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Abstract

Boundary characteristic orthogonal polynomials proposed by the author in 1985 have been used in the Rayleigh Ritz method extensively in order to obtain natural frequencies of vibrating plates with different boundary conditions. The method used products of the characteristic orthogonal polynomials along the two directions of the plate. The first member of the boundary characteristic orthogonal polynomials set satisfied all the boundary conditions of the vibrating beam, including the natural conditions. However, the higher members of the set satisfied only the geometry boundary conditions. In this study, a modified Gram–Schmidt orthogonalization method is presented where all the members of the orthogonal set of polynomials satisfy all the boundary conditions including the natural boundary conditions. Furthermore, the exact solution of the beam differential equation is expressed in the form of a generalized Fourier series in terms of the set of new boundary characteristic orthogonal polynomials which forms an eigenvalue problem that can provide the natural frequencies and the corresponding normal modes of the beam more accurately.

Keywords

Boundary characteristic orthogonal polynomials vibration of beams modified Gramm–Schmidt method

Introduction

Boundary characteristic orthogonal polynomials (BCOPs) were proposed as assumed deflection shape functions in the Rayleigh Ritz method to study the vibrations of beams and plates by Bhat.¹ Several publications resulted using the BCOPs, and some of them can be found in Singh and Chakraverty,² Lam et al.,³ and Kim,⁴ while Chakraverty et al.⁵ give research using BCOPs until 1999. Application of the BCOPs to study plate vibration problems can be found in Chakraverty.⁶ The advantage of using the orthogonal polynomials as assumed shape functions was that the “mass matrix” was diagonal and the “stiffness matrix” had large diagonal terms, making it easier to compute the natural frequencies and the natural modes.

The first member of the set of BCOPs as constructed in Bhat¹ satisfied all the geometrical boundary conditions; however, it did not guarantee satisfaction of the natural boundary conditions of the higher members of the set. Although the deflection can be expressed as a linear combination of the BCOPs, which is a Generalized Fourier series representation of the deflection, the solution is still approximate since the natural boundary conditions are not satisfied. However, Kim⁴ suggested the use of a generating function which will help in the higher polynomials of the set also satisfying the boundary conditions, which can construct the exact solution. His formulation had to be modified by replacing his generating function with an evolution function that is different for different members of the set of BCOPs.

Construction of novel BCOPs satisfying all the boundary conditions

The first member of the BCOP set is chosen as the simplest polynomial that can satisfy four arbitrary beam boundary conditions and can be expressed as

ϕ (η) = a_{0} (1 + a_{1}^{*} η + a_{2}^{*} η^{2} + a_{3}^{*} η^{3} + a_{4}^{*} η^{4})

(1)

where $a_{k}^{*} = a_{k} / a_{0}$ , with $k = 1, 2, 3, 4$ , and $η = x / L$ is the non-dimensioned length coordinate of the beam.

The second and the higher members of the BCOP set are obtained as

\begin{matrix} ϕ_{1} (η) = [g_{0} (η) - B_{0}] ϕ_{0} (η) \\ ϕ_{k + 1} (η) = [g_{k} (η) - B_{k}] ϕ_{k} (η) - C_{k} ϕ_{k - 1} (η) \end{matrix}

(2)

where

g_{k} (η) = A_{k 1} η + A_{k 2} η^{2} + A_{k 3} η^{3} + A_{k 4} η^{4} + η^{5}

(3)

While the member-specific constants $B_{k}$ and $C_{k}$ ensure orthogonality of the newly constructed polynomial with respect to the two previous member polynomials, the member-specific constants $A_{kj}$ ensure that the boundary conditions of the newly constructed polynomial are satisfied. The number of constants $A_{kj}$ depends on the number and nature of the boundary conditions to be satisfied as can be seen later when specific boundary conditions are considered.

