DYNAMIC STABILITY OF A VISCOELASTIC PLATE WITH CONCENTRATED MASSES

B. Kh. Éshmatov and D. A. Khodzhaev UDC 539.374

The paper addresses the geometrically nonlinear problem of dynamic stability of a viscoelastic plate with

concentrated masses. The Bubnov–Galerkin method based on polynomial approximation is used to

reduce the problem to a system of nonlinear Volterra-type integro-differential equations with singular

relaxation kernels. This system is solved by numerical method based on quadrature formulas. The

critical loads are found and their dependence on the arrangement and number of concentrated masses is

studied for a wide range of mechanical and geometrical parameters of the plate. The choice of a

relaxation kernel for dynamic problems for viscoelastic thin-walled plate-like structures is justified.

Results produced by different theories are compared

Keywords: viscoelastic plate, concentrated mass, rapidly increasing compressive load, dynamic stability,

Bubnov–Galerkin method, relaxation kernel

Introduction. Composite plates and shells on which additional masses are mounted are widely used owing to high

viscoelasticity and strength. In designing structural members made of composite materials, it is important to predict their

dynamic properties depending on their configuration, mass distribution, viscoelastic properties, etc.

In most cases, additional masses are longitudinal and transverse ribs, cover plates, fasteners, units of devices and

machines [2, 4, 5]. In theoretical treatment of such problems, it is convenient to interpret attached elements as additional masses

rigidly fixed to systems and concentrated at points. There are published studies on vibrations and dynamic stability of elastic

systems with concentrated masses [3, 15, 16, 20, 22, 34] where problems were solved either in linear formulation or only

separate properties of materials of structures were taken into account. The vibrations of thin-walled structural members with

geometrical nonlinearity and initial deflections are studied in a great number of publications reviewed in [17, 25, 27, 28, 31]. The

nonlinear vibrations and dynamic stability of elastic plates and shells with concentrated masses are addressed only in [10].

Most composite materials are known to possess strongly pronounced viscoelastic properties [9, 12, 14]. The wide

industrial use of new viscoelastic materials and analysis of their dynamic behavior suggest that inhomogeneities such as attached

masses have a strong effect on their strength.

Despite the studies in this field, little attention was given to the behavior of inertially nonuniform viscoelastic systems.

These studies employed either the Voight model [26] or the Boltzmann–Volterra model, which are based on exponential

relaxation kernels that cannot describe the real processes occurring in shells and plates at the initial stage [14].

One of the features of this problem is that the Bubnov–Galerkin method reduces the problem, either linear or nonlinear,

to indecomposable systems of integro-differential equations with singular kernels, which involves additional difficulties. The

numerical method, developed in [8], based on quadrature formulas makes it possible to solve these systems. This method is

highly accurate, universal, efficient (does not need much computer time), and capable of solving a wide class of dynamic

viscoelastic problems [5, 33]. Many numerical results obtained in [5, 6, 8, 23, 24, 33] with this method are in good agreement

with theoretical and experimental data [1, 11, 13, 18, 19, 29, 30, 32].

The objective of the present paper is to analyze the dynamic stability of a viscoelastic plate with concentrated masses

under rapidly increasing loads.

International Applied Mechanics, Vol. 44, No. 2, 2008

208 1063-7095/08/4402-0208 ©2008 Springer Science+Business Media, Inc.

Tashkent Institute of Irrigation and Amelioration, Tashkent, Uzbekistan. Translated from Prikladnaya Mekhanika,

Vol. 44, No. 2, pp. 109–118, February 2008. Original article submitted February 27, 2006.

1. Mathematical Model for Dynamic Stability of a Viscoelastic Plate with Concentrated Masses. Consider a

rectangular viscoelastic plate with side lengths a and b and constant thickness h. It is made of a homogeneous isotropic material

and is subjected to dynamic loading P(t) along the side b (Fig. 1). There are concentrated masses Mp located at points (xp, yp), p =

1, 2, 3, …, I, of the plate.

