!!!!!VIBRATION SUPPRESSION IN A PLATE USING IN -PLANE FORCES By Mahmoud Nabil Abdullatif A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering - Master of Science 2015!ABSTRACT VIBRATION SUPPRESSION IN A PLATE USING IN -PLANE FORCES By Mahmoud Nabil Abdullatif Vibration suppression of flexible structures continues to be a subject of considerable interest due to the increasing demands of high precision space application s. Large phased array antennas, for example, which are used for space -based radar system to tra ck and identify moving objects, require high dimensional stability. The Air Force is interested is exploiting tension mechanics in maintaining dimensional stability of large array structures. To this end, this research investigates the problem of vibration suppression in a thin plate using in -plane tensile forces acting on the boundary. Earlier work has shown that the vibration of a cantilever beam can be effectively suppressed by end -forces and this work investigates the extension of the control strategy t o the plate problem. Two scenarios are considered: one where the end -forces are uniformly distributed, and the other where the end -forces are applied at discrete locations using a web -of-cables structure. For the problem with uniformly distributed end -forc es, both the Rayleigh -Ritz method and the finite element method are used to obtain the state space model of the system in modal coordinates. Simulations are then used to show the efficacy of the control strategy in reducing vibration. The plate with web -of-cables structure is modeled using the finite element method alone. The effect of the web -of-cables structure on in -plane stress distribution in the plate is determined and the pre -stress information is used to obtain the out -of-plane vibration model. The control method requires the tension in the corner cables of the web -of-cable structure to be switched and the efficacy of the control method is investigated for different switching algorithms. Numerical simulations for these different algorithms show that the control method is effective in suppressing the vibration of all modes except the first mode. This is due to the fact that change in the corner tensions is not effective in producing a significant change in the fundamental frequency. To suppress the vib ration of the first mode, future work will have to focus on redesign of the web -of-cables structure relative to the plate dimensions. iii TABLE OF CONTENTS LIST OF TABLES .......................................................................................................................... v LIST OF FIGURES ...................................................................................................................... vii Chapter 1 Introduction .................................................................................................................... 1 1.1. Motivation .......................................................................................................................... 1 1.2. Literature review ................................................................................................................ 3 1.3. Scope and content .............................................................................................................. 5 Chapter 2 Dynamics Of A Thin Rectangular Plate Ð A Review .................................................... 6 2.1 Classical Plate Theory ......................................................................................................... 6 2.2 Plate Dynamics with in -plane forces ................................................................................ 13 Chapter 3 Dynamic Behavior of a Thin Rectangular Plate Subjected to In -Plane Forces ........... 18 3.1 A Simply -Supported, Thin, Rectangular Plate Subjected To Uniformly Distributed In -Plane Forces ............................................................................................................................... 19 3.1.1 Analytic Solution ....................................................................................................... 19 3.1.2 Rayleigh -Ritz Approximation .................................................................................... 21 3.1.3 Finite Element Analysis ............................................................................................. 28 3.1.4 Comparisons ............................................................................................................... 31 3.2 A Simply -Supported Thin Rectangular Plate Subjected To In -Plane Forces At A Few Discrete Locations ..................................................................................................................... 35 3.2.1 Finite Element Analysis ............................................................................................. 36 3.2.2 Commercial Software ................................................................................................. 41 3.2.3 Comparisons ............................................................................................................... 41 Chapter 4 Vibration Control Of A Thin Rectangular Plate Using Uniformly Distributed In -Plane Forces ............................................................................................................................................ 46 4.1 Control Design .................................................................................................................. 46 4.2 Control Scheme For A Simply -Supported Plate ............................................................... 51 4.2.1 Control Scheme Using Rayleigh -Ritz Approximation .............................................. 52 4.2.2 Control Scheme Using Finite Element Analysis ........................................................ 58 4.2.3 Comparisons ............................................................................................................... 63 4.3 Control Scheme For A Plate With Elastic Boundary Conditions ..................................... 64 4.3.1 Verify The Solution Using Elastic Boundary Conditions .......................................... 65 4.3.2 Control Scheme Using Finite Element Analysis ........................................................ 66 Remark ...................................................................................................................................... 77 Chapter 5 Vibration Control Of A Thin Rectangular Plate Using A Web -Of-Cables ................. 78 5.1 A Web -Of-Cables Structure Supporting A Plate .............................................................. 79 5.2 Investigating The Changes in The Corner Cables Tensions ............................................. 84 5.3 In-Plane Stress Distribution .............................................................................................. 96 5.4 Out -Of-Plane Vibration Analysis For Varying Boundary Conditions ........................... 101 5.5 Control Scheme Using The Web -OF-Cables Structure .................................................. 109 5.5.1 Adding !! To The Four Corner Cables ................................................................... 111 iv 5.5.2 Adding !! To Two Corner Cables And Subtracting !! From The Other Two Corner Cables ................................................................................................................................. 116 5.5.3 Using Multiple ! T Values With Randomly Switching ............................................ 120 5.5.4 Using Multiple ! T Values With Systematic Switching ........................................... 125 5.5.5 Summary .................................................................................................................. 131 Chapter 6 Conclusion ................................................................................................................. 132 BIBLIOGRAPHY ....................................................................................................................... 135 !v !LIST OF TABLES !Table 3.1. Comparing the solutions for thin rectangular plate without in -plane forces ........... 32 !Table 3.2. Comparing the solutions for thin rectangular plate with in -plane distributed forces .............................................................................................................................. 34 !Table 3.3. The non -uniform stress distribution frequency results ............................................ 42 Table 4.1. Specifications of the rectangular plate and the controller ........................................ 52 Table 4.2. Frequencies in hz using rayleigh -ritz method .......................................................... 52 Table 4.3. Modified control constants ...................................................................................... 55 Table 4.4. Frequencies in hz using finite element analysis ....................................................... 58 Table 4.5. Frequencies in hz using fem method for elastic boundary condition with !!!!"!!!! ................................................................................................................. 65 Table 4.6. Frequencies in hz using finite element analysis for elastic boundary condithins and !!!!""!!! ........................................................................................................ 66 Table 4.7. Frequencies in hz using finite element analysis for elastic boundary condithins and !!!!"!!! .......................................................................................................... 71 Table 5.1. Preset parameters for the design problem ................................................................ 82 Table 5.2. Cables tension results for the design problem ......................................................... 82 Table 5.3. Cables length results for the design problem ........................................................... 82 Table 5.4. Cables angles and length results for the design problem ......................................... 83 Table 5.5. Designed parameters for the web -of-cables of the deformation problem ................ 94 Table 5.6. The web -of-cables tension and the corner cables length results after the deformation ............................................................................................................. 94 !Table 5.7. The web -of-cables angle results after the deformation .......................................... 94 Table 5.8. Evaluating the coordinate of the centre of the thin plate after the deformation ..... 95 Table 5.9. Comparing ansys and matlab frequency analysis when !! = 0 N ........................ 101 !vi !Table 5.10. Comparing ansys and matlab frequency analysis when !! = 3 N ........................ 104 Table 5.11. Comparing matlab frequency analysis when !! = 0 and 3 N ............................... 104 Table 5.12. Matlab frequency analysis for different !! values ................................................ 109 !!!!vii !LIST OF FIGURES !!Figure 2.1 . Thin Rectangular Plat ............................................................................................. 7 Figure 2.2 . An infinitesimal element of the thin rectangular plate ........................................... 8 Figure 2.3 . Rotation of a normal line after the deformation .................................................. 10 Figure 2.4 . Thin rectangular plate with in -plane forces .......................................................... 13 Figure 2.5 . ............................. 14 An element of the thin rectangular plate with in -plane forces Figure 2.6 . .................................. 14 Side view for an element of the plate with in -plane forces Figure 3.1 . Simply supported plate with uniformly distributed in -plane forces ..................... 20 Figure 3.2 . Four -node quad plate element with 12 DOFÕs ..................................................... 29 Figure 3.3 . Thin rectangular plate dimensions ......................................................................... 31 Figure 3.4 . Finite element meshing for the plate .................................................................... 32 Figure 3.5 . Thin rectangular plate with uniformly distributed in -plane forces ....................... 33 Figure 3.6 . First five mode shapes for thin simply supported rectangular plate with uniformly distributed in -plane forces ................................................................... 35 Figure 3.7 . Dimensions of the plateÕs element and coordinate transformation ...................... 36 Figure 3.8 . Thin Rectangular plate with in -plane stresses ...................................................... 39 Figure 3.9 . Meshing of thin rectangular plate with in -plane prescribed forces ...................... 41 Figure 3.10 . In-plane normal stress distribution in the x direction ........................................... 43 Figure 3.11 . In-plane normal stress distribution in the y direction ........................................... 43 Figure 3.12 . In-plane share stress distribution .......................................................................... 43 Figure 3.13 . Difference between ANSYS and MATLAB Code for the x direction stress distribution ............................................................................................................ 44 Figure 3.14 . Difference between ANSYS and MATLAB Code for the y direction stress distribution ............................................................................................................ 44 !viii !Figure 3.15 . Difference between ANSYS and MATLAB Code for the share stress distribution ............................................................................................................................... 44 Figure 3.16 . First Five modes shapes for thin plate with non -unifor m stress distribution ....... 45 Figure 4.1 . Control design based on filtered outputs .............................................................. 48 Figure 4.2 . Basic system to control the vibration of the rectangular plate ............................. 51 Figure 4.3 . Simply supported thin rectangular plate with in -plane distributed forces ............ 51 Figure 4.4 . Energy deception for simply -supported plate without the controlled in -plane forces using Rayleigh -Ritz approximation ............................................................ 53 Figure 4.5 . Modal displacements for simply supported plate without the controlled in-plane forces using Rayleigh -Ritz approximation ............................................................ 53 Figure 4.6 . Energy deception for simply supported plate with controlled in -plane forces using Rayleigh -Ritz approximation ................................................................................ 54 Figure 4.7 . Modal displacements for simply supported plate with controlled in -plane forces using Rayleigh -Ritz approximation ....................................................................... 54 Figure 4.8 . !" inputs for simply supported plate using Rayleigh -Ritz approximation ........... 55 Figure 4.9 . !" inputs for simply supported plate using Rayleigh -Ritz approximation ........... 55 Figure 4.10 . Energy deception for simply supported plate with modified controlled in -plane forces using Rayleigh -Ritz approximation ............................................................ 56 Figure 4.11 . Modal displacements for simply supported plate with modified control using Rayleigh -Ritz approximation ................................................................................ 56 Figure 4.12 . Modified control inputs !"#$% for simply supported plate using Rayleigh -Ritz approximation ........................................................................................................ 57 Figure 4.13 . Modified control inputs !"#$% for simply supported plate using Rayleigh -Ritz approximation ........................................................................................................ 57 Figure 4.14 . Modified control filter input for simply supported plate using Rayleigh -Ritz approximation ........................................................................................................ 57 Figure 4.15 . Energy deception for simply supported plate with controlled in -plane forces using FEM approximation ............................................................................................... 60 !ix !Figure 4.16 . Modal displacements for simply supported plate with controlled in -plane forces using FEM approximation ..................................................................................... 60 Figure 4.17 . !" input for simply supported plate using FEM approximation .......................... 61 Figure 4.18 . !" input for simply supported plate using FEM approximation .......................... 61 Figure 4.19 . Energy deception for simply supported plate with modified controlled in -plane forces using filter and saturation for FEM approximation .................................... 61 Figure 4.20 . Modal displacements for simply supported plate with modified controlled in -pane forces using FEM approximation .......................................................................... 62 Figure 4.21 . Modified control inputs !"#$% for simply supported plate for FEM approximation ............................................................................................................................... 62 Figure 4.22 . Modified control inputs !"#$% for simply supported plate for FEM approximation ............................................................................................................................... 62 Figure 4.23 . Filter inputs for simply supported plate with modified controlled in -plane forces using for FEM approximation ............................................................................... 63 Figure 4.24 . Thin rectangular plate with in -plane distributed forces and elastic boundary conditions .............................................................................................................. 64 Figure 4.25 . Energy deception for elastic boundary condition without in -plane forces and !"!!""!!!! using FEM approximation ........................................................... 66 Figure 4.26 . Modal displacements for elastic boundary condition without in -plane forces and !"!!""!!!! using FEM approximation ........................................................... 67 Figure 4.27 . Energy deception for elastic boundary condition with controlled in -plane forces and !"!!""!!!!!using FEM approximation .................................................... 67 Figure 4.28. Modal displacements for elastic boundary condition with controlled in -plane forces and !"!!""