The kth member is dependent on two members immediately below, thus establishing a three-term recurrence relation. Employing the condition that the BCOPs are orthogonal to each other, the constants $B_{0}$ , $B_{k}$ , and $C_{k}$ are obtained as

\begin{matrix} B_{0} & = \frac{\int_{0}^{1} w (η) g_{0} (η) ϕ_{0}^{2} (η) d η}{\int_{0}^{1} ϕ_{0}^{2} (η) d η} \\ B_{k} & = \frac{\int_{0}^{1} w (η) g_{k} (η) ϕ_{k}^{2} (η) d η}{\int_{0}^{1} ϕ_{k}^{2} (η) d η} \\ C_{k} & = \frac{\int_{0}^{1} w (η) g_{k} (η) ϕ_{k} (η) ϕ_{k - 1} (η) d η}{\int_{0}^{1} ϕ_{k - 1}^{2} (η) d η} \end{matrix}

(4)

where $w (η)$ is a weight function and $g_{k} (η)$ is an evolution function. If the beam is uniform, $w (η) = 1$ , and hereafter for simplicity, the beam would be assumed to be uniform. The set of orthogonal polynomials will have infinite members with k = 1, 2, 3, …, ∞. The member polynomials and their derivatives with respect to the independent variable $η$ are given as

\begin{matrix} ϕ_{k + 1} (η) = [g_{k} (η) - B_{k}] ϕ_{k} (η) - C_{k} ϕ_{k - 1} (η) \\ ϕ'_{k + 1} (η) = [g_{k} (η) - B_{k}] ϕ'_{k} (η) \\ + g'_{k} (η) ϕ_{k} (η) - C_{k} ϕ'_{k - 1} (η) \\ ϕ_{k + 1}^{″} (η) = [g_{k} (η) - B_{k}] ϕ_{k}^{″} (η) + 2 g'_{k} (η) ϕ'_{k} (η) \\ + g_{k}^{″} (η) ϕ_{k} (η) - C_{k} ϕ_{k - 1}^{″} (η) \\ ϕ ‴_{k + 1} (η) = [g_{k} (η) - B_{k}] ϕ ‴_{k} (η) + 3 g'_{k} (η) ϕ ″_{k} (η) \\ + 3 g ″_{k} (η) ϕ'_{k} (η) + g ‴_{k} (η) ϕ_{k} (η) - C_{i} ϕ ‴_{k - 1} (η) \end{matrix}

(5)

The different combinations of the boundary conditions and the corresponding conditions on $g_{k} (η)$ and its derivatives in order to satisfy all the boundary conditions by the BCOPs are given in Table 1. The evolution function $g_{k} (η)$ differs from the generating function defined by Kim.⁴ While Kim’s generating function is constant for all the members, the evolution function depends on the previous member of the BCOP set.

Table 1.

Combinations of beam boundary conditions and conditions on $g_{k} (η)$ and its derivatives.

