A unified solution for the in-plane vibration analysis of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions

Xiuhai Lv1 , Dongyan Shi2 , Qingshan Wang3 , Qian Liang4

1, 2, 3, 4College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, P. R. China

1Department of Electrical and Mechanical Engineering, Heilongjiang Agricultural Engineering Vocational College, Harbin, P. R. China

3Corresponding author

Journal of Vibroengineering, Vol. 18, Issue 2, 2016, p. 1071-1087.
Received 19 August 2015; received in revised form 2 November 2015; accepted 12 November 2015; published 31 March 2016

Copyright © 2016 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

A unified solution for the in-plane vibration analysis of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions by combining with the improved Fourier series method and Rayleigh-Ritz technique is presented in this paper. Under the current framework, regardless of boundary conditions, each of displacements and rotations of the curved Timoshenko beams is represented by the modified Fourier series consisting of a standard Fourier cosine series and several closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series representation. All the expansion coefficients are determined by the Rayleigh-Ritz technique as the generalized coordinates. The convergence and accuracy of the present method are tested and validated by a lot of numerical examples for multi-span curved Timoshenko beams with various boundary restraints and general elastic coupling conditions. In contrast to most existing methods, the current method can be universally applicable to general boundary conditions and elastic coupling conditions without the need of making any change to the solution procedure.

Keywords: unified solution, in-plane vibration, multi-span curved Timoshenko beams, general boundary conditions, elastic coupling conditions.

1. Introduction

As one of the important structural components, curved Timoshenko beams has abundant engineering applications such as bridges, aircraft structures, space vehicles, turbo-machines and other industrial applications owing to their excellent engineering characteristics. Notably, these beams frequently work in complex environments and may suffer to arbitrary boundary restraints. Therefore, a good understanding of the vibration behavior of curved Timoshenko beams subjected to dynamic loads and general boundary conditions is of particular importance for satisfying the design requirements of strength and stiffness in practical designs.

In the last few decades, a number of computational techniques have been proposed and developed, such as Differential Quadrature method, the Galerkin method, meshless method, the Ritz method, Finite element method, dynamic stiffness method and discrete singular convolution method. An interesting review of the subject can be found in the review articles [1-3]. Culver and Oestel [4] presented a new method for determining the natural frequencies of multi-span horizontally curved girders. Lin and Lee [5] used closed-form solutions to analyze dynamic response of extensional circular Timoshenko beams with general elastic boundary conditions. Kang et al. [6] presented a systematic approach for the free in-plane vibration analysis of a planar circular curved beam system. Issa et al. [7] extended the dynamic stiffness matrix method to analyze the vibration of continuous circular curved beams with the clamped-clamped boundary condition. Chen [8, 9] developed an analytical technique to study the in-plane vibration of continuous curved beams. Wang [10] investigated the effects of rotary inertia and shear on natural frequencies of continuous circular curved beams undergoing in-plane vibrations by using the dynamic stiffness matrix method. Kawakami et al. [11] performed the in-plane and out-of-plane free vibration of horizontally curved beams with arbitrary shapes and variable cross-sections by an approximate method. Riedel and Kang [12] employed wave propagation techniques to study the free vibration of elastically coupled dual-span curved beams subject to classical boundary conditions. Lee [13] applied the pseudospectral method to analyze the free vibration of circularly curved multi-span Timoshenko beams with classical boundary and rigid coupling conditions. Huang et al. [14, 15] derived the in-plane and the out-of-plane transient response of a hinged-hinged and a clamped-clamped non-circular Timoshenko curved beam by using the dynamic stiffness matrix method and the numerical Laplace transform. Leung and Zhu [16] used finite element method to analyze the in-plane vibration of thin and thick curved beams with classical boundary conditions. Krishnan and Suresh [17] utilized a simple cubic linear beam element to study static and free vibration analysis of curved beams using finite element method. Chen [18] applied the differential transform method to investigate the in-plane vibration of arbitrarily curved beam structures. Yang et al. [19] studied free in-plane vibration of uniform and non-uniform curved beams with variable curvatures, including the effects of the axis extensibility, shear deformation and the rotary inertia by using the Galerkin finite element method. Ozturk [20] introduced the reversion method and finite element method to predict in-plane free vibration of a large deflected pre-stressed cantilever curved beam. Eisenberger and Efraim [21] presented an exact dynamic stiffness matrix for a circular beam with pinned-pinned and clamped-clamped boundary conditions.

