4-5 displays the model predictions and experimental measurements, from the setup
shown in Figure 3-1, of the force-displacement response of the beam under incremental quasistatic axial force loading. A comparison with the study developed by Chai (1998) is presented in
Figure 4-6. The study was summarized in section 2-4. Chai’s model can adequately capture the
system stiffness but the mode transitions are incrementally overestimated and the model becomes
increasingly inaccurate as the number of buckled waves increases. The developed static model,
which is based on the minimization of the potential energy, predicts better the loading forces and
displacements at which transitions occur and has a better agreement with the experimental results.

85

3
Experimental
Theoretical

Axial Force (KN)

2.5

Ø7

2
1.5
Ø5

1
Ø3

0.5
0
0

1
2
3
4
Axial Displacement (mm)

5

Figure 4-5: Theoretical and experimental Force-displacement response under force-control
loading.

7
6

Axial Force (KN)

Ø7

Experimental
Theoretical
Chai's Model

5
4

Ø5

3
2
1
0
0

Ø7
Ø3
Ø5
Ø3

2

4
6
8
Axial Displacement (mm)

10

12

Figure 4-6: Comparison between the presented static model and Chai’s model (Chai 1998).

86

4.5.2. Displacement-control formulation
In this section the beam shown in Figure 4-1 is subjected to a gradually-increasing axial
displacement 𝛥. The axial force 𝑝̂ due to the prescribed axial displacement can be expressed as

𝑝̂ =

𝐸𝐴
1 𝐿 𝑑𝑤 2
(𝛥 − ∫ ( ) 𝑑𝑥)
𝐿
2 0 𝑑𝑥

(4-51)

Substituting Equation (4-18) into Equation (4-51), the non-dimensional axial force is expressed by
∞

12
1
𝑃 = 2 (𝑑 − ∑ A2j Nj2 )
𝑄
4

(4-52)

j=1

Substituting Equation (4-52) into Equations (4-39) - (4-42), the non-dimensional midplane
stretching and potential energies are expressed as
∞

1
𝛿 = 𝑑 − ∑ A2j Nj2
4

(4-53)

j=1

∞

6
𝑑
𝑉𝑃 = − 2 (𝑑 2 − ∑ A2j Nj2 )
𝑄
4

(4-54)

j=1

∞

1
𝑉𝑏 = ∑ A2j Nj4
4

(4-55)

j=1

∞

6
𝑑
𝑉𝑐 = 2 (𝑑2 − ∑ A2j Nj2 )
𝑄
2

(4-56)

j=1

Therefore the total potential energy can be expressed in terms of d and Aj as
∞

∞

1
3 𝑑
Π = ∑ A2j Nj4 −
∑ A2j Nj2
4
2 𝑄2
j=1

j=1

87

(4-57)

Theoretical and experimental force-displacement responses of the beam subjected to
incremental quasi-static axial displacement loading are displayed in Figure 4-7. The response
predictions from the developed model correlate with the actual observed experimental behavior.

3

Axial Force (KN)

2.5

Experimental
Theoretical

Ø7

2
1.5
Ø5

1
0.5
0
0

Ø3

1
2
3
4
Axial Displacement (mm)

5

Figure 4-7: Theoretical and experimental force-displacement response under displacementcontrol loading.

4.6.

Dynamic analysis
Snap-through buckling from one mode into a higher or lower mode configuration is

induced when the total energy transitions through an unstable path to a lower value represented by
a different stable geometric configuration. The difference in energies is released as kinetic energy.
A dynamic model was then developed to investigate the released kinetic energy as well as
accelerations of the beam during transitions. The presented model was developed based on the
force-control formulation presented in Section 3.5.1, but displacement-control can also be used for
this study.
88

In this analysis friction is neglected. Therefore, the total energy at any equilibrium state is
the sum of the kinetic and the potential energies. The normalized kinetic energy, K, is expressed
in Equation (4-22) and the total potential energy, Π, is the sum of the potential energy due to
external force and the axial compression and bending strain energies expressed, respectively, in
Equations (4-25), (4-26) and (4-28). The constrained minimization problem of the total energy can
be expressed as
𝑀𝑖𝑛 K + Π
{
0 ≤ 𝑊(𝑥) ≤ 1

(4-58)

The beam is subjected to a linearly increasing and decreasing load, 𝑝̂ , such that

𝑝̂ = ||

𝑡̂
𝑇/2

𝑖𝑓 0 ≤ 𝑡̂ ≤ 𝑇/2

𝑡̂
𝑝̂𝑚𝑎𝑥 (2 −
)
𝑇/2

𝑖𝑓 𝑇/2 ≤ 𝑡̂ ≤ 𝑇

𝑝̂𝑚𝑎𝑥

(4-59)

where T is the loading period, 𝑝̂𝑚𝑎𝑥 is the maximum applied load and 𝑡̂ is the time coordinate. The
variation of the axial loading force, 𝑝̂ , with respect to time, 𝑡̂ is displayed in Figure 4-8.

0
0

T/2

T

Figure 4-8: Variation of axial loading force, 𝑝̂ .

89

3.5

V-VII

Axial Force (KN)

3
2.5

VII-V

2

III-V

1.5
1

I-III

0.5
0
0

V-III

III-I
1

2
3
4
Axial Displacement

5

6

Figure 4-9: Dynamic force-displacement response under force-controlled cyclic loading.
Figure 4-9 displays the simulated dynamic force-displacement response of the beam under
cyclic force loading. The loading period, T, is 10 s and the maximum applied force, 𝑝̂𝑚𝑎𝑥 , is 3500
N. Results show that, during unloading, the beam returns to a lower buckling mode through a
different equilibrium path. The symbols I-III, III-V, V-VII, VII-V, V-III and III-I denote the
transitions between the buckling modes; for example I-III denotes the transition from the first into
the third mode. The normalized axial displacement, 𝑑, kinetic energy, 𝐾, and accelerations, 𝐴,
over a loading cycle are presented in Figure 4-10. The normalized acceleration is given by

𝐴=

𝑚 𝐿4
𝜕 2 𝑊(𝑋, 𝑡)
𝐴̂ =
ℎ𝐸𝐼
𝜕𝑡 2

(4-60)

̂ (𝑥, 𝑡̂)/𝜕𝑡̂ 2 is the transverse acceleration. Snap-through transitions are witnessed
where 𝐴̂ = 𝜕 2 𝑊
by the jumps in the axial displacement response and the spikes in the kinetic energy and
accelerations.