C-C	$\begin{matrix} ϕ_{k} (0) = ϕ'_{k} (0) = 0 \\ ϕ_{k} (1) = ϕ'_{k} (1) = 0 \end{matrix}$	$\begin{matrix} ϕ_{0} (η) = η^{2} - 2 η^{3} + η^{4} \\ g'_{k} (0) = 0, g'_{k} (1) = 0 \\ g_{k} (η) = - 1.5 η^{2} + η^{3} \end{matrix}$
C-S	$\begin{matrix} ϕ_{k} (0) = ϕ'_{k} (0) = 0 \\ ϕ_{k} (1) = ϕ ″_{k} (1) = 0 \end{matrix}$	$\begin{matrix} ϕ_{0} (η) = 1.5 η^{2} - 2.5 η^{3} + η^{4} \\ g'_{k} (0) = 0, g'_{k} (1) = 0 \\ g_{k} (η) = - 1.5 η^{2} + η^{3} \end{matrix}$
S-S	$\begin{matrix} ϕ_{k} (0) = ϕ ″_{k} (0) = 0 \\ ϕ_{k} (1) = ϕ ″_{k} (1) = 0 \end{matrix}$	$\begin{matrix} ϕ_{0} (η) = η - 2 η^{3} + η^{4} \\ g'_{k} (0) = 0, g'_{k} (1) = 0 \\ g_{k} (η) = - 1.5 η^{2} + η^{3} \end{matrix}$
C-F	$\begin{matrix} ϕ_{k} (0) = ϕ'_{k} (0) = 0 \\ ϕ ″_{k} (1) = ϕ ‴_{k} (1) = 0 \end{matrix}$	$\begin{matrix} ϕ_{0} (η) = η^{2} - 2 η^{3} + η^{4} \\ 2 g'_{k} (1) ϕ'_{k} (1) + g ″_{k} (1) ϕ_{k} (1) = 0, 3 g ″_{k} (1) ϕ'_{k} (1) + g ‴_{k} (1) ϕ_{k} (1) = 0 \\ D_{k 0} = ϕ_{k} (0), E_{k 0} = ϕ'_{k} (0), D_{k 1} = ϕ_{k} (1), E_{k 1} = ϕ'_{k} (1) \\ A_{k 1} = 0 \\ T_{22} = 2 E_{n 1} + D_{n 1} \\ T_{23} = 3 E_{n 1} + 3 D_{n 1} \\ T_{32} = E_{n 1} \\ T_{33} = 3 E_{n 1} + D_{n 1} \\ f_{2} = - (4 E_{n 1} + 6 D_{n 1}) \\ f_{3} = - (6 E_{n 1} + 4 D_{n 1}) \\ A_{k 2} = \frac{f_{2} T_{33} - f_{3} T_{23}}{T_{22} T_{33} - T_{32} T_{23}} \\ A_{k 3} = \frac{T_{22} f_{3} - T_{32} f_{2}}{T_{22} T_{33} - T_{32} T_{23}} \\ g_{k} (η) = A_{k 2} η^{2} + A_{k 3} η^{3} + η^{4} \end{matrix}$
F-F	$\begin{matrix} ϕ ″_{k} (0) = ϕ ‴_{k} (0) = 0 \\ ϕ ″_{k} (1) = ϕ ‴_{k} (1) = 0 \end{matrix}$	$\begin{matrix} ϕ_{0} (η) = 1 \\ 2 g'_{k} (0) ϕ'_{k} (0) + η g ″_{k} (0) ϕ_{k} (0) = 0, 3 g ″_{k} (0) ϕ'_{k} (0) + g ‴_{k} (0) ϕ_{k} (0) = 0 \\ 2 g'_{k} (1) ϕ'_{k} (1) + g ″_{k} (1) ϕ_{k} (1) = 0, 3 g ″_{k} (1) ϕ'_{k} (1) + g ‴_{k} (1) ϕ_{k} (1) = 0 \\ D_{k 0} = ϕ_{k} (0), E_{k 0} = ϕ_{k} (0), D_{k 1} = ϕ_{k} (1), E_{k 1} = ϕ'_{k} (1) \\ T_{11} = E_{n 1} - \frac{E_{n 0}}{D_{n 0}} (2 E_{n 1} + D_{n 1}) + {(\frac{E_{n 0}}{D_{n 0}})}^{2} (3 E_{n 1} + 3 D_{n 1}) \\ T_{12} = 4 E_{n 1} + 6 D_{n 1} \\ T_{21} = - 2 (\frac{E_{n 0}}{D_{n 0}}) + {(\frac{E_{n 0}}{D_{n 0}})}^{2} (6 E_{n 1} + 2 D_{n 1}) \\ T_{22} = 12 E_{n 1} + 8 D_{n 1} \\ f_{1} = - (5 E_{n 1} + 10 D_{n 1}) \\ f_{2} - (20 E_{n 1} + 20 D_{n 1}) \\ A_{k 1} = \frac{T_{22} f_{1} - T_{12} f_{2}}{T_{11} T_{22} - T_{21} T_{12}}, A_{k 2} = - \frac{ϕ'_{k} (0)}{ϕ_{k} (0)} A_{1} \\ A_{k 3} = - \frac{ϕ'_{k} (0)}{ϕ_{k} (0)}, A_{2} = {(\frac{ϕ'_{k} (0)}{ϕ_{k} (0)})}^{2}, A_{k 4} = \frac{T_{22} f_{1} - T_{12} f_{2}}{T_{11} T_{22} - T_{21} T_{12}} \\ g_{k} (η) = A_{1} η + A_{2} η^{2} + A_{3} η^{3} + A_{4} η^{4} + η^{5} \end{matrix}$

The set of BCOPs must be constructed such that the evolution function $g_{k} (η)$ satisfies the conditions in Table 1. The determination of the evolution function $g_{k} (η)$ is illustrated below for the different cases of the boundary conditions given in Table 1.

