An improved Fourier series solution for vibration analysis of elastically restrained beams in Rayleigh

Abstract

A combination of Fourier cosines series and polynomials (4 terms, 4th order) in Rayleigh–Ritz method is proposed to solve the vibration problem of a generally restrained beam. The characteristics of auxiliary polynomials, which actually are determined by homogeneous boundary conditions of the beam under consideration, can directly change the convergence properties and numerical instabilities of vibration results significantly. Fast convergence and reliability of the proposed polynomials are illustrated with numerical examples in comparisons with available results.

Keywords

improved Fourier series auxiliary functions Rayleigh-Ritz method

Highlights

➢ A combination of Fourier cosines series and polynomials (4 terms, 4th order) in Rayleigh–Ritz method is proposed.

➢ For a generally restrained beam, auxiliary function with higher derivatives than 4th order converges slowly.

➢ Larger number of polynomial terms causes numerical instabilities and ill-conditioning.

1. Introduction

Vibration of flexible structures, such as beams, plates, and shells, is of great interest in community of structural dynamics due to their wide applications in various engineering branches. The vibration problem of beams with elastic boundary conditions is relatively common in engineering, such as riveted and bolted connections of steel structure, reinforced concrete foundation beam placed on a certain elastic foundation. It is of interest to investigate the vibration of beams with generally restrained boundary. For several decades, a huge amount of research effort has been devoted to vibration analysis of beam structure with various boundary conditions (Dowell, 1971; Mittendorf and Greif, 1977; Rao and Mirza, 1989; Kim and Kim, 2001). Although the beam displacement is usually assumed as a linear combination of trigonometric and hyperbolic functions, the corresponding coefficients and frequency parameters have to be determined from the boundary conditions. However, it is normally tedious and difficult to solve a transcendental equation containing trigonometric and hyperbolic functions in practice, especially for the elastically restrained beams. In the meanwhile, the beam displacement can also be expanded in terms of polynomials (Grossi and Arenas, 1996; Gutierres and Laura, 1996; Bhat, 2015), Fourier series (Abdullah, 2006; Li, 2016), and other functions (Auciello, 1996; Ruta, 1999). Among these, Fourier series methods have been commonly chosen to perform the vibration analysis of generally restrained beams.

Greif (1974) presented a general method for the vibration response prediction of beam with traditional boundary conditions subjected to steady-state displacement excitation. This method involves the application of a simple Fourier expansion and Stokes' transformation, which may be further extended to the vibration problem of plate and shell structures. Later, Greif and Chung (1975) utilized the Rayleigh–Ritz procedure in terms of Fourier series expansion to establish the vibration solution of traditional constrained cylindrical shells with satisfactory analysis accuracy. Further, Greif and Mittendorf (1976) introduced Rayleigh–Ritz method involving simple Fourier sine or cosine series expansions for vibration analysis of a wide class of beam, plate, and shell problems. Chung (1981) employed the Fourier series representations of the displacement function and its derivatives accounting for homogeneous boundary conditions for the free vibration of cylindrical shells, while Wang and Lin (1996) later utilized it to simulate the S-S beams with rotational and/or translational springs at each end. In such cases, the Fourier series method is basically confined to the S-S boundary conditions because of the potential discontinuity problems of the displacement and its derivatives at boundary points. Zhou (1996) developed a simple and straightforward approach consisting of a set of static beam functions, which is a combination of Fourier sine series and polynomial functions (4 terms, 3rd order) in Rayleigh–Ritz method to study the vibration characteristics of plates with various boundary conditions. In a nano/micro sense, atomic size-dependent structures with elastically support have also been studied (Naghinejad and Ovesy, 2018, 2019). Civalek et al. (2022) conducted buckling analysis of FGM microbeams via Rayleigh–Ritz method. The admissible function is a series of polynomials or Fourier sine series and the influences of nonclassical boundary conditions are examined.

