a general solution to vibrations of beams on variable winkler elastic foundation

Com,xtcrs & Srrucrures Vol. 47, No. I, pp. 83-90. 1993 Printed in Great Britain.

0045-7949193 S6.00 + 0.00 0 I993 Pergamon Press Ltd

A GENERAL SOLUTION TO VIBRATIONS OF BEAMS ON VARIABLE WINKLER ELASTIC FOUNDATION

ZHOU DING

Department of Mechanical Engineering, East China Institute of Technology, Nanjing, Jiangsu 210014, People’s Republic of China

(Received 2 January 1992)

Abstract-A general solution to vibrations of beams on variable Winkler elastic foundation is presented. The exact solution of the dynamic response of the beam is obtained by considering the reaction force of the foundation on the beam as the external force acting on the beam, which is an integral equation including the displacement of the beam. The four unknown constants in the solution are decided by the boundary conditions of the beam. The integrals in the solution are approximately and numerically calculated by means of the trapezoidal rule. By letting the right-hand side of the solution equal its left-hand side only at those discrete nodes of the quadrature, the frequency equation is obtained which is described by a determinant whose order is equal to the number of the discrete nodes. The mode shape functions are represented by a series of unified analytical functions. The analysis and programing are very simple. It is possible to find the natural frequencies and mode shapes of vibrations by using a small number of the discrete nodes in the trapezoidal quadrature and it is concluded that the use of the method yields better convergence at lower computation costs. Finally, several examples are given for simply supported beams on variable Winkler elastic foundation.

1. INTRODUCTION

Beams and columns supported along their length are very common in structural configurations. This problem has been treated by numerous authors, and the well-known text by Hetenyi [l] provides a thorough treatment of the Winkler model for elastic foun-

dation. Many other models have heen used [2], but the Winkler model is often adopted. Most of the work on this subject is for the problem of a uniform beam resting on a constant Winkler foundation.

In recent years, the problem of static analysis of beams on variable Winkler foundations was solved. Franklin and Scott [3] presented a closed-form solution for a linear variation of the foundation modulus, using contour integrals. For higher order variation in x (the coordinate along the beam), they presented a partial solution, which is applicable to infinite beams (or piles). Lentini [4] presented a finite difference method to solve the problem when the foundation stiffness varies along x as a power of x. Clastornik er al. [5] presented a solution for finite beams resting on Winkler elastic foundations with stiffness variation that can be represented as a general polynomial of x.

Recently, the problems of vibrations and stability of beams on constant and variable Winkler foundations were respectively solved by Eisenberger and Clastomik [6-81 using the same general approach. The solutions are based on the finite element with exact stiffness, consistent mass and geometric stiffness matrices. In this paper, a general solution to vibrations of beams on a variable Winkler elastic foundation is presented. The exact solution rep-

resented by an integral equation is obtained by regarding the reaction force of the foundation on the beam as the external force acting on the beam. A discrete method is given and the frequency equation is obtained which is described by a determinant whose order is equal to the number of the discrete nodes. The proposed method is expected to give better convergence from coarser discrete nodes, thus reducing data preparation effort and computation costs.

2. THE DIFFERENTIAL EQUATION OF MOTION AND ITS SOLUTION

A beam resting on a variable Winkler elastic foundation is shown in Fig. 1. According to the theory of structure vibrations [9], the differential equation for the displacement of a beam with constant cross-sectional area and flexural rigidity supported on variable Winkler elastic foundation is

EIg+pA$+k(x)y =o, OQX 91, (1)

where ~(x, t) is the displacement of the beam, t is the time, EI is the flexural rigidity, p is the mass density per unit length of the heam, A is the cross- sectional area of the beam, k(x) is the elastic coefficient of Winkler foundation, I is the length of the beam.

When the beam resting on a Winkler elastic foundation vibrates freely, each particle describes a simple harmonic motion of radian frequency o, and if a non-dimensional length coordinate { is defined by

83

84 ZHOU DING

x = l<, the solution of eqn (1) may be taken to be of the form

Y(5, f) = Yw@‘, (2)

where j = ,/ - 1. Then the solution of eqn (1) reduces to that of

$-k4Y= -K(r)Y, (3)

where K(t) describes the elastic coefficient of a Winkler foundation in dimensionless form and A4 = pA14w2/EI is the dimensionless frequency par- ameter.

