In The Name of GOD in Heaven
Nonlinear Vibration
Analysis of Isotropic
Rectangular Plate with
Viscoelastic Laminate
Morad Nazari
Supervisor: Prof. Firooz Bakhtiari Nejad
Introduction
• Importance of usage of composites and polymers is
being more appreciated day to day.
damping
strength
lightness and
etc.
• various properties of these materials are mechanical
bridges
submarines
vibration
structuresabsorbers
of vehicles
• These materials are applicable in aeroplanes
• 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
Literature Review
Literature Review
Developments in mechanics which where used as the
foundation for treating vibration of structures
Literature
Review
Kapania (1989)
Qatu (1992)
Theory of
plates
Viscoelastic
materials
Linear
analysis
Nonlinear
analysis
stability
analysis
Lagrange (1811)
Germain (1821)
Rayleigh (1877)
Love (1892)
Reissner (1945)
Mindlin (1951)
Ferry (1950-55)
Maxwell (1956)
Leissa (1973)
Jensen (1982)
Nayfeh (1994)
Ganapathi (1999)
Ganapathi (1999)
Krys’ko (2004)
Teng (1999)
Ilyasov (2007)
comment
Investigation
of
the
non-linear
Considering
linear
Linear
Influence
Reduced
review
introduction
investigated
Transformed
mechanical
vibration
vibration
articles
the
of rotary
rotary
shear
of
problem
regular
behavior
analysis
ain
analysis
the
complete
the
inertia
deformation
inertia
Von-Karman
and
field
ofof
dynamic
of
and
chaotic
terms
rectangular
polymers
of
model
in
instability
for
The
first
accurate
treatment
of plates
instability
behavior
ofboundary
the
composite
the
plates
cantilevered
shear
of
composite
continuous
vibrations
viscoelastic
equations
analysis
fundamental
on
with
flexural
and
various
layered
2D
of
plates
ofplates
vibrating
bifurcations
plates
systems,
equations
motions
plates
tointo
the
systems
considering
of
Duffing
of
solutions
of
of a
laminates
subjected
tothem
periodic
plates
conditions
isotropic
set
nonlinearities
flexible
ODEs
of ODEs
plate-strips
elastic
andand
solved
inplates
analyzed
stiffness
and
form and size
inplane-axial
ofdamping
instability load
domains
Ilyasov (2007)
method and/or verification
Bubnov-Galerkin
Their
Lindsted-Poincare
Laplace
comparison
His
multiple
research
36-term
finite
results
integral
evolutional
element
evolutional
of
scales
beam
covered
the
were
and
transform
perturbation
Ritz
finite
method
function
more
method
much
method
difference
and
complete
ofaveraging
method;
elementary
method
the
The
method
andarticles
results
than
experimental
rayleigh’s
written
sufficiently
in
data
decades
correlated with
prior to of
results
1990
finite element method
4
Viscoelastic Materials
Mechanical Properties
Step strain
stress input:
input:
solid material
fluid material
6
Classical Viscoelastic Models
Standard
Maxwell model
models
multiple Kelvin-Voigt
standard element
modelmodel
(S1)
(S2)
Basic discrete systems: Maxwell and Kelvin-Voigt models
Distributions of infinite numbers of elastic and viscous elements.
7
Classical Viscoelastic Models
Maxwell model
using complex modulus:
Maxwell model, simulates the
behavior of fluid viscoelastic
materials.
Relaxation:
Creeping:
fluid material
8
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
Dynamic Modeling
Dynamic Modeling
equation of motion in x & y direction:
11
Dynamic Modeling
taking moment
in x&y directions:
equation of motion in z direction
12
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
Dynamic Modeling
Equations of motion in terms of displacements:
(1)
(3)
14
Linear Analysis
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
0
0
SFFF
1
0
CFFF
2
0
SSFF
1
1
CSFF
2
1
CCFF
2
2
16
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
Ritz Method
A can be M or K and a can be m or k
eigenvector
corresponding mode-shape√
18
Ritz Method
Convergence characteristics can be improved by considering
planes of symmetry:
SFFF
CSFF
SSFF
CCFF
FFFF CFFF
19
Ritz Method
Fundamental computed frequency parameters vs. the number of terms in the
approximation series for SSFF and CCFF 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 SFFF and CFFF plates
B.C.