According to the Boltzmann–Volterra model, the stresses �x , � y , �xy and the strains �x , � y , � xy in the mid-surface are

related as follows [12]:

�

�

� ��x x y

ER�

�

� �

1

12

( )( )*

, �

�

� ��y y x

ER�

�

� �

1

12

( )( )*

,

�

�

�xy xy

ER�

�

�

2 11

( )( )

*, (1)

where E is the elastic modulus; � is Poisson’s ratio; and R*

is an integral operator with relaxation kernel R(t):

R R t d

t

*( ) ( ) � � �� �

0

.

The strains �x , � y , � xy and the displacements u, v, w along the axes x, y, z are related, in view of the initial deflections

w w x y0 0� ( ), , as follows [11]:

�x

u

x

w

x

w

x�

�

�

�

�

�

�

�

�

�

� �

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

1

2

2

0

2

, � y

v

y

w

y

w

y�

�

�

�

�

�

�

��

�

�

���

�

�

�

��

�

�

��

�

�

�

�

�

�

�

�

1

2

2

0

2

,

� xy

u

y

v

x

w

x

w

y

w

x

w

y�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

0 0. (2)

We introduce the Dirac delta function into the equation of motion to describe the effect of the concentrated masses on

the viscoelastic plate [16]:

� �m x y h M x x y yp p p

p

I

, ( ) ( )� � � �

�

�� � � ,

1

where � is the density of the plate material.

Substituting (1) and (2) into the following equations [11]:

�

�

�

�

�

�

�

�

�

�

�

�

� � � �x xy xy y

x y x y0 0, ,

209

Fig. 1

x

M1

P(t)

z

a

yP(t)

b

M2 M3

Mp

� � � � �

�

�

�

�

�

�

�

�

��

�

�

���

�

�

D

hR w w

x

w

x

w

y yx xy xy( ) ( )

*1

40 � � �

�

�

�

�

�

�

��

�

�

��

w

x

w

yy�

� � � � �

�

�

�

�

�

�

�

�

�

�

�

�

�q

h hM x x y y

w

tp p p

p

I

� � �

10

1

2

2( ) ( ) ,

� ��

�

�

�

�

�

�

� �

� � � �

2

2

2

2

2

0 0

1

21

� � �x y xy

y x x yR L w w L w w( ) ( , ) ( , )

*

and introducing a stress function in the form

� � �x y xy

y x x y�

�

�

�

�

�

� �

�

� �

2

2

2

2

2

, , ,

we obtain the following system of von Kármán-type nonlinear integro-differential equations for the deflection w and stress

function :

D

hR w w

hM x x y yp p p

p

I

( ) ( ) ( ) ( )*

114

0

1

� � � � � � �

�

�

�

�

�

�

�

��

�� � �

�

�

� �

2

2

w

t

q

hL w( , ) ,

� �1 1

21

40 0

ÅR L w w L w w� � � � � ( ) ( , ) ( , )

*, (3)

where DEh

�

�

3

212 1( )�

is the flexural stiffness of the plate; q is the external load;

L w ww

x

w

y

w

x y( , ) �

�

�

�

�

�

�

� �

�

��

�

�

��

�

�

�

�

�

�

�

�

2

2

2

2

2

22

, L ww

x y x

w

y

w

x y x y( , )

�

�

�

�

�

�

�

�

�

�

�

�

� �

�

� �

2

2

2

2

2

2

2

2

2 2

2 .

The system of partial differential equations (3) with appropriate boundary and initial conditions is a geometrically

nonlinear model for the dynamic stability of a viscoelastic plate with concentrated masses.

To compare the solution of this problem based on the Kirchhoff–Love hypothesis with the solution based on the Berger

hypothesis, we write the equation for the transverse deflection w w x y t� ( , , )of a viscoelastic plate with concentrated masses in

the form following from the Berger hypothesis [6, 19]:

� � � � � �D

hR w w P t

w

x hM x x y yp p p

p

I

( ) ( )*

114

0

2

21

� � � �

�

�

� � � �

�

�

�

�

�� � �

�

�

�

�

�

�

�

2

2

w

t

� �� � � �

�

�

�

�

�

�

� �

�

�

�

��

�

�

���

�

�

�

q

h

D

h ab

w Rw

x

w

y

w

x

61

3

2

2 2

0*�

�

�

� �

�

�

�

��

�

�

��

�

�

�

�

�

�

�

�

!