!!!! using FEM approximation ......................................... 68 Figure 4.29 . !" inputs for a plate with elastic boundary condition and !"!!""!!!! using FEM approximation ............................................................................................... 68 Figure 4.30 . !" inputs for a plate with elastic boundary condition and !"!!""!!!! using FEM approximation ............................................................................................... 69 Figure 4.31 . Energy deception for plate with elastic boundary condition and !"!!""!!!! using modified controlled in -plane forces ............................................................. 69 !x !!Figure 4.32 . Modal displacements for plate with elastic boundary condition and !"!!""!!!! using filtered and saturated in -plane controlled forces ......................... 70 Figure 4.33 . !"#$% inputs for a plate with elastic boundary condition and !"!!""!!!! ..... 70 Figure 4.34 . !"#$% inputs for a plate with elastic boundary condition and !"!!""!!!! ..... 70 Figure 4.35 . Filter input for a plate with elastic boundary condition and !"!!""!!!! using modified control logic ........................................................................................... 71 Figure 4.36 . Energy deception for elastic boundary condition without in -plane forces and !"!!"!!!!! using FEM approximation ............................................................ 72 Figure 4.37 . Modal displacements for elastic boundary condition without in -plane forces and !"!!"!!!! using FEM approximation ............................................................. 72 Figure 4.38 . Energy deception for elastic boundary condition with !"!!"!!!! using controlled in -plane forces ...................................................................................... 73 Figure 4.39 . Modal displacements for elastic boundary condition with !"!!"!!!! using controlled in -plane forces ...................................................................................... 73 Figure 4.40 . Controlled !" forces for plate with elastic boundary condition and !"!!"!!!! ............................................................................................................................... 74 Figure 4.41 . Controlled !" forces for a plate with elastic boundary condition and !"!!"!!!! .................................................................................................................. 74 Figure 4.42 . Energy deception for plate with elastic boundary condition and !"!!"!!!! using modified control logic .................................................................................. 74 Figure 4.43 . Modal displacements for elastic boundary condition and !"!!"!!!! using modified control logic ........................................................................................... 75 Figure 4.44 . !"#$% inputs for a plate with elastic boundary condition and !"!!"!!!! using modified control logic ........................................................................................... 75 Figure 4.45 . !" inputs for a plate with elastic boundary condition and !"!!"!!!! using modified control logic ........................................................................................... 75 Figure 4.46 . Filter inputs for a plate with elastic boundary condition and !"!!"!!!! using modified control logic ........................................................................................... 76 Figure 5.1 . Rectangular plate with a web -of-cables structure .................................................. 79 Figure 5.2 . Quarter of a plate with web -of-cables structure .................................................... 80 !xi !Figure 5.3 . Plate with the with-of-cables before the deformation, (b), after deformation Ð the forces in the web -of-cables structure are shown in (a), and the angles of deformations are shown in (b) ............................................................................... 85 Figure 5.4 . Coordinate Transformation .................................................................................. 91 Figure 5.5 . Thin rectangular plate with the components of the side forces Ð plate is comprised of eight squares of dimension 0.5m for each side. ................................................ 95 Figure 5.6 . In-plane normal stress distribution in x direction when !! = 0 N ........................ 96 Figure 5.7 . In-plane normal stress distribution in y direction when !! = 0 N ........................ 97 Figure 5.8 . In-plane shear stress distributions when !! = 0 N ................................................ 97 Figure 5.9 . Difference between ANSYS and MATLAB code for the x direction stress distribution when !! = 0 N ................................................................................... 97 Figure 5.10 . Difference between ANSYS and MATLAB code for the y direction stress distribution when !! = 0 N ................................................................................... 98 Figure 5.11 . Difference between ANSYS and MATLAB code for the shear stress distribution when !! = 0 N ....................................................................................................... 98 Figure 5.12 . In-plane normal stress distribution in x direction when !! = 3 N ........................ 99 Figure 5.13 . In-plane normal stress distribution in y direction when !! =3 N ......................... 99 Figure 5.14 . In-plane shear stress distributions when !! = 3 N ................................................ 99 Figure 5.15 . Difference between ANSYS and MATLAB code for the x direction stress distribution when !! = 3 N ................................................................................. 100 Figure 5.16 . Difference between ANSYS and MATLAB code for the y direction stress distribution when !! = 3 N ................................................................................. 100 Figure 5.17 . Difference between ANSYS and MATLAB code for the shear stress distribution when !! = 3 N ..................................................................................................... 100 Figure 5.18 . ANSYS frequency analysis when !! = 0 N ........................................................ 102 Figure 5.19 . MATLAB frequency analysis when !! = 0 N .................................................... 102 Figure 5.20 . ANSYS frequency analysis when !! = 3 N ........................................................ 103 Figure 5.21 . MATLAB frequency analysis when !! = 3 N .................................................... 103 !xii !Figure 5.22 . Web-of-cables stiffness evaluation ...................................................................... 105 Figure 5.23 . Out -of-plane vibration results when !! = 0 N .................................................... 107 Figure 5.24 . Out -of-plane vibration results when !! = 1 N .................................................... 107 Figure 5.25 . Out -of-plane vibration results when !! = 2 N .................................................... 108 Figure 5.26 . Out -of-plane vibration results when !! = 3 N .................................................... 108 Figure 5.27 . Energy dispassion for the system without control ............................................... 110 Figure 5.28 . Modal displacement for the system without control ........................................... 110 Figure 5.29 . Energy dissipation using the control logic in (4.7) - adding !! = 3 N to the four corner cables ........................................................................................................ 111 Figure 5.30 . Input force using the control logic in (4.7) - adding !! = 3 N to all four corner cables ................................................................................................................... 111 Figure 5.31 . Modal displacements using the control logic in (4.7) - adding !! = 3 N to all four corner cables ........................................................................................................ 112 Figure 5.32 . Energy dissipation using the modified control logic (4.12) with !!! - adding !! = 3 N to all four corner cables ............................................................................. 113 Figure 5.33 . Forcing input using the modified control logic (4.12) with !!! - adding !! = 3 N to all four corner cables ................................................................................... 113 Figure 5.34 Modal displacement using the modified control logic (4.12) with !!! - adding !! = 3 N to all four corner cables ....................................................................... 113 Figure 5.35 . Energy dissipation using the modified control logic (4.12) - adding !! = 3 N to all four corner cables ........................................................................................... 114 Figure 5.36 . Forcing input using the modified control logic (4.12) - adding !! = 3 N to all four corner cables ........................................................................................................ 114 Figure 5.37 . Modal displacements using the modified control logic (4.12) - adding !! = 3 N to all four corner cables ........................................................................................... 115 Figure 5.38 . Energy dissipation using the control logic (4.7) - adding and subtracting !! = 3 N to two opposite corner cables respectively .......................................................... 116 Figure 5.39 . Forcing input using the control logic (4.7) - adding and subtracting !! = 3 N to two opposite corner cables respectively .............................................................. 116 !xiii !Figure 5.40 . Modal displacements using the control logic (4.7) - adding and subtracting !! = 3 N to two opposite corner cables respectively ...................................................... 117 Figure 5.41 . Energy dissipation using the modified control logic (4.12) with !!! - adding and subtracting !! = 3 N to two opposite corner cables respectively ................ 117 Figure 5.42 . Forcing input using the modified control logic (4.12) with !!! - adding and subtracting !! = 3 N to two opposite corner cables respectively ....................... 118 Figure 5.43 . Modal displacements using the modified control logic (4.12) with !!! - adding and subtracting !! = 3 N to two opposite corner cables respectively ................ 118 Figure 5.44. Energy dissipation using the modified control logic (4.12) - adding and subtracting !! = 3 N to two opposite corner cables respectively ....................... 119 Figure 5.45 . Forcing input using the modified control logic (4.12) - adding and subtracting !! = 3 N to two opposite corner cables respectively ................................................ 119 Figure 5.46 . Modal displacements using the modified control logic (4.12) - adding and subtracting !! = 3 N to two opposite corner cables respectively ....................... 120 Figure 5.47 . Energy dissipation using the control logic (4.7) - multiple ! T values with randomly switching ............................................................................................. 121 Figure 5.48 . Forcing inputs using the control logic (4.7) - multiple ! T N values with randomly switching .............................................................................................................. 121 Figure 5.49 . Modal displacements using the control logic (4.7) - multiple ! T values with randomly switching ............................................................................................. 122 Figure 5.50 . Energy dissipation using the modified control logic (4.12) with !!! - multiple ! T values with randomly switching .................................................................... 122 Figure 5.51 . Forcing inputs using the modified control logic (4.12) with !!! - multiple ! T N values with randomly switching .......................................................................... 123 Figure 5.52 . Modal displacements using the modified control logic (4.12) with !!! - multiple ! T values with randomly switching ...................................................... 123 Figure 5.53 . Energy dissipation using the modified control logic (4.12) - multiple ! T values with randomly switching ..................................................................................... 124 Figure 5.54 . Forcing inputs using the modified control logic (4.12) - multiple ! T N values with randomly switching ............................................................................................. 124 !xiv !Figure 5.55 . Modal displacements using the modified control logic (4.12) - multiple ! T values with randomly switching ..................................................................................... 125 Figure 5.56 . Energy dissipation using the control logic (4.7) - multiple ! T with systematic switching .............................................................................................................. 126 Figure 5.57. Forcing inputs using the control logic (4.7) - multiple ! T N with systematic switching .............................................................................................................. 126 Figure 5.58 . Modal displacements using the control logic (4.7) - multiple ! T with systematic switching .............................................................................................................. 127 Figure 5.59 . Energy dissipation using the modified control logic (4.12) with !!! - multiple ! T with systematic switching .............................................................................. 127 Figure 5.60 . Forcing inputs using the modified control logic (4.12) with !!! - multiple ! T N with systematic switching .................................................................................... 128 Figure 5.61 . Modal displacements using the modified control logic (4.12) with !!! - multiple ! T with systematic switching ............................................................... 128 Figure 5.62 . Energy dissipation using the modified control logic (4.12) - multiple ! T with systematic switching ............................................................................................ 129 Figure 5.63 . Forcing Inputs using the modified control logic (4.12) - multiple ! T with systematic switching ............................................................................................ 129 Figure 5.64 . Modal displacements using the modified control logic (4.12) - multiple ! T with systematic switching ............................................................................................ 130 !"!!!!!Chapter 1 ! Introduction Vibration suppression of flexible structure s has been investigated for decades due to its widespread engineering applications . The objective of vibration suppression is to prevent the structure from being damaged or prevent compromising its desired usability . Space structures , which may undergo vibration due to differential heating, can be seriously limited in their ability to acquire images . Therefore, it is desirable to suppress the undesired vibration and stabili ze the space structure motions. Vibration suppression of flexibl e structure s is an active fi eld to research and various methods have been proposed to address the problem . 1.1. Motivation Vibration suppression of space structure has become an important problem due to the increasing demand of high precision space application s. To reduce the payload, space structure s are desired to have low mass an d therefore low rigidity . In general, space structures are characterized by low damping; therefore, they are highly sensitive to external disturbances and are prone to #!!mechanical vibrations . Consequently, elimination or reduction of structural vibrati on ha s been extensively studied; t hese studies have focused on disturbance rejection controllers, which can eliminate the structural vibrations. Both passive and active controllers have been proposed . In passive vibration suppression approach , passive elements are mounted on the structure to change the structural characteristics such as damping and stiffness. This method is effective for attenuation of high frequencies, but is not effective for suppression of low frequency vibration . In addition, it is expensive and it increases the overall structure weight, which is unfavorable for space applications . In contrast, active vibration controllers are based on feedback of the vibration in the structure . In traditional active vibration control, the structure is modeled as a linear system using modal coordinates ; certain modes of the structure are estimated and controlled using piezoelectric actuators and sens ors. For a large flexible structure, a large number of actuators and sens ors are required, which means the addition of a large number of hardware components such as amplifiers and data acquisition devices. This hardware adds significant weight and cost to the structure. To reduce the hardware requirements of active vibration control, Nudehi [48] proposed to us e end forces in a flexible beam . This thesis presents an extension o f the work done in [48] to control the vibration of a rectangular plate using piezoelectric transducers as sens ors to estimate the modal displacements and in -plane cables that act as actua tors. These in -plane cables provide in-plane tensions that alter the structura l stiffness. By switching in-plane forces, the modal energy can be reduced and the vibration of the structure can be suppressed . $!!1.2. Literature review The problem of thin plate vibration has been studied for several decades . Most of these studies have focus ed on investigating the dynamic behavior of the structure for a certain restraint condition along the edges. Leissa [ 38], for instance , provided very usef ul and comprehensive review for the plate vibration with different type of boundary conditions. The first observation of nodal patterns for a flat square plate is done by the German physicist, Ernst Chladni (published in 1787 in his book "Discoveries in th e Theory of Sound" ), by conducting experiments using sand on vibrating plate. Several years later (in the early 1800s), a French mathematician (Sophie Germain) obtained a differential equation for transverse deforma tion on plates. Her equation had an erro r due to neglecting the strain energy associated with warping of the plate midplane. LangrangeÕs notes , found in 1813 , reveal ed the correct version of the governing equation for a thin plate. However, the differential equation of rectangular plates with fle xural resistance is derived by Navier (1785 Ð 1836). Navier also obtain ed the exact solution for simply supported rectangular plates