90

Nondimensional Axial Displacement, d

600
500
400
300
200
100
0
0

2

4
6
8
Nondimensional Time, t

10

(a)

Nondimensional Kinetic Energy, K

1500

1000

500

0
0

2

4
6
8
Nondimensional Time, t

10

(b)

Figure 4-10: (a) Displacement response, (b) released kinetic energy and (c) generated
accelerations during one loading cycle.
91

Nondimensional Acceleration, A

Figure 4-10 (cont’d)

50

0

-50
0

2

8
6
4
Nondimensional Time, t

10

(c)

The deflected shapes of the beam before and after each transition are shown in Figure 411. It can be noticed that the deflected beam is not symmetric and the snap does not always occur
at the same location. Figure 4-12 displays the distance, 𝑊𝑅 = 𝑊𝑓 −𝑊𝑖 , that the beam travels at
each transition where 𝑊𝑖 and 𝑊𝑓 represent, respectively, the deflected shapes before and after
transitions. The normalized values and locations of the maximum relative displacement at
transitions are presented in Table 4-1. Opposite transitions occur at the same location with opposite
maximum displacement sign except for I-III and III-I. For example, in transitions V-VII and VIIV, maximum displacements are located at 0.37 X-coordinate, but with opposite sign. Similarly,
transitions III-V and V-III have maximum displacements located approximately at 0.68 Xcoordinate. However, transitions I-III and III-I occur at different locations (0.21 and 0.5,
respectively) with the same maximum displacement sign.
92

Normalized Deflection

Normalized Deflection

1

0.5

0
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

1

0.5

0
0

1

0.5

0
0

1

0.5

0.2
0.4
0.6
0.8
Normalized Coordinate

1

0
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

(d) after transition III-V.

1

Normalized Deflection

Normalized Deflection

(c) before transition III-V.
1

0.5

0.5

0
0

1

(b) after transition I-III.
Normalized Deflection

Normalized Deflection

(a) before transition I-III.

0.2
0.4
0.6
0.8
Normalized Coordinate

0.2
0.4
0.6
0.8
Normalized Coordinate

1

(e) before transition V-VII.

0
0

0.2
0.4
0.6
0.8
Normalized Coordinate

(f) after transition V-VII.

Figure 4-11: Deflected shapes before and after each mode transition.

93

1

1

Normalized Deflection

Normalized Deflection

Figure 4-11 (cont’d)

0.5

0
0

1

0.5

0.8
0.6
0.4
0.2
Normalized Coordinate

1

0
0

1

0.5

0
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

1

0.5

0
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

(j) after transition V-III.

1

0.5

Normalized Deflection

Normalized Deflection

(i) before transition V-III.

0
0

1

(h) after transition VII-V.

Normalized Deflection

Normalized Deflection

(g) before transition VII-V.

0.2
0.4
0.6
0.8
Normalized Coordinate

1

0.5

0.2
0.4
0.6
0.8
Normalized Coordinate

1

(k) before transition III-I.

0
0

0.2
0.4
0.6
0.8
Normalized Coordinate

(l) after transition III-I.

94

1

Relative Displacement

1
0.5
0
-0.5
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

Relative Displacement

(a) Relative displacement, WR, at transition I-III.

1
0.5
0
-0.5
-1
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

Relative Displacement

(b) Relative displacement, WR, at transition III-V.

1
0.5
0

-0.5
-1
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

(c) Relative displacement, WR, at transition V-VII.
Figure 4-12: Relative displacement, 𝑾𝑹 = 𝑾𝒇 − 𝑾𝒊 , at mode transitions where 𝑾𝒊 and 𝑾𝒇
represent, respectively, the deflected shapes before and after transitions.
95

Relative Displacement

Figure 4-12 (cont’d)

1

0.5
0

-0.5
-1
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

Relative Displacement

(d) Relative displacement, WR, at transition VII-V.

1
0.5
0

-0.5
0

0.2
0.4
0.6
0.8
Normalized Coordinate

1

Relative Displacement

(e) Relative displacement, WR, at transition V-III.

1
0.5
0

-0.5
0

0.2
0.4
0.6
0.8
Normalized Coordinate

(f) Relative displacement, WR, at transition III-I.

96

1

Table 4-1: Value and location of the maximum displacement at transitions.
Maximum displacement

Normalized X-coordinate of

during transitions

the maximum displacement

I-III

0.997

0.21

III-V

-0.984

0.69

V-VII

1

0.37

VII-V

-1

0.37

V-III

0.895

0.67

III-I

0.942

0.5

Mode transitions

The points located at 0.21, 0.37, 0.5 and 0.69 exhibit the maximum displacements during
mode transitions. Their accelerations over the loading cycle are displayed in Figure 4-13. Results
show that the accelerations generated during loading are much higher than unloading. Moreover,
the accelerations generated during the transitions III-V and V-VII are comparable (Figure 4-13 (b)
and (d)) but higher than I-III. The location X = 0.37 snaps only in the transitions between the fifth
and seventh mode and the location X = 0.21 provide low accelerations during mode switching.
Therefore locations X = 0.5 and X = 0.69 are better candidates to attach the energy scavenger to
the bilaterally constrained beam.

97

Nondimensional Acceleration, A

10
0
-10
-20
0

2
4
6
8
Nondimensional Time, t

10

Nondimensional Acceleration, A

(a) Accelerations at X=0.21.

50

0

-50
0

2
4
6
8
Nondimensional Time, t

10

(b) Accelerations at X=0.37.
Figure 4-13: Non-dimensional accelerations at the locations (a) X = 0.21, (b) X = 0.37,
(c) X = 0.5 and (d) X = 0.69 during a loading cycle of 10 s.

