Nonlinear Vibration Analysis of Isotropic Rectangular Plate with Viscoelastic Laminate

Morad Nazari

Supervisor: Prof. Firooz Bakhtiari Nejad

In The Name of GOD in Heaven

• Importance of usage of composites and polymers is being more appreciated day to day.

• These materials are applicable in

• various properties of these materials are lightness and etc.mechanical strengthdamping

structures of vehiclesaeroplanessubmarinesbridges

Introduction

• Nonlinear vibration analysis of these materials is considered in order to obtain modal characteristics of those systems.

vibration absorbers

• Referring to properties of polymeric materials, they have nonlinearities in stiffness, damping and inertia.

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Literature Review

Literature Review

Developments in mechanics which where used as the foundation for treating vibration of structures

Theory ofplates

Viscoelasticmaterials

Linear analysis

Nonlinearanalysis

stabilityanalysis

Lagrange (1811)

Germain (1821)

The first accurate treatment of platescomment method and/or verification

elementary

Rayleigh (1877)Love (1892)

Reissner (1945)

Mindlin (1951)

Ferry (1950-55)Maxwell (1956)

Leissa (1973)

Jensen (1982)

Literature Review

Kapania (1989)Qatu (1992)

Nayfeh (1994)

Krys’ko (2004)

Teng (1999)

Ganapathi (1999)

Ilyasov (2007)

Ilyasov (2007)

Considering rotary inertia terms in the analysis of vibrating systems

evolutionalConsidering shear deformation in the fundamental equations of plates

evolutionalInfluence of rotary inertia and shear on flexural motions of isotropic elastic plates

His results were more complete than rayleigh’s

mechanical behavior of polymers evolutionallinear vibration analysis of rectangularplates with various boundaryconditions

36-term beam functioncomparison of the Ritz method and experimental data

Linear vibration analysis of cantilevered platesreview articles in the field of composite layered plates

Their research covered much of the articles written in decades prior to 1990

introduction of a complete model forcontinuous 2D systems, consideringnonlinearities in stiffness and damping

multiple scales methodInvestigation of the non-linear instability behavior of the composite laminates subjected to periodic inplane-axial load

finite element methodTransformed the Von-Karman equations of plates into Duffing ODEs and solved them

Lindsted-Poincare perturbation method; The results sufficiently correlated with results of finite element method

investigated regular and chaotic vibrations and bifurcations of flexible plate-strips

Bubnov-Galerkin and finite difference method

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

Laplace integral transform and averaging method

Ganapathi (1999)

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Viscoelastic Materials

Mechanical Properties

Step strain input:

fluid materialsolid material

Step stress input:

solid material fluid material

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Classical Viscoelastic Models

multiple standard element modelMaxwell model Kelvin-Voigt modelStandard models

Basic discrete systems: Maxwell and Kelvin-Voigt modelsDistributions of infinite numbers of elastic and viscous elements.

(S2)

(S1)

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using complex modulus: Relaxation:Creeping:

Maxwell model, simulates the behavior of fluid viscoelastic materials.

Classical Viscoelastic Models

Maxwell model

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fluid material

(sudden stress decrement to zero)

Second standard model shows viscoelastic behavior of solids, properly

Kelvin-Voigt model

second standard model

(sudden stress increment to zero)

Classical Viscoelastic Models

√

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solid material

Dynamic Modeling

equation of motion in x & y direction:

Dynamic Modeling

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taking moment in x&y directions:

equation of motion in z direction

Dynamic Modeling

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By considering nonlinear geometry of Von-Karman, the relation between strains and displacements are:

stress-strain relations for homogeneous viscoelastic material in plane-stress case is:

Regarding to Love-Kirchhoff hypothesis:

Dynamic Modeling

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forces and moments in

terms of displacements √

Dynamic Modeling

Equations of motion in terms of displacements:

(1)

(3)

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Linear Analysis

Ritz Method

The displacement trial functions, in terms of the non-dimensional coordinates x* and y*, are taken as:

B.C. m0 n0

FFFF 0 0

SFFF 1 0

CFFF 2 0

SSFF 1 1

CSFF 2 1

CCFF 2 2

Values for m0 and n0

(XXFF Plate)

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Ritz Method

flexural strain energy:

Minimization of functional (TMax-Ufmax) with respect to coefficient

(M-m0+1)(N-n0+1) simultaneous linear homogeneous equations

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kinetic energy:

Ritz Method

A can be M or K and a can be m or k

eigenvector

corresponding mode-shape

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√

Ritz Method

Convergence characteristics can be improved by considering planes of symmetry:

FFFFSSFF CCFFSFFF CFFFCSFF

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Ritz Method

Fundamental computed frequency parameters vs. the number of terms in the approximation series for and plates

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

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

CFFF4.472 3.533 3.517 3.514 3.492

------ 8.984 8.597 8.521 8.525

SSFF CCFF

SFFF CFFF

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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

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Ritz Method

The first five modes of cantilever beam; numerical (- -), exact(--—)

(XF Beam)

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Nonlinear Analysis

A.H. Nayfeh

tT 1

tT 0fast time scale

slow time scale

Multiple Scales Method

In this method, different time scales are Introduced to obtain uniform expansions and increase infinitely with time.

Multiple Scales Method

Perturbation Method

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Perturbation Method

dimensionless parameters:

time derivatives will be:

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

(1*)

…

(3*)

Perturbation Method

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In-plane displacements are of higher order with respect to lateral displacements:

(5)

(6)

(4)

Perturbation Method

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(A. Shushtari, S. Esmaeil Zadeh Khadem)

Step 1 Substitution of Eqs (4), (5) and (6) in Eqs (1*), (2*) and (3*)2

Perturbation Method

Step 21 Equalizing summation of multipliers of to zero in (1*) and (2*):

B.C.