in 1820 . In 1850, Kirchhoff found the extended plate theory by combin ing the bending and stretching actions of the plate. The natural frequencies of vibrating structure s using single admissible function were obtained by Rayleigh (1877). Ritz (1909) improved Rayleigh Õs theory by including more than a single admissible function and performing a minimization with respect to the unknown coefficients of these admissible functions. He obtained an approximate solution of a freely vibrating plate, which does not have an exact solution. This method is known as the Rayleigh -Ritz method. Finite element method s were developed in t he mid -1950s by Tuner, Clough, Martin, and Topp . Their method obtained the numerical solution of complex plate problems in an efficient way. %!! Plate modeling (vibration analysis) has been studied for vari ous boundary conditions using deferent method (e.g., Williams, [33]; Zhou and Ji, [34]; Farrel and Ryall, [35]; Frendi and Raliss, [36]; Chien -Ching Ma, [27]; Israr, [22]; Werfalli and Karound, [29]; Gupta, [28]). Some of these results used FEM to obtain the numerical solution. The vibration of plate with e lastic foundations is documented in the literature, for instance, Hsu, [24], and Hatami, [31]. The dynamics of flexible structures (beams and plates) have been investigated under the application of end forces ; these include the work by : Herrmann, [12]; Cel ep, [1]; Sasaki and Chonan, [2]; Higuchi and Dowell, [3]; Adali, [4]; Zuo and Schreyer, [5]; Kim and Park, [9]; Kim and Park, [10]; Langthjem and Sugiyama, [6]; and Arreol -Lucas, [13]. Most of these results are related to flutter instability due to the eff ects of the follower forces that have non -conservative nature. Vibration suppression of plate s has been achieved using traditional methods (active and passive vibration control) and they are well documented in the literature. For instance, Shirazi and Owj i [21] investigated the vibration control of simply supported plate. Carra and Ambilli [20] control led the vibration of a rectangular aluminum pla te in contact with air or water. Some of the vibration control methods and stability analysis are presented in the following references: Khorshidi, [14]; Djanan, [15]; Hu and Jia, [16]; Qiu, [17]; Kucuk, [18]; Falangas, [23]; Ashour, [25]; Kovarova, [26]; Yaman, [30] ; and Braghin and Cinquemani [41] . Active vibration control is used in the most of these studies . In general, cables are used to increase the stiffness of lightweight structures. However, using cables to control the vibration has not been studied extensively . Achire and Preument [37] introduce d multiple control strategies to control the vibration in ca ble -stayed bridges. Preumont and Bossens [ 48] propose tendons to create an active damping in truss structures. Skidmore and Hallauer [7] used multiple -actuator theory to actively control individual vibration modes of &!!beam -cable structure. Nudehi [Ref] used a cable to apply an end -force on a cantilever beam to suppress the vibration. Sakamoto and Park [43] and [44] present ed structural and controller design s to attenuate vibration in large membrane using web -like perimeter cables around they membrane. The co ntrol of deployment of space structure s using web -like structure has been studied by G−rdsback and Tibert , [42]; Lan e A Murphey, [45]; and Keon and Murphey , [46]. Greschik and Mejia -Ariza [47] present ed a numerical and theoretical study of control out -of-plane flatness and thermal warping via suspension tensions. Following NudehiÕs work [48], in -plane forces are used in this thesis to control t he vibration of a rectangular plate. The approach is extended by applying the in -plane forces through a web -of-cabl es structure. 1.3. Scope and content This thesis is organized as follows. In chapter 2, a brief review of the dynamic s of a thin plate is presented. Following [39], the theory of the elastic plate is used to derive the equation of motion for a plate, with and without in -plane boundary forces. In chapter 3, some analytical and approximate solution s are obtained for a plate subjected to different types of in -plane forces. Chapter 4 introduces a control strategy, following the work presented by Nudehi [48], to reduce the lateral vibration on a plate. We assumed that the plate is subjected to uniformly distributed in-plane for ces. In chapter 5 a web -of-cables around the plate is proposed as a mechanism to suppress the vibration. This web -of-cables provides the boundary of the plate with in -plane forces that act at few discrete locations. As a result , non -uniform stress distribu tions are created. Using the pre -stress data, an out -of-plane vibration analysis is done to obtain the d ynamic model of the structure. Simulations are carried out to examine the efficacy of the control strategy. '!! Chapter 2 Dynamics Of A Thin Rectangular Plate Ð A Review !A plate is a structure that consists of two surfaces separated by a small distance . The small distance is called the thickness of the plate , which is assumed to be smaller than the other dimensions of the two surfaces , namely the length and the width of the plate . A plate is considered to be thin if the ratio of its thickness to the other dimensions is less than 1/20 . A two -dimensional approximation to the elasticity theory is called the theory of the elastic plate , which considers the deformation of every point in the plate in terms of the midplane deformation. The theory of the elastic plate is used to derive the equation of motion for a thin rectangular plate , with and without in -plane forces acting at the boundary . 2.1 Classical Plate Theory The Cartesian coordi nate system (x, y and z) is used to define the coordinates of a thin rectangular plate that has has a length a, width b, and thickness h as shown in Fig. 2.1. The displacements in the x and y direction are defined as u and v respectively , and the transverse deformation of the midplane is defined as w. The derivation of the equation of motion for the transverse vibration is based on the following assumptions [39] : (!!!Figure 2. 1. Thin Rectangular Plat !1. The thickness of the plate h is smaller than other physical dimensions, namely the length ! and the width !; and usually the ratio of the thick nesses to the smaller dimension (length or width) is less than 1/20 . 2. The material of the plate is homogenous and linear elastic , and the stress -strain relationship follows Hook eÕs Law. 3. The mid plane of the plate remains a neutral plane, i.e., the mid plane does not experi ence stretching or contractions . 4. The transverse deformation of the midplane w is small compared to the thickness h. 5. The transverse shear deformation is negligible , i.e., a straight line normal to the midplane remains normal after the deformation. As a result, the transverse shear st rains can be neglected. 6. The plate is assumed to have negligible r otary inertia. 7. The plate resists the i n-plane forces through bending , and in-plane and out -of-plane displacement. yxzhba)!! Figure 2. 2. An infinitesimal element of the thin rectangular plate Consider an infintismal element of the plate as shown in the Fig. 2.2, which has dimensions dx, dy and h in the x, y and z direction respectively . The element is subjected to bending moments, !!! and !!!, twisting moment s,!!!" and !!!", and shear forces , !! and !!! due to internal stresses .!!! and !!! denote the bending moments per unit length about the x and y axes, respectively; !!!" and !!!"!are the twisting moments per unit leng th about the x and y axes respectively and !! and !!! are shear forces per unit length parallel to y and x axes respectively. Following the derivation in [39], the summation of the forces in the z direction yields !!!!!"!!!!!"!!"!"!"!!!!!"!!!!!"!!"!"!"!!!!!"!!"!!!!!"! (2.1) dyyxzdx4h*!! where !!!is the density of the material of the plate. Dividing both sides by the area we get !!!!!"!!!!!"!!!!!!!"!!(2.2) The sum of the moments of all the forces about the y axis gives !!!!"!!!!!!!!"!"!"!!!"!"!!!"!!!!"!"!"!"!!!!"!"!!!!!!!!!"!"!"!"!!!!!!"!"!# !!!"!!"!"!"#$%!& !!"!#$% !!"#!!"!!"#$#%"& !(2.3) By dropping the inertia moment term (assumption 6) , the equation can be reduced to !!!!!!!!"!!!!"!"!(2.4) The sum of the moments about the x axis gives !!!!"!!!!!!!!"!"!"!!!"!"!!!"!!!!"!"!"!"!!!!"!"!!!!!!!!!"!"!"!"!!!!!!"!"!# !!!"!!"!"!"#$%!& !!"!#$% !!"#!!"!!"#$#%"& !(2.5) and it can be reduced to !!!!!!!!"!!!!"!"!(2.6) Following the method used in [39], the strain -displacement relationship s are derived next. Assuming that the midplane is the xy plane, we consider the straight line AB oriented in the z direction normal to the midplane - see Fig. 2.3. "+!!!Figure 2. 3. Rotation of a normal line after the deformation After deformation, AB rotates to !!!! due to bending and shear deformation. A point c at a distance z from the midplane on line AB will have a displacement u in the x direction after the deformation and could be defined as !!!!!!"!"!(2.7) where !!!"!!"! is the slope of the deformation surface in the xz plane. Similarlly, the displacement v in the y direction can be expressed !!!!!!"!"!!(2.8) The normal strains using strain -displacement relationship can be defined as !!!!!!"!"!!!!!!!!!!!!!!!!!!!"!"!!(2.9) The shear strain is !!!"!!"!"!!"!"!(2.10) Substituting both Equations (2.7) and (2.8) into equations (2. 9) and (2.10), we get !!!!!!!!!!!"!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!"!!!!!!!!!"!#!!!(2.11) According to Hook eÕs law, the strain -stress relationship can be expressed as follow s: !!!!!!!!!!!!!!!!!(2.12) !ABzxzcBefore DeformationAfter DeformationA'AB'Bc'ucxMidplane Line z""!!!!!!!!!!!!!!!!!!!(2.13) !!!!"!!!"!!!!(2.14) where !!! and !!! are the normal stresses, !!" is the shear stress, and !!!!!!!!!!!!!(2.15) where G is the s hear modulus , E is the Young Modulus and ! is the PoissonÕs ratio . Solving the above equations for the stresses results in !!!!!!!!!!!!!!!!!!!!(2.16) !!!!!!!!!!!!!!!!!!!!!(2.17) !!!!"!!!!!!!!!!"!!!(2.18) By integrating the moment of the in -plane stresses over the plate thickness we get the bending and the twisting moments as follows: !!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!"!!!"!!!!"!!!!!!!! !(2.19) Substituting the stress relationships given by (2.16) to (2.18) into the (2.19) and evaluating the integrals, we get !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!"! (2.20) where D is the flexural rigidity that is defined as !!!!!!"!!!!!!!(2.21) and !!!!!!"!and !!!!are the curvature of the deflected midplane that can be expressed by !!!!!!!!"!! (2.22) "#!!!!"!!!!!"!#!! (2.23) !!!!!!!!"! (2.24) Substituting (2.4) and (2.6) into (2.2) and replacing the moments using (2.20); produces !!!"!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!"!#!!!!!!!"!!!!!!!!"! (2.25) Replacing !!!!, !!!!and !!" in (2.25) by (2.22) to (2.24) can reduce it to !!!!!"!!!!!!!"!!"!!!!!!"!!!!!!!!!!"!!!!!!!! (2.26) where !! is the biharmonic operator and can be found from Laplacian operator !!!. Equation (2.26) describes of the motion for a thin rectangular plate of thickness h. "$!!2.2 Plate Dynamics with in -plane forces In this section, all the assumptions in section 2.1 are valid except assumption 3. The plate is subjected to time dependent in -plane distributed forces that act on the boundary. These forces are !!!!!!!!!!, !!!!!!!!! and !!"!!!!!!!!"!!!!! - see Fig. 2.4 . The equation of mot ion for a thin rectangular plate with in -plane forces can be derived following [39]. Figure 2. 4. Thin rectangular plate with in -plane forces xyNxNyNxyNyx"%!! Figure 2. 5. An element of the thin r ectangu lar plate with in -plane forces Figure 2. 6. Side view for an element of the plate with in -plane forces dyyxzdxh16dx22xzyzdy161"&!!For an infinitesimal element as shown in Figs. 2.5 and 2.6, the sum of the forces in the x directions gives !!!!!!!!"!"!!"!!"#!!!!!!!"!!!!"!"!"!"!!"#!!!!!!!!!!!!!"!!"#!!!!!!"!!"!!"#!!!!!!!!!!! (2.27) For a small deflection (2.27) can be reduced to !!!!"!!!!"!"!! (2.28) Similarly , summing the forces in the y direction gives !!!!!!!!"!"!!"!!"#!!!!!!!"!!!!"!"!"!"!!"#!!!!!!!!!!!!!"!!"#!!!!!!"!!"!!"#!!!!!!!!!!! (2.29) and for a small deflection the equation can be reduced to !!!!"!!!!"!"!! (2.30) The s um of the forces in the z direction gives !!!!!!!!"!"!!!"!!"#!!!!!!!!!"!!"#!!!!!!!!!!!"!"!"!!"#!!!!!!!!!"!!"#!!!!!!"!!!!"!"!"!"!!"#!!!!!!!"!!"!!"#!!!!!!!"!!!!"!"!"!"!!"#!!!!!!!"!!"!!"#!!!!!!!!!!!"!"!"!!!!!"!!!!!!!!"!"!"!!!!!"!! (2.31) For a small deflection and using the following approximations !"#!!!!!!!"!"!!!!!!!!!"#!!!!!!!!!!!!!!!"!!"!!"!"!!!!!"!!!" (2.32) "'!! !"#!!!!!!!"!"!!!!!!!!"#!!!!!!!!!!!!!!!"!!"!!"!"!!!!!"!!!"! (2.33) !"#!!!!!!!"!"!!!!!!!!!"#!!!!!!!!!!!!!!!"!!"!!"!"!!!!!"!!!"! (2.34) !"#!!!!!!!"!"!!!!!!!!!!!"#!!!!!!!!!!!!!!!"!!"!!"!"!!!!!"!!!"!! (2.35) Equation (2.31) can be reduced to !!!!"!!!!!"!!!!!!!"!!!!!!!!"!!!!!"!!!!"!# (2.36) Equation (2.3 6) represents a static equilibrium . By putting the inertia force in (2.36), i.e., dÕAlembertÕs Principle ; gives !!!!"!!!!!"!!!!!!!!"!!!!!!!!"!!!!!!!!"!!!!!"!!!!"!# (2.37) Equation (2.37) represents the dynamic equilibrium in the z direction . An important observation should be mentioned that the in -plane forces do not have any contribution to the moment along the edges of the element. Therefore, using the same derivation presented in the section 2.1 for summing the moment s about the x and y axes, the equation of motion for a rectangular plate with in-plane forces can be expressed as !!!!!!"!!!!!!!"!!"!!!!!!"!!!!!!!!!"!!!!!!!!"!!!!!!!!"!!!!!"!!!!"!# (2.38) As mentioned in assumption 5 of section 2.1 for deriving the dynamic equation of a thin rectangular plate, the direct stress in z direction !!! is smaller than other stresses, which is usually neglected. Therefore, the nonzero stress componen ts are only considered for bending "(!!moments for the case of the thin rectangular plate without in-plane forces - see equation ( 2.19). On the other hand, in th e case of a thin rectangular plate with in -plane forces the resulting forces in x, y , and in-plane share forces could be evaluated as !!!!!!!!!"!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!!"!!"!!!!!!!!! (2.39) By introducing the Laplacian operator !!!, equation (2.38) could b e reduced to the compact form !!!!!!!!!!!!!!!"!!!!!!!!"!!!!!!!!"!!!!!"!!!!"!#! (2.40) ")!! Chapter 3 Dynamic Behavior of a Thin Rectangular Plate Subjected to In -Plane Forces !In the previous chapter, the equation of motion of a thin rectangular plate, with and without in-plane forces, was presented . In this chapter some analytical and approximate solutio ns are obtained for the plate subjected to defferent types of in -plane forces . An analytical solution for the plate, with and without in -plane force, is reviewed from the literature for simply supported boundary condition s. An approximate solution is obtai ned using the Rayleigh -Ritz method, which requires assumed mode shapes that satisfy the geometric boundary conditions. Another approximate solution is obtained using FEM analysis. In this method, a polynomial shape function is used. The methods are compare d for a simply supported plate, with and without in -plane forces. This chapter is divided into two sections based on the nature of the in-plane forces. The first section assumes that the plate is subjected to uniformly distributed in -plane forces that act on the boundar y; this results in a uniform stress distribution in the plate . The second section deals with a plate that is subjected to in -plane forces acting at a few discrete locations; this results in a non -uniform stress distribution. "*!!3.1 A Simply -Supported, Thin, Rectangular Plate Subjected To Unifor mly Distributed In -Plane Forces 3.1.1 Analytic Solution Consider the plate shown in Fig. 3.1. The solution to the dynamics of this plate is available in the literature [ 39]; it is reviewed here for convenience. The transverse vibration of a thin rectangular plate was presented in the last chapter in (2.43). Assuming zero shear forces (!!"!!!, the equation can be written as: !!!!!!!!!"!!!!!!!"!!"!!!!!!"!!!!!!!!!!!"!!!!!!!!"!!!!!!!!"!! (3.1) where !! and !! are the in-plane forces, shown in Fig. 3.1. The solution requires two boundary conditions at each edge. For instance, a simply supported boundary condition on all sides (as in Fig. 3.1 ) requires the deflection and bending moment to be zero. !!!!!!!!!!!! (3.2) !!!!!!!!!!!! (3.3) !!!!!!!!!!!! (3.4) !!!!!!!!!!!! (3.5) !!!!!!!!!!!!!"!!!!!!!"!!!!!!!!! (3.6) !!!!!!!!!!!!!"!!!!!!!"!!!!!!!!! (3.7) !!!!!!!!!!!!!"!!!!!!!"!!!!!!!!! (3.8) !!!!!!!!!!!!!"!!!!!!!"!!!!!!!!! (3.9) #+!! Figure 3.1. Simply supported plate with uniformly distributed in -plane forces !The solution to the governing equation in (3.1) is assumed to be of the form: !!!!!!!!!!!!!!"# (3.10) Substituting (3.10) into (3.1) gives !!!!!!"# !!!!!!!!!!!!!"# !!!!!!!!"!!!!"# !!!!!!!!"!!!!"# (3.11) Dividing both sides by !!!!"# ! and assuming the !!!!!!!!!! yields !!!!!!!!!!!!!!!"!!!!!!!!!"! (3.12) In general, the solution for a simply supported plate that is not subjected to in -plane forces can be expressed as [ 39] xysimply supportedabNxNxNyNy#"!! !!!!!!!"!"#!"!!!!!!!!!!!!"#!"!!! (3.13) where !!!" is a constant represents the amplitude. Because the in -plane forces will not affect the bending moment at the edges, we can substitute (3.1 3) into (3.1 2) to get (3.14) !!"!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!"!!!!!!! !"#$# !!!!!!!!!!!!!! (3.14) Equation (3.16) can be used to compute the frequencies of a thin rectangular plate with in -plane forces. When !! and !! are zero, (3.1 4) reduces to the form !!"!!!!!!!!!!!!!!!!! (3.15) 3.1.2 Rayleigh -Ritz Approximation In the Rayleigh -Ritz method, we assumed the solution of the equation of motion in the following form !!!!!!!!!!!!!!!"#!" (3.16) The maximum kinetic and potential energies for the plate shown in Fig. 3.1 can be evaluated from !!"# !!!!!!!!!!!"!!" (3.17) !!"# !!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!"!!" (3.18) The Rayleigh quotient, is defined as ##!! !!!!!!!"# !!"# !!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!"!!"!!!!!!!!"!!"!!!!!!!!!!! (3.19) where !!!!!!!!!!"!!!!!!!!,!!!!!!!!!"!,!!!!!!!!!!"!and !!!!"!!!!!"!#. The n-terms approximation for !!!!!in (3.1 6) can be expressed as !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!! (3.20) where !!!!! is a mode shape that satisfies the geometric boundary conditions , and !! are the constants that will be determined. Letting !!!"!!!!!!!!! (3.21) Equation (3.19 ) can be reduced to !"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!"!!"!!!!!!!"!!"!!!!!!!!!!!!!! (3.22) Substituting (3.20) into (3.22), we get #$!! !"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!" (3.23) where !!!!!!!!!!!!!!!"!!"!!!!!!!!!!! (3.24) and !!!!!!!!!!!!"!!!!!!!!!!!!"!!!"