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**Morad Nazari Supervisor: Prof. Firooz Bakhtiari Nejad**

In The Name of GOD in Heaven Nonlinear Vibration Analysis of Isotropic Rectangular Plate with Viscoelastic Laminate Morad Nazari Supervisor: Prof. Firooz Bakhtiari Nejad 1 1

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Introduction Importance of usage of composites and polymers is being more appreciated day to day. various properties of these materials are damping lightness and etc. mechanical strength These materials are applicable in bridges aeroplanes submarines vibration absorbers structures of vehicles Referring to properties of polymeric materials, they have nonlinearities in stiffness, damping and inertia. Nonlinear vibration analysis of these materials is considered in order to obtain modal characteristics of those systems. 2 2

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Literature Review

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**method and/or verification**

Literature Review Developments in mechanics which where used as the foundation for treating vibration of structures Literature Review Theory of plates Viscoelastic materials Linear analysis Nonlinear analysis stability analysis Kapania (1989) Qatu (1992) Lagrange (1811) Germain (1821) Ferry ( ) Maxwell (1956) Leissa (1973) Nayfeh (1994) Ganapathi (1999) Jensen (1982) Ganapathi (1999) Krys’ko (2004) Rayleigh (1877) Teng (1999) Ilyasov (2007) Love (1892) Reissner (1945) Ilyasov (2007) Mindlin (1951) comment method and/or verification Investigation of the non-linear instability behavior of the composite laminates subjected to periodic inplane-axial load linear vibration analysis of rectangular plates with various boundary conditions Linear vibration analysis of cantilevered plates Reduced the problem of dynamic stability of viscoelastic plates to the solutions of a set of ODEs and analyzed form and size of instability domains Influence of rotary inertia and shear on flexural motions of isotropic elastic plates Considering rotary inertia terms in the analysis of vibrating systems The first accurate treatment of plates Considering shear deformation in the fundamental equations of plates investigated regular and chaotic vibrations and bifurcations of flexible plate-strips review articles in the field of composite layered plates introduction of a complete model for continuous 2D systems, considering nonlinearities in stiffness and damping Transformed the Von-Karman equations of plates into Duffing ODEs and solved them mechanical behavior of polymers Bubnov-Galerkin and finite difference method Laplace integral transform and averaging method Lindsted-Poincare perturbation method; The results sufficiently correlated with results of finite element method Their research covered much of the articles written in decades prior to 1990 comparison of the Ritz method and experimental data His results were more complete than rayleigh’s multiple scales method 36-term beam function finite element method elementary evolutional evolutional evolutional 4

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**Viscoelastic Materials**

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**Mechanical Properties**

Step stress input: Step strain input: solid material solid material fluid material fluid material 6 6 6

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**Classical Viscoelastic Models**

Standard models Maxwell model multiple standard element model Kelvin-Voigt model (S1) (S2) Basic discrete systems: Maxwell and Kelvin-Voigt models Distributions of infinite numbers of elastic and viscous elements. 7 7 7

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**Classical Viscoelastic Models**

Maxwell model fluid material using complex modulus: Relaxation: Creeping: Maxwell model, simulates the behavior of fluid viscoelastic materials. 8 8 8

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**Classical Viscoelastic Models**

Kelvin-Voigt model √ solid material (sudden stress increment to zero) (sudden stress decrement to zero) second standard model Second standard model shows viscoelastic behavior of solids, properly 9

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Dynamic Modeling

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Dynamic Modeling equation of motion in x & y direction: 11 11 11

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**Dynamic Modeling taking moment in x&y directions:**

equation of motion in z direction 12

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Dynamic Modeling stress-strain relations for homogeneous viscoelastic material in plane-stress case is: By considering nonlinear geometry of Von-Karman, the relation between strains and displacements are: Regarding to Love-Kirchhoff hypothesis: forces and moments in terms of displacements √ 13 13 13

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**Dynamic Modeling Equations of motion in terms of displacements: (1)**

(3) 14

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Linear Analysis

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**Ritz Method (XXFF Plate)**