SFFF
CFFF
r=1
r=4
r=9
r=49
Leissa
------
6.463
6.439
6.646
6.648
------
23.252
14.398
14.395
15.023
4.472
3.533
3.517
3.514
3.492
------
8.984
8.597
8.521
8.525
20
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
Ritz Method (XF Beam)
The first five modes of cantilever beam; numerical (- -), exact(--—)
22
Nonlinear Analysis
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 T0  t
Multiple Scales Method
A.H. Nayfeh
slow time scale T1  t
24
Perturbation Method
time derivatives will be:
dimensionless parameters:
25
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
Perturbation Method
(4)
In-plane displacements are of higher order with respect to lateral displacements:
(A. Shushtari, S. Esmaeil Zadeh Khadem)
(5)
(6)
27
Perturbation Method
Step 21
Equalizing summation
of multipliers
of  2(1*),
to zero
(1*)(3*)
and (2*):
1 Substitution
of Eqs (4), (5)
and (6) in Eqs
(2*) inand
(a)
(b)
B.C.
28
Perturbation Method
2
3
Step 22 Equalizing the summation of multipliers of  and  in (3*) to zero:
(c)
(d)
w  w1  w2
29
Perturbation Method
Step 3 Solving PDEs:
(c)
A1 is the complex conjugate of A1
(a),(b)
(d)
Linear self-adjoint stiffness
differential operator
Expansion Theorem
qmn(t):
modal coordinates
30
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
Perturbation Method
Secular Terms:
It is easy to show that:
This equation:
can be expressed as:
(SSSS)
Equalizing
(d):
(CFFF)
to zero is sufficient to eliminate all of secular terms.
and discretizing the real and imaginary parts:
(CFFF)
(SSSS)
32
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
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
Finite Difference Method
implicit method
B(r ) for SSSS plate
B(r , v) for CFFF plate
CFFF plate:
Left hand boundary (x=0)
Right hand boundary (x=1)
35
Perturbation Method
Step 3 Solving PDEs:
(c)
linear vibration analysis
w1(A1) √
Indeterminate Factors Method (only for SSSS plate)
u1 and v1 √
(a),(b)
Finite Difference Method
(d)
Finite Difference Method
w2√
36
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
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
Nonlinear Analysis
Transverse displacement of center of SSSS plate;
neglecting higher order term (left)
considering higher order term (right)
39
Nonlinear Analysis
To be shown in large size
Response at the center of the cantilever plate
40
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
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
Stability Analysis and
Chaotic Behavior
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
Airy Stress Function (φ)
Definition
φ(x,y) is a piece-wise continuous function and the sequence
of its partial derivatives is changeable.
By eliminating
The
strain tensor
thecan
displacements
be written as:
from
compatibility conditions can be found.
45
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
Duffing Equation
SSSS plate
multiplying two sides by
domain of rectangular plate:
and integrating over the
CFFF plate
47
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.”
• This
The movie
effect is
The
observed
ButterflyinEffect
various branches of science.
• Mathematically, chaos is sensitivity to initial conditions.
48
Route to Chaos (SSSS Plate)
Period 16
2
bifurcation for SSSS plate, ε=0.329, Q=4, f=70.6
48
f=63.8
f=68.8
f=70.4
phase portrait(top left), poincare section(top right),
time history(bottom)
49
Route to Chaos
Chaotic behavior of SSSS plate, ε=0.329, Q=4, f=76.0
phase portrait(top left), poincare section(top right), time
history(bottom)
50
Route to Chaos
Bifurcation diagram of SSSS
plate with ε=0.329, Q=4
Universality of period doubling
51
Lyapunov Exponents
Lyapunov exponents criteria for SSSS plate, ε=0.329, Q=4
(period doubling route to chaos)
52
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 = , f = 6.6
(chaos via quasi periodicity); phase portrait(top left),
53
poincare section(top right), time history(bottom)
Route to Chaos (Cantilever Plate)
Quasi
periodic
route
to chaos for
cantilever
plate,
Chaotic
behavior
of cantilever
plate,
ε=-0.01,
Q=4, f=78
ε=-0.01,
Q=4,
at f=70;
phase portrait(top
left), left),
(chaos via
quasi
periodicity);
phase portrait(top
poincare
poincare section(top
section(top right),
right), time
time history(bottom)
history(bottom)
54
Lyapunov Exponents
Lyapunov exponents criteria for cantilever plate,
ε=-0.01, Q=4 (quasi periodic route to chaos)
55
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),
56
time history(bottom)
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
Fractal Dimension
Finding the fractal dimension of strange attractor for
SSSS plate with Q = 4, ε= 0.329 and f = 76.0
There is another way to estimate fractional
dimension called Lyapunov dimension DL
58
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
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
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), 257-293.
•
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).
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
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
Indeterminate Factors Method
64
• 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
Hysteresis
Parameter values of square plate
Parameter values
Solutions to the q(t) in forced vibration;
cantilever plate (right)
SSSS plate (left)
49