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THESIS

  
   
   

LMRABY
Michigan $15.9: are
University

I. 5.2,:

This is to certify that the
thesis entitled
ACI'IVE CONTROL OF MECHANICAL
VIBRATION UTILIZING PNEUMATIC FORCES

presented by

YOSSI CHAIT

has been accepted towards fulfillment
of the requirements for

M. 5- dPgTPP in M. E

%%

Major professor

Dr. Clark Radcliffe
DateML

     

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ACTIVE CONTROL OF MECHANICAL VIBRATION UTILIZING PNEUMATIC FORCES

By

Yossi Cheit

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirement
. for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1984

 

ABSTRACT
ACTIVE CONTROL OF MECHANICAL VIBRATION UTILIZING PNEUIATIC FORCES
By

Yossi Chait

Rotating discs are common machine elements found in steam and gas
turbines. grinding wheels, circular saws. and computer disc memories.
Large and/or unstable transverse vibration of such discs can impair the
Operation of these machines. Many vibration control mechanisms that are
currently utilised, such as electromagnets and induction heaters, are
not practical for certain industrial applications. Presented here are
analytical and experimental results for a pneumatic control
force-generating mechanism. Practicality and economy of the mechanism

were demonstrated with a laboratory prototype.

DEDICATION

To my parents and brother. Tova, Samuel and Arnon Chait

ii

ACKNOWLEDGEMENT

The author wishes to express his sincere appreciation to his major
professor Dr. Clark Radcliffe. for his guidance and assistance during
the period of research and graduate study.

Special thanks to Stan. Bill and Bob from the machine shOp for
their valuable aid in the construction of the experimental setup.

Finally, many thanks to my parents Tova and Samuel for their
constant love and support, to my brother Arnon for the inspiration, and

to Susan for sharing the experience with me.

iii

TABLE OF CONTENTS

Page
LIST OF FIGURES v
NOMENCLATURE vi
INTRODUCTION 1
PNEUMATIC SYSTEM 5
Electromechanical subsystem 6
Fluid subsystem 9
EXPERIMENTAL RESULTS 2 14
Frequency response 18
CONCLUSIONS 22
REFERENCES 23
APPENDIX A 25
APPENDIX B ' 28

APPENDIX C 30

iv

Figure

Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure

Figure

10.
11.

12.

BI.

C1.

C2.

LIST OF FIGURES

Time relation between the transverse vibration
displacement and velocity to control forces

Control system schematic

Cross section of a single pneumatic force generator
A bond graph model of a solenoid valve

A bond graph model of a fluid subsystem

A bond graph model of the reduced-order fluid subsystem
Laboratory prototype

Force and volume flow rate vs. discharge clearance
Pressure distribution on the plate

Step input response

System response to a periodic response

Frequency response

Single wave approximation

Strain gage configuration

Force transducer

Force transducer bending frequency spectrum

Force transducer torsion frequency spectrum

Orifice flow meter

Empirical valve flow coefficient
Empirical discharge flow coefficient

V

P‘89

10

11

14

16

17

18

19

21

21

25

25

27

27

29

31

31

"U
B

H 8 on” W

NOMENCLATURE

cross section

spring stiffness or compliance
valve flow coefficient
energy

voltage

pneumatic force
electromagnetic force
inertia

current

discharge flow coefficient
specific heat ratio
inductance

mass

mass flow rate
pressure

momentum

resistance

universal gas constant
specific gravity
temperature

time

vi

 

displacement
volume
velocity
flux linkage

density

vii

 

INTRODUCTION

Rotating circular discs are common machine elements found in steam
and gas turbines, grinding wheels, circular saws and computer disc
memories. Large and/or unstable transverse vibration [1] in rotating
discs can result in failure of turbine wheels due to wheel-to-case
contact. cutting inaccuracy in grinding wheels and circular saws and
head tracking errors in computer disc memories. For example, in the
forest products industry worldwide, 25-30% of material is lost in the
cutting process [2].