98

Nondimensional Acceleration, A

Figure 4-13 (cont’d)

20

0

-20
0

2
4
6
8
Nondimensional Time, t

10

Nondimensional Acceleration, A

(c) Accelerations at X=0.5.

50

0

-50
0

2
4
6
8
Nondimensional Time, t
(d) Accelerations at X=0.69.

99

10

4.7.

Summary
The post-buckling response of a bilaterally constrained beam subjected to axial force and

displacement was theoretically investigated using energy methods. In contrast to previous studies
that are based on geometric elliptic integrations, this study was based on the minimization of
energy under constraints. The theoretical model presented in this work shows that energy methods,
based on the physics of the device, significantly improve understanding of the system behavior
and improve the correlation with respect to experimental measurements. The model accounts for
longitudinal deformations and friction between the beam and its lateral constraints.
Using the Galerkin discretization method, the deflection is expressed as a linear
combination of the buckling eigen modes. The total energy is then minimized under the
confinement constraint in order to determine the unknown coefficients multiplying the buckling
modes. The static response of the beam was investigated using force- and displacement-control
formulations. The force-displacement response as well as the deflected shapes under axial loadings
were determined. Results show that the model is able to adequately predict the beam geometry at
every loading stage, including the flattening behavior just before the snap buckling transitions, and
simultaneously determines the mode transition events.
A dynamic analysis based on a force-control formulation was carried out to investigate the
released kinetic energy as well as accelerations of the beam during transitions. The beam was
subjected to a cyclic triangular axial force and the constrained minimization problem of the total
energy was solved. It was shown that the beam returns to lower buckling configurations through a
different equilibrium path. Moreover, the buckling transitions create high kinetic energy and
acceleration spikes. However, the location of maximum acceleration differs from one transition to
another. Hence, placement of the piezoelectric energy harvester should be carefully chosen
100

depending on the application. The dissipated energy due to friction was formulated but not
included in the analysis because the computation becomes very expensive. Future developments
will address the optimization of the algorithm to investigate friction effects.

101

CHAPTER 5: ENERGY HARVESTING AND SENSING BASED
ON QUASI-STATIC MECHANICAL ENERGY
CONCENTRATORS AND TRIGGERS 3

5.1.

Introduction
Many real world-physical processes occur at very low frequencies and acceleration levels.

Examples of these processes comprise, among others, temperature and pressure variation between
day and night as well as bone healing and spinal fusion in biomedical engineering. These processes
induce a stress/strain response that occur at frequencies lower than 1 mHz. Monitoring the
evolution of quasi-static strain is important and could provide significant benefits. For instance,
the history of mechanical loading inside different structures could be used to predict the life
expectancy of the structure. This chapter presents a battery-less sensor design that can self-power
by harvesting energy directly from the quasi-static events being sensed.
The technical challenge is illustrated for a piezoelectricity driven self-powered sensor
whose generic architecture is shown in Figure 5-1 (a). The sensor comprises of a piezoelectric
transducer that powers a minimal set of electronic modules by harvesting energy from ambient
strain variations. Typically the sensor electronics comprises of: (a) a rectification module to extract
energy from the transducer; (b) a triggering module that detects events of interest; and (c) a datalogging module that records events either on a non-volatile memory or using remote data

3

Results shown in this chapter were published in Lajnef et al. (2013, 2014a, 2014b, 2014c,

2014d and 2014e).
102

transmission. When the piezoelectric transducer is excited quasi-statically (as shown in Figure 51 (b) by the triangular wave) the load voltage generated by the transducer is approximately constant
as shown in Figure 5-1 (b). Assuming a nominal sized piezoelectric transducer, the power levels
that can be scavenged from mHz strain-signal would be in the order of pico-watts (Sarkar et al.
2013). Unfortunately, the majority of the charge generated by the transducer is lost as leakage
through the electronics (for example through diode leakage) and the residual energy is insufficient
to drive the rest of the sensor modules (triggering and data-logging modules).

Figure 5-1: (a) System architecture of a generic piezoelectricity driven self-powered sensor; and
(b) adverse effect of electronic leakage on energy scavenging for quasi-static excitation.

Figure 5-2: (a) System architecture of the proposed self-powered sensor comprising of
mechanical energy concentrators; and (b) reduction of electronic leakage during the impulsive
energy release of the energy concentrators.
This chapter presents the use of mechanical energy concentrators and triggers as a frontend to circumvent the challenge of quasi-static self-powering. The architecture of the proposed
sensor is shown in Figure 5-2 (a) and its principle of operation is illustrated for a simplified loading
condition in Figure 5-2 (b). The role of the mechanical energy concentrator is to harvest potential

103

energy over a long time-span and release the stored energy as bursts and impulses as shown in
Figure 5-2 (b). The impulsive nature of the energy-release not only results in piezoelectric output
voltage levels that are significantly higher than the triggering but also reduces the effect of energy
leakage as shown in Figure 5-2 (b).
This chapter is organized as follows: Section 5.2 presents the principle of operation of the
proposed self-powered sensor and Section 5.3 describes the design and characterization of the
front-end mechanical energy concentrator. Section 5.4 briefly describes the sensor electronics used
for self-powered data-logging and present measurement results obtained from fabricated
prototypes.

5.2.

Principle of operation
The physics and principle of operation of the sensor is illustrated in Figure 5-3 (a)-(c) using

a bilaterally constrained elastic beam which is subjected to a quasi-static force F. Figure 5-3 (d)
shows an energy diagram schematic that indicates the potential energy stored in the elastic beam
as the magnitude of the quasi-static force (F) is increased. Note that under quasi-static loading
conditions no energy is dissipated as kinetic energy or heat except during snap buckling events.
Assuming that the beam is initially in an energy configuration denoted by state A, increasing the
magnitude of the input signal shifts the configuration to a higher potential energy state B. The state
B corresponds to the position of the beam shown in Figure 5-3 (a). Because the beam is bilaterally
constrained (at point Y), increasing the magnitude of the input signal (F) buckles the beam to
another state shown in Figure 5-3 (b). On the energy-diagram shown in Figure 5-3 (d), the
mechanical buckling results in a negative-energy slope at state B, where part of the stored potential
energy is released instantly as kinetic energy, before the beam buckles to a lower energy state C.