BY the theory of ordinary differential equations [lo], the general solution of eqn (3) is found to be of integral equation form

Y(r) = C, sin X + C2 cos X + C, sinh nt

+ C, cash I.( + F(r)

F(t)=& I

‘K(s)Y(s)[sini(5. -3) 0

- sinh d(< - s)] ds, (4)

where F(l) is the particular solution of eqn (3), others are the homogeneous solution of eqn (3) and C, (i = 1,2,3,4) are the unknown constants.

3. THE DECISION OF UNKNOWN CONSTANTS

The unknown four constants in eqn (4) may be determined by the boundary conditions of the two ends of the beam. Next, the analytical representations of Ci (i = 1,2, 3,4) are given under several most common boundary conditions respectively.

3.1. Simply-simply supported beam

The boundary conditions are

Y(0) = 0, d2Y(5> Tip-

= 0, C=O

Y(1) = 0, d2Y(5) d52

= 0. 5=1

(5)

Substituting eqn (4) into the above equations gives

c, = c4 = 0,

C, = -$, ’ K(s)Y(s) s sin I(1 -s) ds,

0 sin 1

c& ‘K(s)Y(s) s

sinh A(1 -s) ds (6)

0 sinhl ’

Fig. I. A beam on a Winkler elastic foundation.

3.2. Simply supported-clamped beam


Y(0) = 0, d2Y(t) o

ygr =v <=O

dY(5) Y(l)=O, -

d5 = 0.

<=I (7)


where

x sinhIs +coshI sin1(1 -s)

- [sinh I cos I. (1 - s)] ds,

c=_” ’ 3

s A21’ ,, K(s) Y(s)

x [sin 1s + cos I sinh A(1 - s)

-sin1 coshL(1 -s)]ds, (8)

A = sin 1 cash 2 - cos I sinh 1. (9)

3.3. Simply supported-free beam


Y(0) = 0, d2Y(5) 7

= 0, 5=0

d2W> d3Y(5) o

dt2 c_,=“’ 7 = . C=!

(10)


c, = c, = 0,

c, = -;A ’ K(s)Y(s) s 0

x [sinhds +sinh1 cos1(1 -s)

-cash 1 sin I(1 -s)] ds,

(73 = -A$

s ’ K(s)Y(s)

0

x [sin Is + sin I cash A(1 - s)

-cos 1 sinh A(1 - s)] ds, (11)

Vibration of beams on elastic foundation 85

where

A = cos 1 sinh 1 - sin 1 cash L.

3.4. Clamped-clamped beam


(12)

dY(5) Y(0) = 0, - dl t=O=o’

dY(O Y(l)=O, -

dl +,=O’ (13)


c,= -c,, c,= -c,,

c, = -&$ s 1 K(s)Y(s) 0

x [-cos ti -cash Lr - sinh L sin A(1 -s)

+sin1 sinh1(1 -s)+cosLcosh1(1 -s)

+cosh 1 cos( 1 - s)] ds,

c*= -$$ s 'K(s)Y(s) 0

x[sinh+sinhIs-sinhLcosL(l-s)

-sin1coshL(l-s)+cosIsinh1(1-s)

+ cash 1 sin 1(1 - s)] ds, (14)

where

A = f(cos I cash 1 - 1).

3.5. Clamped-free beam


(15)

Wt) Y(0) =o, -

d5 c=o=o’

d2Y(0 7

=o d3Y(0 3 =o.

C=I ’ dt {=I (16)

Substituting eqn (4) into the above equations gives 4. THE NUMERICAL CALCULATION OF INTEGRALS

c,= -c,, c,= -c,, The differential equation (3) in its present form with general variable coefficients of a Winkler elastic foundation is complicated and is difficult to solve analytically. However, from the point of view of applications, such a closed-form solution to eqn (3) is not always necessary. In any given case, and with account taken of the design engineers’ requirements, it may be appropriate to construct approximate solutions and determine the natural frequencies by successive approximations.

c, = -is s ‘K(s)Y(s) 0

x (-cosJ..s -coshLr -sin1 sinh1(1 -s)

+sinh1 sin1(1 -s)-cos1cosh1(1 -s)

-cosh1 cos1(1 -s)]ds,

s 'K(s)Y(s) 0

x [sin Is + sinh Is + sin 1 cash ,l(l - s)

+sinh 1 cos A(1 -s) - cos 1 sinh A(1 -s)

-cash 1 sin A(1 -s)] ds, (17)

where

A= -2(cosLcosh1 + 1).