Clamped–Clamped Conditions (C-C)

The boundary conditions are

ϕ_{k} (0) = ϕ'_{k} (0) = 0, ϕ_{k} (1) = ϕ'_{k} (1) = 0

The first polynomial is the lowest order polynomial that satisfies these conditions and is given by

ϕ_{0} (η) = η^{2} - 2 η^{3} + η^{4}

(6)

Corresponding to these boundary conditions on the evolution function are $g'_{k} (0) = 0, g'_{k} (1) = 0$ .

The evolution function, $g_{k} (η)$ , is the simplest polynomial that satisfies these conditions and is given by

g_{k} (η) = - 1.5 η^{2} + η^{3}

(7)

Clamped–Simply Supported Conditions (C-S)

The corresponding boundary conditions are

\begin{matrix} ϕ_{k} (0) & = ϕ'_{k} (0) = 0 \\ ϕ_{k} (1) & = φ ″_{k} (1) = 0 \end{matrix}

(8)

The first polynomial is the lowest order polynomial that satisfies all these conditions and is given by

ϕ_{0} (η) = 1.5 η^{2} - 2.5 η^{3} + η^{4}

(9)

The corresponding conditions on the evolution function, $g_{k} (η)$ , are given by

g'_{k} (0) = 0, g'_{k} (1) = 0

(10)

The evolution function, $g_{k} (η)$ , is the simplest polynomial that satisfies these conditions and is given by

g_{k} (η) = - 1.5 η^{2} + η^{3}

(11)

Simply Supported–Simply Supported Conditions (S-S)

The corresponding boundary conditions are

\begin{matrix} ϕ_{k} (0) & = ϕ ″_{k} (0) = 0 \\ ϕ_{k} (1) & = ϕ ″_{k} (1) = 0 \end{matrix}

(12)

The first polynomial is the lowest order polynomial that satisfies all these conditions and is given by

ϕ_{0} (η) = η - 2 η^{3} + η^{4}

(13)

The corresponding conditions on the evolution function, $g_{k} (η)$ , are given by

g'_{k} (0) = 0, g'_{k} (1) = 0

(14)

and the evolution function is the simplest polynomial that satisfies these conditions and is given by

g_{k} (η) = - 1.5 η^{2} + η^{3}

(15)

Clamped–Free Conditions (C-F)

The associated boundary conditions are

\begin{matrix} ϕ_{k} (0) & = ϕ'_{k} (0) = 0 \\ ϕ ″_{k} (1) & = ϕ ‴_{k} (1) = 0 \end{matrix}

(16)

The first polynomial is the lowest order polynomial that satisfies all these conditions and is given by

ϕ_{0} (η) = 6 η^{2} - 4 η^{3} + η^{4}

(17)

The corresponding conditions on the evolution function, $g_{k} (η)$ , are given by

\begin{matrix} 2 g'_{k} (1) ϕ'_{k} (1) + g ″_{k} (1) ϕ_{k} (1) = 0, \\ 3 g ″_{k} (1) ϕ'_{k} (1) + g ‴_{k} (1) ϕ_{k} (1) = 0 \end{matrix}

(18)

The evolution function, $g_{k} (η)$ , is given by the simplest polynomial that satisfies these conditions and is given by

\begin{matrix} D_{k 0} & = ϕ_{k} (0), E_{k 0} = ϕ'_{k} (0), D_{k 1} = ϕ_{k} (1), E_{k 1} = ϕ'_{k} (19) \\ A_{k 1} & = 0 \\ T_{22} & = 2 E_{n 1} + D_{n 1} \\ T_{23} & = 3 E_{n 1} + 3 D_{n 1} \\ T_{32} & = E_{n 1} \\ T_{33} & = 3 E_{n 1} + D_{n 1} \\ f_{2} & = - (4 E_{n 1} + 6 D_{n 1}) \\ f_{3} = - (6 E_{n 1} + 4 D_{n 1}) \begin{matrix} A_{k 2} & = \frac{f_{2} T_{33} - f_{3} T_{23}}{T_{22} T_{33} - T_{32} T_{23}} \\ A_{k 3} & = \frac{T_{22} f_{3} - T_{32} f_{2}}{T_{22} T_{33} - T_{32} T_{23}} \\ g_{k} (η) & = A_{k 2} η^{2} + A_{k 3} η^{3} + η^{4} \end{matrix} \end{matrix}