104

If a piezoelectric transducer is attached to the structure, part of the kinetic energy is transferred to
the transducer which is then scavenged for sensing and computation. The impulsive nature of the
energy-release not only results in higher piezoelectric output voltage (due to the high frequency
transition) but also significantly reduces the effect of electronic leakage.

Figure 5-3: Principle of operation of the proposed self-powered sensor: (a)-(c) different buckling
states of a bilaterally constrained elastic beam; (d) energy diagram illustrating how the released
potential energy induces hot-electrons in a floating-gate transistor channel.
The engineering challenge is to achieve an optimal tradeoff between the energy released
during buckling (which is related to the material properties of the beam, the end boundary
conditions and the lateral constraints) and the energy required for sensing and computation. One
possible way to optimize this trade-off is to exploit the physics of mechanical buckling and the
physics of piezoelectricity driven impact-ionized hot electron injection. This principle is also
illustrated using the energy-band diagram in Figure 5-3 (d) where the piezoelectric transducer
converts the released kinetic energy into high-energy electrons (or hot-electrons) in a floating-gate
transistor (Huang at al. 2010). If the energy of some of these electrons (with the right momentum
vector) exceeds the energy barrier (3.2eV) of the silicon, silicon-di-oxide interface (as shown in
105

Figure 5-3 (d)), these electrons surmount the barrier and get trapped onto a floating-gate. The
concentration of the trapped electrons can be measured to estimate the number of times the
magnitude of the input signal exceeded a mechanical threshold (for instance L1 or L2 in Figure 53(d)). Huang at al. 2010 showed that the piezoelectricity driven impact ionized hot-electron
injection (p-IHEI) process can be initiated at pico-Joule energy density, which is well matched to
the energy released during mechanical buckling of a nominally sized elastic beam. The beauty of
this physics-based sensing approach is that it completely eliminates the need for voltage regulation,
energy storage, analog-to digital converters (ADCs), micro-controller units (MCUs) and randomaccess memories (RAMs) and hence can be used to push the fundamental limits of self-powered
sensing using sub-Hz signals.

5.3.

Mechanical energy concentrators and triggers
Mode jumping between stable configurations is related to a partial release of the strain

energy stored in the compressed beam. The energy drop is converted primarily into kinetic energy
allowing the beam to move from one buckled configuration to another more stable configuration
under the increasing axial loading. The snap-through response between the equilibrium positions
induces an acceleration, which creates an input excitation to an energy harvesting oscillator
transducer. Thus, the low-rate strain variations in the axial direction of the beam are transferred
into high-rate transverse acceleration that constitutes the input for the transducer element.
The design of the mechanical energy concentrator follows the test setup described in
Section 3.2.1. The capability of the system to harvest energy under quasi-static excitations was
experimentally investigated. In order to harvest the kinetic energy released during snap-through
transitions, A PVDF Piezoelectric energy-harvester was attached at the buckled beam’s mid-span

106

perpendicular to its longitudinal axis (see Figure 3.1) using an off-the-shelf 5 minute epoxy (J-B
weld 5 min epoxy). The harvester was configured in a cantilever configuration and only one
piezoelectric element was used for the design. However, multiple transducer elements with
different properties as well as other piezoelectric materials (PZT, Microfiber Composites, etc.) can
be used for enhancing the energy transfer. The snap-through buckling of the axially loaded beam
between equilibrium positions induces acceleration at the base of the cantilever, thus transferring
the low-rate strains in the buckled beam’s axial direction into high-rate accelerations input for the
harvester. The post-buckling behavior of the axially loaded bilaterally constrained beam was
described in Chapter 3.The properties of the PVDF film are detailed in Table 5-1. The length and
tip mass of the oscillator were varied to modify its natural frequency. The polycarbonate beam was
subjected to displacement-controlled low-frequency cyclic loading tests. The input frequency was
varied to investigate its effect on the levels of the extractable energy. A 10 MΩ - impedance
oscilloscope was used to record the output voltage of the piezoelectric harvester.
Table 5-1: Properties of the piezoelectric vibrator.
Properties (Piezoelectric PVDF / Substrate )
Elastic Modulus

2 GPa / 2.4 GPa

Density

1780 kg/m3/ 1390 kg/m3

Thickness

28 µm / 205 µm

Width

12 mm / 16 mm

Piezoelectric Constant (d31)

23e-12m/V

Capacitance

2.8nF

Electrical permittivity

115 e-12F/m

107

Figure 5-4 displays the variation of the axial force in the compressed beam and the output
piezoelectric voltage of a full loading/unloading cycle. The first voltage oscillation is caused by
the first contact between the beam and the rigid constraints, which correspond to the first buckling
mode. The other successive voltage output events are generated by the snap-through of the beam
at mode transitions. The high-rate events induce free-vibration in the piezoelectric element. Thus,
the output frequency is the natural frequency of the piezoelectric energy harvester controlled by
the PVDF film stiffness and the lumped mass. This figure also shows the frequency up-conversion
capability of the mechanism where the frequency of the input displacement is 0.05 Hz and the

Axial Force
Output Voltage

8

Axial Force (N)

2000

4

1000

0

0

-4

-1000
0

5

10

15

Piezoelectric Output Voltage (V)

output frequency of the generated voltage is 6.667 Hz.

20

Time (s)

Figure 5-4: Piezoelectric output voltage (output frequency = 6.6667Hz) generated for a full load
cycle at 0.05 Hz.

108

Figure 5-5 shows the computed energy generated by a cantilever piezoelectric PVDF beam
measured across a 10 MΩ resistor. The natural frequency of the vibrator was varied from 10 to 90
Hz by modifying the effective length of the PVDF film and the tip mass. The input deformation
(axial displacement) rate was varied by increasing the loading/unloading cycle period from 6 s to
150 s, which corresponds to lowering the frequency of the input deformation from 0.16 Hz to 0.006
Hz. The obtained results confirm that energy can be transferred at extremely low-rate deformation
input. The input frequency has little influence on the levels of harvested energy. The energy is
mainly controlled by the natural frequency and electro-mechanical properties of the scavenger.
The levels of harvested energy as well as the output signal frequency can be tuned (optimized) by
controlling the piezoelectric element and the constrained beam properties. It should be restated
that other piezoelectric materials (PZT, MFC) can be used for enhanced energy transfer.