Making use of an auxiliary polynomial, Li (2000) adopted a simple and unified Fourier series method to represent the free vibration of beams, regardless of boundary conditions. The beam displacement is assumed as the linear superposition of a Fourier series and an auxiliary polynomial function (5 terms, 4th order), mathematically guaranteeing the continuity of the displacement and its derivatives. Then Li (2002) presented the accuracy and convergence of Fourier series theoretically and numerically for beams with arbitrary boundaries. Li claimed that the cosine series expansion converged faster than its sine counterpart in Fourier series method. However, as pointed out later in this study, such a technique did not consider the influence of convergence of the given auxiliary polynomial function, comparing with its Fourier cosine or sine series counterpart. Similarly, Du et al. (2007, 2011, 2012) proposed an improved and modified Fourier series method using the supplementary functions (4 terms, trigonometric) which are infinitely differentiable to solve several vibration problems of beams, plates, and shells in Rayleigh–Ritz procedure. Recently, Monterrubio and Ilanko (2015) set admissible functions to be a combination of Fourier cosine series and low order polynomials (3 terms, 2nd order) in Rayleigh–Ritz method, and demonstrated accurate results without ill-conditioning for any number of terms. Zheng et al. (2021, 2023a, 2023b) use simple polynomials (2 terms, 3rd order and/or 4th order) and trigonometric functions to construct vibration displacements of shells for the elastic boundary conditions. Although excellent accuracy and convergence are demonstrated in the above literatures, few investigations have been primarily focused on the physical meaning and comparison of convergence of auxiliary functions with different terms and orders. It is still not clear that whether the auxiliary function has an influence on the convergence and precision of the Fourier series solutions and how much influence the auxiliary function could have. In fact, the auxiliary functions play an important role in the stiffness and mass matrixes assembly and numerical calculation for a variety of relative references (Sun et al., 2013; Wang et al., 2019, 2023; Lyu et al., 2019; Yang et al., 2023; Zhao et al., 2023).

Motivated by the current deficiency in the literature, this work presents a discussion on the characteristics of auxiliary functions, which shows that the orders of auxiliary polynomials can directly change the convergence properties of Fourier series significantly and the number of polynomials’ terms may cause numerical instabilities. Based on the homogeneous boundary conditions at each end, a combination of Fourier cosines series and polynomials (4 terms, 4th order) in Rayleigh–Ritz method is proposed to solve the vibration problem of generally restrained beam. The general equations determining the convergence is obtained considering the auxiliary function. Results are compared with previously published results and fast convergence and reliability of the proposed technique are demonstrated.

2. Theoretical formulations

For bending vibration of an Euler–Bernoulli beam (Figure 1), spatial component of the displacement satisfies

\frac{D d^{4} w (ξ)}{d ξ^{4}} - ρ A ω^{2} w (ξ) = 0 0 \leq ξ \leq 1

(1)

\frac{d^{4} w (ξ)}{d ξ^{4}} - ρ_{D} ω^{2} w (ξ) = 0 (ρ_{D} = ρ A / D),

(2)

where D, ρ, and A are, respectively, the flexural rigidity, the mass density, and the cross-sectional area of the beam, ξ is normalized coordinate, ω is frequency in radian per second, and w is the flexural displacement. Note that the flexural rigidity D = EI where E and I are the modulus of elasticity and moment of inertia of the beam, respectively.

The boundary conditions for a generally restrained beam can be assumed as

k_{0} w = - {D w}^{‴}, K_{0} w^{'} = D w^{″} a t ξ = 0

(3)

and

k_{1} {w = D w}^{‴}, K_{1} w^{'} = - D w^{″} a t ξ = 1,

(4)

where k₀ and k₁ are the stiffnesses of the translational springs, and K₀ and K₁ are the stiffnesses of the rotational springs at ξ = 0 and ξ = 1, respectively.

Equations (3) and (4) represent a set of elastically restrained boundaries. When the stiffnesses of those springs are set to be some special values such as zero and infinity, the classical boundary conditions can be obtained immediately.

Although Fourier sine series expansion have their fastest rate of convergence for a S-S beam, cosine series expansion would converge faster than its sine counterpart for most geometric boundary conditions, especially for guiding conditions [19]. For this reason, the solution of equation (2) in this work is represented in forms of the linear combination of a Fourier cosine series and an auxiliary function

w (ξ) = \sum_{m = 0}^{\infty} A_{m} \cos (λ_{m} ξ) + p (ξ), 0 \leq ξ \leq 1, λ_{m} = m π,

(5)

where p(ξ) should satisfy

p^{'} (0) = w^{'} (0) = β_{0}, p^{'} (1) = w^{'} (1) = β_{1}, p^{‴} (0) = w^{‴} (0) = α_{0}, p^{‴} (1) = w^{‴} (1) = α_{1} .