Free–Free Conditions (F-F)

The associated boundary conditions are

\begin{matrix} ϕ ″_{k} (0) & = ϕ ‴_{k} (0) = 0 \\ ϕ ″_{k} (1) & = ϕ ‴_{k} (1) = 0 \end{matrix}

(20)

The first polynomial is the lowest order polynomial that satisfies all these conditions and is given by

ϕ_{0} (η) = 1

(21)

The corresponding conditions on the evolution function, $g_{k} (η)$ , are given by

\begin{matrix} 2 g'_{k} (0) ϕ'_{k} (0) + η g ″_{k} (0) ϕ_{k} (0) = 0, \\ 3 g ″_{k} (0) ϕ'_{k} (0) + g ‴_{k} (0) ϕ_{k} (0) = 0 \\ 2 g'_{k} (1) ϕ'_{k} (1) + g ″_{k} (1) ϕ_{k} (1) = 0, \\ 3 g ″_{k} (1) ϕ'_{k} (1) + g ‴_{k} (1) ϕ_{k} (1) = 0 \end{matrix}

(22)

The evolution function, $g_{k} (η)$ , is given by the simplest polynomial that satisfies these conditions and is given by

\begin{matrix} D_{k 0} = ϕ_{k} (0), E_{k 0} = ϕ'_{k} (0), D_{k 1} = ϕ_{k} (1), E_{k 1} = ϕ'_{k} (1) \\ T_{11} = E_{n 1} - \frac{E_{n 0}}{D_{n 0}} (2 E_{n 1} + D_{n 1}) + {(\frac{E_{n 0}}{D_{n 0}})}^{2} (3 E_{n 1} + 3 D_{n 1}) \\ T_{12} = 4 E_{n 1} + 6 D_{n 1} \\ T_{21} = - 2 (\frac{E_{n 0}}{D_{n 0}}) + {(\frac{E_{n 0}}{D_{n 0}})}^{2} (6 E_{n 1} + 2 D_{n 1}) \\ T_{22} = 12 E_{n 1} + 8 D_{n 1} \\ f_{1} = - (5 E_{n 1} + 10 D_{n 1}) \\ f_{2} - (20 E_{n 1} + 20 D_{n 1}) \\ A_{k 1} = \frac{T_{22} f_{1} - T_{12} f_{2}}{T_{11} T_{22} - T_{21} T_{12}}, A_{k 2} = - \frac{ϕ'_{k} (0)}{ϕ_{k} (0)} A_{1} \\ A_{k 3} = - \frac{ϕ'_{k} (0)}{ϕ_{k} (0)} A_{2} = {(\frac{ϕ'_{k} (0)}{ϕ_{k} (0)})}^{2}, A_{k 4} = \frac{T_{22} f_{1} - T_{12} f_{2}}{T_{11} T_{22} - T_{21} T_{12}} \\ g_{k} (η) = A_{k 1} η + A_{k 2} η^{2} + A_{k 3} η^{3} + A_{k 4} η^{4} + η^{5} \end{matrix}

(23)

Recurrence relations among orthogonal polynomials

Considering orthogonality between polynomials $ϕ_{k}$ and $ϕ_{k + 1}$ , we have

\begin{matrix} \int_{0}^{1} ϕ_{k + 1} ϕ_{k} d η = \int_{0}^{1} g_{k} (η) ϕ_{k}^{2} d η \\ - B_{k} \int_{0}^{1} ϕ_{k}^{2} d η - C_{k} \int_{0}^{1} ϕ_{k - 1} ϕ_{k} d η = 0 \end{matrix}

(24)

The term on the left hand side (LHS) and the last term on the right hand side (RHS) in equation (24) are zero in view of orthogonality between $ϕ_{k}$ and $ϕ_{k + 1}$ and between $ϕ_{k}$ and $ϕ_{k - 1}$ . Hence, we have the relation