(a)

(b)

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Perturbation Method

Step 22 Equalizing the summation of multipliers of and in (3*) to zero:2 3

(c)

(d)

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21 www

(c)

Step 3 Solving PDEs:

Perturbation Method

(a),(b)

1A is the complex conjugate of 1A

Expansion Theorem

Linear self-adjoint stiffness differential operator

qmn(t): modal coordinates

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(d)

multiplying by the first mode-shape p11(x, y) derived by Ritz method,

integrating over the domain D

using following orthogonality relations

(d)

fr : summation of the multipliers of in the right hand side of equation (d).

Modal Equations

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Perturbation Method

Secular Terms:

It is easy to show that:

This equation: can be expressed as:

Equalizing to zero is sufficient to eliminate all of secular terms.

and discretizing the real and imaginary parts:

(CFFF) (SSSS)

(SSSS) (CFFF)

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(d):

Perturbation Method

Natural frequency of cantilever plateThe variation of maximum amplitude of vibration for SSSS plate with

• Natural frequency of viscoelastic plate, reduces with time.

• increasing damping of the plate, natural frequency will be decreased.

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Finite Difference 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

explicit method

implicit method

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Finite Difference Method

implicit method

),( vrB

)(rB

for CFFF platefor SSSS plate

Left hand boundary (x=0)

Right hand boundary (x=1)

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CFFF plate:

(c)

Step 3 Solving PDEs:

Perturbation Method

(a),(b)

Indeterminate Factors Method (only for SSSS plate)

Finite Difference Method

(d) Finite Difference Method

w1(A1) √

u1 and v1 √

linear vibration analysis

w 2 √

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

indeterminate factors method (left)

finite difference method (right)

Nonlinear Analysis

combination method of multiple scales andfinite difference (central difference method)

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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:

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Transverse displacement of center of SSSS plate;

neglecting higher order term (left)

considering higher order term (right)

Nonlinear Analysis

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To be shown in large size

Response at the center of the cantilever plate

Nonlinear Analysis

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Nonlinear response at the center of cantilever plate;

h/a=1/8 (left) h/a=1/20 (right)

Nonlinear Analysis

• Increasing viscoelastic parameter damping, the amplitude of transverse vibration reduces. (trivial)

• For thicker plates the damping parameter shows itself better.

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Nonlinear Analysis

• 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.

Transverse displacement function at the center of the cantilever plate for h/a=1/8;

η=0.1 (left) η=0.8 (right)

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Stability Analysis and Chaotic Behavior

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

Assumptions

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Airy Stress Function (φ)

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.

Definition

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

Airy Stress FunctionSSSS plate

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Duffing Equation

CFFF plate

SSSS plate

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

Very few people are afraid of butterflies … but maybe more should be.

Butterfly Effect

• Edward N. Lorenz

• The movie The Butterfly Effect• This effect is observed in various branches of science.

: “Predictability: Does the flap of a butterfly’s wings in Brazil set off a Tornado in Texas.”

• Mathematically, chaos is sensitivity to initial conditions.48

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)

f=63.824 f=68.88 f=70.416 f=70.6

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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

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Route to Chaos

Bifurcation diagram of SSSS plate with ε=0.329, Q=4

Universality of period doubling

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Lyapunov Exponents

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

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Route to Chaos

Chaotic behavior of SSSS plate, ε=0.329, Q = , (chaos via quasi periodicity); phase portrait(top left), poincare section(top right), time history(bottom)

Quasi Periodic route to chaos for SSSS plate, ε=0.329, Q = , f = 6.5

f = 6.6

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Route to Chaos (Cantilever Plate)

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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)

Lyapunov Exponents

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

Route to Chaos

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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)

Fractal Dimension

Db: box-counting dimension

Definition

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

There is another way to estimate fractional dimension called Lyapunov dimension DL

Fractal Dimension

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Finding the fractal dimension of strange attractor for SSSS plate with Q = 4, ε= 0.329 and f = 76.0

Conclusion

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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.

• 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.

• Lyapunov criteria was employed to verify results of Poincare section and …

Suggestions

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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.

• 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.

• 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.

• Stability analysis and routes to chaos for this continuous system can be treated as an open problem by itself.

References

• A.H. Nayfeh and D.T. Mook, Nonlinear oscillations, Wiley, 1979.

• A. Shushtari, Nonlinear vibration analysis and stability of viscoelastic rectangular plates, PhD Thesis, University of Tarbiat Modarres, Mechanical Engineering Department, 1385 (2006) (in Persian).

• A.W. Leissa, The free vibration of rectangular plates, Sound and Vibration, 31(3), (1973), 257-293.

• M.S. Qatu, Vibration of laminated shells and plates, 2nd ed., Elsevier, Oxford, 2004.

• T.W. Kim and J.H. Kim, Nonlinear vibration of viscoelastic laminated composite plates, Solids and Structures 39 (2002), 2857–2870.

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

Thanks for your patience

• Deflections are small when compared with the plate thickness.

• The normal stresses in the direction transverse to the plate can be ignored.

• Any straight line normal to the middle plane before deformation remains a straight line normal to the neutral plane during deformation.

• The middle plane of the plate does not undergo deformations during bending and can be regarded as a neutral plane.

• Shear strain can be neglected.

Love-Kirchhoff Hypothesis

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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.

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Parameter values

Hysteresis

Solutions to the q(t) in forced vibration;

SSSS plate (left)

cantilever plate (right)

Parameter values

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Parameter values of square plate