In view of the aforementioned issues and concerns, it should be emphasized that most of the existing contributions were restricted to a single or two-span curved beam subjected to a limited set of classical supports. Little research has been devoted to the in-pane vibration problem of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions. However, in practical engineering applications, the boundary and coupling conditions of multi-span curved Timoshenko beams may not always be classical boundary and rigid coupling conditions in nature, and there will always be some elastic boundary and coupling conditions. The in-plane vibration behaviors of multi-span curved Timoshenko beams with general boundary and coupling conditions have remained unsolved until now. Moreover, to the best of the authors’ knowledge, no unified, efficient and accurate solution is available in the literature for the in-plane vibration analysis of multi-span curved Timoshenko beams subjected to general elastic boundary and coupling conditions.

Recently, a modified Fourier series technique proposed by Li [22, 23] is widely used in the vibrations of plates and shells with general boundary constraints by Ritz method, e.g., [24-32]. Therefore, the present work can be considered as an extension of the method and attempts to provide a unified solution method for the in-plane vibration of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions. Under the current framework, the modified Fourier series method together with the Rayleigh-Ritz procedure and the artificial stiffness-like spring technique is adopted to derive the theoretical formulation. The general elastic boundary and coupling constraints of the multi-span curved Timoshenko beams are realized by applying the artificial stiffness-like spring technique. Each of displacements and rotations of each curved Timoshenko beam is represented by the modified Fourier series consisting of a standard Fourier cosine series and several closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series representation. Thereby, all the Fourier expanded coefficients are treated equally and independently as the generalized coordinates and are solved directly by using the Rayleigh-Ritz procedure. The convergence and accuracy of the present formulation are checked by a considerable number of convergence tests and comparisons. A variety of numerical examples are presented for the in-plane vibration of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions, which may serve as benchmark solutions for validating new computational techniques in future.

2. Theoretical formulations

2.1. Geometrical configuration

Fig. 1 shows a multi-span curved Timoshenko beam system, which consists of multiple curved beams coupled together via a set of joints, which are modeled by two groups of linear springs and one group of rotational springs. The use of the coupling springs between two adjacent curved beams allows accounting for the effects of some non-rigid or resilient connectors. The conventional rigid connectors can be considered as a special case when the stiffnesses of these springs become substantially large with reference to the bending rigidities of the involved curved beams. Each curved beam may also be independently supported on a set of elastic restraints at both ends. All the traditional intermediate supports and classical boundary conditions (i.e., the combinations of the simply supported (S), free (F), and clamped end conditions (C)) can be readily obtained from these general boundary conditions by accordingly setting the stiffness constants of the restraining springs to be equal to zero or infinity.

Fig. 1. A multi-span curved Timoshenko beam subjected to general elastic boundary and coupling conditions

 A multi-span curved Timoshenko beam subjected to general elastic boundary  and coupling conditions

2.2. General boundary and coupling conditions

The Timoshenko model for a curved beam consists of three partial differential equations for the curved beam radial displacement w, the tangential displacement u and the rotation θ due to the bending of a cross section. Thus, The differential equations for the vibration of the ith curved Timoshenko beam can expressed as[16]:

(1)
k i G i A i 2 w i θ i 2 - u i R i θ i - ψ i θ i - E i A i R i u i θ i + w i R i = ρ i A i ω 2 w i ,
(2)
k i G i A i R i w i θ i - ψ i - u i R i + E i A i 2 u i θ i 2 + w i R i θ i = ρ i A i ω 2 u i ,
(3)
k i G i A i w i θ i - ψ i - u i R i + E i I i 2 ψ i θ i 2 = ρ i I i ω 2 ψ i ,

where ω is the angular frequency of the curved beam, ki, Gi, Ai, Ii, ρi and Ri are the shear correction factor, the shear modulus, the cross-sectional area, the second moment of the area, the density of the beams and radius of curvature, respectively. The extensional strain εi, flexural strain κi, and shear strain component γi in the co-ordinate system are expressed in:

(4)
ε i = u i R i θ i + w i R i ,
(5)
κ i = ψ i R i θ i ,
(6)
γ i = w i R i θ i - u i R i - ψ i .

According to the linearly elastic theory, the normal force Ni is linearly related to εi, while the bending moment is proportional to the change in curvature as in the technical theory of beams. The shear force-shear strain relation is the familiar one from curved Timoshenko beam theory. Thus:

(7)
N i = E i A i u i R i θ i + w i R i ,
(8)
Q i = k G i A i w i R i θ i - u i R i - ψ i ,
(9)
M i = E i I i ψ i R i θ i .

From the previous reviews, in this study, the artificial stiffness-like spring technique is adopted to simulate the arbitrary boundary conditions and continuity conditions. With this method, the boundary and continuity conditions can be expressed as follows.

At θi= 0:

(10)
k i , i - 1 c u u i - u i - 1 + k θ 0 u i u i = - N i ,
(11)
k i , i - 1 c w w i - w i - 1 + k θ 0 w i w i = - Q i ,
(12)
K i , i - 1 c θ ψ i - ψ i - 1 + K θ 0 θ i ψ i = M i .