3.6. Free-free beam


(18)

d*Y(t> Yip-

=o d3YK) 3

<co ’ d5 = 0,

5=0

d2Y(t-) 7

=. d’Y(t) 3 =o. (19)

t=, ’ dt <=I

Substituting eqn (4) into above equations gives

c3=c,, c,=c*,

c, = -;; i k(s)Y(s) 0

x [-cosIs +coshLs -sin1 sinh1(1 -s)

-sinhA sin1(1 -s)-cos1 cosh1(1 -s)

fcosh1 cos1(1 -s)Jds,

c2 = -&$ I 'K(s)Y(s) 0

x [sin Is - sinh t + sin 1 cash A(1 - s)

-sinhA cosL(1 -s)-cos1 sinh1(1 -s)

+cosh1 sin1(1 -s)]ds, (20)

where

A = 2(cos L cash 1 - 1). (21)

Similarly, the analytical representations of C1 (i = 1,2,3,4) may be obtained for the elastically supported boundaries, and are not listed here.

86 Znou DING

The integrals in eqn (4) may be approximately 4.4. Clamped-clamped beam and numerically calculated by the trapezoidal rule under different boundary conditions, respectively. The quadrature nodes are taken to be

C, = -i& ,$ G(C)Y(L) t-0

{0,5,,52,..., (,, _ , , 1} from the interval [O, 1] and the x [ - cos A& - cash I{, - sinh I sin A(1 - &) length of every discrete interval is equal to l/n. By the use of the n-point quadrature, eqn (4) gives approxi- + sin 1 sinh I( 1 - &) + cos I cash A( 1 - ci)

mately + cash I cos A(1 - &)I

F(5) = &j ,$ aiK(ti)Y(5t)

r-l C2 = -kjj$ $ aiK(5i)y(5i)

I-0

[sin1(5 - 5,) - sinh A(5 - ti)lfJ(t - 5i), (22) x [sin Ati + sinh & - sinh 1 cos d( 1 - &)

where -sinI cosh1(1 -&)+cos1 sinhJ.(l -5,)

a0 = 01” = 1 + cash 1 sin I(1 - &)I. (26) 2, cri=l,i=l,2 ,...) n-l,

tOzo3 5n= lt u(t -ti)= O 5<5i 4.5. Clamped-free beam * ( ,c'

In what follows the constants Ci (i = 1,2,3,4) Cl = -tj$j $ cciK(5i)Y(li)

I-0

under different boundary conditions are, respectively, x [ -cos A& - cash ,I{, - sin 1 sinh A( 1 - &) written as approximate representations.

+sinh I sin ,I(1 - 5,) - cos 1 cash d(1 - &) 4.1. Simply-simply supported beam

-cash I cos I(1 - &)I

c, = - 2niJsinA,$ 4K(ti)Y(S,)sin1(1 - 4,)

t-0

c, = - 2n13iinhI ,$ G(ti)Y(SibinhW - (0.

I 0

c2 = -t&,$ atK(ti)Y(5i)

IO

x [sin d& + sinh I<, + sin I cash A(1 - ri)


x [sinh Ali + cash i sin I( 1 - &)

-sinh 1 cos I.(1 - &)]

‘3 = -t& ,$ aiK(5i>y(5i) I 0

(23) +sinhIcosl(l-~i)-coslsinhl(l-~i)

-cash 1 sin I(1 - t,)]. (27)

Free-free beam

C,= -- i &,$ aiK(5i)y(5i)

I-0

x [ -cos A& + cash Iti - sin Iz sinh I

x (1 - 5,) - sinh 1 sin 1(1 - 5,)

-cos 1 cash I(1 - &) + cash 1 cos A(1 - &)I

x [sin Xji + cos 1 sinh I(1 - &)

- sin 1 cash A(1 - &)I.


G = -g ,i %K(5i)Y(5i) t-0

x [sinh I.& + sinh I cos I(1 - 5,)

- cash I sin A(1 - &)J

‘3 = -t& ,$ aiK(5i)Y(Ci) I 0

x [sin Iti + sin I cash A( 1 - &)

- cos I sinh I(1 - &)I.