B_{k} = \frac{\int_{0}^{1} g_{k} (η) ϕ_{k}^{2} d η}{\int_{0}^{1} ϕ_{k}^{2} d η}

(25)

In view of orthogonality between the members of polynomials $ϕ_{k - 1}$ and $ϕ_{k + 1}$ , we have

\begin{matrix} \int_{0}^{1} ϕ_{k + 1} ϕ_{k - 1} d η = \int_{0}^{1} g_{k} (η) ϕ_{k} ϕ_{k - 1} d η \\ - B_{k} \int_{0}^{1} ϕ_{k} ϕ_{k - 1} d η - C_{k} \int_{0}^{1} ϕ_{k - 1}^{2} d η = 0 \end{matrix}

(26)

Due to orthogonality between members $ϕ_{k + 1}$ and $ϕ_{k - 1}$ and between $ϕ_{k}$ and $ϕ_{k - 1}$ , the LHS term and the second term on the RHS of equation (26) are zero, and hence, we have

\int_{0}^{1} g_{k} (η) ϕ_{k} ϕ_{k - 1} d η - C_{k} \int_{0}^{1} ϕ_{k - 1}^{2} d η = 0

(27)

Accordingly, we have

C_{k} = \frac{\int_{0}^{1} g_{k} (η) ϕ_{k} ϕ_{k - 1} d η}{\int_{0}^{1} ϕ_{k - 1}^{2} d η}

(28)

Considering any two consecutive members of the set of orthogonal polynomials, we have

\begin{matrix} \int_{0}^{1} ϕ_{k} ϕ_{k + 1} d η & = \int_{0}^{1} ϕ_{k} [(g_{k} - B_{k}) ϕ_{k} - C_{k} ϕ_{k - 1}] d η \\ = \int_{0}^{1} g_{k} ϕ_{k}^{2} d η - B_{k} \int_{0}^{1} ϕ_{k}^{2} d η - C_{k} \int_{0}^{1} ϕ_{k - 1} ϕ_{k} d η \end{matrix}

(29)

The first two terms on the RHS of equation (29) are zero and hence we have

\int_{0}^{1} ϕ_{k} ϕ_{k + 1} d η = - C_{k} \int_{0}^{1} ϕ_{k - 1} ϕ_{k} d η

(30)

Extending this recurrence relation, we can write

\begin{matrix} \int_{0}^{1} ϕ_{k} ϕ_{k + 1} d η = - C_{k} \int_{0}^{1} ϕ_{k - 1} ϕ_{k} d η \\ = - C_{k} [- C_{k - 1} \int_{0}^{1} ϕ_{k - 2} ϕ_{k - 1} d η] \\ \dots = {(- 1)}^{k - 1} C_{k} C_{k - 1} C_{k - 2} \dots C_{2} \int_{0}^{1} ϕ_{1} ϕ_{2} d η = 0 \end{matrix}

(31)

in view of the orthogonality between $ϕ_{1}$ and $ϕ_{2}$ . Hence, two consecutive polynomials are orthogonal to each other. Extending this result, we have

\begin{matrix} \int_{0}^{1} ϕ_{k} ϕ_{k + 2} d η & = \int_{0}^{1} ϕ_{k} [(g_{k + 1} - B_{k + 1}) ϕ_{k + 1} - C_{k} ϕ_{k}] d η \\ = \int_{0}^{1} g_{k + 1} ϕ_{k} ϕ_{k + 1} - B_{k + 1} \\ \int_{0}^{1} ϕ_{k} ϕ_{k + 1} d η - C_{k + 1} \int_{0}^{1} ϕ_{k}^{2} d η \end{matrix}

(32)

Replacing the subscript k by k + 1 in equation (28) and substituting in equation (32), we have the RHS of equation (32) zero. Hence, three consecutive polynomials are orthogonal to each other. Extending this argument, it is possible to show that n consecutive polynomials are orthogonal to each other.

