the vibration of thin plates
TRANSCRIPT
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Abstract One of the most important dynamic analyses is modal analysis. When there is no external force on the structure,
the analysis would be modal case. By using this method, we can
study Natural Frequencies and Mode Shapes. This paper presents a
finite element model for a simply supported rectangular plate. The
study uses ABAQUS (v.6.7) software to derive the finite element
model of the rectangular plate. Finally, the obtained results through
FEM would be compared with an exact solution.
Keywords Modal analysis; Natural frequencies; Mode shapes; Finite element method.
1. INTRODUCTION
In engineering practice, however, many components of
machines and structures are subjected to dynamic effects,
produced by time-dependent external forces or
displacements [1].
Dynamic loads may be created by moving vehicles, wind
gusts, seismic disturbances, unbalanced machine vibrations,
flight loads, sound, etc. Dynamic effects of time-dependent
loads on structures are studied in structural dynamics.
Structural dynamic s deals with time-dependent motions of
structures, primarily , with vibration of structures, and
analyses of the internal forces associated with them. Thus ,
its objective is to determine the effect of vibrations on the
performance of the structure or machine [2].
The dynamics of plates , which are continuous elastic
systems, can be modeled mathematically by partial
differential equations based on Newton's laws or by integral
equations based on the considerations of virtual work. In
practical applications only the lateral vibration is of interest,
and the effects of extensional vibrations in the middle plane
may be neglected. Therefore , the inertia forces , associated
with the lateral translation of the plate, are considered. In
this study only the simplified theory of plate vibrations is
introduced; some physical phenomena, associated with, for
instance, damping effects, are not considered [2,3].
Damping effects are caused either by internal friction or
by the surrounding media. Although structural damping is
theoretically present in all plate vibrations, it has usually
little or no effect on (a) the natural frequencies and (b) the
steady-state amplitudes; consequently, it can be safely
ignored in the initial treatment of the problem [4].
Ali Reza Pouladkhan: Ardestan Branch, Islamic Azad University,
Ardestan, Iran. E-mail: [email protected] Jalil Emadi: Ardestan Branch, Islamic Azad University, Ardestan, Iran.
E-mail:[email protected] and [email protected]
Majid Safamehr: Ardestan Branch, Islamic Azad University, Ardestan, Iran. E-mail: [email protected]
Hamed Habibolahiyan: Shahre Majlesi Branch, Islamic Azad
University, Shahre Majlesi, Iran.
The derivation of the governing differential equation of
motion is, in most cases, a simple extension of the static
case by adding effective forces to the plate that result from
accelerations of the mass of the plate. These are the inertia
forces.
We consider various kinds of motion of plates. There is a
free vibration, which occurs in the absence of applied loads
but may be initiated by applying initial conditions to the
plate. The free vibration deals with natural characteristics of
the plates, and these natural vibrations occur at discrete
frequencies, depending only on the geometry and material
of the plates. Then, there is a forced vibration, which results
from an application of time-dependent loads. Forced
vibrations come in two kinds: a harmonic response, when a
periodic force is applied to the plate; and a transient
response, when the applied force is not a periodic force [5].
The differential equation of forced, undamped motion of
plates has the form
, , , , , , Where both and are functions of time, as well as
space; is the mass density of the material, and is the plate thickness. In the forced vibrations , , causes the dynamic response.
2. FREE FLEXURAL VIBRATIONS OF
RECTANGULAR PLATES
Consider a rectangular plate with arbitrary supports. Let
us assume that certain transverse surface loads distributed
on the surface cause the particles, located in the middle
surface, to attain the deflections and velocities directed
perpendicularly to the initial (undeformed) middle surface.
At a certain time, which is assumed to be the initial, the
plate is suddenly released from all external loads. The
unloaded plate, which has initial deflection and velocity,
begins to vibrate. The particles located in the middle surface
move in the direction perpendicular to the plate and, as a
result, the plate becomes curved [6].
Such vibrations are called free or natural transverse
vibrations. The plate will execute free or natural lateral
vibrations. As stated previously, natural vibrations are
functions of the material properties and the plate geometry
only, and are inherent properties of the elastic plate,
independent of any load .Thus, for natural or free vibrations, , , is set equal to zero,and the above equation becomes
, , , , 0 Deflection must satisfy the boundary conditions at the
plate edge and the following initial conditions:
Ali Reza Pouladkhan, Jalil Emadi, Majid Safamehr, Hamed Habibolahiyan
The Vibration of Thin Plates by using Modal
Analysis
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When 0 : , , , Where and are the initial deflection and initial
velocity for point , . The above equation is the governing , fourth-order
homogeneous partial differential equation of the undamped,
free, linear vibrations of plates.
