Tamkang Journal of Science and Engineering, Vol. 12, No. 2, pp. 113-122 (2009)
113
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 a, b, 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 characteristics 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 using the Rayleigh-Ritz scheme. Hodges et al. [4] computed the fundamental frequencies and the corresponding modal shapes using a discrete transfer matrix scheme.
Lee and Kuo [5] solved the problem of bending vibrations 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 boundary conditions for modeling beams on elastic foundations.
*Corresponding author. E-mail: [email protected]
Their results were dependent upon foundation parameters.
Sharma and DasGupta [20] examined the bending problem of axially constrained beams on nonlinear Winklertype elastic foundations using Green’s functions. Beaufait and Hoadley [21] solved the problem of elastic beams
on a linear foundation using the midpoint difference technique. Kuo and Lee [22] investigated the deflection of
non-uniform beams resting on a nonlinear elastic foundation using the perturbation method. Chen [23] generated
the numerical solutions for beams resting on elastic foundations using the differential quadrature element approach. 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 eigenvalue 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 numerical results acquired using the finite element method.
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Ming-Hung Hsu
2. Formation
(7)
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 = A0f(x), and r is the density of
beam material. Function f(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:
(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)
(2)
(12)
where I is the second moment of area of the non-uniform beam, I = I0y(x), kf is the elastic foundation constant, and E is Young’s modulus of non-uniform beam
material. The function y(x) is dependent upon the crosssection shape. Parameter Io is the second moment of
area at the end, where x = 0. Hamilton’s principle is
given by
Consider the following clamped-clamped non-uniform
beam resting on an elastic foundation. The corresponding boundary conditions are
u (0, t) = 0
(13)
(14)
(3)
u (L, t) = 0
where dW is virtual work. Substituting Eqs. (1) and (2)
into Eq. (3) yields the equations of motion. The transverse motion, u, of the non-uniform beam resting on an
elastic foundation is governed by
(15)
(16)
Let the displacement response be
(4)
Consider the clamped-free beam to be clamped at the
end, where x = 0. The corresponding boundary conditions are as follows:
u (0, t) = 0
(17)
where w is the natural frequency of a non-uniform beam
resting on an elastic foundation. Substituting Eq. (17)
into Eq. (4) yields
(5)
(18)
(6)
Vibration Analysis of Non-Uniform Beams Resting on Elastic Foundations Using the Spline Collocation Method
Equation (18) can be rewritten as
115
The corresponding boundary conditions of the pinnedpinned non-uniform beam resting on an elastic foundation are
beams resting on elastic foundations have been efficiently solved using fast computers with a range of numerical methods, including the Galerkin method, finite element technique, differential quadrature approach, differential transform scheme, boundary element method, and
Rayleigh-Ritz method [24-27]. In this study, the spline
collocation method is employed to formulate discrete eigenvalue 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. ElHawary 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:
(24)
(32)
(25)
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.
(19)
The corresponding boundary conditions of the clampedfree non-uniform beam resting on an elastic foundation
are
(20)
(21)
(22)
(23)
(26)
(27)
(33)
The corresponding boundary conditions of the clampedclamped non-uniform beam resting on an elastic foundation are
where the distance, hk, between two adjacent knots remains constant. The spline function is given as follows
[28-30]:
(28)
(29)
(30)
(31)
3. The Spline Collocation Model
The numerous complex problems of non-uniform
(34)
where k is the element number, and Bk,-2 (xk), Bk,-1 (xk),
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Ming-Hung Hsu
Bk,0 (xk), …, Bk,N+1 (xk), Bk,N+2(xk) are the basis for the
function defined over the region of a k £ x k £ bk . The
deflection of the kth beam element at the knots is given
by the following equation:
(38)
(35)
where M is the total number of elements, ak,i is a coefficient to be determined and Bk,i (xk) is the spline function.
The domain contains N + 5 collocation points. The equations of motion of a non-uniform beam can be rearranged
into the spline collocation method formula, yielding,
(39)
(40)
The spline collocation method can be applied to rearrange the boundary conditions of a clamped-free nonuniform beam resting on an elastic foundation into matrix form as follows:
(41)
(36)
(37)
(42)
(43)
Vibration Analysis of Non-Uniform Beams Resting on Elastic Foundations Using the Spline Collocation Method
117
(51)
(52)
(44)
The spline collocation scheme can be utilized to rearrange the boundary conditions of a pinned-pinned nonuniform 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 nonuniform beam resting on an elastic foundation into the
following matrix form:
(49)
(50)
The MATLAB program is used to obtain the solution of
the eigenvalue problem. The following figures summarize 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 13 show the non-dimensional natural frequencies of the
non-uniform beams resting on elastic foundations with
an area of A = Ao (1 + ax / L), second moment of area of
I = Io (1 + ax / L) and taper ratio of a = -0.5. The
non-dimensional natural frequency is defined as w =
w rA0 L4 / ( EI 0 ). The non-dimensional elastic foundation stiffness is defined as k f = k f L4 / ( EI 0 ). To verify
the correctness of the spline collocation method, nonuniform 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 compared with the numerical results obtained using the finite
element method. Obviously, the foundation stiffness increases 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 + bx / L), second moment of area of I = Io (1 +
bx / L)3 and taper ratio of b = -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 using the spline collocation method closely follows the
curve obtained using the finite element method. The frequencies of vibrations of non-uniform beams resting on
elastic foundations increase as the foundation stiffness
increases. Figures 7-9 plot the non-dimensional natural
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Ming-Hung Hsu
Figure 1. The non-dimensional natural frequencies of clamped-free non-uniform beams resting on elastic foundations with an
area of A = Ao (1 + ax / L), second moment of area of I = Io (1 + ax / L) and taper ratio of a = -0.5.