"

#

$#

%

&

#

'#

2

0

2

00

w

ydxdy

ba

. (4)

2. Numerical Algorithm for Solving Nonlinear Dynamic Problems for Viscoelastic Systems with Concentrated

Masses. We first apply the numerical method proposed in [8] to systems of nonlinear integro-differential equations describing

one-dimensional dynamic problems for viscoelastic systems:

� �a w w X t w w t w w dkn n k k k N k N

t

�� ( ) ( )� �

( � � � �2

1 1

0

, , , , , , , ,� �

�

�

�

�

�

�

�

�

�

n

N

1

,

w w w w k Nk k k k( ) , � ( ) � , , , ,0 0 1 20 0� � � � , (5)

210

where w w tk k� ( ) are unknown functions of time; X k and k are given functions continuous within the range of their

arguments; akn and (k are constant numbers.

A system of equations of the form (5) arises in many nonlinear dynamic problems of viscoelasticity, including vibration

and dynamic-stability problems for viscoelastic structures such as rods and beams with concentrated masses.

System (5) can be written in matrix form:

� �Aw w X t w t w d

t

�� , , , , ( )� �

�

�

�

�

�

�

�( � � �

2

0

, w w w w( ) � ( ) �0 00 0� �, , (6)

where

A

a a a

a a a

a a a

N

N

N N NN

�

�

�

�

�

�

�

�

�

�

�

�

11 12 1

21 22 2

1 2

�

�

� � � �

�

, w

w

w

wN

�

�

�

�

�

�

�

�

�

�

�

�

1

2

�

, (

(

(

(

2

1

2

2

2

2

0 0

0 0

0 0

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

� � � �

�

N

,

X

X

X

X N

�

�

�

�

�

�

�

�

�

�

�

�

1

2

�

,

�

�

�

�

�

�

�

�

�

�

�

�

1

2

�

N

.

Resolving system (6) for w, we obtain the following quite simple recurrent formula for the calculation of the unknowns

at the points ti = ih, i = 0, 1, 2, … (h is the interpolation step):

� �w w t w A A t t w X t w B ti i j i j

j

i

j j j k j� �

�

�

�

� � � � ��1 0 1 01

1

0

2� , ,( � �, ,t wk k

k

j

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�0

, (7)

where A–1

is the inverse matrix to A; Aj, j = 0, 1, …, i; Bk, k = 0, 1, …, j are the interpolation points.

The two-dimensional dynamic problem for viscoelastic systems is reduced to the system of nonlinear

integro-differential equations

a w w X t w w t wklnm nm kl kl kl NM kl�� ( )� �( � �

211 11, , , , , , , ,� �� �w dNM

t

m

M

n

N

( )� �

011

��

�

�

�

�

�

�

�

��

,

w w w w k N l Mkl kl kl kl( ) , � ( ) � ,0 00 0� � � �, 1, 2, , 1, 2, ,� � .

Introducing matrices A,(2

, X, and , we again arrive at the matrix equation (6) for w = w(t) whose numerical solutions

can be found from the recurrent formula (7). The error of the numerical method is the same as the error of the quadrature formulas

and is of the same order of smallness with respect to the interpolation step [5, 8].