Previous research has dealt with two approaches to reducing
transverse vibration: increasing disc stiffness and increasing disc
damping. Both approaches can be classified into two groups: active and
passive techniques. Passive techniques that result in increasing
stiffness include pretensioning or prestressing, radial edge slots. and
floating collars. Active techniques include increasing stiffness with
an induction heating mechanism for feedback control of thermal membrane
stresses [2]. and increasing damping with active control applied with an
electromagnetic mechanism as an external force generator [5]. However.
many of these mechanisms could impair the Operation of the controlled
system. For example. the magnetic information stored in a computer disc
cannot be prOperly maintained in the presence of external magnetic
fields or high temperatures. The need for an additional practical
mechanism for industrial applications is the motivation for the present
research.

Transverse vibration of a circular disc can be viewed as a
superposition of disc vibration modes. Mote [3] showed that in cases of
small plate damping. the disc motion is often dominated by one mode

1

2
only. This theory is implemented in spectral control techniques [4.6].
Disc displacement measurements are transformed to a frequency domain and
one dominant vibration mode is identified and selected for control.
Controlling one mode only, generally of a frequency less then 40 hertz,

allows for the use of a controller with limited frequency response.

 

VELOCITY DISPLACEMENT CONTROL FORCE

 

 

 

 

 

 

Figure 1: Time relation between the transverse vibration

displacement and velocity to control forces

A pneumatic control mechanism. consisting of inexpensive solenoid
valves, for Radcliffe's active control technique [4] has been developed.
External force generators are utilized to dissipate the transverse
energy from the controlled vibration mode (Figure l). The energy
interaction between the force generator and the disc is defined as the

time integral of the product of the external force and transverse

3
velocity. Energy dissipation from the disc occurs when the external
force and the vibration velocity are of an Opposite sign as illustrated
in figure 1. The feasibility of this mechanism for wood industry
applications is demonstrated by the passive nonrcontacting pneumatic
guides, presently used by the industry. Vibration control of the above
mentioned systems is feasible with pneumatic forces. however. magnetic

forces or high temperature generating mechanisms are not.

 

 

 

I a:

{3R

    

Transducer

 

.B

 

j

 

 

[41
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For e 1 r

Figure 2: Control system schematic

The structure of an active control system for transverse vibration
employing an opposing pair of non-contacting pneumatic force generators
is shown in Figure 2 [5]. Displacement sensor output is utilized to
evaluate the magnitude and frequency of the controlled active vibration

mode and its velocity. Velocity information is sent to the controller

which determines the required activation frequency Of the force
generators. At that frequency, the controller turns on the force
generator Opposing the velocity Of the controlled mode (Figure 1).
Phase-locked loop control continues until the control forces are in
phase with the mode negative velocity. At that state. the control

system maximizes the dissipation Of energy from the disc.

PNEUMATIC SYSTEM

A lumped parameter model of the pneumatic system (Figure 3) was
developed to calculate the pneumatic force generated by the system. A
bond graph [7] model of a system consists of lumped parameter elements
linked together by lines representing power bonds. These power bonds
are connected with common effort and flow variables. Efforts and flows
are associated with physical quantities of a system. Effort and flow
are force and velocity in mechanical translational systems, pressure and
volume flow rate in hydraulic systems, and voltage and current in
electrical systems. Bond graph modeling is advantageous for several
reasons. First, it provides a unified model for mixed systems, e.g.
electromechanical systems. Also. it provides a systematic method to
derive system state equations and to define to correct system inputs and

outputs.