(6)

The auxiliary function is introduced to tackle all the relevant discontinuities with the original displacement and its derivatives at the boundaries. It should be noted that α₀, α₁, β₀, and β₁ are simply obtained from boundary conditions, at ξ = 0 and ξ = L.

For generally restrained beams, the first and third derivatives of displacements at each end (represented by β₀, β₁, α₀, and α₁) are not generally equal to zero, which means the auxiliary function must have at least 4 terms and 4th derivatives. As is known, simple polynomials are widely used to express the auxiliary function

p (ξ) = \sum_{i = 1}^{n} C_{i} ς_{i} (ξ), i = 1,2, . . ., n,

(7)

while the ideal and simplest polynomials are

ς_{i} (ξ) = ξ^{i}, i = 1,2, . . ., n .

(8)

Although other sets of auxiliary functions can be built by orthogonal polynomials using the Gram–Schmidt process, the purpose of this article is to propose a simple set in forms of polynomial function. Next, the auxiliary polynomial functions with different orders and terms are discussed.

2.1. The influence of auxiliary function orders

Substituting equation (5) into equation (2), one is able to obtain

\sum_{m = 0}^{\infty} {λ_{m}}^{4} A_{m} \cos (λ_{m} ξ) + p^{(4)} (ξ) - ρ_{D} ω^{2} (\sum_{m = 0}^{\infty} A_{m} \cos (λ_{m} ξ) + p (ξ)) = 0 .

(9)

Multiplying equation (9) with cos (λ_mξ) and integrating it from 0 to 1¸ results in

\frac{1}{2} {λ_{m}}^{4} A_{m} + \int_{0}^{1} p^{(4)} (ξ) \cos (λ_{m} ξ) d ξ - ρ_{D} ω^{2} (\frac{1}{2} A_{m} + \int_{0}^{1} p (ξ) \cos (λ_{m} ξ) d ξ) = 0, m = 1,2, \dots

(10)

and

\int_{0}^{1} p^{(4)} (ξ) d ξ - ρ_{D} ω^{2} (A_{0} + \int_{0}^{1} p (ξ) d ξ) = 0, m = 0 .

(11)

Obviously, the convergence can actually be determined in a direct procedure using equation (10). The Fourier cosine coefficients are given by

A_{m} = \frac{2}{({λ_{m}}^{4} - ρ_{D} ω^{2})} {ρ_{D} ω^{2} [\frac{{(- 1)}^{m} p^{'} (1) - p^{'} (0)}{{λ_{m}}^{2}} - \frac{{(- 1)}^{m} p^{‴} (1) - p^{‴} (0)}{{λ_{m}}^{4}}] + \frac{ρ_{D} ω^{2} - {λ_{m}}^{4}}{{λ_{m}}^{4}} \int_{0}^{1} p^{(4)} (ξ) \cos (λ_{m} ξ) d ξ}

(12)

The most important observation from equation (12) is that the convergence of the Fourier series solutions is decided by the boundary conditions of auxiliary function p(ξ) and its derivatives at both ends. In order to simplify the presentation, three letter symbols are used to define various boundary conditions, that is, the symbol C denotes clamped boundary condition, S denotes simply supported conditions, and F denotes free boundary without restraints. A two-letter symbol C-S denotes a beam with the clamped and simply supported conditions at ξ = 0 and ξ = 1, respectively. Now two typical conditions are considered below.

(1) If p⁽⁴⁾ (ξ) = const, it is easy to confirm that

A_{m} = \frac{2}{({λ_{m}}^{4} - ρ_{D} ω^{2})} {ρ_{D} ω^{2} [\frac{{(- 1)}^{m} p^{'} (1) - p^{'} (0)}{{λ_{m}}^{2}} + \frac{{(- 1)}^{m + 1} p^{‴} (1) + p^{‴} (0)}{{λ_{m}}^{4}}]} .