3. Analysis of the Dynamic Stability of a Viscoelastic Plate with Concentrated Masses. Bubnov–Galerkin

Method. Let the plate be hinged along the edges and be subjected to a compressive force Ð(t) = vt (v is the loading rate) along the

side b (Fig. 1). Then the solution of (3) that would satisfy the boundary conditions will be sought in the form of polynomial

approximation of deflections:

w x y t w tn x

a

m y

bw x y wnm

m

M

n

N

( ) ( )sin sin , ( ), , ,� �

��

��) )

11

0 0nm

m

M

n

Nn x

a

m y

bsin sin

) )

��

��

11

. (8)

Substituting (8) into the second equation of (3) and equating the coefficients of like harmonics of the trigonometric

functions on both sides of this equation, we find the following force function [11]:

211

( , , ) ( )( ) cos(

,

*x y t E R w w w w C

i

i j

N

ir js ir js irjs� � �

�

�

1

0 01� ��

��

�

�j x

a

r s y

br s

M)

cos( )

,

) )

1

�

� �

�

�

Ai j x

a

r s y

bD

i j x

a

r

irjs irjscos( )

cos( )

cos( )

cos() ) ) � s y

b

))

�

� � �

���B

i j x

a

r s y

b

P t y

irjs cos( )

cos( ) ( )) )

2

2, (9)

where

� �

� � � �� �

Cir ir js

i j r s

irjs � �

�

� � �

*

*

2

2 2 22

4

,� �

� � � �� �

Air ir js

i j r s

irjs �

�

� � �

*

*

2

2 2 22

4

,

� �

� � � �� �

Dir ir js

i j r s

irjs �

�

� � �

*

*

2

2 2 22

4

,� �

� � � �� �

Bir ir js

i j r s

irjs � �

�

� � �

*

*

2

2 2 22

4

, * �

a

b.

Substituting (8) and (9) into the first equation of (3), applying the Bubnov–Galerkin procedure, and using the properties

of the Dirac delta function [16], we obtain the following system of nonlinear integro-differential equations for the unknowns

w w tkl kl� ( ):

�

) )

) ) )b

Eh

wb

ah E

Mk x

a

n x

a

l y

bkl p

p p p4

2 2

3

2 3

4�� sin sin sin s� in ��

m y

bw

p

p

I

m

M

nm

n

N)

���

���

�

�

�

�

�

�

�

�111

� �

��

�

�

�

�

�

�

�

�

� �

�

�

�

�

�

� �

�

�

�

�

�k P

E

b

hw

klkl

*

)

� *

2 2 2

2

2

2

12 1� �� �

�

�

�

� �

2

01 R w wkl kl*

��

�

�

�

� � �

16 11

4

4

2

+

)

kl

k mirjs nm ir j

hq

kl E

b

h h

a w R w wln ( )(*

s ir js

m r s

M

n i j

N

w w�

��

�� 0 0

11

)

, ,, ,

,

k N l M� �1 1, , ,2, , 2, ,� � , (10)

where+ kl = 1 if k and l are odd and+ kl = 0 if at least one of these indices is even; the coefficients ak mirjsln are defined as in [6].

Introducing the following dimensionless variables into (10):

w

h

w

h

M

M

x

a

y

bP

P

E

b

hq

q

E

b

h

kl kl p p p, , , ,

0

0

2

; ;* *�

�

�

�

�

� ��

�

�

�

� � � � �

4

;*

*

*t

P

P

vt

P

t

S

P

Pcr cr cr

(

,

S PcEh

vb

PP

E

b

h�

�

��

�

�

��

��

�

�

�

� �

�

cr crcr* *

;

(

33

4

2 2 2

3 1

) )

� (2

)

; ( )S

R t

and keeping the notation, we obtain

1 1

411

2 2

SB w

ktw

klnmkl nm

m

M

n

N

kl��

��

�� ��

�

�

�

� ��

�

�

�

� �

* *

� � � �2

2

01

�

�

�

�

�

�

�

�

� �R w wkl kl*

212

� �

��

��

16 1

411

+

)

kl

k mirjs

m r s

M

n i j

N

n

q

P kl P

a w

cr cr* *

, ,, ,

ln � �� �m ir js ir jsR w w w w1 0 0� �*

,

w w w w k N l Mkl kl kl kl( ) � ( ) � , , , , ,0 0 1 2 1 20 0� � � �, , , ,� � , (11)

where c = E / � is the speed of sound in the plate material; Pcr = [ / ( ( ))] ( / )) �2 2 2

3 1� E h b is the Eulerian critical load;

( ) �� ( ) / ( )*2 4

Eh P b2

cr is the fundamental frequency of vibrations; M0 = ab�h is the mass of the plate; if k n, , l m, , then

B M k x n x l y m yk m p p p p p

p

I

ln �

�

�4

1

sin sin sin sin) ) ) ) ; otherwise, B M k x l yklkl p p p

p

I

� �

�

�1 42 2

1

sin sin) ) .