(24) ‘2 = -k j-&j ,$ aiK(ti)Y(5i)

I-0

x [sin At, - sinh L& + sin 1 cash I( 1 - &)

- sinh 1 cos a( 1 - &) - cos 1 sinh A( 1 - &)

+cosh 1 sin A(1 - &)I. (28)

In the above equations, A under different boundary conditions was already derived in the last section.

5. THE REPRESENTATIONS OF DISPLACEMENT FUNCTIONS

By substituting the results from the last section into eqn (4), the displacement function Y(r) under differ-

(25) ent boundary conditions can be obtained.



Y(r) = -&

sinle n x

{ xiFO 4K(ti)Y(tiM A(1 - &>

- %$ $0 giK(ti)Y(tiNnh A(1 - ti)

- ,$O aiK(6)YCti)Lsin l(t - Ti)

- sinh I(( - &)]U(t - t) 1

. (2%


* Y(r)= -Tjj$ . * ,; 1 s ,To aiK(&)Y(&)[sinh I&

+cosh 1 sin ,I( 1 - &) - sinh 1 cos A( 1 - &)I

+F i cr,K(&)Y(&)[sin I& i=O

+cos 1 sinh A(1 - &) -sin L cash A(1 - &)I

- i 4K(ti)Y(ti)[Sin A(( - C) i=O

(30) -sinh n(c - &)]U(< - 5,) 1

.


Y(C)= --A {

y ,i cr,K(&)Y(&)[sinh At, t==O

+sinh1cos1(1 -&)-cosh1 sinIZ(1 -&)I

+yiio aJ(&)Y(&)[sin J.&

+sin1 coshL(1 -&)-cosLsinhJ(1 -&)I

- f: %K(ti)Y(t)W Act - C) i=O

-sinh I.(( - &)]U(c - &) >

.

5.4. Clamped-clamped beam

Y(r)= -&

(31)

sin 25 - sinh kt n X

{ A iFo aiK(C)Y(ti)

x[-cosl&-coshl&-sinh1sin1(1 -&)

+sin 1 sinh A(1 - 5,) + cos 1 cash A(1 - &)

+cosh 1 cos ,I(1 - &)I + cos It - cash It

A

I

x 1 aiK(&) Y(ti)[sin Ati + sinh Ati i=O

-sinh 1 cos d(l - 5,) - sin 1 cash A(1 - &)

+cos 1 sinh A(1 - &) + cash 1 sin 1(1 - &)I

-i$o aiK(C) Y(ti)[sin A(5 - <i)

-sinh I(< - r,)]U(l - &) I

. (32)


Y(f)= -A sin Jr - sinh I< ”

X A iFo GWY(5,)

x [ -cos Ati - cash A& - sin 1 sinh d(1 - &)

+sinh I sin I(1 - &) - cos 1 cash 1(1 - &)

-cash 1 cos I(1 - &)I + cos At - cash Jy

A

x i aiK(&)Y(&)[sin Agi + sinh JL$ is0

+sin 1 cash A(1 - &) + sinh 1 cos I(1 - &)

-cos 1 sinh 1(1 - &) -cash A sin A(1 - &)J

- i$o aiK(C)Y(&)bin J(t - 5,)

- sinh J(l - &)]U(t - &) >

. (33)

5.6. Free-free beam

y(r)= -& sin 2r + sinh I[ ”

A iFo aiK(&)Y(L)

x [ -cos Al, + cash & - sin It sinh A( 1 - ti)

- sinh 1 sin ,I( 1 - &) - cos 1 cash A( 1 - &)

+cosh 1 cos 1(1 - &)I + cos nt + cash X

A

x i aiK(&)Y(&)[sin At, - sinh Xi i=O

+sinLcoshL(l-&)-sinh1cos1(1-ri)

-cos 1 sinh A(1 - &) + cash 1 sin A(1 - &)I

- i aiK(C)Y(L)[sin J(r - C) i-0

-sinh I(r - &)]U(r - &) . (34)

88 ZHOU DING

6. THE FREQUENCY EQUATION 6.4. Clamped-clamped beam

Letting the right-hand side of eqn (4) equal its left-hand side only at those discrete nodes, the homo- B, = ocjK(tj)

{

sin Ati - sinh Ati

geneous linear algebra equation whose order is n + 1 A

about Y(&) (i = 0, 1,2,. . . , n) is x[-cosl{,-coshl<,-sinhIsink(l-ti)

2n13+B,, B,, ... B,,

(35)

where the coefficients of eqn (35) are related to the boundary conditions of the beam.