(13)

For beams with generally elastic restraints (representing the displacements and its derivatives are not generally equal to zero at ξ = 0 and ξ = 1), it follows that $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ . If a beam is guided at both ends with only translational springs (representing the first derivatives of displacements are both equal to zero at ξ = 0 and ξ = 1), then one sees that $A_{m} \sim O (\frac{1}{{λ_{m}}^{8}})$ . In the case when a beam is clamped at both ends (representing the displacements and its first derivatives are both equal to zero at ξ = 0 and ξ = 1), then one is able to obtain $A_{m} \sim O (\frac{1}{{λ_{m}}^{8}})$ . If a beam is simply supported at both ends with only rotational springs (representing the displacements and its second derivatives are both equal to zero at ξ = 0 and ξ = 1), then it provides that $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ . For other classical boundary conditions like C-S, C-F, F-F, S-F, or F-F cases, when a beam is restrained at each end with translational/rotational restraint (representing the first derivatives of displacements are not equal to zero at ξ = 0 and ξ = 1 simultaneously), then one sees that $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ .

(2) If p^(t) (ξ) = const (t = 5,6,7,…,n,…,∞), it is clear from equation (12) that

A_{m} = \frac{2}{({λ_{m}}^{4} - ρ_{D} ω^{2})} (\begin{array}{l} ρ_{D} ω^{2} [\frac{{(- 1)}^{m} p^{'} (1) - p^{'} (0)}{{λ_{m}}^{2}} + \frac{{(- 1)}^{m + 1} p^{‴} (1) + p^{‴} (0)}{{λ_{m}}^{4}}] \\ + \frac{ρ_{D} ω^{2} - {λ_{m}}^{4}}{{λ_{m}}^{4}} [\sum_{j = 2}^{\infty} {(- 1)}^{j} \frac{{(- 1)}^{m} p^{(2 j + 1)} (1) - p^{(2 j + 1)} (0)}{{λ_{m}}^{2 (j - 1)}}] \end{array})

(14)

where 2j+1 ≤ t.

Obviously, it is convenient to obtain the convergence of Fourier cosine coefficients. For beams with generally elastic restraints, it follows that $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ . When a beam is guided at both ends with only translational springs, one knows that $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ . If a beam is clamped at both ends, one is able to obtain $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ . For other classical boundary conditions like S-S, C-S, C-F, F-F, S-F, or F-F cases, one sees that $A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})$ .

To investigate the convergence speed of the Fourier series solutions, the results of cosine coefficients are illustrated below for different boundary conditions in Table 1.

Table 1.

Convergence of Fourier cosine coefficients, A_m, for beams with different boundary conditions.

Derivatives	Generally restrained	Guided	Clamped
p⁽⁴⁾ (ξ) = const	$O (\frac{1}{{λ_{m}}^{6}})$	$O (\frac{1}{{λ_{m}}^{8}})$	$O (\frac{1}{{λ_{m}}^{8}})$
p^(t) (ξ) = const (t = 5,6,7,…,n,…,∞)	$O (\frac{1}{{λ_{m}}^{6}})$	$O (\frac{1}{{λ_{m}}^{6}})$	$O (\frac{1}{{λ_{m}}^{6}})$

As can be seen, the auxiliary function with 4^th derivatives converges much faster than its counterpart with higher derivatives for guided and clamped conditions. For beams with general elastic restraints, it is clear that the convergence rate of Fourier series is same. However, in the real calculations, it is generally true that the auxiliary function with 4th derivatives converges much faster than its counterpart with higher derivatives as illustrated later.

2.2. The influence of auxiliary function terms

In this work, natural frequencies of an elastically restrained beam are obtained in Rayleigh–Ritz method, which is convenient as described below.

The displacement function can be expressed as

w (ξ) = \sum_{m = 0}^{\infty} A_{m} \cos (λ_{m} ξ) + \sum_{i = 1}^{k} C_{i} ς_{i} (ξ), 0 \leq ξ \leq 1, λ_{m} = m π,

(15)

where ζ_i (ξ) is a set of auxiliary functions.

The kinetic and strain energies of the beam are given by

T = \frac{1}{2} ω^{2} ρ A \int_{0}^{1} w^{2} (ξ) d ξ

(16a)

U_{b e a m} = \frac{1}{2} D {\int_{0}^{1} (\frac{\partial^{2} w (ξ)}{\partial ξ^{2}})}^{2} d ξ .

(16b)

The strain energy of the springs is expressed as

U_{s p r i n g} = \frac{1}{2} k_{0} w {(0)}^{2} + \frac{1}{2} K_{0} w^{'} {(0)}^{2} + \frac{1}{2} k_{1} w {(1)}^{2} + \frac{1}{2} K_{1} w^{'} {(1)}^{2} .