2

Energy (µJ)

1.6

1.2

0.8
Harvester's Natural
Frequency

6.66Hz
37.88Hz
89.29Hz

0.4

-2

10

-1

Input Frequency (Hz)

10

Figure 5-5: Harvested energy for different piezoelectric elements and input loading frequencies.
109

In a second set of experiments the constrained beam was loaded past its fifth buckling mode
(Φ5) and the attained equilibrium position was selected as the initial configuration for this set of
tests. Loading cycles were then applied between a maximum load past the seventh mode (Φ7)
equilibrium position and a minimum load below the third stable configuration (Φ3). The applied
axial load and the voltage output from the PVDF oscillator are shown in Figure 5-6 as function of
time. In the figure, the labels Φ3 to Φ7 indicate the positions of snaps or mode transitions. The
input axial load was varied at a frequency of 0.11 Hz. At this very slow rate, the voltage response
is driven by the natural frequency of the piezoelectric beam (free vibration between transitions),
tuned at 6 Hz for this test. This test clearly shows that the system can be initially centered
(positioned) so as to harvest energy from compressive as well as from tensile input global loading

5

3,000

Output Voltage
Axial Force

Axial Force (N)

Ø7
2,000
0

Ø5

1,000

Ø5
Ø3
0
0

2

4

Time (s)

6

8

Piezoelectric Output Voltage (V)

deformations.

-5
10

Figure 5-6: Frequency up-conversion principle for alternating loading between the third and the
seventh modes at 0.11 Hz. The fifth mode was used as the initial equilibrium position.

110

5.4.

Self-powered piezo-floating-gate sensing and data logging
The circuit level schematic and the principle of operation of a linear p-IHEI based data

logger is shown in Figure 5-7. The circuit consists of a floating-gate pMOS transistor Mfg whose
source is driven by a constant current source Iref that is powered by either a piezoelectric transducer
or by some other energy source Vdda. Note that both the energy sources are isolated by a diode,
which allows Vdda to supersede the signal generated by the piezoelectric transducer. Also note that
a full-bridge rectifier (formed by four diodes) is used for extracting energy from the transducer
and drive the constant current source Iref. The polysilicon gate of the pMOS transistor is electrically
insulated by silicon-dioxide (hence the name “floating-gate”); therefore, any electron injected onto
the gate is retained for a long period of time (8 bits precision for 8 years) (Diorio et al. 1996).
Electrons are injected onto the floating-gate using an impact-ionized hot-electron injection (IHEI)
process that involves applying Vsd > 4.2V (in 0.5-μm CMOS process) across the source and the
drain terminal. The large electric field near the drain of the pMOS transistor creates impact-ionized
hot-electrons. When the energy of these electrons exceeds the gate-oxide potential barrier
(≈3.2eV) they can get injected onto the floating-gate. IHEI current, Iinj, in a pMOS transistor is
dependent on the transistor source current Is, the source to- drain voltage Vsd and the gate-to-drain
voltage Vgd across the transistor. This dependence can be expressed in functional form as:
𝐼𝑖𝑛𝑗 = 𝑓(𝐼𝑠 , 𝑉𝑠𝑑 , 𝑉𝑔𝑑 )

(5-1)

where f (.) is an arbitrary function that could be empirically determined (Huang et al. 2011).
However, the circuit in figure. 5-7 (a) achieves stable and ultra-linear injection using a negative
feedback loop formed by the opamp A and the floating-gate transistor Mfg. The source current is
held constant at Iref, which ensures that the source-to-gate voltage Vsg remains constant during
injection. When switch SP is open, the feedback is enabled and opamp A ensures that the source111

to-drain voltage Vsd is held constant to Vref. Thus, according to Equation (5-1) the injection current
Iinj remains constant. The amount of charge injected onto the floating-gate and hence the decrease
in floating-gate voltage Vfg is proportional to the duration for which the source current Is is
activated and SP is open. This can be expressed as:

Δ𝑉𝑓𝑔 =

𝐼𝑖𝑛𝑗
1 𝑇
∫ 𝐼𝑖𝑛𝑗 𝑑𝑡 =
𝜏(𝑇)
𝐶𝑇 0
𝐶𝑇

(5-2)

where 𝜏 is the duration of injection and 𝐶𝑇 is the total floating-gate capacitance, which includes
the capacitance Cfg, tunneling capacitance and other parasitic capacitances associated with the
floating node. The change in floating-gate voltage Δ𝑉𝑓𝑔 could be measured by closing the switch
SP, as shown in Figure 5-7 (b), which breaks the feedback loop by shorting the other terminal of
Cfg to ground. Because the source current Iref is constant, Δ𝑉𝑠 = Δ𝑉𝑓𝑔 which is read-out through a
unity-gain buffer. Figure 5-8 shows the measured response of a linear injector where the source
voltage Vs is first initialized to 4.3V (using FN tunneling), Vref = 4.8V and Iref = 30nA. It shows
that the response follows the linear model almost throughout the range. The piezoelectric
transducer is emulated by applying a 50ms long pulse signal (amplitude Vdd =6.5V) after which
the switch SP is turned ON and the source voltage Vs is measured. Figure 5-8 shows that the change
in Vs is linear with respect to the number of applied pulses. The deviation from the linear injection
model occurs at the end points of the operating voltage and is due to the finite operating range of
the amplifier A. This shows that the linear injector has a linear range of almost 4V. The resolution
of the linear injector was measured to be greater than 13 bits (Huang et al. 2011).

112

Figure 5-7: Schematic of a linear p-IHEI based data logging circuit: (a) when operating in the
data logging mode; and (b) when operating in a data read-out mode (Lajnef et al. 2014b).

113

Figure 5-8: Measured output range and the linearity for linear injection circuit (Sarkar et al.
2013).