1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

__l

‘I
I
I
I
I
I
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I
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I
|
I
I
I
I
I
I
I
I
I

      

 

r—
I
I
I
I
I
I
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I

L
r.
I
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Figure 3: Cross section of a single pneumatic force generator

6

The pneumatic system (Figure 3) consists of a solenoid valve, short
pipe and a diffuser. The pneumatic system analysis was simplified by
separating it into two subsystems: An electromechanical subsystem to
study the solenoid valve behavior and a fluid subsystem to study the air
flow behavior.
Electromechanical Subsystem

The bond graph model of a solenoid valve [7] is shown in Figure 4.
Due to lack of sufficient data from the valve manufacturer. system
parameters were obtained experimentally. The valve analysis is
simplified by separating the electrical behavior from the mechanical
behavior. The coupling between electrical and mechanical behavior is
represented by a mixed IC field [7] where I denotes the coil inductance
and C denotes the spring stiffness. That is. the inductance is a
function of the plunger position within the magnetic core. and this
position depends on the spring stiffness and the magnetic force. Valve
input is voltage e and the model calculates the required current i. The
electrical momentum A is termed flux linkage. Mechanical parameters are
spring stiffness C3, plunger mass 13, and viscous damping R5. Is and KO
represents spring deflection and plunger position within the coil
respectively. Another input to the system is a pneumatic force F. a
result of the pressure difference across the valve.

The flow across the valve is a function Of the valve opening. The
valve Opening, which is the plunger displacement. is calculated from

Newtons's second law

dXs Pm
.. (l)
M

I F
I{]- |
I
I ___________
I O I. O _l

I r. P
——>¥I 2L>IMC€;*4I-£Q¥I3

I

I- I X '
Li----JI Cfisx I
'Ca R5 "

I, _________ J
PLUNGER

Figure 4: A bond graph model of a solenoid valve

where the differential equation for the plunger momentum can be
extracted from Figure 4. This is done by summing all the efforts
connected with a common flow variable which is the plunger velocity.
The efforts are the magnetic force. spring and damping forces and the

force introduced by the pressure difference across the valve. Doing so,

we get
de Pm
-- = - r - c . x. - as t —- — F (2)
dt M

the force produced by the magnetic field is obtained from the energy

8
state of the electrical system. The total energy stored in the system,
assuming no energy transfer between flow and mechanical momentums, is

the sum of the mechanical and the electrical energies

t A.Xc
EIx.xc) = I (it. + f‘v)dt = I (i‘dl + ftdxc) (3)

0
where the electrical energy is derived from the total energy expression

by holding the plunger displacement constant

A A”
E(A.Xc) = I i‘dk = ——_———— (4)
O 2'L(Xc)
and the expression for the flux linkage dynamics is obtained from Figure
4. We sum all the efforts connected by a common flow variable which is
the coil current. These efforts are the input voltage and the voltage

drOp across the coil resistance. Neglecting capacitance dynamics in the

coil at low frequencies, we get

d1
n—go—i‘R4 (5)
dt

where the current in the coil is derived from the definition that

relates flux linkage and current in a coil

1

 

(6)

i:
L(Xc)

the force produced by the magnetic field is in the direction of the

9

plunger displacement. From Equations (3) and (4) it is clear that

as A“ a
f = -— = — -———- *-L(Xc) (7)
axe 2‘L’(Xc) dt

Fluid Subsystem

A bond graph model of a fluid subsystem consisting of incoming flow
from the solenoid valve and discharging flow from the diffuser is shown
in Figure 5. System inputs are input pressure P1 and atmospheric
pressure Pa, and the system model computes the required input flowrate
Q1. C1 and C2 denote compliance in the pipe and diffuser. Friction
losses, losses associated with area reduction at inlet and discharge
planes are represented by R1. R3 and R2. 11 and 12 denote pipe and
diffuser inertias. The valve opening/flow rate restriction is
represented by Rx. Q2 denotes volume flow rate from pipe to diffuser,
and Q3 denotes discharge volume flow rate. P2 and P3 correspond to
pressures in the pipe and diffuser.

The behavior Of the force exerted by the flow system on the plate
can be obtained by solving four differential equations as defined by
Figure 5. Due to nonlinearities such as choked flow and nonlinear flow
resistances. it is impossible to derive an analytical solution to the
problem. The problem can be solved numerically, using a digital

integration algorithm.