(17)

Substituting the displacement function (15) into the kinetic and strain energy expressions and minimizing the Rayleigh quotient with respect to the coefficients A_m and C_i yields the eigenvalue equation

(K - w^{2} M) [\begin{array}{l} A \\ C \end{array}] = {0} .

(18)

where K and M are the stiffness matrix and mass matrix, respectively. It is noted that for given truncation numbers M and K, the dimension of both are (M+K+1) × (M+K+1). Correspondingly, the column vectors of the modified Fourier series expansion coefficients are arranged as

{\begin{cases} A = {A_{0}, A_{1}, \dots, A_{m} \dots, {A_{M}}}^{T} \\ C = {C_{1}, \dots, C_{m} \dots, {C_{K}}}^{T} \end{cases}

(19)

Solution of the eigenvalue equation will yield the natural frequencies and eigenvectors of the beam. The mode shapes are readily obtained by substituting the expansion coefficients of the Fourier series, namely the eigenvectors into the displacement function.

Here, the auxiliary functions can also be polynomials satisfying the geometric boundary conditions of the beam. The equation (6) suggests that the simplest set of polynomials can be given as

ς_{i} (ξ) = ξ^{i} (i = 1,2,3,4)

(20)

which guarantee that the first and third derivatives of displacements at each end remains continuous. To explain the reason, a comparison between 4 terms and 6 terms with the same 4^th order in auxiliary functions is carried out for a free–free beam to show that larger number of terms may cause numerical instabilities. The auxiliary functions with 4 terms are set as equation (20), while the auxiliary functions with 6 terms are given in reference [24] as

ς_{1} (ξ) = 1, ς_{2} (ξ) = ξ, ς_{3} (ζ) = ξ^{2}, ς_{4} (ξ) = ξ^{2} - ξ, ς_{5} (ξ) = ξ^{3} - ξ^{2}, ς_{6} (ξ) = ξ^{4} - 2 ξ^{3} + ξ^{2} .

(21)

Exact expressions of the stiffness matrix elements can be easily derived where the Fourier series is truncated to M = 20. Therefore, the matrices resulting from the terms are regularly structured sparse matrices which consist of only one diagonal with Fourier cosine series. Note that such regular structure is partially destroyed following the introduction of auxiliary functions but the sparsity of the matrix is preserved. For the sake of clarity, the patterns of the matrix are shown in Figure 2, where the black squares denote non-zero entries.

A condition number for a matrix measures how sensitive the answer is to perturbations in the input data and to roundoff errors made during the solution process. The main diagonal matrices above (M+1)th row and column are the same, while the other matrices decided by the auxiliary functions are different. It is easy to get that

C o n d (b) > C o n d (a),

(22)

which means that larger terms of polynomials are easier to cause numerical instabilities and ill-conditioning. Thus to keep the solution as simple as possible and minimize the numerical problems, polynomial functions with 4 terms and 4th order is proposed in this work.

3. Results and discussions

In this section, the convergence characteristics of Fourier series solution will be checked.

The auxiliary polynomials with 4th order and 4 terms are assumed as equation (20), while the auxiliary polynomials with 6th order and 4 terms are assumed as

ς_{i} (ξ) = ξ^{i} i = 3,4,5,6

(23)

First consider a S-S beam with rotational springs at both ends, as shown in Figure 1. Let K₁L/D = 0 (representing the rotational spring stiffness is equal to zero) at ξ = 1 and all the others are equal to infinity (set to be a very large number, 1.0E10, in the following calculations), which actually becomes C-S boundary condition.

Table 2 shows the first five frequency parameters,

μ_{i} = \frac{L}{π} {(ω_{i} \sqrt{\frac{ρ A}{D}})}^{1 / 2}

, obtained using both the 4th and 6th order polynomials, for the various rotational stiffnesses at ξ = 1. Note that the Fourier series is truncated to M = 20 in Table 2. In particular, for K₁L/D = 1.0E10, this situation essentially becomes the classical clamped–clamped boundary. Here, the calculation results of frequency parameters are considered to be correct enough while the frequency parameters are the same as the exact results in the second decimal place. The current solution shows an excellent agreement with those presented in references (Blevins, 1979; Li, 2000). It can be seen that convergence between numerical results decreases, especially in high mod numbers.

Table 2.

First five frequency parameters for different stiffnesses of the rotational springs.