Figure 5-9: Micrograph of a fabricated prototype integrating an array of linear floating-gate
injectors along with digital command-control and programming circuits (Lajnef et al., 2014b).
114

Figure 5-9 shows the micrograph of a p-IHEI based data logging circuit which was
fabricated in a 0.5 μm CMOS process. The prototype integrates an array of linear floating-gate
injectors that can be programmed using on-chip high-voltage charge pumps. Programming,
selection and read-out of the injectors can be controlled using digital command and control, which
is implemented on-chip by an integrated digital processing module.
Two sets of experiments were conducted to validate the behavior of the described quasistatic sensing functionalities. In the first runs, the system was cycled between only two equilibrium
positions (loading then unloading). The actuator, which applied the external force, was moved at
a frequency of 0.1 Hz and then 0.5 Hz while the induced oscillations at the PVDF beam were
measured at 12.82 Hz. Figure 5-10 shows the applied input deformation (for a 0.1 Hz loading) and
the rectified voltage output generated by the vibrating piezoelectric element. It is noted from the
levels of generated voltage that the released energy is higher during loading when the applied force
(added potential) is being increased. The combined piezoelectric-floating-gate event counting
device was calibrated to detect and record voltages higher than 4V. The impact ionized hotelectron injection process at the floating gate memory cell is activated only when the voltage
exceeds the preset threshold. For each event the gate is thus active for about 0.2 seconds. The
voltage variations at the floating gate was measured periodically and is shown in figures 5-11 and
5-12. The results in Figure 5-11 were recorded for cyclic input loading events applied at 0.1 Hz
while the measurements in Figure 5-12 were obtained for an input loading applied at 0.5 Hz. It is
observed that the system clearly implements a linear injector allowing to directly relate the number
of applied events to the measured voltage variation at the floating gate output.

115

Figure 5-10: Measured output piezoelectric voltage generated for loading cycles at 0.1 Hz,
showing the system’s ability for energy generation under quasi-static loading.

Figure 5-11: Recorded variations of the voltage at the analog floating gate for input cyclic
loading events at 0.1Hz.
116

Figure 5-12: Recorded variations of the voltage at the analog floating gate for input cyclic
loading events at 0.5 Hz.
In the next set of experiments the restrained beam was initially loaded past its fifth buckling
mode which was selected as the starting configuration. Loading cycles were then applied between
a maximum load past the seventh mode equilibrium position and a minimum load below the third
stable configuration. The mode shapes are shown in Figure 3-2. The applied axial load, axial
deformation, and the rectified voltage output from the PVDF transducer are shown in Figure 5-13
as function of time. The input load varies at a frequency of about 0.1 Hz. At this rate the voltage
response is driven by the natural frequency of the piezoelectric harvester. For this case the PVDF
beam was tuned to 6 Hz. The combined piezoelectric-floating- gate event counting device was
again calibrated to detect and record voltages higher than 4 V. For these specific experimental
runs, the preset threshold is exceeded only during the snap-buckling from the third to the fifth

117

mode as clearly shown in Figure 5-13. This translates into a calibrated system that records only
these transitions. The difference in voltage output between transitions is due to the levels of stored
strain energy in the system before the transition events and can be tuned and calibrated as shown
experimentally in Chapter 3. Figure 5-14 shows the recorded voltage variations at the linear
injector. The sensor clearly implements a linear recording of events allowing direct
correspondence between the number of applied events and the measured sensor output voltage.

Figure 5-13: Illustration of the energy conversion principle under cyclic loading at 0.1 Hz for the
bilaterally constrained beam configuration.

118

Figure 5-14: Recorded variations of the voltage at the analog floating gate for input cyclic
loading events at 0.1 Hz.

5.5.

Summary
This chapter presented a novel self-powered sensing mechanism that combines the physics

of mechanical buckling in bilaterally constrained elastic beams with the physics of piezoelectricity
driven hot-electron injection. During the snap-through events, part of the stored strain energy is
released as kinetic energy. The piezoelectric transducer that is attached to the buckling element
converts the released kinetic energy into high-energy electrons in a floating-gate transistor. The
concentration of the trapped electrons can be measured to determine the number of times the
magnitude of the input signal exceeded a mechanical threshold.
Integrating mechanical energy concentrators and triggers with the piezo-floating gate
sensor allows for harvesting the released energy as bursts and impulses. The impulsive nature of
119

the energy-release not only results in piezoelectric output voltage levels that are significantly
higher than the triggering but also reduces the effect of energy leakage.
The piezo-floating-gate data logging is used as a counting device. The impact ionized hotelectron injection process at the floating gate memory cell is activated only when the voltage
exceeds the preset threshold. The levels of the generated voltage differ from one mode transition
to another. Therefore the system can be calibrated to record some specific events. The system
implements a linear injector allowing to directly relate the number of applied events to the
measured voltage variation at the floating gate output.
As shown in Chapter 3, the input frequency has little influence on the occurrence of the
snap-buckling events. Therefore, the generation and levels of energy bursts are independent of the
input frequency. Results confirm that the energy can be transferred at extremely low-rate
deformation input. Moreover, the mechanism allows for frequency up-conversion from the low
input frequency (in the order of mHz) to the natural frequency of the piezoelectric scavenger
determined by its stiffness and the attached tip mass.
As a result, the proposed sensor was demonstrated to self-power at mHz frequencies and
with the potential to scale down to μHz frequencies, a feature that is not possible with other selfpowered sensing approaches.

120

CHAPTER 6: CONCLUSIONS

6.1.