The displacement trial functions, in terms of the non-dimensional coordinates x* and y*, are taken as: Values for m0 and n0 B.C. m0 n0 FFFF SFFF 1 CFFF 2 SSFF CSFF CCFF 16

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**Ritz Method kinetic energy: flexural strain energy:**

Minimization of functional (TMax-Ufmax) with respect to coefficient (M-m0+1)(N-n0+1) simultaneous linear homogeneous equations 17

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**Ritz Method √ corresponding mode-shape**

A can be M or K and a can be m or k eigenvector √ corresponding mode-shape 18

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**Ritz Method Convergence characteristics can be improved by considering**

planes of symmetry: SSFF CCFF SFFF CFFF FFFF CSFF 19

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Ritz Method Fundamental computed frequency parameters vs. the number of terms in the approximation series for and plates SSFF CCFF B.C. r=1 r=4 r=9 r=16 Leissa SSFF 3.550 3.330 3.538 3.494 3.369 CCFF 10.435 7.230 6.965 6.948 3.942 Fundamental computed frequency parameters Ω11Ω12 vs. the number of terms in the approximation series for and plates SFFF CFFF B.C. r=1 r=4 r=9 r=49 Leissa SFFF ------ 6.463 6.439 6.646 6.648 23.252 14.398 14.395 15.023 CFFF 4.472 3.533 3.517 3.514 3.492 8.984 8.597 8.521 8.525 20

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Ritz Method The two lowest computed frequency parameters vs. the number of terms in the approximation series, for cantilever plate Counter-plot of first six modes of cantilever plate 21

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Ritz Method (XF Beam) The first five modes of cantilever beam; numerical (- -), exact(--—) 22

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Nonlinear Analysis

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**Multiple Scales Method**

Perturbation Method Multiple Scales Method In this method, different time scales are Introduced to obtain uniform expansions and increase infinitely with time. fast time scale Multiple Scales Method A.H. Nayfeh slow time scale 24

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**Perturbation Method time derivatives will be:**

dimensionless parameters: 25

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Perturbation Method Substituting the dimensionless values into equations of motion and neglecting the symbol * for simplification, one can get following equations from Eqs (1) to (3): … (3*) (1*) 26

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**Perturbation Method (4)**

In-plane displacements are of higher order with respect to lateral displacements: (A. Shushtari, S. Esmaeil Zadeh Khadem) (5) (6) 27 27

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Perturbation Method Step 1 Substitution of Eqs (4), (5) and (6) in Eqs (1*), (2*) and (3*) Step 21 Equalizing summation of multipliers of to zero in (1*) and (2*): (a) (b) B.C. 28

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Perturbation Method Step 22 Equalizing the summation of multipliers of and in (3*) to zero: (c) (d) 29 29

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**Perturbation Method Step 3 Solving PDEs: (c)**

is the complex conjugate of (a),(b) (d) Linear self-adjoint stiffness differential operator Expansion Theorem qmn(t): modal coordinates 30

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Modal Equations (d) multiplying by the first mode-shape p11(x, y) derived by Ritz method, integrating over the domain D using following orthogonality relations fr : summation of the multipliers of in the right hand side of equation (d). 31

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**Perturbation Method Secular Terms: It is easy to show that:**

This equation: can be expressed as: (SSSS) (CFFF) Equalizing to zero is sufficient to eliminate all of secular terms. (d): and discretizing the real and imaginary parts: (CFFF) (SSSS) 32

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**Natural frequency of cantilever plate**

Perturbation Method Natural frequency of viscoelastic plate, reduces with time. increasing damping of the plate, natural frequency will be decreased. Natural frequency of cantilever plate The variation of maximum amplitude of vibration for SSSS plate with 33

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**Finite Difference Method**

explicit method Discrete form of equation of motion and boundary conditions based on central difference method : A(r): Amplification matrix Definition (stability condition). A finite difference scheme is known to be stable if: no answers for implicit method 34

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**Finite Difference Method**

implicit method for SSSS plate for CFFF plate CFFF plate: Left hand boundary (x=0) Right hand boundary (x=1) 35