Mode	K₁L/D = 0 (C-S)			K₁L/D = 1			K₁L/D = 100			K₁L/D = 1.0E10 (C-CS)
Mode	4th	6th	Exact (Blevins, 1979)	4th	6th	Ref. (Li, 2000)	4th	6th	Ref. (Li, 2000)	4th	6th	Exact (Blevins, 1979)
1	1.24988	1.24989	1.24988	1.28656	1.28657	1.28656	1.49137	1.49139	1.49137	1.50563	1.50564	1.50562
2	2.25000	2.25009	2.25	2.27077	2.27086	2.27081	2.47681	2.47693	2.47681	2.49987	2.49987	2.49975
3	3.25001	3.25027	3.25	3.26479	3.26506	3.26491	3.46884	3.46914	3.46884	3.50012	3.50033	3.50001
4	4.25002	4.25060	4.25	4.26146	4.26206	4.26175	4.46105	4.46176	4.46108	4.50067	4.50067	4.5
5	5.25004	5.25112	5.25	5.25936	5.26047	/	5.45380	5.45497	/	5.50042	5.50124	5.5

Table 3 shows the first five frequency parameters obtained using both the 4th and 6th order polynomials for the various translational stiffnesses at ξ = 1. That is, the boundary conditions vary from C-F to C-S. From this table, it can be seen that the frequency parameters exhibit a more obvious change than that in Table 2. High precision of the current method is again illustrated.

Table 3.

First five frequency parameters for different stiffnesses of the translational springs.

Mode	k₁L/D = 0 (C-F)			k₁L/D = 1		k₁L/D = 100		k₁L/D = 1.0E10 (C-S)
Mode	4th	6th	Exact (Blevins, 1979)	4th	6th	4th	6th	4th	6th	Exact (Blevins, 1979)
1	0.59686	0.59686	0.59686	0.63980	0.63964	1.15882	1.15883	1.24988	1.24989	1.24988
2	1.49418	1.49420	1.49418	1.49726	1.49728	1.78763	1.78765	2.25000	2.25009	2.25
3	2.50026	2.50038	2.50025	2.50090	2.50097	2.57324	2.57336	3.25001	3.25027	3.25
4	3.50001	3.50030	3.49999	3.50023	3.50037	3.52522	3.52555	4.25002	4.25060	4.25
5	4.50005	4.50067	4.50000	4.50011	4.50043	4.51155	4.51226	5.25004	5.25112	5.25

To examine the convergence of the solution, the results for a guided-guided beam using both the 4th and 6th order polynomials are compared in Table 4 as the number of truncated terms in the Fourier series varies. In this case, as mentioned earlier, the 4th order series, say,

A_{m} \sim O (\frac{1}{{λ_{m}}^{8}})

, outperforms the 6th series, say,

A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})

, which is seen from the results in Table 4.

Table 4.

First five frequency parameters for a guided-guided beam with translational springs, $k_{0} L^{3} / D = k_{1} L^{3} / D = 1$ , at both ends.

Mode	M = 3		M = 5		M = 10		Ref. (Li, 2000)
Mode	4th	6th	4th	6th	4th	6th	Ref. (Li, 2000)
1	0.378274	0.378277	0.378274	0.378275	0.378274	0.378274	0.378274
2	1.01011	1.01011	1.01011	1.01011	1.01011	1.01011	1.01011
3	2.00128	2.00128	2.00128	2.00128	2.00128	2.00128	2.00128
4	3.00038	3.00038	3.00038	3.00038	3.00038	3.00038	3.00038
5	4.23764	4.15609	4.00016	4.00016	4.00016	4.00016	/

As represented earlier, for a C-S beam both Fourier coefficients with the 4th and 6th order polynomials are

A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})

. However, in the real calculations, the 4th order polynomials converge faster than the 6th polynomials, as evidenced in Table 5. The same result is illustrated using absolute tolerances of the first frequency parameters in Figure 3. Meanwhile, as aforementioned, the mode shapes of the C-S beam can be calculated readily. In Figure 4, the convergence speed of the first mode shapes with different truncated number of Fourier cosine terms M plotted for a C-S beam. The first mode shape can be accurately obtained by taking only one term in the Fourier cosine series with 4th order polynomials. It also indicates that the 4th order polynomials converge faster than the 6th polynomials. For small values of M, one can observe clearly that convergence between numerical results decreases in high mode numbers.

Table 5.

First five frequency parameters for a C-S beam.

