Vibration Analysis of Non-Uniform Beams Resting on Elastic

Vibration Analysis of Non-Uniform Beams Resting

on Elastic Foundations Using the Spline

Collocation Method

Ming-Hung Hsu

Department of Electrical Engineering, National Penghu University,

Penghu, Taiwan 880, R.O.C.

Abstract

The natural frequencies of non-uniform beams resting on elastic foundations are numerically

obtained using the spline collocation procedure. The spline collocation method is a numerical

approach effective at solving partial differential equations. The boundary conditions that accompanied

the spline collocation procedure were used to convert the partial differential equations of non-uniform

beam vibration problems into a discrete eigenvalue problem. The beam model considers the taper

ratios �, �, the boundary conditions, and the elastic foundation stiffness, kf, all of which impact the

dynamic behavior of non-uniform beams resting on elastic foundations. This work developed the

continuum mechanics and combined with the spline collocation method to simulate the dynamic

properties of non-uniform beams resting on elastic foundations.

Key Words: Elastic Foundation, Vibration Analysis, Non-Uniform Beam, Spline Collocation Method,

Taper Ratio

1. Introduction

Non-uniform beams resting on elastic foundations

are important structural elements. The dynamic charac-

teristics of such non-uniform beams are of considerable

importance in many designs. Abrate et al. [1�3] solved

vibrations problems in non-uniform rods and beams us-

ing the Rayleigh-Ritz scheme. Hodges et al. [4] com-

puted the fundamental frequencies and the correspond-

ing modal shapes using a discrete transfer matrix scheme.

Lee and Kuo [5] solved the problem of bending vibra-

tions in non-uniform beams with an elastically restrained

root. Tsai et al. [6�9] studied the static behaviors of

beams resting on a tensionless elastic foundation. Akbarov

et al. [10�19] who conducted a static analysis of thick,

circular and rectangular plates resting on a tensionless

elastic foundation, generated positive solutions for a

fourth-order differential equation with nonlinear bound-

ary conditions for modeling beams on elastic foundations.

Their results were dependent upon foundation parameters.

Sharma and DasGupta [20] examined the bending prob-

lem of axially constrained beams on nonlinear Winkler-

type elastic foundations using Green’s functions. Beau-

fait and Hoadley [21] solved the problem of elastic beams

on a linear foundation using the midpoint difference tech-

nique. Kuo and Lee [22] investigated the deflection of

non-uniform beams resting on a nonlinear elastic founda-

tion using the perturbation method. Chen [23] generated

the numerical solutions for beams resting on elastic foun-

dations using the differential quadrature element ap-

proach. However, the spline collocation method has not

been used to solve the problem of non-uniform beams

resting on elastic foundations. In this study, the spline

collocation method is applied to formulate discrete ei-

genvalue problems of different non-uniform beams resting

on elastic foundations. The spline collocation approach

is easily implemented and should prove interesting for

designers. Simulation results are compared with numeri-

cal results acquired using the finite element method.

Tamkang Journal of Science and Engineering, Vol. 12, No. 2, pp. 113�122 (2009) 113

*Corresponding author. E-mail: [email protected]

2. Formation

The kinetic energy of a beam with a non-uniform

cross-section resting on an elastic foundation is as follows:

(1)

where L is the length of the non-uniform beam, u is the

transverse bending deflection, A is the cross-sectional

area of the beam, A = A0�(x), and � is the density of

beam material. Function �(x) is dependent on the shape

of the cross-section. Parameter Ao is the area of the

cross section at the end, where x = 0. The strain energy

of a non-uniform beam resting on an elastic foundation

can be derived as follows:

(2)

where I is the second moment of area of the non-uni-

form beam, I = I0�(x), kf is the elastic foundation con-

stant, and E is Young’s modulus of non-uniform beam

material. The function �(x) is dependent upon the cross-

section shape. Parameter Io is the second moment of

area at the end, where x = 0. Hamilton’s principle is

given by

(3)

where �W is virtual work. Substituting Eqs. (1) and (2)

into Eq. (3) yields the equations of motion. The trans-

verse motion, u, of the non-uniform beam resting on an

elastic foundation is governed by

(4)

Consider the clamped-free beam to be clamped at the

end, where x = 0. The corresponding boundary condi-

tions are as follows:

u (0, t) = 0 (5)

(6)

(7)

(8)

Consider the following pinned-pinned non-uniform beam

resting on an elastic foundation. The corresponding

boundary conditions are

u (0, t) = 0 (9)

(10)

u (L, t) = 0 (11)

(12)

Consider the following clamped-clamped non-uniform

beam resting on an elastic foundation. The correspond-

ing boundary conditions are

u (0, t) = 0 (13)

(14)

u (L, t) = 0 (15)

(16)