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**Perturbation Method Step 3 Solving PDEs: (c) linear vibration analysis**

w1(A1) √ Indeterminate Factors Method (only for SSSS plate) (a),(b) u1 and v1 √ Finite Difference Method (d) Finite Difference Method w 2 √ 36

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**Second in-plane modes for SSSS plate in x direction;**

Nonlinear Analysis combination method of multiple scales and finite difference (central difference method) Second in-plane modes for SSSS plate in x direction; indeterminate factors method (left) finite difference method (right) 37

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**Nonlinear Analysis Fundamental in-plane modes for cantilever plate;**

u (left) v (right) Fundamental in-plane modes of cantilever plate obtained by fourth order curve fitting: 38

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**Transverse displacement of center of SSSS plate;**

Nonlinear Analysis Transverse displacement of center of SSSS plate; neglecting higher order term (left) considering higher order term (right) 39

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**Nonlinear Analysis To be shown in large size**

Response at the center of the cantilever plate 40

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**Nonlinear response at the center of cantilever plate;**

Nonlinear Analysis Increasing viscoelastic parameter damping, the amplitude of transverse vibration reduces. (trivial) For thicker plates the damping parameter shows itself better. Nonlinear response at the center of cantilever plate; h/a=1/8 (left) h/a=1/20 (right) 41

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Nonlinear Analysis Transverse displacement function at the center of the cantilever plate for h/a=1/8; η=0.1 (left) η=0.8 (right) Considering higher order terms, causes the response to be damped faster. Increasing viscoelastic parameter, amax of higher order terms of w will approach the amax of first order term. 42

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**Stability Analysis and Chaotic Behavior**

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Assumptions We assume that the plate can be excited vertically with external driving force with distributed force P and the previous governing equations takes the form: f † is of an acceleration type 44

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**Airy Stress Function (φ)**

Definition φ(x,y) is a piece-wise continuous function and the sequence of its partial derivatives is changeable. The strain tensor can be written as: By eliminating the displacements from compatibility conditions can be found. 45

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**Airy Stress Function SSSS plate**

Considering the fundamental mode-shape as the most important mode-shape in vibration analysis of plate and solving the PDE: 46

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**Duffing Equation SSSS plate**

multiplying two sides by and integrating over the domain of rectangular plate: CFFF plate 47

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Butterfly Effect Very few people are afraid of butterflies … but maybe more should be. Edward N. Lorenz : “Predictability: Does the flap of a butterfly’s wings in Brazil set off a Tornado in Texas.” The movie The Butterfly Effect This effect is observed in various branches of science. Mathematically, chaos is sensitivity to initial conditions. 48

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**Route to Chaos (SSSS Plate)**

Period bifurcation for SSSS plate, ε=0.329, Q=4, phase portrait(top left), poincare section(top right), time history(bottom) 4 2 16 8 f=70.6 f=68.8 f=70.4 f=63.8 49

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**Route to Chaos Chaotic behavior of SSSS plate, ε=0.329, Q=4,**

phase portrait(top left), poincare section(top right), time history(bottom) f=76.0 50

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**Route to Chaos Bifurcation diagram of SSSS plate with ε=0.329, Q=4**

Universality of period doubling 51

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Lyapunov Exponents Lyapunov exponents criteria for SSSS plate, ε=0.329, Q=4 (period doubling route to chaos) 52

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Route to Chaos Quasi Periodic route to chaos for SSSS plate, ε=0.329, Q = , f = 6.5 Chaotic behavior of SSSS plate, ε=0.329, Q = , (chaos via quasi periodicity); phase portrait(top left), poincare section(top right), time history(bottom) f = 6.6 53

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**Route to Chaos (Cantilever Plate)**

Quasi periodic route to chaos for cantilever plate, ε=-0.01, Q=4, at f=70; phase portrait(top left), poincare section(top right), time history(bottom) Chaotic behavior of cantilever plate, ε=-0.01, Q=4, f=78 (chaos via quasi periodicity); phase portrait(top left), poincare section(top right), time history(bottom) 54