Bg = ajK(rj) i

sin Ati sin sin A(1 - tj>

- s sinh A(1 - tj) - [sin d(& - r,)

- sinh a(ti - tj)lu(t’i - 6) 1

i,j = 0, 1,2, . . . , n. (36)


Bij = ajK(tj) {

y [sinh 16 + cash I sin A( 1 - 6)

- sinh 1 cos 1(1 - rj)] + E!hAS [sin ltj

+ cos I sinh I( 1 - t;,) - sin I cash A(1 - s)]

- [sin I(5, - Cj) - sinh d(& - 5j)]U(<i - S) 1

.

(37)


Bii = uj K(tj) {

y [sinh lej + sinh 1 cos I (1 - 4)

-cash 1 sin I(1 - tj)] + E!+% [sin Jjj

+sin Iz cash A(1 - t;i) - cos 1 sinh I(1 - tj)]

-[sin I(ri- C) - sinh A(& - t,)]U({, - C) >

.

(38)

+ sin 1 sinh A(1 - tj) + cos 1 cash L(l - {,)

+ cash 1 cos I( 1 - tj)] + cos A(, - cash Lyi

A

x [sin II& + sinh A[, - sinh 1 cos A( 1 - S)

- sin 1 cash I( 1 - t;,) + cos I sinh I (1 - <,)

fcoshi sin1(1 -<,)I-[sin1(t,-5,)

- sinh ,I(<, - ~,)]U(I$ - tj) 1

. (39)


Bu = qK(5j) 1

sin I” dSinh Iti [ -_cos A<, _ cash itj

-sin1sinh1(1-<,)+sinh1sini(l-&)

-cosIcoshI(l-{,)-coshIcosL(l-c,)]

+ cos ,I< - cash A[,

A [sin 16, + sinh At;,

+ sin 1 cash A( 1 - tj) + sinh 1 cos A( 1 - ci)

-cosIsinhd(l-<,)-coshIsink(l-<,)I

-[sin I(& - C) - sinh I(& - c,)]V(& - tj) 1

.

(40)

6.6. Free-free beam

Bg= ajK(lj) I

sin ,I[, + sinh @Ti

A [ - cos A$ + cash It/

-sin 1 sinh ,I(1 - {,) - sinh I. sin A(1 - tj)

-cosIcoshd(l -<,)+coshIcosL(l -<,)I

+cos I& + cash &

A [sin ltj - sinh ,I{,

+ sin L cash A( 1 - tj) - sinh 1 cos I (1 - r,)

- cos Iz sinh A(1 - <,)+cosh 1 sin A(1 - S)]

- [sin I(l, - rj)-sinh A(<, - t,)]U(& - C)

(41)


The frequency equation is given by setting the examples. The first eight natural frequencies of determinant of the coefficient matrix of eqn (35) equal simply supported beams resting on Winkler elastic to zero as follows: foundations with constant, linear and parabolic

2n1’ + B, 4, 42 ... 41,

40 2nl’+ B,, B,* ‘. . B,,

B20 B21 2nl’+B,, ... B2,,

BMI B,, B,,2 . . . 2nl’ + B,,

= 0. (42)

From the above equation, the natural frequencies can be calculated numerically by means of a computer. Substituting the natural frequencies into eqn (35), the relative values of li (i = 0, 1,2,. . . , n) can be obtained, thus giving the mode shapes.

7. NUMERICAL EXAMPLES

In order to illustrate the application of the proposed method, the simply supported beams on variable Winkler elastic foundations are taken as

moduli are listed in Table 1, Table 2 and Table 3, respectively.

8. CONCLUSIONS

A general method has been given for the vibration analysis of beams on variable elastic foundations. The numerical examples show that good results can be achieved with a small number of the discrete nodes in quadrature, thus reducing the compu- tational cost.