PIPE DIFFUSER

Pa

Figure 5: A bond graph model of a fluid subsystem

The physical elements in a model are associated with the
eigenvalues Of a linearized model of the system. In a system with
widely spread time constants; time constant being the inverse of an
eigenvalue. dynamics with small time constants have already decayed
while the dynamics with larger time constants are yet to start. Also.
when using a digital integration algorithm, it is necessary to integrate
with a time step associated with the smallest time constant. I As a
result. the problem cannot be run to completion with a reasonable number
Of steps. For these reasons, we generally drOp terms associated with
small time constants from the system model, and approximate the system
behavior using slower eigenvalues only. The inertia terms derived from
the pipe and diffuser lengths were drOpped from the model for the above
reasons. Pipe and diffuser compliances can be combined into one term in

the absence of flow rate dynamics between them. Furthermore. since the

11
pipe is very short. the friction term can be neglected considering the
predicted pressure recovery in the diffuser. A reduction to a first

order model is shown in Figure 6.

Figure 6: A bond graph model of the reduced order fluid subsystem

The total force applied on the disc is essentially a product of
static pressure only. A diffuser recovers potential energy (static
pressure) from kinetic energy (flow velocity). However, increasing the
diffuser angle eventually results in a stall. or in other words.
negative pressure on the disc face. Kline's [12] work on Optimum design
of straight-walled diffusers provides us with the proper dimensions for
maximum pressure recovery. Several expressions [13.14] for the total
force on the disc exist. Yet. in the absence of stall regions. the
pressure distribution in the diffuser is nearly uniform. which allows

for the following derivation

12

A
FORCE = I psdA = p2,3 . A (8)
o

where the effective area is the exit cross-section of the diffuser.

From the equation Of state for a perfect gas. it can be shown that

dP2,3 R OT
= (min ' moat) I 8
dt V

 

 

(9)

where the input mass flow rate is a function of the valve Opening. This
function takes on different characteristics based on the shape of the
orifice and the plunger. For a quick-Opening valve. a straight line
relation through 70% of maximum valve Opening is suggested [8]. The
flow regions are defined in the system: free and choked. Choked flow
occurs when the inlet/exit pressure ratio reach a critical ratio of 1.89
[9]. For pressure ratios above 1.89 an increase in the upstream
pressure results in a proportional increase in the mass flow rate with
fixed volume flow rate. while downstream pressure changes have no effect
on both Of the above [9]. At free flow conditions. the mass flow rate
across the valve is a product of the volume flow rate multiplied by the
inlet air density. The expression for the volume flow rate across the

solenoid. as proposed by The National Fluid Power Association [10] is

 

P1 - P2,3

 

Q1 = 22.48 ‘ Cv * (10)

T ‘ SG

and for choked flow conditions, inlet volume flow rate is calculated

from the following analytical formula [11]

13
1
m = 0.532 ‘ Cv ‘ A ‘ P1 ‘ -—- (11)
T
The discharge mass flow rate. for one dimensional and isentrOpic
flow, is a product of the volume flow rate and the air density at the
diffuser. Discharge volume flow rate is Obtained from the orifice
equation, with the flow coefficient adjusted for compressible flow. The
geometry at discharge plane is similar to an orifice geometry with the
exception of a radial flow turn at the exit plane. The diffuser can be
viewed as a plenum in the presence of large area ratio between diffuser
and discharge plane. which allows the assumption of zero downstream

velocity. Doing so. we get

 

Q3 = K ‘ A ‘ (12)
p

 

and by assuming an isentrOpic model for a real adiabatic process we can

write the pressure-density relation as follows

P ‘ p1]k = constant . (13)

although there is a temperature increase for an isentrOpic process. as
the entropy-temperature diagram indicates [9]. we shall assume that it
is constant. This assumption is valid for small temperature changes. A
5 degree change from an average temperature of 68 degrees Fahrenheit
results in less than one percent error in density calculations. The
density is calculated from Equation (13) based on 72 degrees Fahrenheit.
Choked mass flow rate is derived from Equation (12) by setting P2,3 to

27.18 psia which is the critical pressure.