Mode	M = 5		M = 10		M = 20		Exact (Blevins, 1979)
Mode	4th	6th	4th	6th	4th	6th	Exact (Blevins, 1979)
1	1.24988	1.25194	1.24988	1.24999	1.24988	1.24989	1.24988
2	2.25000	2.26055	2.25000	2.25066	2.25000	2.25009	2.25
3	3.25013	3.28612	3.25001	3.25194	3.25001	3.25027	3.25
4	4.25054	4.29244	4.24999	4.25448	4.25002	4.25060	4.25
5	5.32034	5.58977	5.25009	5.25805	5.25004	5.25112	5.25

The results for a clamped–clamped beam are compared in Table 6. As mentioned earlier, for this case, the 4^th order polynomials, say,

A_{m} \sim O (\frac{1}{{λ_{m}}^{8}})

, is better suited than the 6th order polynomials, say,

A_{m} \sim O (\frac{1}{{λ_{m}}^{6}})

, which is seen from the results in Table 6. The same result is depicted using absolute tolerances of the first frequency parameters in Figure 5. Figure 6 shows the convergence speed of the first mode shapes with different truncated terms M for a clamped–clamped beam. The results indicate that the mode shapes with 4th order polynomials become virtually identical to the exact solution sooner and the remarkable convergence of the current solution is illustrated again.

Table 6.

First five frequency parameters for a clamped–clamped beam.

Mode	M = 10		M = 20		M = 40		M = 80		Exact (Blevins, 1979)
Mode	4th	6th	4th	6th	4th	6th	4th	6th	Exact (Blevins, 1979)
1	1.50556	1.50581	1.50563	1.50564	1.50562	1.50563	1.50562	1.50562	1.50562
2	2.50010	2.50064	2.49987	2.49987	2.49978	2.49979	2.49976	2.49976	2.49975
3	3.50004	3.50244	3.50012	3.50033	3.50007	3.50009	3.50002	3.50002	3.50001
4	4.50210	4.50519	4.50067	4.50067	4.50017	4.50019	4.50002	4.50002	4.5
5	5.50011	5.50941	5.50042	5.50124	5.50022	5.50032	5.50004	5.50004	5.5

For a generally restrained beam with translational and rotational springs at both ends, assuming

k_{0} L^{3} / D = k_{1} L^{3} / D = 1

and

K_{0} L / D = K_{1} L / D = 100

, the frequency parameters,

μ_{i} = L {(ω_{i} \sqrt{\frac{ρ A}{E I}})}^{1 / 2}

are listed in Table 7. It can be seen that the 4^th order polynomials has obviously outperformed the 6th order polynomials indeed.

Table 7.

First five frequency parameters for a beam with general elastic restraints, $k_{0} L^{3} / D = k_{1} L^{3} / D = 1$ and $K_{0} L / D = K_{1} L / D = 100$ .

Mode	M = 5			M = 10			M = 20			Ref. (Rao and Mirza, 1989)
Mode	4th	6th	Ref. (Li, 2000)	4th	6th	Ref. (Li, 2000)	4th	6th	Ref. (Li, 2000)	Ref. (Rao and Mirza, 1989)
1	1.188301	1.188343	1.188301	1.188301	1.188341	1.188301	1.188301	1.188341	1.188301	1.188301
2	3.144180	3.158704	3.14418	3.144180	3.158704	3.14418	3.144180	3.158704	3.14418	3.14418
3	6.227220	6.257073	6.22726	6.227220	6.257073	6.227224	6.227220	6.257073	6.227221	6.22722
4	9.336970	9.381270	9.33717	9.336970	9.381269	9.33717	9.336970	9.381269	9.336975	9.336969
5	12.44992	12.50813	12.4514	12.44988	12.50812	12.45001	12.44988	12.50812	12.44990	12.44988

Further, the convergence speed of the polynomials with 4th order and 4 terms is compared to infinitely differentiable auxiliary functions from reference (Du et al. 2007) for a general supported beam. The results with the 4th order polynomials are better than those obtained with infinitely differentiable trigonometric functions, as provided in Table 8.

Table 8.

First five frequency parameters for a beam with general elastic restraints, $k_{0} L^{3} / D = k_{1} L^{3} / D = 10$ and $K_{0} L / D = K_{1} L / D = 0.1$ .