Research contributions
6.1.1. Post-buckling response of bilaterally constrained beams using an energy method
The models developed based on geometric assumptions and geometric elliptic integrations

are able to capture the system stiffness. However, they fail to accurately predict the levels of forces
and displacement at which transitions occur, especially as the number of buckled waves increases.
The developed static model, which is based on the minimization of the potential energy, is able to
adequately predict the mode transition events as well as the beam geometry at every loading stage,
including the flattening behavior just before the snap buckling transitions.
6.1.2. Dynamic response and transitions in multiple bifurcation points mechanical energy
concentrators
This research presented a study of the mechanical energy concentrators’ dynamic response
under cyclic axial loading. The analysis was based on the minimization of the total energy under
the prescribed boundary conditions and constraints. The developed model assesses the dynamic
force-displacement response of the system as well as the released kinetic energy and generated
transverse accelerations at buckling mode transitions.
6.1.3. Energy transfer at quasi-static excitations using mechanical energy concentrators
This research presented a technique to harvest energy at very low frequencies (less than 1
Hz) using the snap-through behavior between the multiple equilibrium positions of axially-loaded
bilaterally-constrained beams. The approach relies on the high-rate motions that are generated
121

during the transitions between the system’s multiple potential energy wells. The input is not a
vibration motion but a slow quasi-static deformation. Unlike the developed vibration-based energy
harvesters, the proposed mechanism converts the slow-rate input strain into high-rate accelerations
generated at buckling mode transitions. During these transitions part of the stored potential energy
is released as kinetic energy. The role of the piezoelectric energy harvester is to convert the
released kinetic energy into voltage that can be stored or used for computation and sensing.
6.1.4. Self-powered sensing at micro Hertz
This research introduced a novel technique for self-powered sensing at very low
frequencies by integrating mechanical energy concentrators and triggers with piezo-floating gate
sensors. Piezo-floating-gate data logging is used as a counting device that records the number of
times the magnitude of the input signal exceeded a mechanical threshold. Since the post-buckling
response of bilaterally constrained beams is independent of the input frequency, the same levels
of energy bursts are always generated regardless the input frequency. Therefore the proposed
sensor is able to self-power at µHz frequencies, a feature that is not possible with other selfpowered sensing approaches.

6.2.

Conducted work
The post-buckling response of an axially loaded bilaterally constrained beam was

investigated experimentally, numerically and theoretically. It was shown that due to lateral rigid
walls, the elastica is able to undergo higher buckling modes as the axial load increases. Under
variable axial loading the beam transitions between the different equilibrium states of the system.
The transitions between the buckling modes always occur at the same axial force and displacement
levels. Furthermore the behavior is independent of the excitation frequency. However, the response
122

of the system is highly dependent of its parameters such as thickness and length of the beam,
modulus of elasticity, gap between the rigid walls, position of the beam and boundary conditions.
These parameters affect the axial displacements and forces at which transitions occur and the
accelerations generated during buckling mode switching. The effect of these parameters was
numerically investigated using an FE model. It was shown that placing the beam adjacent to one
of the lateral walls with fixed-end supports generates higher accelerations. Furthermore the
accelerations amplify by increasing the thickness, Young’s modulus or gap between the walls, or
by decreasing the beam length. However, the combination between the length and one of the other
parameters may have the opposite effect. Therefore to reduce the beam length, the thickness,
Young’s modulus and gap should be upper-bounded in order to allow the development of higher
modes and hence the generation of high accelerations.
Although tuning the parameters of the system affects dramatically the accelerations
generated in snap-through transitions, the number and the spacing between these events was shown
to be non-sensitive to the studied parameters. In order to control the number and the spacing
between the mode transitions, multiple slender beams were stacked into a system in a parallel
configuration. It was shown that any modification of the beams’ thicknesses, Young’s moduli and
gaps between the walls affects the locations of the transitions and the overall stiffness of the
system. More important is that the number and the spacing between these events can be controlled
by tuning the aforementioned parameters without sacrificing the levels of the accelerations.
A theoretical model using energy methods was developed to investigate the post-buckling
response of a bilaterally constrained beam subjected to axial force or displacement. The model
accounts for longitudinal deformations and friction. The total energy was minimized under the
confinement constraint in order to determine the deflected shape, transitions states and energy
123

balance at transitions.

The static response of the beam was investigated using force- and

displacement-control formulations. Results show that the model is able to adequately predict the
beam geometry at every loading stage, including the flattening behavior just before the snap
buckling transitions, and simultaneously determines the mode transition events.
The released kinetic energy as well as accelerations of the beam during transitions were
studied using a dynamic analysis. It was shown that the beam returns to lower buckling
configurations through a different equilibrium path. Moreover, the buckling transitions generate
high kinetic energy and acceleration spikes. However, the location of the maximum acceleration
differs from one transition to another. Hence, placement of the piezoelectric energy harvester
should be carefully chosen depending on the application.
During the snap-through events, part of the stored strain energy is released as kinetic
energy. The piezoelectric transducer that is attached to the buckling element converts the released
kinetic energy into high-energy electrons in a floating-gate transistor. Integrating mechanical
energy concentrators and triggers with the piezo-floating gate sensor allows for harvesting the
released energy as bursts and impulses. The impulsive nature of the energy-release not only results
in piezoelectric output voltage levels that are significantly higher than the triggering but also
reduces the effect of energy leakage.
The piezo-floating-gate data logging was used as a counting device. The impact ionized
hot-electron injection process at the floating gate memory cell is activated only when the voltage
exceeds the preset threshold. The levels of the generated voltage differ from one mode transition
to another. Therefore the system can be calibrated to record specific events. The concentration of
the trapped electrons can be measured to determine the number of times the magnitude of the input
signal exceeded a mechanical threshold. Energy can be transferred at extremely low-rate
124

deformation input. The generation and levels of energy bursts are independent of the input
frequency. The mechanism allows for frequency up-conversion from the low input frequency (in
the order of mHz and μHz) to the natural frequency of the piezoelectric scavenger determined by
its stiffness and the attached tip mass.

6.3.