#!!!!!"!!!!!!"!#. By minimizing (3.23) with respect to !!!as !"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.25) we obtain #%!! !!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!"!!!!!!!!"!# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.26) Substituting (3. 23) back into the (3.26), we get #&!! !!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!"!!!!!!!!"!# !!!!!!!!!!!!!!!!!!"!!"!!!! (3.27) Dividing both sides by !!!!!!!!!!!"!!", we get !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!"!!!!!!!!!!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!!!!"!!"!!! (3.28) Denoting the first term in (3.31) as !!" and the second term as !!", and !!!!!! we get !!"!!!!!"!!!!!!!! (3.29) The eigenvalue problem (3.29) is used to solve the dynamic equation. To illustrate this method, an example is provided below: The Rayleigh quotient can be expressed as #'!! !!!!!!!"# !!"# !!! (3.30) where N and D can be expressed as follows: !!!!"# !!!!!!!!!"!!!!!!!!!!!!!!! (3.31) !!!!"# !!!!!!!!!"!!!!!!!!!!!!!!! (3.32) where !!!!!" is the stiffness matrix, !!!!" is the mass matrix and !!!!!!!!!!!. The elements of the stiffness and mass matrices are defined below: !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!"!!"! (3.33) !!!!!!!!"!!" (3.34) where !!!!!!!!!"!, !!!!!!!!"! and ! is the assumed mode shapes. We can express (3.30) as !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.35) Minimizing (3.35) with respect to !!!!! !!!!!!!, we get !"!!!!!!"!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.36) #(!!!!!"!!!!!!!"!!!!! !"!!!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!! where !!!!! are the eigenvalues. Form (3.31) and (3.32) we have !"!!!!!!!!and !"!!!!!!!, (3.36) can be reduced to !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.37) The free vibration equation can be expressed as !!!!!!!!"#$% !!!!!!"#$% !!! (3.38) Dividing both sides by !!!"#$% ) gives !!!!!!!!! (3.39) where the mass matrix, !!!, can be evaluated from !!!!!!!!!!!!!!"!!" (3.40) and the stiffness matrix, !!! can be obtained from !!!!!!!"!!!!!"!! (3.41) where !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!"!!"!!!! (3.42) !!"!!!!!!"!!"!!!! !!"!!!!!!"!!"!!!! (3.43) #)!!For the plate without in -plane forces, equation s (3.43) are neglected. From (3.39) Ð (3.43) we can obtain the dynamics of the plate with and without the uniformly distributed in -plane forces using Rayleigh -Ritz method. 3.1.3 Finite Element Analysis !! The finite element method use variation and interpolation methods to obtain the dynamics of a structure efficiently . Usually, the FEM method is used for complex structures with complex boundary conditions. In this method , the actual structure is divided into small finite parts that are called finite elements. These elements have nodes that surround and connect the m to the neighbor ing elements. This combination is called the finite element mesh. The solution of the governing equation is found for each element individually using polynomial shape function that satisfies the geometrical boundary condition s of the structure. This solution is called a local solution. Then , the local solution s for all elements are assembled to find the global solution. In this section , the FEM analysis uses the kinetic energy and the potential energy to obtain the dynamics o f the plate using the global solution after we applied the boundary conditions. The plate is discretized into finite quad elements as shown in Fig. 3.2 has four nodes and each no de has three degree of freedoms, namely, one translational degree of freedom i n the z direction ,!!!", and two rotational degrees of freedom , !!"# and !!"# , about the x and y axes respectively. #*!! Figure 3.2. Four -node quad plate element with 12 DOFÕs For the element shown in Fig. 3.2, the polynomial shape function can be assumed in the following form [49]: !!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!"# !!!!!!!!!!!!!!!"# ! (3.44) where the functions in the x and y direction s are !!"!!!!!!!!! (3.45) !!"!!!!!!!!!!! (3.46) !!"!!!!!! (3.47) !!"!!!!!!! (3.48) xyzw21w!11w!12w!21w!22wy22wy21wy11 wy12w11w12w22ba$+!! !!"!!!!!!!!!!!!!!!!!!!!!! (3.49) !!"!!!!!!!!!!!!!!!!!!!!!! (3.50) !!"!!!!!!!!!!!!!!!!!!!! (3.51) !!"!!!!!!!!!!!!!!!!!!!! (3.52) !!!!!!!!!!!!!!!!!!!!!!!!! (3.53) !!!!!!!!!!!!!!!!!!!!!!!!! (3.54) !!"!!!!!!!!!!!!!!! (3.55) !!"!!!!!!!!!!!!!!! (3.56) The polynomial shape function (3.44) is used to obtain the stiffness and mass matrices for each element using the potential and kinetic energies as presented in section 3.1.2. Equation (3.33) and (3.3 4) can be used to express the elemental stiffness and mass matrices as follows: !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!"!!"!!!!!!!!!!!!!!!"!!" (3.57) !!!!!!!!!"!!"! (3.58) After we obtained the stiffness and mass matrices for each element, we assembled them to get the global stiffness and mass matrices. Then , the boundary conditions are applied to $"!!obtain the reduced global matrices. These reduced matrices are used to obtain the frequencies and mode shapes of the plate. !!!!!!!!!!!! (3.59) where !!!!! is the eigenvalues and ! are the eigenvectors . !! and !! are the reduced global stiffness and mass matrices. 3.1.4 Comparisons In order to examine the accuracy of the approximation methods, a comparison with the analytical solution is made for a simply supported plate , with and without in -plane forces. A plate with the dimensions shown in Fig. 3.3 is considered. Figure 3.3. Thin rectangular plate dimensions 2m1m0.5mm0.5mmxyz$#!!The material properties for the plate are assumed to be: Material Aluminum YoungÕs modulus 70 Gpa Mass density 2730 kg/!! Table 3.1 shows the first five natural frequencies for the plate without in -plane forces and both approximation methods that presented in sections 3.1.2 and 3.1.3. In the finite element method, we used !"!!" elements meshing as shown in the Fig. 3.4 . Figure 3.4. Finite element meshing for the plate Table 3. 1. Comparing the Solutions for Thin Rectangular Plate without In -Plane Forces Mode Analytical in Hz FEM in Hz Rayleigh -Ritz in Hz % Error in FEM % Error in Rayleigh -Ritz 1 1.532 1.5531 1.53088 2.22 0.073 2 2.4512 2.5045 2.44912 3.52 0.084 3 3.9832 4.0590 3.98001 3.19 0.08 4 6.1279 6.2190 6.12353 2.27 0.071 5 6.1279 6.2193 6.12353 2.60 0.071 2m1m20 Elements10 Elements$$!!Table 3.1 shows that the Rayleigh -Ritz approximation is closer to the analytical solution than the EFM method because it uses mode shape s instead of polynomial shape functions. However, the FEM solution could be improved by increasing the number of elements, i.e., by using fine r meshing. The results show that the maximum error using FEM method is around 3.5%, however, the maximum error for the Rayleigh -Ritz method is 0.084%. In conclusion, using these approximations, we obtained solutions close to the analytical solution with acceptable level of accuracy. Figure 3.5. Thin rectangular plate with uniformly dist ributed in -plane forces For a plate with similar material and geometrical properties, we added uniformly distributed in-plane forces along the boundaries as shown in Fig. 3.5 . The forces along the x axis is denoted as !!!!!"!!! and the forces along the y axis is!!!!!!!!. The results are shown in Table 3.2 using the analytical and both approximation methods. 2m1m0.5mm0.5mmxyzNx = 10 N/mNx = 10 N/mNy = 5 N/mNy = 5 N/m$%!!Table 3. 2. Comparing the Solutions for Thin Rectangular Plate with In -Plane Distributed Forces Mode Analytical in Hz FEM in Hz Rayleigh -Ritz in Hz % Error in FEM % Error in Rayleigh -Ritz 1 1.9289 1.9456 1.92765 1.415 0.064 2 2.959 3.0032 2.9568 2.45 0.074 3 4.5719 4.6380 4.56859 2.44 0.072 4 6.5609 6.6461 6.55641 1.99 0.068 5 6.7671 6.8495 6.76247 2.15 0.068 As shown, the Rayleigh -Ritz method is more accurate compared with FEM. However, using the Rayleigh -Ritz method has the limitation that it depends on mode shapes that should satisfy the geometrical boundary conditions. In contrar y, FEM method can solve more complex geometry and boundary conditions. Therefore, the FEM can handle problems where the in -plane forces are not uniformly distributed; in other words it can solve the problem where these in-plane forces are prescribed at arb itrary locations on the plate . With the maximum error reach around 2.4% using FEM, the results lie in the acceptable range of accuracy and can be considered as close to the analytical solution. Figure 3.6 shows the mode shapes associated with the frequenci es listed in Table 3.2 . $&!! Figure 3.6. First five mode shapes for thin simply supported rectangular plate with uniformly distributed in -plane forces 3.2 A Simply -Supported Thin Rectangular Plate Subjected To In -Plane Forces At A Few Discrete Locations In section 3.1, we presented three different ways to solve the governing equation, namely, the analytical method, the Rayleigh -Ritz method , and the finite elements method. For the case where there are in -plane forces at a few discrete locations, the analytical solution is not available in the literature and the Rayleigh -Ritz method become cumbersome because the mode shapes for such problems with non-uniform stress distributi ons are not readily $'!!available. Therefore, the FEM is used to numerically solve the governing equation. We use a code we created using MATLAB and verify the results using commercial FEM software. 3.2.1 Finite Element Analysis The analysis is divided into two steps: evaluating in -plane stress distributions and out -of-plane vibration analysis. The quad element shown in the Fig. 3.7 is used for in -plane stress distributions. The element has four nodes: one at each corner, and two degrees of freedom at each node. These degrees of freedom are the components of in -plane displacement, u and v, in the x and y directions respectively. Figure 3.7. Dimensions of the plateÕs element and coordinate transformation Each component of in-plane displacement can be defined in non -dimensional coordinates as !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.60) where the shape function !! can be written using the coordinate transformation shown in Fig. 3.7 as follows: !!!!!!!!!!! (3.61) 1243xy1234Coordinate Transformation abbauvuuuvvv$(!! !!!!!!!!!!! (3.62) !!!!!!!!!!! (3.63) !!!!!!!!!!! (3.64) The displacement vector for each element can be written as !!!!!!!!!!!!!!!!!!!! (3.65) and !!!!!!!!!!!!!!!!!!!!!!!!!! (3.66) Combining (3.65) and (3.66), the in -plane displacements for each element can be expressed as !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.67) The kinetic energy for a plate element can be obtained from !!!!!!!!!!!"!!!!!!!!!! (3.68) where !! is the element al mass matrix and can be evaluated from !!!!!!!!!!" (3.69) The potential energy for each element is given by the expression !!!!!!!!!!!!"# !!!!!" (3.70) where $)!! !!!!"!"!"!"!"!"!!"!"! (3.71) and !!!"# is the constituent matrix for in -plane stress analysis . !!"# can be obtained from the relation !!"# !!!!!!!!!!!!!!!!!!! (3.72) Equation (3.7 0) can be expressed as !!!!!!!!!!!!!! (3.73) where !!! is the element al stiffness matrix and has the from !!!!!!!!!"# !!!!" (3.74) where !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.75) For non -dimensional coordinates of node j, the elemental mass matrix can be found from !!!!!!!!!!!!!!!!!!"#!!"!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#!!"!!"!!!!!!!!!!!!!!!!!!!!! (3.76) By evaluating (3.7 6) the elemental mass matrix is obtained as $*!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (3.77) The elemental stiffness matrix for non-dimensional coordinates can be obtained from !!!!!!!!!!!!!!!"# !!!!"#!!"!!"!!!!!! (3.78) where ! is the Jacobian matrix and can be evaluated from !!!"!"!"!"!"!"!"!" (3.79) For stress distributions of a thin isotropic rectangular plate (see Fig. 3. 8), HookeÕs law is used to evaluate the stress -strain relationships as !!!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!" (3.80) Figure 3.8. Thin Rectangular plate with in -plane stresses yx%+!!The elemental forcing vector for exterior (boundary) forces can be obtained from !!!!!!!!!!!!!! (3.81) where !! is the boundary over which tractions are specified, and !!is the boundary traction. After creating the global stiffness and forces matrices, the boundary condition s were applied to get the reduced global matrices. The boundary condition s were achieved by fixing three nodes in the central element: 6 DOFÕs, in the x and y directions. After applying the boundary conditions, the displacement, strain and stresses can be evaluated as follows: !!"#$ !!!"#$ !!!!!!!!!!"#$ !!!!!"#$ (3.82) where !!is the total displacements and has two components of the displacement in the x and y directions Ð see (3.65) for one element . Then the strain and stress can be obtained from !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"# !!! (3.83) Comparing (3.8 3) with (3.80), the stress vector has the following elements !!!!!!!"#!!!". These elements represent the normal and shear stresses . They can be used in out -of-plane vibration analysis by evaluating !!!!!!!"#!!!" (using the expressions below ) for each elemental potential energy using (3.57). !!!!!!!!"!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!!"!!"!!!!!!!!! (3.84) Instead of using the same in -plane forces and uniform stress distributions for all elements, we used non -uniform pre -stress distribution information to evaluate the in -plane forces for each element. %"!!3.2.2 Commercial Software There are many commercial software based on finite element analysis that solve complex structure problem s. One of the best -known commercial software is ANSYS. For out -of-plane vib ration analysis, we used pre-stress analysis to examine the non -uniform stress distribution effects on the plate. A 3D shell element was used to mesh a simply supported plate of 2m length and 1m width, and a thickness of 0.5mm . The plate was subjected to in-plane forces as shown in the Fig. 3.9. Figure 3.9. Meshing of thin rectangular plate with in -plane prescribed forces 3.2.3 Comparisons The first five frequencies for the plate shown Fig. 3.9 are listed in Table 3.3 . The table presents the results for the case where the in -plane forces are not uniformly distributed. Instead, few discrete in-plane loads are considered. We used commercial FE M software, ANSYS, to verify the results that we got from our MATLAB FEM code for the plate with the meshing shown in Fig. 3.4. 2m1m0.5mm0.5mmxyz10 N5 N0.5m10 N10 N10 N10 N10 N5 N5 N 5 N5 N5 N5 N5 N 5 N5 N%#!!Table 3. 3. The Non -Uniform stress distribution frequency results Mode ANSYS Hz MATLAB Hz % Difference 1 2.3964 2.2915 4.37 2 3.6437 3.5697 2.03 3 5.4468 5.3826 1.17 4 6.2627 5.9567 4.88 5 7.3659 7.0454 4.35 From Table 3.3, we can conclude that the maximum difference between the modal frequencies results was around 4.8% . This could be due to the fact that more DOFÕs are used in ANSYS discretization. Therefore, the results using our FEM MATLAB code has been proven using multiple methods and the errors remain in the low range, less than 5%, errors. In table 3.2 , we saw that our FEM results were higher than the analytical solution. However, table 3.3 shows that our FEM results lower than what we got from the commercial software, which is a good indication that our FEM code has solved the problem with a good accuracy. As presented in sections 3.2.1 and 3.2.2 the solution procedure divided into two parts. In the first part, evaluations for the stress distributions are obtained for both FEM methods and the results are shown in figures below. %$!! Figure 3.10. In-plane normal stress distribution in the x direction !Figure 3.11. In-plane normal stress distribution in the y direction Figure 3.12. In-plane share stress distribution !4.64E+03+1.32E+04+2.22E+04+4.01E+04+5.79E+04+7.58E+04+9.37E+04+1.12E+05+1.29E+05+1.47E+05+1.65E+05+1.83E+05+2.01E+05Normal stress distribution in x direction using ANSYS !4.64E+03+1.33E+04+2.22E+04+4.01E+04+5.80E+04+7.59E+04+9.38E+04+1.12E+05+1.30E+05+1.47E+05+1.65E+05+1.83E+05+2.01E+05Normal stress distribution in x direction using MATLAB !4.61E+03!1.12E+03+5.87E+03+1.28E+04+1.98E+04+2.68E+04+3.38E+04+4.08E+04+4.78E+04+5.47E+04+6.17E+04+6.87E+04+7.57E+04Normal stress distribution in y direction using ANSYS !4.61E+03!1.12E+03+5.87E+03+1.28E+04+1.98E+04+2.68E+04+3.38E+04+4.08E+04+4.78E+04+5.47E+04+6.17E+04+6.87E+04+7.57E+04Normal stress distribution in y direction using MATLAB !9.92E+04!9.06E+04!7.33E+04!5.61E+04!3.88E+04!2.16E+04!4.31E+03+1.29E+04+3.02E+04+4.74E+04+6.47E+04+8.19E+04+9.92E+04Shear stress distribution using ANSYSShear stress distribution using MATLAB !9.92E+04!9.06E+04!7.33E+04!5.61E+04!3.88E+04!2.16E+04!4.31E+03+1.29E+04+3.02E+04+4.74E+04+6.47E+04+8.19E+04+9.92E+04%%!!The difference between our MATLAB Code and the commercial software are shown in the figures below Figure 3.13. Difference between ANSYS and MATLAB Code for the x direction stress distribution Figure 3.14. Difference between ANSYS and MATLAB Code for the y direction stress distribution Figure 3.15. Difference between ANSYS and MATLAB Code for the share stress distribution !4.03E+02!3.85E+02!3.50E+02!3.15E+02!2.80E+02!2.45E+02!2.10E+02!1.75E+02!1.40E+02!1.05E+02!6.97E+01!3.46E+01+4.95E!01 !4.99E!01!4.56E!01!3.69E!01!2.83E!01!1.97E!01!1.11E!01!2.42E!02+6.21E!02+1.48E!01+2.35E!01+3.21E!01+4.07E!01+4.94E!01 !4.65E!01!4.23E!01!3.39E!01!2.56E!01!1.73E!01!8.94E!02!5.99E!03+7.74E!02+1.61E!01+2.44E!01+3.27E!01+4.11E!01+4.94E!01%&!!Figures 3.12, 3.12 a nd 3.14 show that the ratio between the maximum difference and the maximum normal stress evaluated in the x and y directions and shear stress in xy plane are !!!!"!!"!!", !!!"!!"!! and !!!"!#!!"!!!respectively. These ratio are so small and can be neglected comparing with the maximum stress values that have been evaluated. Using these stress data, an out -of-plane vibration analysis has been made as discussed before and the mode shapes for the frequencies listed in table 3.3 are shown in Fig 3.1 6. Figure 3.16. First Five modes shapes for thin plate with non -uniform stress distribution %'!! Chapter 4 !!Vibration Control Of A Thin Rectangular Plate Using Uniformly Distributed In -Plane Forces !In the previous chapters we reviewed the dynamics of a thin rectangular plate with and without in-plane forces, and we showed some techniques to solve the dynamic equations. In this chapter, we introduce a control strategy to reduce the lateral vibration on the plate; it is assume d that the plate is subjected to uniformly distributed in -plane forces. The control strategy uses state feedback and switches the in -plane forces to suppress the vibration. The method presented is an extension of the work by Nudehi [ 48] where vibration of a beam was suppressed using an end force. 