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Lyapunov Exponents Lyapunov exponents criteria for cantilever plate, ε=-0.01, Q=4 (quasi periodic route to chaos) 55

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Route to Chaos Quasi periodic route to chaos for cantilever plate, Q = , f=1.34 Chaotic behavior of cantilever plate, Q = , f=1.35; phase portrait (top left), poincare section(top right), time history(bottom) 56

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**Fractal Dimension Definition Db: box-counting dimension**

R : the length of unit contour (square) N(R): the number of boxes covering the strange attractor 57

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Fractal Dimension Finding the fractal dimension of strange attractor for SSSS plate with Q = 4, ε= and f = 76.0 There is another way to estimate fractional dimension called Lyapunov dimension DL 58

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Conclusion Linear vibration of undamped isotropic rectangular cantilever plate was investigated first by Ritz method and mode-shapes of plates were obtained. Results obtained had acceptable convergence and were in a good agreement with previous researches. Nonlinear equations of motions were obtained based on Kelvin-Voigt viscoelastic model and nonlinear geometry of Von-Karman. Then dimensionless equations were derived and a combination method of multiple scale and finite difference was employed to solve these equations and the time history of nonlinear natural frequencies and nonlinear response of plate were obtained. The equations of continuous system were discretized by Galerkin method to obtain the discrete equations. Numerical integration schemes were applied to the resulting ODEs to construct the phase portrait and etc. Lyapunov criteria was employed to verify results of Poincare section and … 59

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Suggestions We suggest that further researches in this direction can be done in following fields: In Ritz method, customarily the basic functions in vibration analysis are also referred to as trial functions or admissible functions. In comparison with the simple algebraic polynomials, the selection of Chebyshev polynomials as the basic functions yields higher accuracy. • Among classical viscoelastic models, the standard model represents mechanical properties (creeping and relaxation functions) of viscoelastic solids in the best manner, and using this model is suggested for future researches. All of the processes in this thesis are applicable for rectangular plates with boundary conditions of type XXFF, only the time wasting computer coding of finite difference method for these boundary types must be done. Also these procedures may be used for plates with exact solution. Stability analysis and routes to chaos for this continuous system can be treated as an open problem by itself. 60

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References T.W. Kim and J.H. Kim, Nonlinear vibration of viscoelastic laminated composite plates, Solids and Structures 39 (2002), 2857–2870. A.W. Leissa, The free vibration of rectangular plates, Sound and Vibration, 31(3), (1973), A.H. Nayfeh and D.T. Mook, Nonlinear oscillations, Wiley, 1979. M.S. Qatu, Vibration of laminated shells and plates, 2nd ed., Elsevier, Oxford, 2004. A. Shushtari, Nonlinear vibration analysis and stability of viscoelastic rectangular plates, PhD Thesis, University of Tarbiat Modarres, Mechanical Engineering Department, 1385 (2006) (in Persian).

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Publications F. Bakhtiari Nejad and M. Nazari, Transverse vibration of plate with at least two sequent free edges – Part I: Linear analysis, The 7th Conference of Iranian Aerospace Society, Sharif University of Technology F. Bakhtiari Nejad and M. Nazari, Transverse vibration of plate with at least two sequent free edges – Part II: Nonlinear analysis of cantilever plate, The 7th Conference of Iranian Aerospace Society, Sharif University of Technology

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**Thanks for your patience**

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**Love-Kirchhoff Hypothesis**

The middle plane of the plate does not undergo deformations during bending and can be regarded as a neutral plane. Deflections are small when compared with the plate thickness. Any straight line normal to the middle plane before deformation remains a straight line normal to the neutral plane during deformation. Shear strain can be neglected. The normal stresses in the direction transverse to the plate can be ignored. 65

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**Indeterminate Factors Method**

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The equations of continuous system were discretized by Galerkin method to obtain the discrete equations. Numerical integration schemes were applied to the resulting ODEs to construct the phase portrait and etc. Parameter values 63

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**Parameter values of square plate**

Hysteresis Parameter values of square plate Parameter values Solutions to the q(t) in forced vibration; SSSS plate (left) cantilever plate (right) 49

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