EXPERIMENTAL RESULTS

A laboratory prototype was constructed (Figure 7) to evaluate the
force on the plate. the required volume flow rate. and the frequency
response of the system. The force was measured with a force transducer
fabricated from four active strain gages in Iheatstone bridge
configuration (Appendix A). A standard ASME flow meter. which consists
of a two inch pipe thin-plate orifice mounted between flanges (Appendix
B). was used to measure volume flow rate. Pressure distribution on the
disc, downstream and upstream pressures. were measured with Entran EPI
pressure transducers. The data acquisition facility consisted of a

PDP-11/02 minicomputer.

 

 

Figure 7:Laboratory prototype

14

15

In converting the pressure difference data [to volume flow rate
data. Equations (10)-(13) were used. The correct flow coefficients for
the valve and the discharge plane were found by calibration with the
flow meter (Appendix C). Tho flow regions were observed: choked valve
flow with free discharge flow. and free valve flow with choked discharge
flow. Initially, the pressure ratio across the valve (80 psia to
atmospheric) is larger than the critical pressure ratio (1.89). and
discharge plane pressure ratio is lower than this critical ratio. If
the clearance is reduced, the pressure in the diffuser is large enough
to choke the discharge flow, and the corresponding increase in the
diffuser pressure results in the valve pressure ratio reduced to a value
below critical. Discharge clearances between 5 and 18 thousands of an
inch produce choked flow across the discharge plane and a constant flow
exists. Increased clearance results in decreased pressure in the
diffuser which results in choked valve flow. These patterns are shown
in Figure 8.

The force behavior was found to be nearly linear in each Of the
above flow regimes (Figure 8). Under free discharge flow, a linear
sensitivity of 66.7 lbf/inch is Observed. A much higher sensitivity of
1000 lbf/inch is observed for a choked discharge flow. This high
sensitivity results from the prOportionality between discharge mass flow
rate and discharge area under choked flow condition. The model
simulation results for the force at small clearances are higher than the
experimental results. This variation is a result Of the flow
streamlines reattachment to the diffuser lip which creates a vacuum
between the lip and the plate. This vacuum is a typical phenomena in

devices with that geometry. such as nozzle-flapper devices [15].

16

 

 

 

 

 

 

ZII 9: 5x 17
: GP ~
_ A I :‘16 A
15- . ~ E
_ INPUT PRESSURE: 80 pSIO ~ g
A " A G : V
g .I a :3 SOLID LINE -—-—-MODEL r-IS :3
w w - POINTS EXPERIMENT Z a:
L) "" \ - 3
a: _ c:
E B A ”'1‘ C:
.. FORCE ~ m
- : :53
5— A ~ ~ 9
- . q . —-13
:1 A a h A a A d g s :
U I I I l I I l 1 I I 12
a w 29 3a 48 so 50

Figure 8: Force and volume flow rate vs. discharge clearance

This problem was reduced with careful design of the exit lip and
diffuser angle. Further reduction of the prototype lip will result in
elimination of vacuum on the plate. With an area increase of 37 percent
over a length of 0.75 inches. there is a 90 percent increase in force
levels compared with a flat pipe discharge. The efficiency of a
diffuser is illustrated in Figure 9. The flow separation was limited to
very small discharge clearances and at the diffuser lip only as the

pressure distribution curve indicates.