To solve this equation and obtain , , in general, one can assume the following solution : , , !,
Which is a separable solution of the shape functions !, describing the modes of the vibration and some harmonic function of a time; is the natural frequency of the plate vibration which is related to vibration period " by the relationship 2$/".
Introducing above equation into the previous equation we
have !, ! 0 For example, in the case of a rectangular, simply
supported plate, the shape function may be taken as
!, & & '() *$+
),-
(,- $.
Where + and . are the plate dimensions and '() is the vibration amplitude for each value of * and .Substitution of the above equation into the previous equation results in
the homogeneous algebraic equation */$/+/ 2 *$+
$. /$/./
0 Solving this equation for gives the natural frequencies
() $*+
.0 The fundamental natural frequency can be obtained by
letting * 1, 1. Note that ,again (as in the buckling analysis), the
amplitude '() cannot be determined from the linear eigenvalue problem.
So; the fundamental natural frequency is :
-- $ 1+ 1.0 For the steel plate as follow properties 7800 45 *6 ; 9 200 :+ ; ; 0.3 ; 1 ** ; + 0.2* ; . 0.1*
96121 ; 18.315 ?. * -- $ @ 10.2 10.1A 0 18.3157800 B 0.001 ; -- 1890.453 E+F G
3. THE FINITE ELEMENT METHOD (FEM)
The finite element method (FEM) is based on the concept
that one can replace any continuum by an assemblage of
simply shaped elements with well-defined force
displacement and material relationships. While one may not
be able to derive a closed-form solution for the continuum,
one can derive an approximate solution for the element
assemblage that replaced it.
According to the FEM, a plate is discretized into a finite
number of elements (usually, triangular or rectangular in
shape), called finite elements and connected at their nodes
and along interelement boundaries. Unknown functions
(deflections, slopes, internal forces, and moments) are
assigned in the form of undetermined parameters at those
nodes. The equilibrium and compatibility conditions must
be satisfied at each node and along the boundaries between
finite elements [7].
In this study we want to compare the FEM with
equilibrium method and obtain Mesh Convergence Curve to
optimize the results. For this investigation we use ABAQUS
software. ABAQUS is one of the most important Finite
Element Softwares. The Modal Analysis is used in this
study.
4. MODAL ANALYSIS
The frequency extraction procedure :
performs eigenvalue extraction to calculate the natural
frequencies and the corresponding mode shapes of a
system.
will include initial stress and load stiffness effects due
to preloads and initial conditions if geometric nonlinearity is
accounted for in the base state, so that small vibrations of a
preloaded structure can be modeled.
is a linear perturbation procedure.
5. EIGENVALUE EXTRACTION
The frequency extraction procedure uses eigenvalue
techniques to extract the frequencies of the current system.
The eigenvalue problem for the natural frequencies of an
undamped finite element model is : H IJ 0 Where
H is the mass matrix (which is symmetric and positive definite).
I is the stiffness matrix (which includes initial stiffness effects if the base state included the effects of nonlinear
geometry).
J is the eigenvector (the mode of vibration). If initial stress effects are not included and there are no
rigid body modes, I is positive definite; otherwise , it may not be.Negative eigenvalues normally indicate an instability.
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6. GEOMETRY AND PROBLEM DESCRIPTION
The model used for this study is a plate with simple
supports with S4R elements.
Fig. 1 Boundary configuration of the simply supported rectangular
plate.
Fig. 2 Typical finite element model of the rectangular plate.
S4R : A 4-node doubly curved thin or thick shell , reduced
integration , hourglass control , finite membrane strains
TABLE 1
NUMBER OF ELEMENTS USED TO ACHIEVE OPTIMU
PLATE.
A.G.S Number of Mesh Natural Frequency (
0.02 50 1959.411
0.01 200 1906.193
0.005 800 1892.684
0.004 1250 1890.799
0.003 2211 1889.228
Where A.G.S is : Approximate Global Size
The nearest result into the Analytical method is :
Approximate Global Size : 0.004
Number of Element : 1250 ; --sec N 300.93 OP Where the above Mesh is optimum because the error is
minimum and : 1892.684 1890.7991890.799 B 100 0.0997
So; we can draw the Mesh Convergence Curve
Fig. 3 Mesh Convergence Curve for the Finite Element
DESCRIPTION
is a plate with simple
1 Boundary configuration of the simply supported rectangular
2 Typical finite element model of the rectangular plate.
node doubly curved thin or thick shell , reduced
ss control , finite membrane strains [8].