Let the displacement response be

(17)

where � is the natural frequency of a non-uniform beam

resting on an elastic foundation. Substituting Eq. (17)

into Eq. (4) yields

(18)

114 Ming-Hung Hsu

Equation (18) can be rewritten as

(19)

The corresponding boundary conditions of the clamped-

free non-uniform beam resting on an elastic foundation

are

(20)

(21)

(22)

(23)

The corresponding boundary conditions of the pinned-

pinned non-uniform beam resting on an elastic founda-

tion are

(24)

(25)

(26)

(27)

The corresponding boundary conditions of the clamped-

clamped non-uniform beam resting on an elastic foun-

dation are

(28)

(29)

(30)

(31)

3. The Spline Collocation Model

The numerous complex problems of non-uniform

beams resting on elastic foundations have been effici-

ently solved using fast computers with a range of numer-

ical methods, including the Galerkin method, finite ele-

ment technique, differential quadrature approach, differ-

ential transform scheme, boundary element method, and

Rayleigh-Ritz method [24�27]. In this study, the spline

collocation method is employed to formulate discrete ei-

genvalue problems for various non-unifrom beams. Prenter

et al. [28�30] investigated spline and variation methods.

Bert and Sheu [31] presented a static analysis of beams

and plates using the spline collocation method. El-

Hawary et al. [32] examined quartic spline collocation

methods for solving linear elliptical partial differential

equations. Archer [33] investigated odd-degree splines

using high-order collocation residual expansions and

adopted nodal collocation methods to solve the problem

with one-dimensional boundary values. Patlashenko and

Weller [34] applied the spline collocation approach to

solve two-dimensional problems, and determined the

post bucking behavior of laminated panels subjected to

mechanical and heat-induced loadings. In this work, the

knots, xk,i, are considered as follows:

(32)

where xk,0, xk,1, xk,2, …, xk,N�1, xk,N are the abscissas of

the knots and xk,�2, xk,�1, xk,N+1, xk,N+2 are the abscissas of

the extended fictitious knots.

(33)

where the distance, hk, between two adjacent knots re-

mains constant. The spline function is given as follows

[28�30]:

(34)

where k is the element number, and Bk,-2 (xk), Bk,�1 (xk),

Vibration Analysis of Non-Uniform Beams Resting on Elastic Foundations Using the Spline Collocation Method 115

Bk,0 (xk), …, Bk,N+1 (xk), Bk,N+2(xk) are the basis for the

function defined over the region of a x bk k k . The

deflection of the kth beam element at the knots is given

by the following equation:

(35)

where M is the total number of elements, ak,i is a coeffi-

cient to be determined and Bk,i (xk) is the spline function.

The domain contains N + 5 collocation points. The equa-

tions of motion of a non-uniform beam can be rearranged

into the spline collocation method formula, yielding,

(36)

(37)

(38)

(39)

(40)

The spline collocation method can be applied to rear-

range the boundary conditions of a clamped-free non-

uniform beam resting on an elastic foundation into ma-

trix form as follows:

(41)

(42)

(43)

116 Ming-Hung Hsu

(44)

The spline collocation scheme can be utilized to rear-

range the boundary conditions of a pinned-pinned non-

uniform beam resting on an elastic foundation into the

following matrix form:

(45)

(46)

(47)

(48)

The spline collocation scheme can be used to rearrange

the boundary conditions of a clamped-clamped non-

uniform beam resting on an elastic foundation into the

following matrix form:

(49)

(50)

(51)

(52)

The MATLAB program is used to obtain the solution of

the eigenvalue problem. The following figures summa-

rize the numerical results obtained.

4. Numerical Results

To determine the validity of the present technique,

several examples of the vibrations of non-uniform beams

resting on elastic foundations are considered. Figures 1�

3 show the non-dimensional natural frequencies of the

non-uniform beams resting on elastic foundations with

an area of A = Ao (1 + �x / L), second moment of area of

I = Io (1 + �x / L) and taper ratio of � = �0.5. The

non-dimensional natural frequency is defined as �

� �A L EI0

4

0/ ( ). The non-dimensional elastic founda-

tion stiffness is defined as k k L EIf f 4

0/ ( ). To verify

the correctness of the spline collocation method, non-

uniform beams resting on elastic foundations are used to

obtain results that can be compared with those acquired

using the finite element method. The simulation results

computed using the spline collocation method is com-

pared with the numerical results obtained using the finite

element method. Obviously, the foundation stiffness in-

creases the frequencies of vibrations of the non-uniform

beams resting on elastic foundations. Unlike the finite

element method, the spline collocation method does not

require a calculation of integrals to generate a solution.