17

 

 

 

 
    
 

 

 

 

 

4n 2
39.: _
A I PRESSURE _ WI P Mr
12 1 J'F
w - 2
3 20- -
E I —1
a - 5—3
Ln _‘ _ <1
E III _ f.
; FORCE RATIO 3;
- 8
0—
‘Ig _ I . I I I I I I n
m ‘0“ 0.2 . oIa 9.4 9.5

RADIUS (inch)

Figure 9: Pressure distribution On the plate

The response of the solenoid valve to a step input is utilized to
obtain Opening and closing times for different excitations. An Opening
time of a 12 milliseconds (msec) with a nominal 12 do volts excitation
(Vdc). was reduced to 6 msec with 44 Vdc excitation (Figure 10). High
voltage is applied for a period of 6 msec and is maintained for an
additional 3 msec to decrease full Opening time (valve completely Open).
then the voltage is drOpped to its nominal value (12 Vdc). Valve
closing time is a strong function of the spring stiffness. The
manufacturers spring of 300 N/m was replaced with a 3000 N/m spring.
This resulted in the closing times being reduced from 32 msec to 14
msec. Eddy current and saturation in the coil. that reduce the flux

linkage. result in the longer opening time Observed in the simulation

18
(Figure 10). The linear approximation for the flow/valve Opening
relation was found to be inaccurate. From Figure 10 we observe a

steeper initial slope which can be explained with an exponential

 

 

   

 

 

 

 

relation.
28
- MODEL
IS—I .
:45 I EXPERIMENTAL
m In—
g .1
E)- ..I
5—
n I I I I r I I I I I I I T 1
8.88 8.81 8.82 8.83

TIME (seconds)

Figure 10: Step input response

Frequency Response

Due to a wide range of Operating pressures (0 to 80 psid) and
nonlinearities in the system. such as choked flow and compressibility,
the frequency response of the system cannot be obtained by
linearization. The frequency response of the system is obtained by
subjecting the valve to a periodic excitation and recording its
response. Negative force on the plate corresponds to a force generated

by the opposing force generator. System output for a periodic

19
excitation is shown in Figure 11. High frequencies in the force plot
resulted from the force transducer bending at natural frequencies around
500 hertz (Appendix A). The transducer sensitivity of 0.25X10-3 results
in a negligible effect on forces due to transducer deflection on the
diffuser clearance. A 6 msec and 14 msec Opening and closing times.

during frequency response testing, are Observed in Figure 11.

 

      
  

 

 

 

 

 

 

 

In so
I """ I FORCE I """ 3
~ I VOLTAGE I
E “—‘s
E; 0‘" """"""" I ""'"'"E 23
E 5 : .1‘
5 E 6'
: 5 >
'16 I I I I
8.8

TIME (seconds)

Figure 11: System response to a periodic excitation

A periodic excitation consists of an on-off excitation of an
Opposing pair of force generators. This results in an alternating
positive force acting on the disc from both sides. The solenoid is
excited with 44 Vdc for 9 msec to reduce the full Opening time. For
longer excitations, the voltage was dropped to 12 Vdc after 9 msec to

prevent Overheating. The resulting force wave takes on two shapes: a

 

20

wave with zero force between forces on each side of the disc (Figure 11)
which corresponds to frequencies up to 36 hertz, and a wave with forces
acting from both sides at the same time which corresponds to frequencies
greater than 36 hertz. At frequencies above 36 hertz. the second valve
is Opened before the first valve is completely closed. This results in
both valves generating forces at the same time and the total force on
the disc is a sum Of both forces. The maximum excitation frequency
tested is 45 hertz, based on a valve closing time Of 14 msec.

In order for the system to Operate on one vibration mode without
exciting other modes Of higher frequencies, the excitation frequency was
constructed such that the force wave shape will possess a large signal
to noise ratio. Signal tO noise ratio is the ratio between the first
discrete harmonic to the second largest discrete harmonic obtained with
spectral analysis. The Observed ratio for different frequencies exceeds
15 decibels (db). A spectral analysis of the force wave shown in Figure
11 is shown in Figure 12. The analysis was performed with a discrete
Fourier transform routine.

The approximation Of the force wave as the largest single wave
Obtained from the spectral analysis is shown in Figure 13. For most
excitation frequencies the magnitude of the largest wave is greater than

of the nominal force for the specific clearance.