4.1 Control Design In absence of damping, the equation of motion of a thin rectangular plate subjected to in -plane forces can be expressed in the form !!!!!!!!! (4.1) where !!!!!!!!!!!! is the mass matrix, !!!!!! is the identity matrix, !!!!!! is the stiffness matrix, and !!!!!!!!!!!!! are the modal displacements. Using Rayleigh -Ritz Approximation (see section 3.1.2), equation (4.1) can be written as %(!!!!!!!!!!!!!!!"!!!!!!"!!!! (4.2) where !! is the stiffness matrix due to bending, and !!" and !!" are the geometric stiffness matrices due to the in -plane forces ( !! and !!). Assuming the presence of proportional damping (Rayleigh damping), the damping matrix can be expressed as !!!!!!!!!! (4.3) Adding the damping effects to (4.2), we get !!!!!!!!!!!!!!!!!"!!!!!!"!!!!! (4.4) For the purpose of designing a controller, equation (4.4) is converted to the state space representation as follows: !!!!! (4.5) !!!!!!!!!!!!!!!!!!!!!"!!!!!!!!"!!!!!! (4.6) where !!!!!!!!!! and !!!!!!!!!!! are the state variables, and !!!!!!!! and !!!!!!!! are the control inputs. The following choice of inputs !!!!!!!"!!!!!!!!!!"!!!!!!!!"!!!!!!!!!!"!!!!! !!!!!!!"!!!!!!!!!!"!!!!!!!!"!!!!!!!!!!"!!!!! (4.7) guarantees asymptotic stability of the equilibrium !!!!!!!!!!. In (4.7), !!!! and !!!! are predefined constant values. To verify asymptotic stability of the equilibrium, consider the energy of the plate as the Lyapunov function candidate; !!!!!!!!!!!!!!!!!! (4.8) Taking the first derivative of (4.8) and substituting (4.7) gives %)!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!"!!!!! !!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!!!! !!!!!!!!!!!!!! (4.9) The D and !!! matrices in equation (4. 9) are positive definite . As a result, it can be shown that !!! and !!! if !!!!. Using LaSalleÕs Theorem we can claim that the origin is asymptotically stable. Furthermore, since V is radially unbounded , the origin is globally asymptotically stable. The control law in (4.7) implies that the in -plane forces will be applied to the system if they reduce the energy of the system; otherwise, they will be zero. The control law in (4.7) can be used to control t he desired modes; however, the control inputs will have high frequencies that can exceed the bandwidth of the actuators. To accommodate this problem following [ 48], a low -pass filter is used in the control loop to filter these high frequencies without affe cting the stability of the system. The outputs of the system !!and !! are first defined as follows (see Fig. 4.1) !!!!!!!!!!!!!"!! !!!!!!!!!!!!!"!! (4.10) Figure 4.1. Control design based on filtered outputs Flexible Plate !(+)Low-Pass FilterMemoryless NonlinearityMemoryless NonlinearityLow-Pass Filter !(+)%*!!The outputs are fed back to the system after it passes through a low -pass filter that has a time constant !!. This filter eliminates the high frequency components of the outputs and provides a signal that does not e xceed the actuator bandwidth. The outputs of the filter are denoted as !!! and !!!; therefore, the filter dynamics can be represented as !!!!!!!!!!!!! !!!!!!!!!!!! (4.11) The outputs of the filters, !!! and !!!, are passed through memoryless nonlinearities !!"#$ ! and !!"#$ ! respectively such that the control forces do not exceed their predefined maximum values. The saturated inputs can be defined as !!"#$ !!!!!!!!!!!"!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!"!!!!!!!!!!!!!!!! !!"#$ !!!!!!!!!"!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!"!!!!!!!!!!!!!!! (4.12) where ! is a positive constant. To prove that the dynamic system will remain asymptotically stable after we add the filter and the saturation, we consider the new Lyapunov function candidate; !!"# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"# !!"!!!!!"#$ !!"!!!!!!!!! (4.13) Using (4.9), its derivative can be found as follows: &+!!!!"# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!!!! !!!!!!!"#$ !!!!!!!!!!!"#$ !!!!!!! (4.14) Using (4.11), equation (4.1 4) can be reduced to Since D and !!!!are positive definite, !!!"# !!. !!"# is equal to zero when !!!!. When !!!!, we have !!!!!!!!!!!!!!!!"!!!!!!!!!!!!!"!!!!, which implies !!!!!!!!! and !!"# !!!!!!"# !!!!!. Using LaSalleÕs Theorem [Ref], we can therefore claim that the origin is asymptotically stable . To illustrate this control scheme, the state space representation can be rewritten as: !!!!! (4.16) where !!!!!!!!!!!!!!!!"!!!!!!!!!"!!!!!!!!! (4.17) Equation (4.1 6) can be expressed as !!!!!!!!!!!!!!!!!!!"!!!!!!!!!"!!!!!!!!!!!!! (4.18) Therefore, the basic system that describes the control scheme is shown below. !!"# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$ !!!!!!!!!!!!"#$ !!!!!!!!!!! !!!!!!!!!!!!!!!!!"#$ !!!!!!!!!"#$ !!!!! !!!!!!!!!!!!! (4.15) &"!!!Figure 4.2. Basic system to control the vibration of the rectangular plate !4.2 Control Scheme For A Simply -Supported Plate !!The control scheme presented in section 4.1 is applied to the simply supported plate depicted in Fig. 4.3 . The plate length, width , and thickness are equal to L, W, and h respectively . Figure 4.3. Simply supported thin rectangular plate with in -plane distributed forces The plate is subjected to time -dependent , in-plane forces that are applied according to the control logic in (4.7) for the case without saturation, and according to the control logic in (4.12) for the case with saturation. The control scheme is applied to the mathematical model of the plate obtained using the approximation methods described in chapter 3 , namely the Rayleigh -Ritz method and the FEM. Logic+zxyxySimply SupporedhLW&#!!4.2.1 Control Scheme Using Rayleigh -Ritz Approximation !We assume the rectangular plate to have the following geometric and material properties: Table 4.1. Specifications of the rectangular plate and the controller Material Aluminium YoungÕs modulus , E 70 Gpa Mass density , " 2730 kg/!! Passion Ratio, # 0.35 Damping Ratio, $ 0 Damping Ratio, % 0.0002 Length, L 2 m Width, W 1 m Thickness, h 0.5 mm !! 10 N !! 5 N Using the Rayleigh -Ritz method, the first five frequencies of the plate where obtained; these are listed in Table 4.2 . The frequencies listed here will not be used in implementation of the controller; it will be compared with the frequencies obtained using FEM for model verification. Table 4.2. Frequencies in Hz using Rayleigh -Ritz method Mode Freq. in Hz for !!!! and !!!! Freq. in Hz for !!!!"! and !!!! Freq. in Hz for !!!! and !!!!! Freq. in Hz for !!!!"! and !!!!! 1 1.53088 1.6748 1.8063 1.92765 2 2.44912 2.8000 2.6314 2.9568 3 3.98001 4.4706 4.0965 4.56859 4 6.12353 6.2525 6.5493 6.55641 5 6.12353 6.2756 6.6199 6.76247 To investigate the efficacy of the control strategy, we first examine the variation of the energy and the first five modal displacements in the absence of control. The simulation results are shown in Figs. 4.4 and 4.5 for the following initial conditions: &$!! !!!!!!!!!!!!!!!!!"!!!"!!!!!! (4.19) Figure 4.4. Energy deception for simply -supported plate without the controlled in-plane forces using Rayleigh -Ritz approximation Figure 4.5. Modal displacements for simply supported plate without the con trolled in-plane forces using Rayleigh -Ritz approximation From Figs. 4.4 and 4.5 we observe that the energy of the system and the modal displacements do not converge to zero in 40 seconds. This is due to the fact that the plate has low damping. We now examine the variation of energy and the first five modal displacements for the controlled system based on the control logic in (4.7). The results are shown in Figs. 4.6 and 4.7. Time, t (Sec.)010203040 Lyapunov Energy 0100200System EnergyTime, t (Sec.)0510152025303540 -101Mode 1 Time, t (Sec.)0510152025303540 -0.200.2Mode 2Time, t (Sec.)0510152025303540 -0.500.5Mode 3 Time, t (Sec.)0510152025303540 -0.0200.02Mode 4Time, t (Sec.)0510152025303540 -0.200.2Mode 5&%!! Figure 4.6. Energy deception for simply supported plate with controlled in -plane forces using Rayleigh -Ritz approximation Figure 4.7. Modal d isplacements for simply supported plate with controlled in-plane forces using Rayleigh -Ritz approximation Figures 4.6 and 4.7 show that the energy and the modal dispalcements are converged to zero in approximately 6 secondes. These results demonstrate the efficacy of the control. The controlled forces that are used to suppress the vibration are shown in Figs. 4.8 and 4.9. Time, t (Sec.)010203040 Lyapunov Energy 0100200System EnergyTime, t (Sec.)0510152025303540 -101Mode 1Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3 Time, t (Sec.)0510152025303540 -0.0200.02Mode 4 Time, t (Sec.)0510152025303540 -0.200.2Mode 5 &&!! Figure 4.8. !! inputs for simply supported plate using Rayleigh -Ritz approximation Figure 4.9. !! inputs for simply supported plate using Rayleigh -Ritz approximation It can be observed form Figs. 4.8 and 4.9 that the control inputs switch with high frequency. Furthermore, the control inputs keep switching even after the plate vibration has subsided. To address these problems, we implement the control logic in (4.12) Ð the results are shown in Figs. 4.10 Ð 4.14. The time constant for the filter and the saturation constant for the modified control are listed in Table 4.3. Table 4.3. Modified control constants Time constant, ! 0.15 !, for saturation 0.5 Time, t (Sec.)010203040 0510Time, t (Sec.)010203040 0246&'!! Figure 4.10. Energy deception for simply supported plate with modified controlled in -plane forces using Rayleigh -Ritz approximation Figure 4.11. Modal d isplacements for simply supported plate with modified c ontrol using Rayleigh -Ritz approximation Time, t (Sec.)010203040 Lyapunov Energy 0100200System EnergyTime, t (Sec.)0510152025303540 -101Mode 1 Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3 Time, t (Sec.)0510152025303540 -0.0200.02Mode 4 Time, t (Sec.)0510152025303540 -0.200.2Mode 5 &(!! Figure 4.12. Modified control inputs !!"#$ for simply supported plate using Rayleigh -Ritz approximation Figure 4.13. Modified control inputs !!"#$ for simply supported plate using Rayleigh -Ritz approximation Figure 4.14. Modified control filter input for simply supported plate using Rayleigh -Ritz approximation From the results shown in Figs. 4.6 Ð 4.13, we observe that the control scheme with filter and saturation (4.12) takes longer time to suppress the vibration of the plate compared with the control scheme without the filter and saturation (4.7). However, the input forces have lower switching frequency and converge to zero as the vibration subsides. Time, t (Sec.)010203040 0510Time, t (Sec.)010203040 0246Time, t (Sec.)010203040 Inputs Of The System-100001000&)!!4.2.2 Control Scheme Using Finite Element Analysis For the plate shown in Fig. 4.3 and the properties listed in Table 4.1 , FEM analysis is used to model the plate dynamic (see chapter 3) . The plate is discretized to 20 elements along the x-direction and 10 elements along the y-direction - see Fig. 3.4. The first five frequencies are listed in Table 4.4 for different combinat ion of !! and !!. Table 4.4. Frequencies in Hz using Finite Element Analysis Mode Freq. in Hz for !!!! and !!!! Freq. in Hz for !!!!"! and !!!! Freq. in Hz for !!!! and !!!!! Freq. in Hz for !!!!"! and !!!!! 1 1.5531 1.6941 1.8242 1.9456 2 2.5045 2.8467 2.6810 3.0032 3 4.0590 4.5383 4.1702 4.6380 4 6.2190 6.3647 6.5070 6.6461 5 6.2193 6.7824 6.5745 6.8495 Comparing Table 4.4 with Table 4.2, we observe that the modal frequeceies obtained using the Rayleigh -Rtiz method are closed those obtained using FEM with reasonable difference (1%). To simulate the control scheme using FEM dynamic model, we should first p erform a model reduction. This is because the mass and stiffness matrices obtained using FEM have a dimension much higher than those obatined using Rayleigh -Ritz method. Model reduction will allow us to control the vibration in a few modes of the system in a computationally efficient manner . Mod al analysis is used to reduce the order of the system and transfer it to the state space representation . The governing equation for the freely vibrating system has the following form with absence of damping: !!!!!!! (4.20) &*!! where !!!! is the vector of the physical coordinate. Solving eigenvalue problem for (4.20), we get !!!!!!!!!! (4.21) We define the physical coordinates ! in terms of the model coordinate x using the relation !!!!!! (4.22) where !!is the matrix of eigenvectors. Substitut ion of (4.22) in to (4.20) gives !!!!!!!!!!! (4.23) Multiply (4.23) by!!!, we get !!!!!!!!!!!!! (4.24) where !!!!!! (4.25) !!!!!!!!!!!!!!!!!!!!!!!! (4.26) Equation (4.23) can be written as !!!!!! (4.27) where !!!"#$ !!, and !!!!!!!!!!. Using (4.27), we decouple our system to multiple equations that have the following form: !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (4.28) The proportional damping is used to obtain the damping matrix as !!!!!!!!! (4.29) Adding damping effect to (4.28), we get !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (4.30) After using the modal reduction, our system is decoupled into n equations. The first five equations from (4.30) are used to form the dynamic model that are used in the control scheme. The results for the control logic in (4.7) using the initial conditions in (4.19) are shown in the '+!!Figs. 4.15 Ð 4.18. Figure 4.15. Energy deception for simply supported plate with controlled in-plane forces using FEM approximation Figure 4.16. Modal displacements for simply supported plate with controlled in -plane forces using FEM approximation Time, t (Sec.)010203040 Lyapunov Energy 0100200System EnergyTime, t (Sec.)0510152025303540 -101Mode 1 Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3 Time, t (Sec.)0510152025303540 -0.0200.02Mode 4 Time, t (Sec.)0510152025303540 -0.200.2Mode 5'"!! Figure 4.17. !! input for simply supported plate using FEM approximation Figure 4.18. !! input for simply supported plate using FEM approximation For the modified control scheme using filter and saturation (4.12), the results are shown in Figs. 4.19 Ð 4.23; the time and saturation constants are listed in Table 4.3. Once again, it can be obse rve that the control logic in (4.12) are taking more time to suppress the vibration. Figure 4.19. Energy deception for simply supported plate with modified controlled in -plane forces using filter and saturation for FEM approximation Time, t (Sec.)010203040 0510Time, t (Sec.)010203040 0246Time, t (Sec.)010203040 Lyapunov Energy 0100200System Energy'#!! Figure 4.20. Modal displacements for simpl y supported plate with modified controlled in -pane forces using FEM approximation Figure 4.21. Modified control inputs !!"#$ for simply supported plate for FEM approximation Figure 4.22. Modified control inputs !!"#$ for simply supported plate for FEM approximation Time, t (Sec.)0510152025303540 -101Mode 1 Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3 Time, t (Sec.)0510152025303540 -0.0200.02Mode 4 Time, t (Sec.)0510152025303540 -0.200.2Mode 5 Time, t (Sec.)010203040 0510Time, t (Sec.)010203040 0246'$!! Figure 4.23. Filter inputs for simply supported plate with modified controlled in-plane forces using for FEM approximation 4.2.3 Comparisons !!In this section, we investigated the efficacy of the control strategy that is presented in section 4.1 for simply supported rectangular plate using two approximation methods (Rayleigh -Ritz method and FEM). The results are shown in Figs. 4.6 Ð 4.14 using Rayleigh -Ritz method and in Figs. 4.15 Ð 4.23 using FEM. Comparing these two sets of results, we observe that they are identical using bot h control logics (with and without modification using filter and saturation) . Therefore, either method can be used to accurately simulate the control scheme in section 4.1. For both method, the energy converge to zero within approximately 5 seconds using t he control logic in (4.7); the time required for convergence jumps up to 15 seconds with the control logic using filter and saturation (4.12). This is a price we have to pay to reduce the bandwidth requirement for the actuator: the in -plane forces used in the modified control logic have lower switching frequency and they converge to zero as the vibration of the system decays out. Time, t (Sec.)010203040 Inputs Of The System-100001000'%!!4.3 Control Scheme For A Plate W ith Elastic Boundary Conditions !!In this section, the effect of changing the boundary conditions is investigated. A plate with uniformly distributed springs around the boundaries is studied Ð see Fig. 4.24. Figure 4.24. Thin rectangular plate with in -plane distributed forces and elastic boundary conditions The plate length, width, and thickness are labeled as L, W, and h respectively , and !! represents the stiffness constant for uniformly distributed boundary springs. Using elastic boundary condition converts the problem boundary into a general case that c an reproduce other boundary conditions. For instance, the previous case (simply supported boundary conditions) can be obtained by assigning a very large stiffness constant. In addition, using different values for the stiffness constant alter the stiffness matrix of the system that can change the dynamic behavior. The plate is subjected to time -dependent , in-plane uniformly distributed forces that are applied according to the control scheme presented in section 4.1. The control scheme is applied after the dynamic model of the system is obtained using an approximation method, i.e., the mass; damping ; and stiffness matrices are obtained. The FEM is used together with model reduction to model these matrices (where the stiffness constants !! are added to their associated degrees of zxyxyelastic boundarieshLWKsKsKsKsKs KsKsKs'&!!freedom in the global stiffness matrix) in this section due to the limitation of the Rayleigh -Ritz method (see chapter 3). 4.3.1 Verify The Solution Using Elastic Boundary Conditions ! To verify the elastic boundary condition results, a special case is considered by assigning a large stiffness constant !!!. Therefore, the problem will be similar to the simply supported boundary condition. For a plate that has a properties listed in Table 4.1 , the frequency analysis using different forcin g combination are shown in the table below. Table 4.5. Frequencies in Hz using FEM method for elastic boundary condition with !!!!"!!!! Mod e Freq. in Hz for !!!! and !!!! Freq. in Hz for !!!!"! and !!!! Freq. in Hz for !!!! and !!!!! Freq. in Hz for !!!!"! and !!!!! 1 1.5531 1.6941 1.8242 1.9456 2 2.5045 2.8467 2.6810 3.0032 3 4.0590 4.5383 4.1702 4.6380 4 6.1290 6.3647 6.5921 6.6461 5 6.1293 6.4782 6.6270 6.8495 Comparing the results listed in Table 4.5 and Table 4.4 , we observe that they are close to each other, which prove our previous claim (the elastic boundary condition is a general case that can reproduce other boundary conditions). ''!!4.3.2 Control Schem e Using Finite Element Analysis !!A plate with an elastic boundary condition as shown in Fig. 4.24 is studied , the stiffness cons tant !! is set to be !""