Table 1. Frequency parameters li (i = 1,2, . . ,8) of a simply supported beam on Winkler elastic foundation with constant modulus, K(T) = &,, n = 10

10 3.220 (3.219)

50 3.485 (3.484)

100 3.748 (3.748)

150 3.966 (3.966)

200 4.152 (4.153)

250 4.318 (4.317)

300 4.465 (4.465)

350 4.599 (4.599)

400 4.723 (4.723)

450 4.837 (4.837)

500 4.945 (4.944)

600 5.138 (5.139)

700 5.713 (5.314)

800 5.473 (5.473)

900 5.620 (5.620)

1000 5.755 (5.756)

1500 6.321 (6.322)

2000 6.766 [6.760] (6.767)

6.293 (6.293) 6.334

(6.333) 6.382

(6.382) 6.429

(6.429) 6.476

(6.476) 6.521

(6.521) 6.566

(6.566) 6.610

(6.610) 6.652

(6.652) 6.695

(6.695) 6.735

(6.736) 6.816

(6.816) 6.893 6.894) 6.970

(6.969) 7.041

(7.042) 7.112

(7.112) 7.437

(7.437) 7.723

[7.716] (7.724)

9.421 (9.428) 9.440

(9.440) 9.454

(9.454) 9.470

(9.469) 9.484

(9.484) 9.499

(9.499) 9.513

(9.513) 9.527

(9.528) 9.541

(9.542) 9.557

(9.556) 9.571

(9.571) 9.599

(9.599) 9.627

(9.627) 9.655

(9.655) 9.682

(9.693) 9.710

(9.710) 9.845

$8;)

[9:968] (9.972)

12.568 (12.568)

12.573 (12.573)

12.579 (12.579)

12.585 (12.585) 12.591

(12.591) 12.598

(12.598) 12.604

(12.604) 12.610

(12.610) 12.616

(12.616) 12.623

(12.623) 12.629

(12.629) 12.641

(12.641) 12.654

(12.654) 12.666

(12.666) 12.679

(12.678) 12.690

(12.691) 12.751

(12.751) 12.812

[12.806] (12.811)

15.709 (15.709)

15.712 (15.711)

15.715 (15.714)

15.718 (15.718)

15.721 (15.721) 15.724

(15.724) 15.727

(15.727) 15.731

(15.730) 15.734

(15.734) 15.737

(15.737) 15.740

(15.740) 15.746

(15.747) 15.752

(15.753) 15.759

(15.759) 15.766

(15.766) 15.773

(15.772) 15.804

(15.804) 15.835

[15.832] (15.835)

18.849 (18.850)

18.851 (18.851)

18.854 (18.853)

18.856 (18.855) 18.857

(18.857) 18.859

(18.859) 18.860

(18.861) 18.863

(18.863) 18.865

(18.864) 18.866

(18.866) 18.868

(18.868) 18.873

(18.872) 18.875

(18.876) 18.879

(18.879) 18.884

(18.883) 18.887

(18.887) 18.906

(18.905) 18.924

[18.926] (18.924)

21.991 (21.991) 21.993

(21.992) 21.993

(21.993) 21.995

(21.995) 21.996

(21.996) 21.998

(21.997) 21.998

(21.998) 21.999

(21.999) 22.001

(22.000) 22.002

(22.002) 22.003

(22.003) 22.005

(22.005) 22.007

(22.008) 22.010

(22.010) 22.013

(22.012) 22.015

(22.015) 22.024

(22.026) 22.038

[22.038] (22.038)

25.134 (25.133)t 25.134

(25.134) 25.134

(25.134) 25.135

(25.135) 25.135

(25.136) 25.137

(25.137) 25.137

(25.137) 25.138

(25.138) 25.138

(25.139) 25.140

(25.140) 25.140

(25.141) 25.141

(25.142) 25.145

(25.144) 25.146

(25.145) 25.148

(25.147) 25.149

(25.149) 25.157

(25.156) 25.168

[25.165]$ (25.164)

7 Exact results. $n =5.

90 ZHOU DING

Table 2. Frequency parameters li (i = 1,2, . . . , , 8) of a simply supported beam on Winkler elastic foundation with linear modulus, K(r) = &( 1 - at), n = 10

500 0.2 4.837 6.695 9.577 12.623 15.737 18.866 22.002 25.140 0.4 4.721 6.652 9.541 12.616 15.734 18.865 22.001 25.138 0.6 4.595 6.610 9.527 12.610 15.731 18.863 21.999 25.138 0.8 4.456 6.568 9.513 12.604 15.727 18.860 21.998 25.137