Solution of the vibrating beam

The differential equation for the beam vibration can be written as

EI \frac{d^{4} y}{{dx}^{4}} + m \frac{d^{2} y}{{dt}^{2}} = 0

(33)

In free vibrations, the response is assumed in the form of a sinusoidal function in time. Accordingly

y (x, t) = Y (x) \sin ω t

(34)

where the axial coordinate x is non-dimensionalized as $η = x / L$ and L is the length of the beam. Accordingly, the beam equation can be written as

y ″″ - λ^{4} y = 0

(35)

where $λ^{4} = m ω^{2} / EI$ and the $()'$ indicates differentiation with respect to $η$ .

Assume the solution of the vibrating beam in the form

y (η) = \sum_{i = 0}^{N} c_{k} ϕ_{k} (η)

(36)

which is a generalized Fourier series representation of the solution in terms of the BCOPs, with $N \to \infty$ . Substituting equation (36) into equation (35), we have

y ″″ (η) = \sum_{i = 0}^{N} c_{i} ϕ_{i}^{″″} (η)

(37)

The fourth derivative of the assumed shape functions is expressed in Fourier series in terms of the BCOPs as

ϕ_{k}^{″″} (η) = \sum_{l = 0}^{i} d_{kl} ϕ_{l} (η)

(38)

Multiplying both sides of equation (38) by $ϕ_{l}$ and integrating from 0 to 1, we get

\int_{0}^{1} ϕ_{k}^{″″} ϕ_{l} d η = d_{kl} \int_{0}^{1} ϕ_{l}^{2} d η

(39)

where

d_{kl} = \frac{\int_{0}^{1} ϕ_{k}^{″″} ϕ_{l} d η}{\int_{0}^{1} ϕ_{l}^{2} d η}

(40)

Substituting equations (36–40) into equation (35) results in

\begin{matrix} c_{0} [d_{00} ϕ_{0} + d_{01} ϕ_{1} + d_{02} ϕ_{2} + d_{03} ϕ_{3} + \dots] - λ^{4} c_{0} ϕ_{0} \\ + c_{1} [d_{10} ϕ_{0} + d_{11} ϕ_{1} + d_{12} ϕ_{2} + d_{13} ϕ_{3} + \dots] - λ^{4} c_{1} ϕ_{1} \\ + c_{2} [d_{20} ϕ_{0} + d_{21} ϕ_{1} + d_{22} ϕ_{2} + d_{23} ϕ_{3} + \dots] - λ^{4} c_{2} ϕ_{2} \\ + c_{3} [d_{30} ϕ_{0} + d_{31} ϕ_{1} + d_{32} ϕ_{2} + d_{33} ϕ_{3} + \dots] - λ^{4} c_{3} ϕ_{3} \end{matrix}

(41)

Equation (41) can be reorganized in the form

\begin{matrix} ϕ_{0} [c_{0} (d_{00} - λ^{4}) + c_{1} d_{10} + c_{2} d_{20} + c_{3} d_{30} + \dots] \\ + ϕ_{1} [c_{0} d_{01} + c_{1} (d_{11} - λ^{4}) + c_{2} d_{21} + c_{3} d_{31} + c_{4} d_{41} + \dots] \\ + ϕ_{2} [c_{0} d_{02} + c_{1} d_{12} + c_{2} (d_{22} - λ^{4}) + c_{3} d_{23} + c_{4} d_{24} + \dots] \\ + ϕ_{3} [c_{0} d_{02} + c_{1} d_{12} + c_{2} (d_{22} - λ^{4}) + c_{3} d_{23} + c_{4} d_{24} + \dots] \\ + \dots \\ + \dots \end{matrix}

(42)

This can be cast in the form of an eigenvalue problem as

[D - λ^{4} I] {c} = [0]

(43)

where

\begin{matrix} [D - λ^{4} I] = \\ [\begin{matrix} (d_{00} - λ^{4}) & d_{01} & d_{02} & d_{0 N} \\ d_{10} & (d_{11} - λ^{4)}) & d_{21} & d_{1 N} \\ d_{20} & d_{21} & (d_{22} - λ^{4}) & d_{2 N} \\ d_{N 0} & d_{N 1} & d_{N 2} & d_{NN} \end{matrix}] \end{matrix}

(43)

and the column vector of generalized coordinates is given by

{c} = {\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{N} \end{matrix}}

(44)

The formulation leads to the same final form as that obtained in a Galerkin solution.