!!!. The frquency analysis for the plate with the properties listed in Table 4.1 are shown in Table 4.6. Table 4.6. Frequencies in Hz using Finite Element Analysis for elastic boundary condithins and !!!!""!!! Mode Freq. in Hz for !!!! and !!!! Freq. in Hz for !!!!"! and !!!! Freq. in Hz for !!!! and !!!!! Freq. in Hz for !!!!"! and !!!!! 1 1.5379 1.6772 1.8041 1.9242 2 2.4633 2.8014 2.6338 2.9525 3 3.9737 4.4446 4.0773 4.5376 4 6.0603 6.1680 6.1247 6.4305 5 6.0289 6.6086 6.2971 6.6678 Comparing Table 4.5 with 4.6, we observe that the frequencies are decreased as we reduced the stiffness constant !!!. The control scheme (section 4.1) is applied after we obtained the model dynamics using FEM to investigate the efficacy of the control logic. The results for the control scheme using the initial conditions in (4.19) are shown in Figs. 4.27 Ð 4.35. Figure 4.25. Energy deception for elastic boundary condition without in -plane forces and !!!!""!!! using FEM approximation Time, t (Sec.)010203040 Lyapunov Energy050100150System Energy'(!! Figure 4.26. Modal displacements for elastic boundary condition without in -plane forces and !!!!""!!! using FEM approximation Comparing Figs. 4.25 and 4.26 with 4.3 and 4.4, we observe that the system total energy is reduced by using the elastic boundary conditions. Figure 4.27. Energy deception for elastic boundary condition with controlled in -plane forces and !!!!""!!!!using FEM approximation Time, t (Sec.)0510152025303540 -101Mode 1Time, t (Sec.)0510152025303540 -0.200.2Mode 2Time, t (Sec.)0510152025303540 -0.500.5Mode 3Time, t (Sec.)0510152025303540 -0.0200.02Mode 4Time, t (Sec.)0510152025303540 -0.200.2Mode 5Time, t (Sec.)010203040 Lyapunov Energy050100150System Energy')!!Figure 4.28. Modal displacements for elastic boundary condition with controlled in-plane forces and !!!!""!!! using FEM approximation Figure 4.29. !! inputs for a plate with elastic boundary condition and !!!!""!!! using FEM approximation Time, t (Sec.)010203040 0510Time, t (Sec.)0510152025303540 -101Mode 1Time, t (Sec.)0510152025303540 -0.200.2Mode 2Time, t (Sec.)0510152025303540 -0.500.5Mode 3Time, t (Sec.)0510152025303540 -0.0200.02Mode 4Time, t (Sec.)0510152025303540 -0.200.2Mode 5'*!! Figure 4.30. !! inputs for a plate with elastic boundary condition and !!!!""!!! using FEM approximation Applying the modified control logic (4.12) using the filter and saturation with the properties listed in Table 4.3, the results are shown in the Figs. 4.31 Ð 4.35. Figure 4.31. Energy deception for plate with elastic boundary condition and !!!!""!!! using modified controlled in -plane forces Time, t (Sec.) 010203040 0246Time, t (Sec.)010203040 Lyapunov Energy050100150System Energy(+!! Figure 4.32. Modal displacements for plate with elastic boundary condition and !!!!""!!! using filtered and saturated in -plane controlled forces Figure 4.33. !!"#$ inputs for a plate with elastic boundary condition and !!!!""!!! Figure 4.34. !!"#$ inputs for a plate with elastic boundary condition and !!!!""!!! Time, t (Sec.)0510152025303540 -1-0.500.51Mode 1Time, t (Sec.)0510152025303540 -0.2-0.100.10.2Mode 2Time, t (Sec.)0510152025303540 -0.500.5Mode 3Time, t (Sec.)0510152025303540 -0.02-0.0100.010.02Mode 4Time, t (Sec.)0510152025303540 -0.2-0.100.10.2Mode 5Time, t (Sec.) 010203040 0510Time, t (Sec.)010203040 0246("!! Figure 4.35. Filter input for a plate with elastic boundary condition and !!!!""!!! using modified control logic To investigate the boundary condition effects on the control logic, we reduced the stiffness constant to !!!!"!!!. The frequency analysis results for different combination of in -plane forces are listed in Table 4.7. Table 4.7. Frequencies in Hz using Finite Element Analysis for elastic boundary condithins and !!!!"!!!! Mode Freq. in Hz for !!!! and !!!! Freq. in Hz for !!!!"! and !!!! Freq. in Hz for !!!! and !!!!! Freq. in Hz for !!!!"! and !!!!! 1 1.4397 1.5664 1.6647 1.7764 2 2.2204 2.5273 2.3485 2.6427 3 3.4896 3.8983 3.5532 3.9574 4 4.9045 5.0257 5.0672 5.1862 5 5.1663 5.6107 5.1983 5.6414 Comparing Tables 4.6 and 4.7, we observe that the frequencies are reduced as we reduce the stiffness constant for the boundary springs. The control scheme results are shown in the Figs. 4.36 Ð 4.46 using K = 10!!! with the initial condition in (4.21). Time, t (Sec.)010203040 Inputs Of The System-100001000(#!! Figure 4.36. Energy deception for elastic boundary condition without in -plane forces and !!!!"!!!! using FEM approximation Figure 4.37. Modal displacements for elastic boundary condition without in -plane forces and !!!!"!!! using FEM approximation Time, t (Sec.)010203040 Lyapunov Energy 050100System EnergyTime, t (Sec.)0510152025303540 -101Mode 1Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3Time, t (Sec.)0510152025303540 -0.0200.02Mode 4Time, t (Sec.)0510152025303540 -0.200.2Mode 5($!! Figure 4.38. Energy deception for elastic boundary condition with !!!!"!!! using controlled in -plane forces Figure 4.39. Modal displacements for elastic boundary condition with !!!!"!!! using controlled in -plane forces Time, t (Sec.)010203040 Lyapunov Energy 050100System EnergyTime, t (Sec.)0510152025303540 -101Mode 1Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3 Time, t (Sec.)0510152025303540 -0.0200.02Mode 4 Time, t (Sec.)0510152025303540 -0.200.2Mode 5 (%!! Figure 4.40. Controlled !! forces for plate with elastic boundary condition and !!!!"!!! Figure 4.41. Controlled !! forces for a plate with elastic boundary condition and !!!!"!!! Figure 4.42. Energy deception for plate with elastic boundary condition and !!!!"!!! using modified control logic Time, t (Sec.)010203040 0510Time, t (Sec.)010203040 0246Time, t (Sec.)010203040 Lyapunov Energy 050100System Energy(&!! Figure 4.43. Modal displacements for elastic boundary condition and !!!!"!!! using modified control logic Figure 4.44. !!"#$ inputs for a plate with elastic boundary condition and !!!!"!!! using modified control logic Figure 4.45. !! inputs for a plate with elastic boundary condition and !!!!"!!! using modified control logic Time, t (Sec.)0510152025303540 -101Mode 1Time, t (Sec.)0510152025303540 -0.200.2Mode 2 Time, t (Sec.)0510152025303540 -0.500.5Mode 3Time, t (Sec.)0510152025303540 -0.0200.02Mode 4 Time, t (Sec.)0510152025303540 -0.200.2Mode 5 Time, t (Sec.)010203040 0510Time, t (Sec.)010203040 0246('!! Figure 4.46. Filter input s for a plate with elastic boundary condition and !!!!"!!! using modified control logic Comparing the results we got for the plate with the elastic boundary condition (Figs. 4.27 Ð 4.35 for !!!!""!!! and Figs. 4.38 Ð 4.46 for !!!!!"!!!!, we observe that the vibration takes longer time to converge to zero as we decrease the stiffness constant. The control logic in (4.7) takes approximately 6 seconds to eliminate the vibration for !!!!!""!!! and 7 seconds for !!!!!"!!!. However, the m odified control logic (4.12) using filter and saturation takes around 18 seconds to do that for !!!!!""!!! and 20 seconds for !!!!!"!!!. In conclusion, using the elastic boundary conditions alters the stiffness matrix and reduces the efficacy of the control logic (section 4.1) that is based on increasing the stiffness of the system using controlled in -plane forces. Time, t (Sec.)010203040 Inputs Of The System-5000500((!!Remark: Form the previous results we can claim that our control logic with and without the low -pass filter and the saturation ( logics in (4.7) and (4.12)) does the designed purpose by controlling the vibration of the thin elastic plate using uniformly distributed in -plane forces. Adding the filter and saturation to the control logic effects the results by increasing the vibrating time from 5 to 15 seconds in the simply supported plate. However, they reduce the switching frequency of the in-plane forces and converge these forces to zero as the vibration subsidies. These results can be improved by assigning different values for the time constant ! and ! for the saturation . For instance, assigning smaller !!, higher bandwidth will speed the vibration suppression, but it will cause more frequent switching in the in -plane forces. This tradeoff process should be considered in real application when we choose the actuator for our control logic. !!!!!!!!!!!!!!!!!!!!!()!! Chapter 5 Vibration Control Of A Thin Rectangular Plate Using A Web -Of-Cables In chapter four, a control strategy was introduced to control the vibration in a plate using uniformly distributed in -plane forces. Two different types of boundary conditions were considered and the control efficacy was proven through simulation. In this chapter, a web -of-cables around the plate is proposed as a mechanism to suppress the vibration of the plate. The web-of-cables structure provides the boundary of the plate with in-plane forces that act at a few discrete locations, which results in a non -uniform stress distribution. A model of the pre -stressed plate is first developed in this chapter. The dynamic model of the plate is then obtained from vibration analysis using t he pre -stress data. Finally, the control logic introduced in section 4.1 (which is an extension of the work presented by Nudehi [ 48]) is applied to suppress the vibration. (*!!5.1 A Web -Of-Cables Structure Supporting A Plate ! In this section, a web -of-cables structure is proposed for supporting a plate and the forces in the cables are investigated. The web structure, which is shown in Fig. 5.1, is comprised of the following: ¥ Corner cables, which their lengths are labeled as !!" and !!!". ¥ Side cables that have a catenary shape and their lengths are labeled as !!!, !!, !!, !!!and !!. ¥ Boundary cables that connects the web structure to the plate edges; their lengths are labeled as!!!, !!, !!, !!, and !!. Figure 5.1. Rectangular plate with a web -of-cables structure LLLKKKKLLLLLLLLLKKKKl l l l l l l l l l LLLLhhhhhhxyCorner cable Side cable Boundary cable XYWLXYTTTT!!1223111223)+!!We first study the static characteristics of the web structure. The tensions in the boundary cables are considered to be equal to !! and perpendicular to the plate edges. Also, the tensions in the corner cables ! are equal and they have the same angle !!with respect the x-axis, which can be obtained from !!!"# !!!! (5.1) where W and L are the width and the length of the plate. Equation (5.1) follows from the assumption that the corner cables are congruent with the diagonal of the plat. To simplify the problem, it is considered that the web -of-cables configuration is symmet ric in both x and y directions. Assuming symmetry, it is sufficient to investigate the top -right quarter of the plate that has three cables on the top edge and two cables on its right edge as shown in Fig. 5.2. Figure 5.2. Quarter of a plate with web -of-cables structure XYT!TTTTTFFFRRxy!!!""KnotL/2W/2x = L/4y = W/2 )"!!In Figure 5.2, !! and !! are the distances from the corner of the plate to the pulley in the x and y directions respectively. The tensions in the side cables are assumed to be all positive and denoted by !!!!!!!!!!!!!!! and !!. Also, the cables are assumed to be inextensible. The plate is considered to be rigid; hence, the web -of-cables structure can have deformation. By considering the static equilibrium at each knot (point where two or more cab les meet), we get 11 equations from the 6 knots Ð see Fig. 5.2. However, in this web structure there are 13 unknowns that are labeled as T,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! and !!!. To simplify the problem, we assume two arbitrar y variables to match the number of the equations with the number of unknowns. We assume !!! and !!! to be known . We observe that by changing either !!! or!!!!the configuration of the web structure is changed as well, which means that the problem has inf inite number of solutions for different values of !!! or!!!. To solve for the unknowns, we write the equation for static equilibrium at the six knots first: !!"#!!!!!"#!"#!!!!!!!!!!!!!"#!"#!!!!!!!!!! (5.2) !!"#!!!!!"#!"#!!!!!!!!!!!!!"#!"#!!!!!!!!! (5.3) !!!"#!"#!!!!!!!!!!!!!"#!"#!!!!!!!! (5.4) !!!"#!"#!!!!!!!!!!!!!"#!"#!!!!!!!!!!! (5.5) !!!"#!"#!!!!!!!!!!!!"#!"#!!!!!!!! (5.6) !!!"#!"#!!!!!!!!!!!!"#!"#!!!!!!!!!!!! (5.7) !!!!!"#!"#!!!!!!!!!!! (5.8) )#!! !!!"#!"#!!!!!!!!!!!!!"#!"#!!!!!!!! (5.9) !!!"#!"#!!!!!!!!!!!!!"#!"#!!!!!!!!!!! (5.10) !!!!!"#!"#!!!!!!!!!!! (5.11) In addition, we have from the geometry !!!"#!!!!!! (5.12) where ! is known - see Equation (5.1). In (5.2) Ð (5.12), we have eleven unknowns since !! and !!! are considered to be known. These equations are solved for the parameters listed in Table 5.1; the results are shown in Table 5.2, and 5.3. Table 5.1. Preset parameters for the design problem !!(N) 1 & (deg.) 26.5651 !!(m) 0.2 !!(m) 1 L (m) 2 W (m) 1 Table 5.2. Cables tension results for the design problem T (N) 14.7208 !!(N) 11.9315 !!(N) 11.7627 !!(N) 11.6774 !!(N) 4.3501 !!(N) 4.1138 Table 5.3. Cables length results for the design problem !!(m) 0.2857 !!(m) 0.2214 !!(m) 0.8163 !!(m) 0.7551 !!(m) 0.5 )$!!From the results in Tables 5.2, and 5.3, the remaining unknowns of the web -of-cables structure are solved using the following kinematic relationships and presented in Table 5.4: Table 5.4. Cables angles and length results for the design problem !!(deg.) 12.0948 !!(deg.) 7.3264 !!(deg.) 2.454 !!(deg.) 20.1707 !!(deg.) 6.9811 !!(m) 1.0227 !!(m) 0.5041 !!(m) 0.5005 !!(m) 0.5327 !!(m) 0.5037 !!! !!!!"#!!!!!!!!! (5.13) !!!!"#!!!!!!!! (5.14) !!!!"#!!!!!!!! (5.15) !!!!!"#!!!!!!!!! (5.16) !!!!"#!!!!!!!! (5.17) !!!!!"#!!!!!!"#!! (5.18) !!!!!"#!!!!!!"#!! (5.19) !!!!!"#!!!!!!"#!! (5.20) !!!"#!!!!!!!!!"#!! (5.21) !!!"#!!!!!!!!!"#!! (5.22) )%!!5.2 Investigating The Changes in The Corner Cables Tensions ! In the previous section, the web -of-cables structure was investigated with constant corner tension. In this section, we investigate the effect of changing the corner cables tensions from their nominal values. Specifically, we increase the tension of two op posite corners by !! and decrease the tension of the other two corners by !!. As a result of these changes, the web -of-cables structure is deformed and the plate undergoes rotation Ð see Fig. 5.3. Due to the way that !! is applied, symmetry about the center of the plate is maintained. In addition, all the side and boundary cables length are considered to remain unchanged from their values in the undeformed configuration. In the new configuration, the side cables will no longer be perpendicular to the edge of the plate. The fixed coordinates of the pulleys are used to determine the angles of the corner cables after the deformation. By analyzing the new web configuration, we obtain 81 equations that include 81 unknowns. The se equations are obtained from both kinematics and kinetic equilibrium equations for the structure. To eliminate the moment that is caused by changing the tensions at the corners, we allowed the plate to translate and freely rotate about its origin. In thi s setup the web cable lengths and the corner tensions T are known from the design configuration - see Tables 5.2 Ð 5.4. We considered adding !!!!! to two corner tensions and subtracting it from the others two corner tensions as shown in Fig. 5.4. )&!! Figure 5.3. Plate with the with -of-cables before the deformation, (b), after deformation Ð the forces in the web -of-cables structure are shown in (a), and the angles of deformations are shown in (b) T + ! TT + ! TT - ! TT - ! TxyTTTTxy(a)(b)FFFFFFFFFFFFRRRRRRRR11234567891011122345678TTTTTTTTTTTTTTT 123456789101112 T13141516""""""""""""""" "1234567891011 1213141516ÕÕ0#x#y$1%1$2%2% 3%40&)'!!To investigate the r otational and translational displacements in the plate, a full plate model is used. The 81 unknown variables of the model are comprised of the following: ¥ The unknown forces, which are labeled as !!, where i = 1, 2,.., 12; !!, where j = 1, 2,.., 8; and !!, where n = 1, 2,.., 16. ¥ The unknown angles of deformation, which are labeled as !!, where i = 1, 2,.., 12; !!, where j = 1, 2,.., 8; !!, where n = 1, 2,.., 16; !!, where s = 1, 2,.., 4; and !. ¥ The unknown corner cables length, which are lab eled as !!!, where r = 1, 2,..,4. The corner cables static equilibrium equations are: !!!!!"#!!!!!!!!"#!!!!!!"#!! (5.23) !!!!!"#!!!!!!!!"#!!!!!!"#!! (5.24) !!!!!"#!!!!!!!!"#!!!!!!"#!! (5.25) !!!!!"#!!!!!!!!"#!!!!!!"#!! (5.26) !!!!!"#!!!!!!!!"#!!!!!!"#!! (5.27) !!!!!"#!!!!!!!!"#!!!!!!"#!! (5.28) !!!!!"#!!!!!!!"!"#!!"!!!!"#!! (5.29) !!!!!"#!!!!!!!"!"#!!"!!!!"#!! (5.30) The static equilibrium of the top edge knots gives: !!!"#!!!!!!"#!!!!!!"#!!! (5.31) !!!"#!!!!!!"#!!!!!!"#!!! (5.32) )(!! !!!"#!!!!!!"#!!!!!!"#!!! (5.33) !!!"#!!!!!!"#!!!!!!"#!!! (5.34) !!!"#!!!!!!"#!!!!!!"#!!! (5.35) !!!"#!!!!!!"#!!!!!!"#!!! (5.36) !!!"#!!!!!!"#!!!!!!"#!!! (5.37) !!!"#!!!!!!"#!!!!!!"#!!! (5.38) !!!"#!!!!!!"#!!!!!!"#!!! (5.39) !!!"#!!!!!!"#!!!!!!"#!!! (5.40) For the bottom edge knots, the static equilibrium equations can be expressed as !!!"#!!!!!!"#!!!!!!"#!!! (5.41) !!!"#!!!!!!"#!!!!!!"#!!! (5.42) !!!"#!!!!!!"#!!!!!"!"#!!"! (5.43) !!!"#!!!!!!"#!!!!!"!"#!!"! (5.44) !!!"#!!!!!"!"#!!"!!!!!"#!!!! (5.45) !!!"#!!!!!"!"#!!"!!!!!"#!!!! (5.46) !!"!"#!!"!!!!!"#!!!!!!"!"#!!"! (5.47) !!!!"#!!!!!!"!"#!!"!!!"!"#!!"! (5.48) !!!!"#!!!!!!"!"#!!"!!!"!"#!!"! (5.49) ))!! !!"!"#!!"!!!!!"#!!!!!!"!"#!!"! (5.50) The static equilibrium for left edge knots yield !!!"#!!!!!!"#!!!!!!"#!! (5.51) !!!"#!!!!!!"#!!!!!!"#!! (5.52) !!!"#!!!!!!"#!!!!!!"#!! (5.53) !!!"#!!!!!!"#!!!!!!"#!! (5.54) !!!"#!!!!!!"#!!!!!!"#!! (5.55) !!!"#!!!!!!"#!!!!!!"#!! (5.56) For the right edge knots, the static equilibrium equations are !!!"#!!!!!!"#!!!!!"!"#!!" (5.57) !!!"#!!!!!!"#!!!!!"!"#!!" (5.58) !!!"#!!!!!!"#!!!!!"!"#!!" (5.59) !!!"#!!!!!!"#!!!!!"!"#!!" (5.60) !!!"#!!!!!!"#!!!!!"!"#!!" (5.61) !!!"#!!!!!!"#!!!!!"!"#!!" (5.62) The geometry of the structure for the top edge gives !!!"#!!!!!!"#!!!!!!"#!!!!! (5.63) !!!"#!!!!!!"#!!!!!!"#!!! (5.64) )*!! !!!"#!!!!!!"#!!!!!!"#!!!!! (5.65) !!!"#!!!!!!"#!!!!!!"#!!! (5.66) !!!"#!!!!!!"#!!!!!!"#!!!!! (5.67) !!!"#!!!!!!"#!!!!!!"#!! (5.68) !!!"#!!!!!!"#!!!!!!"#!!!!! (5.69) !!!"#!!!!!!"#!!!!!!"#!! (5.70) For the bottom edge, the geometry of the structure gives !!!"#!!"!!!!"#!!!!!!"#!!!! (5.71) !!!"#!!"!!!!"#!!!!!!"#!! (5.72) !!!"#!!!!!!!"#!!!!!!"#!!"!! (5.73) !!!"#!!!!!!!"#!!!!!!"#!!" (5.74) !!!"#!!"!!!!"#!!"!!!!"#!!!!! (5.75) !!!"#!!!!!!!"#!!"!!!!"#!!" (5.76) !!!"#!!"!!!!"#!!!!!!!"#!!"!! (5.77) !!!"#!!"!!!!"#!!!!!!!"#!!" (5.78) For the right edge of the plate, the geometry of the web structure can be expressed as !!!"#!!!!!!"#!!!!!!!!"#!! (5.79) !!!"#!!!!!!"#!!!!!!"#!! (5.80) *+!! !!!"#!!!!!!"#!!!!!!!!"#!! (5.81) !!!"#!!!!!!"#!!!!!!"#!! (5.82) For the left edge of the plate, the equations are !!!"#!!"!!!!"#!!!!!!!!"#!!" (5.83) !!!"#!!"!!!!"#!!!!!!"#!!" (5.84) !!!"#!!"!!!!"#!!!!!!!!"#!!" (5.85) !!!"#!!"!!!!"#!!!!!!"#!!" (5.86) Finally, for the corner cells, the kinematic relationships are !!!"#!!!!!!"