 

21

 

 

 

 

 

 

 

 

 

 

10—
w .J
8
E 5-4
O
<
5 ..
.J
u FIT—llTTITllllLrllliilitlri‘ $11.1]
100 200 300 400 500
FREQUENCY (hertz)
Figure 12: Frequency spectrum
2?,
g __
O
u.
‘1' 1Tri W—TTI IIjjllIlf.i";..II1llllI
0.00 0.01 0.02 0.03 0.04 0.05 0.06

TIME (seconds)

Figure 13: Single wave approximation

 

CONCLUSIONS

A pneumatic force generating mechanism was develOped for control Of
transverse vibration in rotating discs. The model results for volume
flow rates and pressures agree with experimrntal results. We are able
to predict the force on the disc at different discharge clearances and
different input pressures.

This mechanism provides an adequate solution to industrial need for
a practical mechanism. Pressurized air is available in many plants.
The solenoid valves are inexpensive, and the controller can be
implemented with a low priced microcomputer.

Frequency response results indicate that this mechanism is capable
of generating forces at frequencies up to 45 hertz. Thus. the mechanism
can be incorporated in an active control technique.

Future work should include an Optimization Of the solenoid valve
design to improve the frequency response, and Optimization of the

diffuser and discharge geometry to increase the force.

22

REFERENCES

 

9.

REFERENCES

Note. C. D.. Jr.. and Szymani, R., "Circular Saw Vibration

Research”, The Shock and Yibration Digest. Vol. 10, No. 6. June
1978. pp. 15—30.

Note, C. D.. Jr.. Schajer, G. 8., and Eflloyen. 8.. "Circular Saw
Vibration Control by Induction Of Thermal Membrane Stresses".

ASME Journal for Engineering for Industry, Vol. 103, Feb. 1981,
pp. 81-89.

Note. C. D.. Jr., and Holdyen, 8., "Confirmation Of the Critical
Speed Theory for Symmetrical Circular Saws", ASME Journal
of Engineering for Industry, Vol. 97, NO. 3, 1975, pp. 1112-1118.

Radcliffe, C. J., and Mote, C. D.. Jr.. "Identification and Control
of Rotating Disc Vibration", ASME Journal Of Dygamic
SystemsI Measurement, and Cogtrol, Vol. 105. March 1983, pp. 39—45.

Ellis. R. W.. and Mote. C. D.. Jr.. "A Feedback Vibration

Controller for Circular Saws". ASME Journal of Dynamic
SystemsI MeasurementI and Control, Vol. 101, March 1979. pp. 44-49.

Ballas, M. J.. ”Feedback Control of Flexible Systems", IEEE
Transactions on Automatic Control, Vol. ac-23. NO. 4. 1978. PP.
673-679.

Karnopp, D., and Rosenberg, R.. System Dynamics: A Unified Approach,
John Wiley and Sons, 1975.

Beard, C. 8.. Final Cogtrol Elements: Yalveg and Actuagors, Chilton
C0... 1969. pp. 6’90

 

Potter, M. C., and Foss, J. F.. Fluid Mechanics, Great Lakes Press
Inc., 1982.

10. Fleischer. H.. Practicg;;Air Yglve Sizing, Numatics Inc., 1973.

 

pp. 9.

ll. Shapiro, A. R., The Dynamics and Thermodynamics of Compressible

Fluid flow. The Ronald Press CO., 1953, Vol I. pp. 82-100.

12. Kline, S. J.. Abbott, D. 8.. and Fox, R. W.. "Optimum Design of

Straight Walled Diffusers”, ASME Journal of Basic Engineering, Sept.
1959. pp. 321-331.

23

24

13. Dmitriyer. V. N.. and Shashkov, A. G., "Force Of the Jet Action on

the Baffle in Pneumatic and Hydraulic Control Units".

Pneumatic and Hydraulic Coptrol Systems. Vol 1, Pergamon press, 1968.
pp. 272—284. ‘

14. Blackburn, J. F., Reethof. 6., and Shearer, J. L.. Fluid

Power Cogtrol. The Technology Press of M.I.T, 1960, pp. 313-318.