Results and discussion

Since the beam characteristic orthogonal polynomials form a complete set and are also linearly independent in view of equation (2), the solution can be expressed as a generalized Fourier series in terms of the BCOPs. When the beam boundary conditions are applied, solution to the resulting eigenvalue problem provides an “exact” solution to the beam differential equation. While the exponential solution provides an “exact” solution in terms of the sinusoidal and hyperbolic sinusoidal functions, the present method provides “exact” solutions in terms of the BCOPs whose order steadily changes monotonically. The exact results due to Young and Felgar⁷ provide up to five modes only, while n = 6 terms are used in computing the results in this study.

Table 2 presents natural frequency coefficients, $Ω = (ml ω^{2} / EI) = λ^{2}$ , for the five cases shown in Table 1. As can be seen, the natural frequency coefficients are very close to the exact solutions, even with the low number of terms used, n = 6. The results are very close to the exact results for the cases of C-S, S-S, and C-F beams.

Table 2.

Natural frequency coefficients for the different cases of beam boundary conditions.

Cases	Exact	Rayleigh Ritz with old BCOPs	Exact with BCOPs satisfying all boundary conditions
C-C	22.373286	22.373286	22.375426
	61.672822	61.672882	61.677967
	120.903391	121.130621	121.006004
	199.859448	201.126049	200.181319
	298.555533	353.433320	299.228923
		531.828852	418.152978
C-S	15.418206	15.418213	15.418733
	49.964862	49.964964	49.969845
	104.247697	104.573236	104.296435
	178.269730	181.682883	178.532507
	272.030971	335.966285	272.772401
		783.178283	387.969555
S-S	9.869604	9.869616	9.869625
	39.478418	39.478501	39.478731
	88.826440	89.474383	88.833054
	157.913671	161.807027	157.927750
	246.740110	454.582023	247.921703
		775.394745	357.336481
C-F	3.5160154	3.516015	3.516037
	22.034492	22.034498	22.036892
	61.697214	61.8224	61.753280
	120.901916	127.349730	121.289338
	199.859530	227.787287	206.176927
		847.556596	335.222642
F-F	0	0	0
	0	0	0
	22.373286	22.564205	22.373352
	61.672822	63.537311	61.673159
	120.903391	223.362552	121.066814
	199.859448	455.678395	223.733524

BCOPs: boundary characteristic orthogonal polynomials.

Conclusion

Beam characteristic orthogonal polynomials are constructed so as to satisfy both the geometrical and natural boundary conditions of the different beam problems. The BCOPs form a complete set that satisfies all the boundary conditions. Expressing the solution in a generalized Fourier series provides an “exact” solution to the beam differential equation. Natural frequency coefficients are obtained for different cases of boundary conditions. The solutions will tend toward the accurate values when sufficiently large number of terms are used in the generalized Fourier series.

Footnotes

Academic Editor: Francisco D Denia

Declaration of conflicting interests

The author declares that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

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. Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. J Sound Vib 1985; 102: 493–499.

Singh

Chakraverty

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Liew

Chow

. Use of two dimensional orthogonal polynomials for vibration of circular and elliptic plates. J Sound Vib 1992; 154: 261–270.

Kim

. The vibration of beams and plates studied using orthogonal polynomials. PhD Thesis, Department of Mechanical Engineering, The University of Western Ontario, London, ON, Canada, 1988.

Chakraverty

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Stiharu

. Recent research on vibration of structures using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Shock Vib Digest 1999; 31: 187–194.

Chakraverty

. Vibration of plates. Boca Raton, FL: CRC Press, 2009.

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Jr . Tables of characteristic functions representing normal modes of vibration of a beam (Engineering research series no. 44). Austin, TX: The University of Texas at Austin, 1949.

Vibration of beams using novel boundary characteristic orthogonal polynomials satisfying all boundary conditions

Abstract

Keywords

Introduction

Construction of novel BCOPs satisfying all the boundary conditions

Recurrence relations among orthogonal polynomials

Solution of the vibrating beam

Results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References