ED TO ACHIEVE OPTIMUM MESH FOR STEEL
Natural Frequency (--) 1959.411
1906.193
1892.684
1890.799
1889.228
Where A.G.S is : Approximate Global Size
into the Analytical method is :
1890.799 E+F/Mesh is optimum because the error is
0997% S 5%
Mesh Convergence Curve :
3 Mesh Convergence Curve for the Finite Element Model.
Several mode shapes are shown in the following figures.
It is clear that with increasing mode number, the natural
frequency is increased. Also, it could be seen that some
mode shapes have the same natural frequency.
words, the rectangular plate has an equal frequency for
mode shapes 5,6 ; 11,12 ; 15, 16. For example, for mode
shapes 5 and 6 we have : /- 1208.1 OP ; Mode
Mode Shape 1:
Mode Shape 2:
Mode Shape 3:
Mode Shape 4:
Several mode shapes are shown in the following figures.
It is clear that with increasing mode number, the natural
frequency is increased. Also, it could be seen that some
mode shapes have the same natural frequency. In other
rectangular plate has an equal frequency for
mode shapes 5,6 ; 11,12 ; 15, 16. For example, for mode
Mode 5 and Mode 6
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Mode Shape 5:
Mode Shape 6:
Mode Shape 7:
Mode Shape 8:
Mode Shape 9:
Mode Shape 10:
Mode Shape 11:
Mode Shape 12:
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Mode Shape 13:
Mode Shape 14:
Mode Shape 15:
Mode Shape 16:
Mode Shape 17:
Mode Shape 18:
Mode Shape 19:
Mode Shape 20:
Fig. 4 Natural mode shapes and frequencies of the rectangular
plate.
4 Natural mode shapes and frequencies of the rectangular
plate.
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Frequency-Mode number Diagram is shown in the
following figure.
Fig. 5 Natural frequencies of simply supported plate vs. mode
numbers.
It is clear that mode numbers 5 and 6 have the same
natural frequency (NT NU 1208.1 OPnumbers 11 , 12 and 15 , 16 have an equal frequency
(N-- N- 2437.8 OP ; N-T N-U 3152
7. CONCLUSION
A finite element model was presented for this study. This
paper reviewed the capability of the shell element (S4R)
provided by commercialized FEA codes, and discussed a
simple case of dynamic finite element analysis.
the finite element modeling technique, the study showed
acceptable results in comparison with exact solution for a
simply supported rectangular plate.
REFERENCES
[1] Ventsel, E., and Krauthammer, T., Thin Plates and Shells, Marcel
Dekker, New York, 2001.
[2] Nowacki, W., Dynamics of Elastic Systems, John Wiley and Sons,
New York, 1963.
[3] Warburton, G., The vibration of rectangular plates, Proc J Mech
Engrs, vol. 168, No. 12, pp. 371384 (1954).
[4] Leissa, A.W., Vibrations of Plates, National Aeronautics and Space
Administration, Washington, D.C., 1969.
[5] Thomson, W.T., Vibration Theory and Applications, Prentice
Englewood Cliffs, New Jersey, 1973.
[6] Dill, E.N. and Pister, K.S., Vibration of rectangular plates and plate
systems, Proc 3rd US Natl Congr Appl Mech, pp. 123
[7] Zienkiewicz, O., The Finite Element Method, McGraw
1977.
[8] HKS, (2005) ABAQUS User's Manual version 6.6, (Providence, RI:
Hibbitt, Karlsson, and Sorenson).
1.
number Diagram is shown in the
5 Natural frequencies of simply supported plate vs. mode
It is clear that mode numbers 5 and 6 have the same OP). Also, mode numbers 11 , 12 and 15 , 16 have an equal frequency 3152.8 OP).
A finite element model was presented for this study. This
paper reviewed the capability of the shell element (S4R)
by commercialized FEA codes, and discussed a
simple case of dynamic finite element analysis. Based on
chnique, the study showed
acceptable results in comparison with exact solution for a
Ventsel, E., and Krauthammer, T., Thin Plates and Shells, Marcel
of Elastic Systems, John Wiley and Sons,
Warburton, G., The vibration of rectangular plates, Proc J Mech
Leissa, A.W., Vibrations of Plates, National Aeronautics and Space
Thomson, W.T., Vibration Theory and Applications, Prentice-Hall,
Dill, E.N. and Pister, K.S., Vibration of rectangular plates and plate
Appl Mech, pp. 123132 (1958).
Zienkiewicz, O., The Finite Element Method, McGraw-Hill, London,
HKS, (2005) ABAQUS User's Manual version 6.6, (Providence, RI:
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