System (11) is integrated using the recurrent formula (7). Singular Koltunov–Rzhanitsyn relaxation kernels with three

rheological parameters, R t Ae tt

( ) �� �- + 1

, 0 1. .+ , are used [14].

As in [11], the critical time and, hence, the critical load is determined from the requirement that the deflection should not

exceed the thickness of the plate. To determine the dynamic critical load, we will use a coefficient KD equal to the ratio of the

dynamic critical load to the upper static critical load.

The results of calculations performed on a PC using the Delphi language are presented in Figs. 2–8. The following will

be used as input data, unless other data are presented: w kl04

10��

, �w kl0 0� , A = 0.05, + = 0.25, - � 0.05, q = 0, * = 1, S = 1,

M1 � 0.1.

The convergence of the Bubnov–Galerkin method was analyzed in all cases examined. The values of N and M in (8) at

which the abrupt increase in deflections begins earliest were found.

Figure 2 shows how the viscoelastic properties of the material affect the behavior of the plate with a mass concentrated

at its center (0.5; 0.5). The abscissa axis indicates a dimensionless parameter t equal to the ratio of the compressive force to the

static force, while the ordinate axis indicates the dimensionless deflection wkl . The coefficient KD is equal to 4.84 in the elastic

case (A = 0) and to 4.67 and 4.53 in the viscoelastic cases (A = 0.05; 0.1). These results show that the viscoelastic properties of the

plate decrease the critical load.

Next, we analyzed the dynamic behavior of the viscoelastic plate for different values of the rheological parameter +

(+ � 0.05, 0.075, 0.25, 0.5) (Fig. 3). The coefficient KD appears equal to 2.82, 3.90, 4.67, and 4.78, respectively. As Fig. 3

indicates, the critical load and critical time increase with +. The rheological parameter + plays the most important role compared

with A and-. For example, when + =0.05 and 0.5, the corresponding critical loads differ by more than 40%. Note also that with

S �1and + = 0.05, the critical number of half waves N appears equal to 4 in the viscoelastic case and to 2 in the elastic case.

Further analysis shows that changes in the third rheological parameter- (0 1. .- ) does not have a substantial effect on

the critical time and critical load.

Figure 4 allows more detailed analysis of the behavior of the viscoelastic plate for different relaxation kernels. It can be

seen that the viscoelastic solution based on the exponential relaxation kernel R t Aet

( ) ��-

(curve 2) is close to the elastic

solution (A = 0 curve 1), while the viscoelastic solution based on the Koltunov–Rzhanitsyn kernel R t Ae tt

( ) �� �- + 1

(A � 0 05. ,

- � 0.05, + � 0.25 curve 3 and + � 0.05 curve 4) differs from the elastic one by more than 40%. These results indicate that the

viscoelastic solutions based on exponential kernels often used by many authors are not new because they coincide with the

213

Fig. 2 Fig. 3

1

1 — + = 0.05

2 — + = 0.075

3 — + = 0.25

4 — + = 0.5

1

0 1 2 3 4 5 6 t 0 1 2 3 4 5 6 t

2

3

wkl

2

33

2

1

1

2

3

wkl1 — A = 0

2 — A = 0.05

3 — A = 0.1

4

elastic solution. Hence, to solve dynamic problems of viscoelasticity, it is necessary to use the Koltunov–Rzhanitsyn relaxation

kernel that describes the processes occurring in viscoelastic structures not only at the initial stage, but also later on.