#!!!!!!"#!!!!!!"#!! (5.87) !!!"#!!!!!!"#!!!!!!"#!!!!!!"#!! (5.88) !!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!" (5.89) !!!"#!!!!!!"#!!!!!!"#!!"!!!!"#!!! (5.90) !!!"#!!!!!!"#!!!!!!"#!!"!!!!"#!!! (5.91) !!!"#!!!!!!"#!!!!!!"#!!"!!!!"#!! (5.92) !!!"#!!"!!!!"#!!"!!!!"#!!!!!!"#!! (5.93) !!!"#!!"!!!!"#!!"!!!!"#!!!!!!"#!! (5.94) Equations (5.23) to (5.94) provide a total of 72 equations. The remaining 9 equations require us to define the coordinate transformation (see Fig. 5.4) associated with the rotation of the plate. *"!! Figure 5.4. Coordinate Transformation The rotated coordinates ! and ! are related to the original coordinates as follows: !!!!!!"#!!!!!"#! !!!!!!"#!!!!!"#! !!!"#$# !!!!!!!!!!!! (5.95) Then !!!!!!!!!! !!!!"#!!!!!!!!!!!!!!!!"#$# !!!!!!!!!!!! (5.96) In (5.95) and (5.96), !! and !! are the distances evaluated from the center of the plate to the corner cables. For instance, !! and !!for the upper -right corner can be obtained from !!!!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#! (5.97) !!!!!!"#!!!!!!"#!!!!!!"#!!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#! (5.98) Similarly, the other three !! and !! distances can be obtained. Using the coordinate transformations formula in (5.95), the distance between the four fixed corner pulleys can be written as yxy'x'!V*#!! !!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!! (5.99) !!!!"#!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!! (5.100) !!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!! (5.101) !!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!!"#!!"!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!! (5.102) *$!! !!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!!!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!! (5.103) !!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!"!"#!!!!!!!!!!"#!!!!!!!"#!!"!!!!"#!!!!!!!"#!!"!"#!!!!!"#!!!!!!!"#!!"!!!!"#!!!!!!!"#!!"!"#!!!!"!"#!!!!!!! (5.104) Balance of moment about the center of mass of the plate gives !!!!!!!!!!"#!!!!!!!!!!!!!!"#!!!!!!!!!!!!!!"#!!!!!!!!!!!!!!!"#!!!!!!!! (5.105) !!!!!"#!!!!!!!!!!!"#!!!!!!!!!!!"#!!!!!!!!!!!"#!!!!!!!! (5.106) !!!!!"#!!!!!!!!!!!"#!!!!!!!!!!!"#!!!!!!!!!!!"#!!!!!!! (5.107) The corner cables forces equilibrium in x and y directions give two more equations Ð see (5.106) and (5.107). Solving Equations (5.23 Ð 5.95) and (5.99 Ð 5.107) for the 81 unknowns, the results are listed in the Tables 5.6 Ð 5.7. *%!!Table 5.5. Designed parameters for the web -of-cables of the deformation problem !!(m) 1.0227 !!(m) 0.2 !!(m) 0.5041 !!(m) 0.5327 !!(m) 0.5005 !!(m) 0.5037 !!(m) 0.2857 !!(m) 0.8163 !!(m) 0.2214 !!(m) 0.7551 Table 5.6. The web -of-cables tension and the corner cables length results after the deformation !!!!!(N) 13.374 !!!!!(N) 2.1842 !!!!!(N) 12.0654 !!!!!"(N) 1.3958 !!!!!(N) 11.2005 !!!!!!(N) 1.3981 !!!!!"(N) 10.3646 !!!!!"(N) 1.0541 !!!!!!(N) 9.8846 !!!!!"(N) 0.2097 !!!!!"(N) 9.8264 !!!!!"(N) 2.3795 !!!!!(N) 5.2929 !!!!!"(N) 1.0179 !!!!!(N) 4.1846 !!!!!"(N) 0.0973 !!!!!(N) 4.0488 !!"!!!"(m) 0.3882 !!!!!(N) 4.05 !!"!!!"(m) 0.6313 Table 5.7. The web -of-cables angle results after the deformation Angle Value (deg.) Angle Value (deg.) !!!!! 16.2928 !!!!! 24.3577 !!!!! 8.3985 !!!!!" 32.5475 !!!!! 2.9975 !!!!!! 36.6389 !!!!!" 2.9654 !!!!!" 32.6817 !!!!!! 8.2794 !!!!!" 24.9902 !!!!!" 9.4504 !!!!!" 7.031 !!!!! 32.8934 !!!!!" 7.6092 !!!!! 7.0395 !!!!!" 7.0398 !!!!! 7.0378 !!!!! 27.7876 !!!!! 8.4143 !!!!! 28.4137 ! 0.2373 The translation of the center of the plate is evaluated (see Table 5.8) from the following equations: *&!! !!!!!!!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#! (5.108) !!!!!!!!!"#!!!!!!"#!!!!!!"#!!!!"#!!!!!"#!!!!!!"#!!!!!!"#!!!"#! (5.109) Table 5.8. Evaluating the coordinate of the centre of the thin plate after the deformation !!(m) !!(m) !!!"#$!!!!" !!!"#$!!!!" The results shown in Table 5.8 indicate that the plate does not move; this is due to symmetric forcing. When the tension and angle of the deformation for each cable around the plate (the boundary cables) are evaluated, the in -plane stress distributions can be obtained. The boundary cable forces on the plate are decomposed into two components, in the !! and !! directions - see Fig. 5.5. Figure 5.5. Thin rectangular plate with the components of the side forces Ð plate is comprised of eight squares of dimension 0.5m for each side. !!!"!"#"$%&"#"'%(")*%+*,-,."*/".0-"+1-2)134-5"/*1)-6")*%+*,-,."*/".0-"+1-2)134-5"/*1)-77*'!!5.3 In-Plane Stress Distribution !!In section 3.2, we used FEM to derive the dynamic model for out -of-plane vibration for plate subjected to forces at discrete locations. The problem was divided in to two parts: evaluating in -plane stress distributions and out -of-plane vibration analysis. Following the first part of the analysis, the in -plane stress distributions are obtained using our FEM code (in MATLAB) and verified with commercial FEM software (A NSYS). Each element has three stress values: two normal stresses in the x and y directions and one shear stress in xy plane. After we evaluated these three values for each element, we plotted the results for the entire plate. The results for both FEM methods for !! = 0 N are shown in the Figs. 5.6 Ð 5.8 and the difference between them are shown in Figs. 5.9 Ð 5.11. Figure 5.6. In-plane normal stress distribution in x direction when !! = 0 N Normal stress distribution in x direction uinsg ANSYS VAL+3.28E+01+8.34E+02+2.44E+03+4.04E+03+5.64E+03+7.25E+03+8.85E+03+1.05E+04+1.21E+04+1.37E+04+1.53E+04+1.69E+04+1.85E+04 VAL+3.28E+01+8.34E+02+2.44E+03+4.04E+03+5.64E+03+7.24E+03+8.85E+03+1.05E+04+1.21E+04+1.37E+04+1.53E+04+1.69E+04+1.85E+04Normal stress distribution in x direction uinsg MATLAB*(!! Figure 5.7. In-plane normal stress distribution in y direction when !! = 0 N Figure 5.8. In-plane shear stress distributions when !! = 0 N Figure 5.9. Difference between ANSYS and MATLAB code for the x direction stress distribution when !! = 0 N Normal stress distribution in y direction uinsg ANSYSNormal stress distribution in y direction uinsg MATLAB VAL!1.73E+02+6.37E+02+2.26E+03+3.88E+03+5.50E+03+7.12E+03+8.74E+03+1.04E+04+1.20E+04+1.36E+04+1.52E+04+1.68E+04+1.85E+04 !1.73E+02+6.38E+02+2.26E+03+3.88E+03+5.50E+03+7.12E+03+8.74E+03+1.04E+04+1.20E+04+1.36E+04+1.52E+04+1.68E+04+1.85E+04VALShear stress distribution uinsg ANSYSShear stress distribution uinsg MATLAB VAL!1.32E+04!1.21E+04!9.77E+03!7.47E+03!5.17E+03!2.87E+03!5.75E+02+1.72E+03+4.02E+03+6.32E+03+8.62E+03+1.09E+04+1.32E+04 VAL!1.32E+04!1.21E+04!9.77E+03!7.47E+03!5.17E+03!2.87E+03!5.75E+02+1.72E+03+4.02E+03+6.32E+03+8.62E+03+1.09E+04+1.32E+04 VAL!3.11E!01!2.80E!01!2.20E!01!1.59E!01!9.80E!02!3.71E!02+2.37E!02+8.46E!02+1.45E!01+2.06E!01+2.67E!01+3.28E!01+3.89E!01*)!!Figure 5.9 shows that the maximum difference between both results for the stress distribution in the x direction is negligible , around 0.389 . This difference is small compared to the maximum stress value of !!!"!!"!!!!!!and can therefore be neglected. Figure 5.10. Difference between ANSYS and MATLAB code for the y direction stress distribution when !! = 0 N Figure 5.11. Difference between ANSYS and MATLAB code for the shear stress distribution when !! = 0 N Figure 5.10 and 5.11 show that the maximum difference between both FEM results for both normal stresses in y direction and shear stresses in xy plane is equal to 0.4 , which are also negligible. For !!!!!!!, the stress distributions are shown in Figs 5.12 Ð 5.14 for both FEM methods. The differences between the two methods are shown in Figs 5.15 Ð 5.17. VAL!3.79E!01!3.45E!01!2.77E!01!2.08E!01!1.40E!01!7.23E!02!4.26E!03+6.38E!02+1.32E!01+2.00E!01+2.68E!01+3.36E!01+4.04E!01 VAL!4.15E!01!3.80E!01!3.08E!01!2.37E!01!1.66E!01!9.45E!02!2.33E!02+4.80E!02+1.19E!01+1.91E!01+2.62E!01+3.33E!01+4.04E!01**!! Figure 5.12. In-plane normal stress distribution in x direction when !! = 3 N Figure 5.13. In-plane normal stress distribution in y direction when !! =3 N Figure 5.14. In-plane shear stress distributions when !! = 3 N Normal stress in x direction using ANSYS VAL!2.06E+03+7.18E+02+6.27E+03+1.18E+04+1.74E+04+2.29E+04+2.85E+04+3.40E+04+3.96E+04+4.52E+04+5.07E+04+5.63E+04+6.18E+04 VAL!2.06E+03+7.20E+02+6.27E+03+1.18E+04+1.74E+04+2.29E+04+2.85E+04+3.40E+04+3.96E+04+4.52E+04+5.07E+04+5.63E+04+6.18E+04Normal stress in x direction using MATLABNormal stress in y direction using ANSYS VAL!3.38E+03!1.45E+03+2.40E+03+6.26E+03+1.01E+04+1.40E+04+1.78E+04+2.17E+04+2.55E+04+2.94E+04+3.33E+04+3.71E+04+4.10E+04Normal stress in y direction using MATLAB VAL!3.38E+03!1.45E+03+2.41E+03+6.26E+03+1.01E+04+1.40E+04+1.78E+04+2.17E+04+2.55E+04+2.94E+04+3.33E+04+3.71E+04+4.10E+04Shear stress distribution using ANSYSShear stress distribution using MATLAB VAL!7.48E+03!5.58E+03!1.77E+03+2.03E+03+5.83E+03+9.64E+03+1.34E+04+1.72E+04+2.10E+04+2.48E+04+2.87E+04+3.25E+04+3.63E+04 VAL!7.47E+03!5.57E+03!1.77E+03+2.03E+03+5.84E+03+9.64E+03+1.34E+04+1.72E+04+2.10E+04+2.49E+04+2.87E+04+3.25E+04+3.63E+04"++!! Figure 5.15. Difference between ANSYS and MATLAB code for the x direction stress distribution when !! = 3 N Figure 5.16. Difference between ANSYS and MATLAB code for the y direction stress distribution when !! = 3 N Figure 5.17. Difference between ANSYS and MATLAB code for the shear stress distribution when !! = 3 N VAL!2.56E+00!2.32E+00!1.84E+00!1.37E+00!8.89E!01!4.12E!01+6.44E!02+5.41E!01+1.02E+00+1.49E+00+1.97E+00+2.45E+00+2.92E+00 VAL!2.96E+00!2.74E+00!2.32E+00!1.89E+00!1.46E+00!1.03E+00!6.06E!01!1.79E!01+2.48E!01+6.76E!01+1.10E+00+1.53E+00+1.96E+00 VAL!3.68E+00!3.34E+00!2.65E+00!1.95E+00!1.26E+00!5.69E!01+1.23E!01+8.15E!01+1.51E+00+2.20E+00+2.89E+00+3.58E+00+4.28E+00"+"!!The maximum difference between ANSYS and MATLAB results when !! = 3 N are 0.2 for the stress in the x direction, 0.196 for the stress in the y direction, and 0.428 for the shear stress in the xy plane. These differences are quite small compared to the maximum stress value of the plate , which is !!!"!!"!!!!!, and can therefore be neglected. 5.4 Out -Of-Plane Vibration Analysis For Varying Boundary Conditions !!An out -of-plane vibration analysis of t he plate will be conducted using the pre -stress data obtained in section 5.3. In our previous vibration analysis (chapter 4), we assumed that the stresses are uniformly distributed around the plate due to uniformly distributed in -plane forces around the pl ate boundaries. In this chapter , the stress distribution is not assumed to be uniformly distributed. The analysis is done using MATLAB FEM code and verified using commercial FEM software (ANSYS). The results for a plat e with simply supported boundaries are first listed in Table 5.9 and the mode shapes are shown in Figs. 5.18 and 5.19 using ANSYS and MATLAB respectively , for !! = 0 N. For !! = 3 N, the results are listed in Table 5.10 and the mode shapes shown in Figs. 5.20 and 5.21. Table 5.9. Comparing ANSYS and MATLAB frequency analysis when !! = 0 N Mode MATLAB Frequency Hz ANSYS Frequency Hz Difference % 1 1.681 1.683 0.118835 2 2.639 2.62 0.719969 3 4.188 4.192 0.09541 4 5.309 5.526 3.92689 5 6.288 6.436 2.572 6 6.295 6.454 2.19 "+#!! Figure 5.18. ANSYS frequency analysis when !! = 0 N Figure 5.19. MATLAB frequency analysis when !! = 0 N "+$!! Figure 5.20. ANSYS frequency analysis when !! = 3 N Figure 5.21. MATLAB frequency analysis when !! = 3 N "+%!!Table 5.10. Comparing ANSYS and MATLAB frequency analysis when !! = 3 N Mode MATLAB Frequency Hz ANSYS Frequency Hz Difference % 1 1.827 1.712 6.2944 2 2.727 2.666 16.6 3 4.231 4.233 0.04724 4 5.478 5.543 0.876 5 6.363 6.476 2.24 6 6.432 6.509 0.679 The changes in the plate modes and frequencies are examined using MATLAB results for !!!! and !!!!!!!. These changes are presented in Table 5.11. !!!!!!!! (5.110) where !! is ith mode with !T = 0 N and !!!is ith mode with !T = 3 N , M is the mass matrix, and ! is the identity matrix. Table 5.11. Comparing MATLAB frequency analysis when !! = 0 and 3 N Mode ( i) Frequency Hz for !T = 0 N Frequency Hz for !T = 3 N Difference % !!!!!! 1 1.681 1.827 7.99124 0.9993 2 2.639 2.727 3.22698 0.999 3 4.188 4.231 1.0872 0.9982 4 5.309 5.478 3.085 0.9987 5 6.288 6.363 1.1786 0.9958 As expected, the entries in the last column are not identically unity since !!!!!. Also, all the natural frequencies increase as ! T is increased. The stiffness matrix of the web-of-cables structure is evaluated and added as elastic boundary conditions for a floating plate to replace the simply supported boundary condition. To evaluate the stiffness matrix of the web -of-cables structure, we model the plate and the surround ing web as shown in Fig. 5.22. "+&!! Figure 5.22. Web-of-cables stiffness evaluation !The points where the web -of-cables attaches to the plate are marked as !!, where i = 1, 2,É,12. The knot points that the web cables join each other are marked as !!!, where j = 1, 2,É,20. The displacement of each of these point ( !! or !!!) in z direction is assumed to be !!, where n = 1, 2,.., 32. Some of these new DOF in z direction are added to the global DOF of the plate structure. Therefore, the web -of-cables and the plate are considered to be a one structure. The stiffness of each cable is labeled as !! (s = 1, 2,.., 40) and can be evaluated by dividing the tension on the cable by its length . Using second order partial derivatives (the Hessian matrix method), the web -of-cables stiffness matrix , !!!!, can be obtained as follows K4qqqqqK5K7K9K11 K13K15qCorner1K16Corner2qqqK1K2K37K39qqqqqqK25K27K29K31K33K35Corner3K36Corner4K24qqqqK17K19K21K23K6K8K10K12K14K18K20K22K26K28K30K32K34K38K40K3pppppppppppp32q45678910111213141516171819201123456789101112"+'!! !!!!!!!!!"!#$ !!!!!!!!"!#$ !!!!!!!!!!!"!#$ !!!!!!!!!!"!#$ !!!!!!!!!!"!#$ !!!!!!!!!"!#$ !!!!!!!!!!!"!#$ !!!!!!!!!!!"!#$ !!!!!!!!!!!!!"!#$ !!!! (5.111) where !!!"!#$ is t he total potential energy of the web -of-cables structure and can be obtained from the summation of the potential energy of each cable. !!"!#$ !!!!!!!!!!!!"!!!! (5.112) where !!! is the difference between the displacements of the two ends of the cable in z direction. After the stiffness matrix of the web -of-cables, !!!!, is obtained, we connected the two structures through their associated DOF (more DOF will be added to the global DOF of the mass and stiffn ess matrices). The structure stiffness, !!!!, increases the stiffness of the boundaries of the global stiffness matrix K of the floating plate in the frequency analysis. In addition, the stiffness matrix K also includes the pre -stress effect that is cau sed by adding the web -cable forces, which is done using pre -stress information to obtain the frequency analysis as presented in section 5.3 . The first five out -of-plane vibration frequencies and modes are listed in Table 5.12 and shown in Figs 5.23 Ð 5.26 respectively for different values of !!. "+(!! Figure 5.23. Out -of-plane vibration results when !! = 0 N Figure 5.24. Out -of-plane vibration results when !! = 1 N "+)!! Figure 5.25. Out -of-plane vibration results when !! = 2 N ! Figure 5.26. Out -of-plane vibration results when !! = 3 N "+*!!Table 5.12. MATLAB frequency analysis for different !! values First Five Eigenvalues Modes ! T = 0 N ! T = 1 N ! T = 2 N ! T = 3 N 1 6.3649 6.3793 6.4128 6.4188 2 17.6285 16.5403 14.4832 12.15 3 21.2724 22.317 24.1678 26.0546 4 44.6023 43.6041 40.8418 36.9578 5 75.4323 76.4714 79.2026 82.835 From Table 5.12, we can observe that the modes 2 and 4 are decreased as the value of !! increased . However, the frequencies are increased as we increased the value of !! for the other modes . 5.5 Control Scheme Us ing The Web -OF-Cables Structure !!In this section, the control scheme introduced in chapter 4 is used to control the vibration of a floating rectangular Aluminum plate using the web-of-cables structure. The in -plane forces that are created by the web -of-cables structure are controlled usin g the control logic (4.7) or (4.12) through variation in the tension of the corner cables above their nominal values . Following section 4.1, the modes (of a plate with similar dimension and properties as in chapter 4) are examined using the initial conditi ons below . !!!!!!!!!!!!!!!!!!!!!!!!"!!!!!!!!!!!!!!!!! (5.113) ""+!!FEM models for different value of corner tensions (see section 5.4 for example), where the system stiffness matrix !!!!!"#$ !!!!!! and the proportional damping matrix !!!!" with!!!!!!!!", are used to investigate the efficacy of the control strategy. The energy dissipation and the modal displacement for the system in absence of the control scheme are shown in Figs. 5.27 and 5.28 below; these figures will be compared with results obtained with different control schemes . Figure 5.27. Energy dispassion for the system without control Figure 5.28. Modal displacement for the system without control Time, t (Sec.)020406080 Lyapunov Energy0246System EnergyTime, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5"""!!5.5.1 Adding !! To The Four Corner Cables The control scheme in section 4.1 is investigated for the floating plate by switching !! in the corner cables tensions between 0 and 3 N . For The initial conditions in (5.11 3), the results of control scheme in (4.7) are shown in Figs. 5.29 Ð 5.31. Figure 5.29. Energy dissipation using the control logic in (4.7) - adding !! = 3 N to the four corner cables Figure 5.30. Input force using the control logic in (4.7) - adding !! = 3 N to all four corner cables A comparison of Fig. 5.29 with Fig. 5.27 shows that the controller is effective in suppressing the vibration of the plate using the web-of-cables structure. The vibration energy is converged to zero in approximately 60 seconds . I n the absence of the control scheme , only 30% of the system energy decay out in 60 seconds due to the damping in the system . The modal displacements results for the controlled system are shown in Fig. 5.31. Time, t (Sec.)01020304050607080 Lyapunov Energy0246System EnergyTime, t (Sec.)01020304050607080 0123""#!! Figure 5.31. Modal displacements using the control logic in (4.7) - adding !! = 3 N to all four corner cables ! Figure 5.31 shows that the modal displacements for the higher modes are converged to zero in approximately 60 seconds; however, in absence of the control the modal displacements for those modes did not converge to zero even after 80 seconds as shown in Fig 5.28. From Fig. 5.31, we also observe that the controller does not suppress the vibration in the first mode. The results of modified control logic in (4.12) are shown in Figs. 5.32 Ð 5.34; the time constant of the low pass filter and the saturation constant were chosen as !!!!!" and !!!. Time, t (Sec.) 01020304050607080 -0.0100.01Mode 1Time, t (Sec.) 01020304050607080 -0.200.2Mode 2Time, t (Sec.) 01020304050607080 -0.500.5Mode 3Time, t (Sec.) 01020304050607080 -0.100.1Mode 4Time, t (Sec.) 01020304050607080 -0.200.2Mode 5""$!! Figure 5.32. Energy dissipation using the modified control logic (4.12) with !!! - adding !! = 3 N to all four corner cables !Figure 5.33. Forcing input using the modified control logic (4.12) with !!! - adding !! = 3 N to all four corner cables ! Figure 5.34. Modal displacement using the modified control logic (4.12) with !!! - adding !! = 3 N to all four corner cables Time, t (Sec.)01020304050607080 Lyapunov Energy0246System EnergyTime, t (Sec.)01020304050607080 0123Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5""%!!From Figs. 5.32 and 5.34, we observe that the energy and modal displacements take more time to converge to zero comparing with the unmodified control (4.7). However, the switching frequencies of the input forces are decreased compare with the unmodified co ntrol logic (4.7) Ð see Fig. 5.33. Adding saturation to the modified control logic (4.12) using !!!!!" (to converge the input forces to zero as the system vibration subsided), the results are shown in Figs. 5.35 Ð 5.37. Figure 5.35. Energy dissipation using the modified control logic (4.12) - adding !! = 3 N to all four corner cables Figure 5.36. Forcing input using the modified control logic (4.12) - adding !! = 3 N to all four corner cables Time, t (Sec.)01020304050607080 Lyapunov Energy 0246System EnergyX: 79.33Y: 0.1188Time, t (Sec.)01020304050607080 0123""&!! Figure 5.37. Modal displacements using the modified control logic (4.12) - adding !! = 3 N to all four corner cables From Fig. 5.36, we can observe that the input forces are converged to zero as the system vibration converged to zero. However, the energy and modal displacements take more time to decay out compared with the unmodified control scheme in (4.7). This is the price that we should pay to get an input force that less frequently added and converge to zero as the vibration subsides. Comparing the modified control logic results with and without the saturation (as shown in Figs. 5.35 Ð 5.37 and Figs. 5.32 - 5.34 respectively), we observe that adding the saturation reduce the efficacy of the c ontrol logic by increasing the vibrating time. This effect can be observed in Fig. 5.35, where the system at time t = 79.33 seconds still has an energy equal to 0.1188. Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5""'!!5.5.2 Adding !! To Two Corner Cables And Subtracting !! From The Other Two Corner Cables In the previous subsection, the control scheme is investigated by switching !! between zero and 3 N in all four corner -cables. That means the web -of-cables structure apply equal perpendicular tensions on the plate boundaries. Therefore, the plate is not subjected to shear forces along the boundaries. In this subsection, we modify the configuration of the web -of-cables structure by adding the !! = 3 N to two corner cables tension, T, and subtracting it form the other two corner cables (the new con figuration presented in section 5.2). This modification will create shear forces in addition to in -plane forces around the boundaries as presented in section 5.4. The control scheme (section 4.1) results are shown below using the initial conditions in (5.1 13). Figure 5.38. Energy dissipation using the control logic (4.7) - adding and subtracting !! = 3 N to two opposite corner cables respectively Figure 5.39. Forcing input using the control logic (4.7) - adding and subtracting !! = 3 N to two opposite corner cables respectively !From Fig. 5.38, we observe that introducing this new switching scheme successfully controlled the vibration within 50 seconds. This m eans the shear forces that are generated by the new configuration lowered the vibrating time by 10 seconds. Once again , we observe that the control Time, t (Sec.)01020304050607080 Lyapunov Energy0246System EnergyTime, t (Sec.)01020304050607080 0123""(!!logic does not suppress the vibration in the first mode, but the modal displacements for the higher modes de cay out within 50 seconds as shown in Fig. 5.40 below. However, the modal displacements for previous configuration (section 5.5.1) takes 60 seconds to do that Ð see Fig. 5.31. The modified control logic (4.12) results are shown in Figs. 5.41 Ð 5.43; the time constant of the low -pass filter and the saturation constant were chosen as !!!!!" seconds and !!!!. Figure 5.40. Modal displacements using the control logic (4.7) - adding and subtracting !! = 3 N to two opposite corner cables respectively ! Figure 5.41. Energy dissipation using the modified control logic (4.12) with !!! - adding and subtracting !! = 3 N to two opposite corner cables respectively !Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5Time, t (Sec.)01020304050607080 Lyapunov Energy0246System Energy"")!! Figure 5.42. Forcing input using the modified control logic (4.12) with !!! - adding and subtracting !! = 3 N to two opposite corner cables respectively Figure 5.43. Modal displacements using the modified control logic (4.12) with !!! - adding and subtracting !! = 3 N to two opposite corner cables respectively !!Once again, adding the saturation effect to the modified control logic (4.12) using the saturation constan t!!!!!!", we observe that the control efficacy decreased as shown in Figs. 5.44 Ð 5.46. Time, t (Sec.)01020304050607080 0123Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5""*!! Figure 5.44. Energy dissipation using the modified control logic (4.12) - adding and subtracting !! = 3 N to two opposite corner cables respectively ! Figure 5.45. Forcing input using the modified control logic (4.12) - adding and subtracting !! = 3 N to two opposite corner cables respectively !!From Fig. 5.44, we observe that introducing the new configuration reduce the system energy at time t = 79.33 seconds by 34% compared with the previous configuration Ð see Fig. 5.35. This proves that introducing the in -plane shear forces increase the control efficacy. However, there is no improvement occurred to suppress the first mode. Time, t (Sec.)01020304050607080 Lyapunov Energy 0246System Energyt : 79.33E: 0.078Time, t (Sec.)01020304050607080 0123"#+!! Figure 5.46. Modal displacements using the modified control logic (4.12) - adding and subtracting !! = 3 N to two opposite corner cables respectively !!!!5.5.3 Using Multiple !T Values With Randomly Switching !!In this subsection, we modify the idea of the switching forces by introducing multiple values for the switching forces, ! T, instead of a single value. This new approach helps in adding lower ! T force to the system when the higher ! T force cannot be applied (to do a positive work by removing energy from the system). These multiple ! T values are; 1, 2, and 3 N that are labeled as!!!!, !!!, and !!! respectively. The control logic (4.7) results are shown in Figs. 5.47 Ð 5.49 using the initial conditions in (5.11 3). Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5"#"!! Figure 5.47. Energy dissipation using the control logic (4.7) - multiple ! T values with randomly switching !! Figure 5.48. Forcing inputs using the control logic (4.7) - multiple ! T N values with randomly switching Time, t (Sec.)020406080 Lyapunov Energy0246System Energy01020304050607080 0123Time, t (Sec.) 24.648525.431726.214926.998127.7812 0123Time, t (Sec.) 1"##!! Figure 5.49. Modal displacements using the control logic (4.7) - multiple ! T values with randomly switching Form Figs. 5.47 - 5.49, we can observe that the new approach is successfully controlled the vibration and the lower inputs forces ( ! T = 1 and 2 N) are applied when the higher ( ! T = 3 N ) value cannot be added. The modified control logic in Equation (4.12) using a low -pass filter with a time constant !!!!!" second s and a saturation with !!!!!"!is then examined and the results are shown in the figures below. Figure 5.50. Energy dissipation using the modified control logic (4.12) with !!! - multiple ! T values with rand omly switching Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5Time, t (Sec.)020406080 Lyapunov Energy0246System Energy"#$!! Figure 5.51. Forcing inputs using the modified control logic (4.12) with !!! - multiple ! T N values with randomly switching !! Figure 5.52. Modal displacements using the modified control logic (4.12) with !!! - multiple ! T values with randomly switching Time, t (Sec.)01020304050607080 0123Time, t (Sec.) 18.515319.171319.772720.4049 0123Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5"#%!! Figure 5.53. Energy dissipation using the modified control logic (4.12) - multiple ! T values with randomly switching Comparing Figs 5.53 and 5.35, we can observe that the dissipation of the energy (using multiple !T values with randomly switching) is improved. For instance, in previous case (section 5.5.1) the energy at time t = 79.33 seconds was 0.1188, howev er, using this scheme the energy converged to zero . Figure 5.54. Forcing inputs using the modified control logic (4.12) - multiple ! T N values with randomly switching Time, t (Sec.)020406080 Lyapunov Energy0246System EnergyTime, t (Sec.) 01020304050607080 0123Time, t (Sec.)37.954238.594439.185739.8277 00.511.522.533.5"#&!! Figure 5.55. Modal disp lacements using the modified control logic (4.12) - multiple ! T values with randomly switching 5.5.4 Using Multiple ! T Values With Systematic Switching In this subsection, we improve the multiple ! T values with the random switching approach by introducing a systematic switching between the multiple ! T values. The new logic is applied to the system starting from ! T = 0 N and then increased up to ! T = 3 N with increment equal to 1 N. Then logic starts to decrease ! T by the same increment until it reaches zero. In addition, between these two increasing and decreasing, the control logic can decrease or increase ! T based on system need. Therefore, using this new scheme is more convenient in the application because instead of instantaneously switches from ! T = 0 N t o 3 N when the force is needed, the controller performing it gradually with smaller increment until the forces reach its maximum values. To investigate the efficiency of this scheme, a simulation is done using same initial condition and properties as in th e previous cases and the results are shown in Figs. 5.56 Ð 5.64. Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5"#'!! Figure 5.56. Energy dissipation using the control logic (4.7) - multiple ! T with systematic switching !!! Figure 5.57. Forcing inputs using the control logic (4.7) - multiple ! T N with systematic switching Time, t (Sec.)01020304050607080 Lyapunov Energy0246System EnergyTime, t (Sec.)01020304050607080 0123Time, t (Sec.)23.371323.555523.739723.923924.108124.2922 00.511.522.533.512"#(!! Figure 5.58. Modal displacements using the control logic (4.7) - multiple ! T with systematic switching !! Figure 5.59. Energy dissipation using the modified control logic (4.12) with !!! - multiple ! T with systematic switching Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5Time, t (Sec.)01020304050607080 Lyapunov Energy0246System Energy"#)!! Figure 5.60. Forcing inputs using the modified control logic (4.12) with !!! - multiple ! T N with systematic switching Figure 5.61. Modal displacements using the modified control logic (4.12) with !!! - multiple ! T with systematic switching Time,t (Sec.) 01020304050607080 0123Time, t (Sec.) 29.288029.434929.581829.728729.875630.022530.169430.3162 01232Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5"#*!! Figure 5.62. Energy dissipation using the modified control logic (4.12) - multiple ! T with systematic switching ! Figure 5.63. Forcing Inputs using the modified control logic (4.12) - multiple ! T with systematic switching Time, t (Sec.)01020304050607080 Lyapunov Energy0246System EnergyTime, t (Sec.) 011.666723.333435.000146.666858.333570.0002 0123Time, t (Sec.) 44.771644.978045.184445.390845.597245.8037 0123"$+!! Figure 5.64. Modal displacements using the modified control logic (4.12) - multiple ! T with systematic switching From previous results (Figs. 5.56 Ð 5.64), we observe that the new approach is succeed ed in control ling the vibration of the plate with similar performance of the randomly switching scheme. Therefore, we can conclude that using systematic switching with multiple ! T values is effectively controlling the vibration of the plate. These results can be improved b y reducing the value of time constant ! that filters the high frequencies or the saturation constant ! to cutoff the added forces. Time, t (Sec.)01020304050607080 -0.0100.01Mode 1Time, t (Sec.)01020304050607080 -0.200.2Mode 2Time, t (Sec.)01020304050607080 -0.500.5Mode 3Time, t (Sec.)01020304050607080 -0.100.1Mode 4Time, t (Sec.)01020304050607080 -0.200.2Mode 5"$"!!5.5.5 Summary !!The in -plane stress distributions are altered as the in -plane forces applied. As a result, the out -of-plan e stiffness matrix is changed (due to the fact that the in -plane stresses can alter the out -of-plane dynamics according to Equations (2.39 ) and (2.40)), which can be observed from the frequency analysis results (section 5.4). Using this fact, the control l ogic that is presented in section 4.1 is proposed to use state feedback algorithm to switch the in -plane forces on and off to suppress the vibration. In this chapter, we propose a web -of-cable structure around the plate as a mechanism to suppress t he out -of-plane vibration. Using the control logic introduced in section 4.1, controlling the vibration of the higher modes is proven. However, the first mode could not be suppressed using multiple switching schemes. In the first scheme, a single switching is done by adding !T to the corner cables tension, T. The second scheme (adding !T to two corner cables and subtracting from the other two corner cables) improves the efficacy of the control strategy, but no sensible improvement on suppression of the vibr ation of the first mode is observed. Introducing the third and the fourth schemes also does not show any improvement to control the first mode. However, the switching strategy that introduced in both schemes improves the results for the higher modes. Intro ducing the web -of-cable structure does not control the vibration on the first mode due to the fact that the rate of change in the first mode frequency is not the same as in the higher modes when !T is applied to the corner cable tensions Ð see Table 5.12. These results can be improved by increasing !T value that is applied to the corner cables tension or by increasing the number of cables around the boundaries (boundary cables). "$#!!!!!!Chapter 6 Conclusion !!In this work, we extend the approach that was presented in [48] to control the vibration of cantilever beam using end -force to a plate structure using in -plane forces. The underlying goal of the work is to control the vibration of space structure s that can be used for space -based image acquisitio n. In chapter 2, we reviewed the dynamics of a thin plate without and with in -plane forces. The se dynamic equations were derived using the equilibrium approach based on classical plate theory with assumptions similar to those in thin beam or Euler -Bernoulli beam theory . After we presented the dynamics of the plate we introduced a few approaches for vibration analysis in chapter 3 . Specifically , we examined two different approximation methods and compared the results with the analytical solutio n for simply supported boundary conditions. The solutions for the dynamic equat ions are divided into two parts: the first part assumes that the plate is subjected to uniform stress distributions , in the second part , the plate is assumed to have non-uniform stress distributions . In chapter 4, we demonstrate a control strategy based on the in -plane uniformly distributed control forces to reduce the lateral vibration of the thin rectangular plate. The control scheme uses state feedback to switch the in -plane forces . These forces were switched on and off "$$!!based on a control logic t o remove energy from the system . The control scheme was implemented for two different boundary conditions, the simply supported boundary conditions and the elastic bounda ry conditions; the Rayleigh -Ritz approximation method and FEM were used for vibration analysis. After implementation of the basic control strategy, it was modified by adding a low -pass filter to reduce the bandwidth requirement of the actuator. The control logic is also modified by adding a saturation, which reduces the amount of added forces when the system vibration has reduced below a certain level . After we established the stability of the system mathematically, we verified it by simulating the results for both the unmodified and modified control logic. In chapter 5, we applied both the unmodified and modified control logic to control the vibratio n of a thin rectangular plate supported by a web -of-cables structure. The web -of-cables structure connects the thin rectangular plate to four fixed pulleys and provides the in -plane forces that are needed to remove the energy from the plate. To achieve tha t, we first design ed the web -of-cables structure and investigated the deformations that are caused by changing the corner cables tension. After we got the information of the deformed and undeformed web -of-cables structures, we used them to perform the in -plane stress distributions analysis. The pre -stress information was used to carryout the out -of-plane vibration analysis. Finally, from the out -of-plane vibration analysis we derived the dynamic models that are used to perform the control scheme. The contro l logic in this chapter was applied to control the switching of the corner cables additional tensions, which affects the web -of-cables structure. The web -of-cables structure adds concentrated forces to the edges of the thin rectangular plate, which create a non -uniform stress distribution inside the plate. The control scheme was verified u sing different cases as follows: increasing the tension of all four corner pulleys, increasing the tension of two corners "$%!!and decreasing the tension of the other two, usin g multiple tension levels with a random switching, and finally using multiple tension levels with systematic switching. Form the simulation results of these cases we were able to show that we successfully controlled the vibration of the higher modes of the thin rectangular plate using the control scheme. However, introducing the web -of-cable structure to suppress the vibration was not successful in suppressing the vibration of the first mode . Clearly, future work will be required to modify the control schem e to suppress the vibration of the fundamental frequency, which has more serious effects on the space -based image acquisition than the high frequencies. For instance, redesign the web-of-cables structure by including more cables around the plate boundaries (boundary cables) worth investigation; it can provide more in -plane forces that can increase the controller efficacy. 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