The Vibration of Thin Plates

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

World Academy of Science, Engineering and Technology 59 2011

2880

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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The Vibration of Thin Plates

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