Figure 5 shows how the concentrated mass affects the behavior of the viscoelastic plate. The mass is concentrated at the

center of the plate; the loading rate S is equal to 0.2. With M1 = 0, 0.1, 0.3, 0.5, the coefficient KD has the following values,

respectively: 6.61, 6.96, 7.20, 7.29. As the concentrated mass increases, the curve shifts to the right, toward large values of t. The

lower the loading rate S, the stronger the effect of the concentrated mass.

Figure 6 demonstrates the influence of the number of masses concentrated at one (0.5, 0.5) (curve 1), two (0.5, 0.5),

(0.3, 0.5) (curve 2), and three points (0.5, 0.5), (0.3, 0.5), (0.1, 0.5) (curve 3) on the behavior of the viscoelastic plate. When

M M M1 2 3� � � 0.1, we have KD = 4.67, 5.01, 5.12. Hence, the critical time and critical load increase with the number of

concentrated masses.

Figure 7 shows the time dependence of the deflection for S = 0.1, 1, 10 (the respective values of KD are 8.18, 4.67, 2.85).

Recall that the parameter S is inversely proportional to v2

. Predictably, the critical load and critical time increase with the loading

rate v in the viscoelastic case, as in the elastic case [11]. However, the abrupt increase in deflections in the viscoelastic case

occurs earlier than in the elastic case.

The effect of a geometrical parameter * equal to the ratio of the side lengths of the viscoelastic rectangular plate was

examined. When *= 1, 2, 3, the coefficient KD is equal to 4.67, which indicates that the values of KD found for a square plate can

be applied to rectangular plates of other configurations in the viscoelastic case.

We also compared the results produced by the linear and nonlinear theories (Berger and Kirchhoff–Love). It was

revealed that these results are mainly dependent on three parameters (initial deflection, loading rate, and additional static load)

and almost coincide within the range of initial deflection (10 104

02� �

/ /w kl ) and within the ordinary ranges of S and q. The

difference increases with the initial deflection (w kl01

100�

), especially when increased together with the loading rate and

external load.

Figure 8 shows the solutions obtained using the linear theory (curve 1), Kirchhoff–Love theory (curve 2), and Berger

theory (curve 3) for S � 0.1, w0 = 10–1

, q = 0. The respective values of KD are 3.51, 3.77, and 4.42. As can be seen, the linear

theory differs from the Kirchhoff–Love and Berger theories by 7% and 21%, respectively. Note that the curve abruptly increases

214

Fig. 4 Fig. 5

1

1 — M1 = 0

2 — M1 = 0.1

3 — M1 = 0.3

4 — M1 = 0.5

1

0 1 2 3 4 5 6 t 0 1 2 3 4 5 6 7 8 9 t

2

3

wkl

2

3

32

1

1

2

3

wkl

4

4

Fig. 6 Fig. 7

1

1 — S = 0.1

2 — S = 1

3 — S = 10

1

0 1 2 3 4 5 6 7 t 0 1 2 3 4 5 6 7 8 9 t

2

3

wkl2 3

3

2

11

2

3

wkl

all the time in the linear case and increases spasmodically in the nonlinear cases, thus demonstrating the nonlinear effect.

Equations (4) were used to obtain the results within the Berger theory.

Conclusions. An analysis of the nonlinear dynamic of viscoelastic plates with concentrated masses lead to the

following conclusions:

– the viscoelastic properties of the material decrease the critical load and critical time;

– the solution based on an exponential kernel coincide with the elastic solution, which means that such kernels cannot be

used as relaxation kernels describing the real viscoelastic properties of the plate material;

– singular Koltunov–Rzhanitsyn kernels should be used as relaxation kernels because they include a sufficient number

of rheological parameters to describe the real processes occurring in structures and agree well with experiments;

– concentrated masses increase the critical load in both elastic and viscoelastic cases;

– the choice of the linear, Berger, or Kirchhoff–Love theory should be based on the geometrical and physical

parameters of plates.

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