Figure 2. The non-dimensional natural frequencies of pinned-pinned non-uniform beams resting on elastic foundations with an
area of A = Ao (1 + ax / L), second moment of area of I = Io (1 + ax / L) and taper ratio of a = -0.5.
Figure 3. The non-dimensional natural frequencies of clamped-clamped non-uniform beams resting on elastic foundations with
an area of A = Ao (1 + ax / L), second moment of area of I = Io (1 + ax / L) and taper ratio of a = -0.5.
Vibration Analysis of Non-Uniform Beams Resting on Elastic Foundations Using the Spline Collocation Method
119
Figure 4. The non-dimensional natural frequencies of clamped-free non-uniform beams resting on elastic foundations with an
area of A = Ao (1 + bx / L), second moment of area of I = Io (1 + bx / L)3 and taper ratio of b = -0.5.
Figure 5. The non-dimensional natural frequencies of pinned-pinned non-uniform beams resting on elastic foundations with an
area of A = Ao (1 + bx / L), second moment of area of I = Io (1 + bx / L)3 and taper ratio of b = -0.5.
Figure 6. The non-dimensional natural frequencies of clamped-clamped non-uniform beams resting on elastic foundations with
an area A = Ao (1 + bx / L), second moment of area of I = Io (1 + bx / L)3 and taper ratio b = -0.5.
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Ming-Hung Hsu
Figure 7. The non-dimensional natural frequencies of clamped-free non-uniform beams resting on elastic foundations with an area A = Ao (1 + ax / L), second moment of area of I = Io (1 + ax / L) and k f = 10.
Figure 9. The non-dimensional natural frequencies of clamped-clamped non-uniform beams resting on elastic
foundations with an area A = Ao (1 + ax / L), second
moment of area of I = Io (1 + ax / L) and k f = 10.
Figure 8. The non-dimensional natural frequencies of pinnedpinned non-uniform beams resting on elastic foundations with an area A = Ao (1 + ax / L), second moment of area of I = Io (1 + ax / L) and k f = 10.
Figure 10. The non-dimensional natural frequencies of clamped-free non-uniform beams resting on elastic foundations with an area A = Ao (1 + bx / L), second moment of area of I = Io (1 + bx / L)3 and k f = 10.
frequencies of the non-uniform beams with an area of A
= Ao (1 + ax / L), second moment of area of I = Io (1 + ax /
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 taper ratio, a, increases. Figures 10-12 list the non-dimensional natural frequencies of the non-uniform beams
with an area of A = Ao (1 + bx / L), second moment of area
of I = Io (1 + bx / L)3 and k f =10. The frequencies of the
non-uniform beams resting on elastic foundations decrease gradually as the taper ratio, b, increases. We conclude that the clamped-clamped boundary conditions give
rise to higher frequencies of the non-uniform beams resting on elastic foundations compared with those for simply supported boundary conditions. Generally, the taper
ratio b has a stronger influence on the frequencies of the
non-uniform beams on elastic foundations than the taper
ratio a.
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-
Vibration Analysis of Non-Uniform Beams Resting on Elastic Foundations Using the Spline Collocation Method
121
non-uniform beams resting on elastic foundations.
References
Figure 11. The non-dimensional natural frequencies of pinnedpinned non-uniform beams resting on elastic foundations with an area A = Ao (1 + bx / L), second moment of area of I = Io (1 + bx / L)3 and k f = 10.
Figure 12. The non-dimensional natural frequencies of clamped-clamped non-uniform beams resting on elastic
foundations with an area A = Ao (1 + bx / L), second
moment of area of I = Io (1 + bx / L)3 and k f = 10.
dations. Numerical results for relatively more complicated vibration problems, including those with complex
geometrical and mixed boundary conditions, will be presented 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 a and
b and the elastic foundation stiffness markedly affect the
frequencies of the non-uniform beams. The values of
taper ratios a and b are inversely related to the frequencies of the non-uniform beams. The spline collocation
scheme effectively elucidates the dynamic behavior of
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Manuscript Received: Sep. 28, 2007
Accepted: Jan. 20, 2009