Future research
6.3.1. Optimization of the piezoelectric energy scavenger
The levels of the harvestable energy are independent of the input frequency. However, they

highly depend on the natural frequency and electro-mechanical properties of the scavenger. The
piezoelectric energy scavenger used in the experiments was made of polymer polyvinylidene
fluoride (PVDF) for its low cost and ease to install. However, PVDF materials have low
mechanical-to-electrical conversion coefficients. Other piezoelectric materials such as Lead
Zirconate Titanate (PZT) and Microfiber Composites can be used to enhance the energy transfer.
Dimensions of the piezoelectric energy scavenger and the attached lumped mass should be
optimized in order to maximize the energy harvested during snap-through events.
6.3.2. Optimization of mechanism’s parameters for different strain ranges
The parametric analysis performed in this work provides confidence that the mechanism
can be scalable for embedment within structures. Strains present the input to the mechanical energy
concentrators and triggers. Therefore, dimensions and material properties of the bilaterally
constrained beam as well as the gap between the lateral rigid walls should be tuned such that
transitions occur under the applied level of strain. A design manual or chart can be developed to
determine the optimized values of the parameters for different strain ranges. It would be of interest
if the mechanism is installed in a pre-compressed configuration such that it transitions between a
125

higher and lower buckling mode as the global input strain alternates between compression and
traction.
6.3.3. Optimization of the algorithm to include friction effects
The theoretical results shown in this work do not include the effect of friction on the postbuckling dynamic response of the bilaterally constrained beam. It would be of interest to
investigate the effect of friction on the post-buckling behavior as well as the levels of the released
energy and generated accelerations. Even though the formulation of the energy dissipated due to
friction was presented in section 4.4.3, the computation was very costly due to the coupling
between numerical constrained minimization and numerical integration. A simplification or
approximation of the dissipated energy would alleviate the minimization problem resolution.

126

APPENDIX

127

Table A-1: Cases in parametric studies. Case 1 corresponds to the experimentally evaluated
system. Parameter values in bold text indicate changes from the values in Case 1.
BC 1

BC 2

= 0 free;

= 0 free;

E

L

t

h0

Beam Position

(MPa)

(mm)

(mm)

(mm)

Side/Center

=1 fixed

=1 fixed

1

1

2250

250

2.34

4

Side

Case 2

0.75

1

2250

250

2.34

4

Side

Case 3

0.5

1

2250

250

2.34

4

Side

Case 4

0

1

2250

250

2.34

4

Side

Case 5

1

0.75

2250

250

2.34

4

Side

Case 6

1

0.5

2250

250

2.34

4

Side

Case 7

1

0

2250

250

2.34

4

Side

Case 8

0.5

0.5

2250

250

2.34

4

Side

Case 9

0

0

2250

250

2.34

4

Side

Case 10

1

1

3375

250

2.34

4

Side

Case 11

1

1

4500

250

2.34

4

Side

Case 12

1

1

6750

250

2.34

4

Side

Case 13

1

1

2250

212.5

2.34

4

Side

Case 14

1

1

2250

187.5

2.34

4

Side

Case 15

1

1

2250

125

2.34

4

Side

Case 16

1

1

2250

76.2

2.34

4

Side

Reference
case 1

128

Table A-1 (cont’d)
Case 17

1

1

2250

250

1.56

4

Side

Case 18

1

1

2250

250

1.17

4

Side

Case 19

1

1

2250

250

0.78

4

Side

Case 20

1

1

2250

250

2.34

4.83

Side

Case 21

1

1

2250

250

2.34

5.66

Side

Case 22

1

1

2250

250

2.34

7.32

Side

Case 23

1

1

2250

250

2.34

4

Center

Case 24

1

1

3375

250

2.34

4

Center

Case 25

1

1

4500

250

2.34

4

Center

Case 26

1

1

6750

250

2.34

4

Center

Case 27

1

1

2250

212.5

2.34

4

Center

Case 28

1

1

2250

187.5

2.34

4

Center

Case 29

1

1

2250

125

2.34

4

Center

Case 30

1

1

2250

250

1.56

4

Center

Case 31

1

1

2250

250

1.17

4

Center

Case 32

1

1

2250

250

0.78

4

Center

case 33

1

1

2250

250

2.34

4.83

Center

Case 34

1

1

2250

250

2.34

5.66

Center

Case 35

1

1

2250

250

2.34

7.32

Center

4

Side

Length and thickness variation
Case 36

1

1

2250

129

212.5

1.56

Table A-1 (cont’d)
Case 37

1

1

2250

212.5

1.17

4

Side

Case 38

1

1

2250

212.5

0.78

4

Side

Case 39

1

1

2250

187.5

1.56

4

Side

Case 40

1

1

2250

187.5

1.17

4

Side

Case 41

1

1

2250

187.5

0.78

4

Side

Case 42

1

1

2250

125

1.56

4

Side

Case 43

1

1

2250

125

1.17

4

Side

Case 44

1

1

2250

125

0.78

4

Side

Case 45

1

1

2250

76.2

1.56

4

Side

Case 46

1

1

2250

76.2

1.17

4

Side

Case 47

1

1

2250

76.2

0.78

4

Side

Length and gap variation
Case 48

1

1

2250

212.5

2.34

4.83

Side

Case 49

1

1

2250

212.5

2.34

5.66

Side

Case 50

1

1

2250

212.5

2.34

7.32

Side

Case 51

1

1

2250

187.5

2.34

4.83

Side

Case 52

1

1

2250

187.5

2.34

5.66

Side

Case 53

1

1

2250

187.5

2.34

7.32

Side

Case 54

1

1

2250

125

2.34

4.83

Side

Case 55

1

1

2250

125

2.34

5.66

Side

Case 56

1

1

2250

125

2.34

7.32

Side

130

Table A-1 (cont’d)
Case 57

1

1

2250

76.2

2.34

4.83

Side

Case 58

1

1

2250

76.2

2.34

5.66

Side

Case 59

1

1

2250

76.2

2.34

7.32

Side

Length and Young’s modulus variation
Case 60

1

1

3375

212.5

2.34

4

Side

Case 61

1

1

4500

212.5

2.34

4

Side

Case 62

1

1

6750

212.5

2.34

4

Side

Case 63

1

1

3375

187.5

2.34

4

Side

Case 64

1

1

4500

187.5

2.34

4

Side

Case 65

1

1

6750

187.5

2.34

4

Side

Case 66

1

1

3375

125

2.34

4

Side

Case 67

1

1

4500

125

2.34

4

Side

Case 68

1

1

6750

125

2.34

4

Side

Case 69

1

1

3375

76.2

2.34

4

Side

Case 70

1

1

4500

76.2

2.34

4

Side

Case 71

1

1

6750

76.2

2.34

4

Side

131

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132

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