Mode	M = 5		M = 10		M = 20		Ref. (Rao and Mirza, 1989)
Mode	4th	Ref. (Du et al. 2007)	4th	Ref. (Du et al. 2007)	4th	Ref. (Du et al. 2007)	Ref. (Rao and Mirza, 1989)
1	2.035385	2.035385	2.035385	2.035385	2.035385	2.035385	2.035385
2	2.788458	2.788459	2.788458	2.788458	2.788458	2.788458	2.788458
3	4.947258	4.947259	4.947253	4.947253	4.947253	4.947253	4.947253
4	7.919302	7.919300	7.919258	7.919258	7.919256	7.919256	7.919256
5	11.030485	11.030313	11.028606	11.028606	11.028597	11.028597	11.028597

A comparison between 4 terms and 6 terms with the same 4th order polynomials as aforementioned is carried out in Table 9 for a free–free beam. Note here that the rigid body modes are not included. As already shown in Figure 2, larger terms of polynomials are easier to cause numerical instabilities and ill-conditioning. When the number of terms of the Fourier series M = 200, the first frequency parameter of 6 terms polynomials is instable and ill-conditioned.

Table 9.

First five frequency parameters for a free–free beam.

Mode	M = 10		M = 20		M = 200		Exact (Blevins, 1979)
Mode	4 terms	6 terms	4 terms	6 terms	4 terms	6 terms	Exact (Blevins, 1979)
1	1.50562	1.50568	1.50562	1.50563	1.50562	1.58793	1.50562
2	2.49977	2.50055	2.49975	2.49986	2.49975	2.49975	2.49975
3	3.50062	3.50068	3.50001	3.50002	3.50001	3.49791	3.50001
4	4.50129	4.50946	4.50003	4.50041	4.50000	4.50000	4.5
5	5.70468	5.70625	5.50007	5.50008	5.50000	5.50000	5.5

The last example involves a comparison between the auxiliary functions with 4 terms and 5 terms, where the fifth term is a constant. As shown in Table 9, The first three frequency parameters of 5 terms polynomials are instable and ill-conditioned for a clamped–clamped beam, when the number of terms of the Fourier series M = 80 (Table 10).

Table 10.

First five frequency parameters for a clamped–clamped beam.

Mode	M = 20		M = 40		M = 80		M = 100		Exact (Blevins, 1979)
Mode	4 terms	5 terms	4 terms	5 terms	4 terms	5 terms	4 terms	5 terms	Exact (Blevins, 1979)
1	1.50563	1.50563	1.50562	1.50562	1.50562	1.59244	1.50562	1.50562	1.50562
2	2.49987	2.49986	2.49978	2.49978	2.49976	2.69675	2.49975	2.49975	2.49975
3	3.50012	3.50019	3.50007	3.50007	3.50002	3.40863	3.50002	3.50002	3.50001
4	4.50067	4.50063	4.50017	4.50017	4.50002	4.49989	4.50001	4.50001	4.5
5	5.50042	5.50068	5.50022	5.50021	5.50004	5.50004	5.50002	5.50002	5.5

4. Conclusions

In this study, the influence of supplementary function on the convergence characteristics of an improved Fourier series solution is formulated and analyzed using a free vibration problem of flexible beam with elastic boundary supports. The convergence of auxiliary polynomials with different orders and terms are compared in the Rayleigh–Ritz method. It is concluded that:

(1) Auxiliary function with higher derivatives than 4th order converges slowly for a generally restrained beam.

(2) Larger number of polynomial terms causes numerical instabilities and ill-conditioning.

(3) Based on the physical conditions, a combination of Fourier cosines series and simple polynomials (4 terms, 4th order) in the Rayleigh–Ritz method is proposed for the free vibration analysis of beams with arbitrary support.

The remarkable convergence and accuracy of the proposed technique are demonstrated with several truncation terms both theoretically and numerically. The proposed method can be easily extended to plates and shells with general boundary conditions to achieve fast convergence and high precision property.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by the Natural Science Foundation of China (No. 51839005) and the Development of High-efficient Tandem Propeller (No. Z2021SFC-11/DZ).

ORCID iD

Jianzheng Wang

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An improved Fourier series solution for vibration analysis of elastically restrained beams in Rayleigh–Ritz method

Abstract

Keywords

Highlights

1. Introduction

2. Theoretical formulations

2.1. The influence of auxiliary function orders

2.2. The influence of auxiliary function terms

3. Results and discussions

4. Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References