15. Wark, C. E.. and Foss, J. F., "Forced Caused by the Radial

Out-Flow Between Parallel Discs". Submitted for publication in
ASME Journal of Fluids Engineering, 1984.

16. Bean, H. S., "Fluid Meters Their Theory and Application".

ASME Research Committee on Fluid Meters, 6th edition, 1971.

17. Miller. R. W., Flow Measurement Engineering Handbook, McGraw-Hill,

1983, chap. 8.

 

APPENDI CE S

APPENDIX A

The force transducer (Figure A1) consists of a Wheatstone bridge with
four active legs. Each resistor is a Micro-measurements 120 Ohms strain

gage with a 2.115 gage factor. The bridge sensitivity can be shown to be

Ebd = 0.25 . E ‘ F . ( -81 +83 -8, +84 ) (A1)
Where: E = Excitation voltage

Ebd = Output voltage

F = Gage factor

8 Strain
strain gages 2 and 4 are subjected to tension while 1 and 3 are subjected
to compresssion. Assuming same strain levels at each gage location all the
strains in Equation A1 possess the same value. Furthermore. with the
bridge configured in Figure A1, four times the sensitivity is achieved,
compared with a single active leg transducer. The strain gage grid line
direction prevents output bias due tO force eccentricity from the axis.
The transducer was calibrated with static loads. Bridge output sensitivity
was found to be 0.2 millivolt/lbf. with static deflection sensitivity Of
0.00025 inch/lbf.

Both bending and torsion frequency spectrum Of the transducer were

Obtained with a HP model 5423A Structural Dynamics Analyzer. Figures A3

and A4 show the frequency spectrums.

25

 

26

 

 

 

 

 

Figure A1: Strain gage configuration

 

Figure A2: Force transducer

 

 

27

 

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Figure A3: Force transducer bending frequency spectrum

 

 

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Figure A4: Force transducer torsion frequency spectrum

 

APPENDIX B

The flow meter utilized in this research is a standard ASME two inch
pipe thin-plate orifice mounted between flanges. The flow coefficient
(0.6) for this conventional flow meter are well established [16].
Variation Of the flow coefficient for Reynolds numbers between 10,000 and
1,000,000 is less than 2 percent [16]. The orifice plate, was made from
stainless steel with 0.3 inch bore. Orifice thickness Of one eight Of an
inch was used to prevent large deflection Of the plate as a result of
differential pressure. This value is based on the ASME value for maximum
allowable deflection [16]. Vena contracts pressure taps are located at one
pipe diameter for inlet pressure tap, and 1.05 pipe diameter [16]. TO hold
bias errors due to upstream and downstream disturbances. without utilizing
flow conditioners. two inch pipe diameter was maintained for fourteen pipe
diameters upstream and downstream lengths are used [17]. The pressure drOp
across the orifice was measured with a differential pressure transducer and
volume flow rate was calculated from the orifice equation [16]. Figure A1

shows the configuration Of the flow meter.

28

 

 

 

Figure

 

29

Bl: Orifice flow meter

 

APPENDIX C

A discharge coefficient Of a flow device is defined as the ratio Of
the actual mass flow rate to the ideal, zero contraction flow rate; the
contraction coefficient being the ratio Of the jet area to the
cross-section in the device. The overall loss coefficient includes the
area ratio effect. nonuniform velocity profile, contraction effect,
compressibility effect and all the losses between downstream and upstream
pressure taps. Although there are numerous analytical formulas for the
overall flow coefficient, best results are obtain by direct calibration Of
each device.

In general, the flow coefficient increases substantially with pressure
ratio increases due to compressibility effect [11]: the pressure ratio
being the pressure ratio across the device. The empirical flow
coefficients for the valve flow Equations (10) and (11) (Figure B1) and for

the discharge plane flow Equation (12) (figure B2) follow the above trend.

30

VALVE FLOW COEFFICIENT

DISCHARGE FLOW COEFFICIENT

31

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