0.2 5.618 7.041 9.682 12.679 15.766 18.884 22.013 25.148 0.4 5.468 6.971 9.656 12.666 15.759 18.879 22.101 25.146 0.6 5.301 6.898 9.627 12.654 15.752 18.876 22.007 25.145 0.8 5.113 6.824 9.599 12.641 15.746 18.873 22.006 25.143

1500

2000

0.2 6.165 7.345 9.864 12.733 15.794 18.899 22.024 25.155 0.4 5.991 7.251 9.765 12.716 15.784 18.894 22.019 25.151 0.6 5.798 7.155 9.724 12.696 15.774 18.888 22.015 25.148 0.8 5.574 7.057 9.684 12.679 15.766 18.884 22.013 25.148

0.2 6.597 7.614 9.921 12.788 15.823 18.917 22.034 25.161 0.4 6.405 7.503 9.870 12.763 15.810 18.909 22.028 25.161 0.6 6.187 7.390 9.819 12.739 15.798 18.901 22.024 25.155 0.8 5.937 7.272 9.767 12.716 15.784 18.894 22.019 25.151

Table 3. Frequency parameters 1, (i = 1,2,. . ,8) of a simply supported beam on Winkler elastic foundation with parabolic modulus, K(t) = &(l - fit*), n = 10

&I B 1, A, 1, & 1, & A, As

500 0.2 4.884 6.710 9.562 12.624 15.737 18.866 22.003 25.140 0.4 4.821 6.683 9.552 12.620 15.735 18.866 22.00 1 25.140 0.6 4.753 6.657 9.542 12.616 15.733 18.864 22.001 25.140 0.8 4.682 6.630 9.534 12.612 15.731 18.862 21.999 25.138

1000 0.2 5.679 7.067 9.692 12.683 15.769 18.884 22.013 25.148 0.4 5.597 7.022 9.675 12.675 15.763 18.882 22.011 25.146 0.6 5.507 6.980 9.657 12.667 15.759 18.880 22.011 25.146 0.8 5.409 6.935 9.638 12.659 15.753 18.878 22.009 25.144

1500 0.2 6.233 7.378 9.817 12.739 15.798 18.901 22.024 25.155 0.4 6.138 7.321 9.792 12.728 15.792 18.898 22.023 25.153 0.6 6.032 7.263 9.767 12.716 15.784 18.894 22.019 25.151 0.8 5.917 7.206 9.741 12.704 15.778 18.892 22.017 25.151

2000 0.2 6.671 7.653 9.939 12.796 15.827 18.919 22.034 25.161 0.4 6.564 7.587 9.905 12.780 15.819 18.913 22.034 25.161 0.6 6.444 7.521 9.874 12.765 15.810 18.909 22.028 25.161 0.8 6.312 7.454 9.841 12.749 15.802 18.903 22.024 25.155

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REFERENCES

M. Hetenyi, Beams on Elastic Foundations. University of Michigan Press, Ann Arbor, MI (1946). A. D. Kerr, Elastic and viscoelastic foundation models. J. appf. Mech., ASME 31, 491-498 (1964). J. N. Franklin and R. F. Scott. Beams equation with variable foundation coefficient. J. Engng Mech. Div., ASCE 105, 81 l-827 (1979).

M. Eisenberger, D. Z. Yankelevsky and M. A. Adin. Vibration of beams fully or partially supported on elastic foundations. Earthquake Engng Struct. Dynamics 13, 651-660 (1985).

M. Lentini, Numerical solution of the beam equation with nonuniform foundation coefficient. J. appl. Mech., ASME 46, 90-904 (1979).

9.

J. Clastomik, M. Eisenberger, D. Z. Yankelevsky and M. A. Adin. Beams on variable Winkler elastic foun- 10. dation. J. appl. Mech., ASME 53, 925-928 (1986).

M. Eisenberger, D. Z. Yankekevsky and .I. Clastornik. Stability of beams on elastic foundations. Compuf. Struct. 24, 135-140 (1986). M. Eisenberger and J. Clastornik. Vibrations and buck- ling of a beam on variable Winkler elastic foundation. J. Sound Vibr. 115, 233-241 (1987). S. Timoshenko, D. H. Young and W. Weaver Jr., Vibration Problems in Engineering. John Wiley, New York (1974). K. Miller, Ordinary Differential Equations. Academic Press, New York (1982).

a general solution to vibrations of beams on variable winkler elastic foundation

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