Figures 4�6 show the non-uniform beam with an area of

A = Ao (1 + �x / L), second moment of area of I = Io (1 +

�x / L)3 and taper ratio of � = �0.5. The frequencies of

vibrations of the non-uniform beams resting on elastic

foundations are computed using the finite element and

the spline collocation methods. The curve obtained us-

ing the spline collocation method closely follows the

curve obtained using the finite element method. The fre-

quencies of vibrations of non-uniform beams resting on

elastic foundations increase as the foundation stiffness

increases. Figures 7�9 plot the non-dimensional natural


118 Ming-Hung Hsu

Figure 1. The non-dimensional natural frequencies of clamped-free non-uniform beams resting on elastic foundations with anarea of A = Ao (1 + �x / L), second moment of area of I = Io (1 + �x / L) and taper ratio of � = �0.5.

Figure 2. The non-dimensional natural frequencies of pinned-pinned non-uniform beams resting on elastic foundations with anarea of A = Ao (1 + �x / L), second moment of area of I = Io (1 + �x / L) and taper ratio of � = �0.5.

Figure 3. The non-dimensional natural frequencies of clamped-clamped non-uniform beams resting on elastic foundations withan area of A = Ao (1 + �x / L), second moment of area of I = Io (1 + �x / L) and taper ratio of � = �0.5.


Figure 4. The non-dimensional natural frequencies of clamped-free non-uniform beams resting on elastic foundations with anarea of A = Ao (1 + �x / L), second moment of area of I = Io (1 + �x / L)3 and taper ratio of � = �0.5.

Figure 5. The non-dimensional natural frequencies of pinned-pinned non-uniform beams resting on elastic foundations with anarea of A = Ao (1 + �x / L), second moment of area of I = Io (1 + �x / L)3 and taper ratio of � = �0.5.

Figure 6. The non-dimensional natural frequencies of clamped-clamped non-uniform beams resting on elastic foundations withan area A = Ao (1 + �x / L), second moment of area of I = Io (1 + �x / L)3 and taper ratio � = �0.5.

frequencies of the non-uniform beams with an area of A

= Ao (1 + �x / L), second moment of area of I = Io (1 + �x /

L) and k f 10. The frequencies of the non-uniform beams

resting on elastic foundations are affected by tapering.

The frequencies of the non-uniform beams resting on

elastic foundations generally decrease rapidly as the ta-

per ratio, �, increases. Figures 10�12 list the non-di-

mensional natural frequencies of the non-uniform beams

with an area of A = Ao (1 + �x / L), second moment of area

of I = Io (1 + �x / L)3 and k f 10. The frequencies of the

non-uniform beams resting on elastic foundations de-

crease gradually as the taper ratio, �, increases. We con-

clude that the clamped-clamped boundary conditions give

rise to higher frequencies of the non-uniform beams rest-

ing on elastic foundations compared with those for sim-

ply supported boundary conditions. Generally, the taper

ratio � has a stronger influence on the frequencies of the

non-uniform beams on elastic foundations than the taper

ratio �.

5. Concluding Remarks

This work develops an efficient algorithm based on

the spline collocation scheme, Euler-Bernoulli beam

theory and Hamilton’s principle for solving eigenvalue

problems of non-uniform beams resting on elastic foun-

120 Ming-Hung Hsu

Figure 7. The non-dimensional natural frequencies of clam-ped-free non-uniform beams resting on elastic foun-dations with an area A = Ao (1 + �x / L), second mo-ment of area of I = Io (1 + �x / L) and k f = 10.

Figure 8. The non-dimensional natural frequencies of pinned-pinned non-uniform beams resting on elastic foun-dations with an area A = Ao (1 + �x / L), second mo-ment of area of I = Io (1 + �x / L) and k f = 10.

Figure 9. The non-dimensional natural frequencies of clam-ped-clamped non-uniform beams resting on elasticfoundations with an area A = Ao (1 + �x / L), secondmoment of area of I = Io (1 + �x / L) and k f = 10.

Figure 10. The non-dimensional natural frequencies of clam-ped-free non-uniform beams resting on elastic foun-dations with an area A = Ao (1 + �x / L), second mo-ment of area of I = Io (1 + �x / L)3 and k f = 10.

dations. Numerical results for relatively more compli-

cated vibration problems, including those with complex

geometrical and mixed boundary conditions, will be pre-

sented elsewhere. Appropriate boundary conditions and

the spline collocation method are applied to transform

the partial differential equations of non-uniform beams

resting on elastic foundations into discrete eigenvalue

problems. Numerical results revel that taper ratios � and

� and the elastic foundation stiffness markedly affect the

frequencies of the non-uniform beams. The values of

taper ratios � and � are inversely related to the frequen-

cies of the non-uniform beams. The spline collocation

scheme effectively elucidates the dynamic behavior of

non-uniform beams resting on elastic foundations.

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Manuscript Received: Sep. 28, 2007

Accepted: Jan. 20, 2009

122 Ming-Hung Hsu

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Vibration Analysis of Non-Uniform Beams Resting on Elastic