At θi=ϕi:

(13)
k i , i + 1 c u u i - u i + 1 + k θ 1 u i u i = N i ,
(14)
k i , i + 1 c w w i - w i + 1 + k θ 1 w i w i = Q i ,
(15)
K i , i + 1 c θ ψ i - ψ i + 1 + K θ 1 θ i ψ i = - M i .

At the left end (of the first curved beam):

(16)
k θ 0 u 1 u 1 = - N 1 ,
(17)
k θ 0 w 1 w 1 = - Q 1 ,
(18)
K θ 0 θ 1 ψ 1 = M 1 .

At the right end (of the Nth curved beam):

(19)
k θ 1 u N u N = N N ,
(20)
k θ 1 w N w N = Q N ,
(21)
K θ 1 θ N ψ N = - M N ,

where, referring to Fig. 1, ki,jcu and ki,jcw denote the stiffnesses of the linear coupling springs in the θi-direction and zi-direction, and Ki,jcθ denote the stiffness of the rotational coupling spring at the junction of beams i and j, respectively; kθ0ui, kθ0wi, kθ1ui, kθ1wi are the stiffnesses of linear boundary springs, and Kθ0θi, Kθ1θi are the stiffnesses of the rotational boundary springs at the left and right ends of the curved Timoshenko beam i, respectively.

All the conventional (homogeneous) curved beam boundary conditions can be considered as the special cases of Eqs. (10)-(21). For example, the simply supported end condition is easily modeled by simply setting the stiffnesses of the linear springs and rotational springs to be infinity and zero, respectively.

2.3. Admissible displacement functions

The admissible function is the essence of the weak formulation such as the Rayleigh-Ritz method to achieve an accurate, convergent and unified solution. The traditional Fourier series, a well-known form of admissible function for its excellent convergence, is limited to some very simple boundary conditions and would result in the discontinuities of the displacements and their derivatives as well. For the titled problem, the admissible functions are required not only to be regular enough to be differentiable, but also satisfy the geometry boundary conditions and continuity conditions at the junction. Recently, a modified Fourier series technique proposed by Li [22, 23] is widely used in the vibrations of plates and shells with different boundary conditions by Rayleigh-Ritz method, e.g., [24-32]. In this technique, each displacement of the structure under consideration is expressed as a conventional cosine Fourier series with the addition of several supplementary terms. The purpose of introducing the supplementary terms, taking the linear vibration of a classical beam for example, is explained here. Though an exact solution generally exists in the form of sine Fourier series when the beam is with the simply supported ends, it cannot be widely applicable for other boundary conditions. This is because that the original displacements and their derivatives of the edges exist potential discontinuities, in other words, the expanded expressions can’t be differentiated through term-by-term, which will make the solution not converge or converge slowly. The detail illustration is given in Ref. [22]. More information about the improved Fourier series can be seen in Refs. [23-32]. In this formulation, the modified Fourier series technique is adopted and extended to investigate the in-plane vibration of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions.

Combining Eqs. (1)-(3) and (10)-(21), it is obvious that each displacement/rotation component of a multi-span curved Timoshenko beam is required to have up to the second derivative. Therefore, regardless of boundary and coupling conditions, each displacement/rotation component of the curved Timoshenko beams is assumed to be a one-dimensional modified Fourier series as:

(22)
u i θ i , t = m = 0 A m i c o s λ m θ i + l = 1 2 a l i ξ l θ i e j ω t ,
(23)
w i θ i , t = m = 0 B m i c o s λ m θ i + l = 1 2 b l i ξ l θ i e j ω t ,
(24)
ψ i θ i , t = m = 0 C m i c o s λ m θ i + l = 1 2 c l i ξ l θ i e j ω t ,

where j2= –1 and λm=mπ/ϕi. ξlx denote the supplementary terms introduced to remove all the discontinuities potentially associated with the first-order derivatives at the boundaries and then ensure and accelerate the convergence of the series expansion of the curved beam displacement. Ami, Bmi and Cmi are the expansion coefficients of standard cosine Fourier series. ali, bli and cli represent the corresponding expansion coefficients of the supplementary terms ξlx. These two supplementary terms are defined as:

(25)
ξ 1 θ i = ϕ i 2 π s i n π θ i 2 ϕ i + ϕ i 2 π s i n 3 π θ i 2 ϕ i ,
(26)
ξ 2 θ i = - ϕ i 2 π c o s π θ i 2 ϕ i + ϕ i 2 π c o s 3 π θ i 2 ϕ i .

It is easy to verify that:

(27)
ξ 1 0 = ξ 1 ϕ i = ξ 1 ' ϕ i = 0 ,         ξ 1 ' 0 = 1 ,
(28)
ξ 2 0 = ξ 2 ϕ i = ξ 2 ' 0 = 0 ,         ξ 2 ' ϕ i = 1 .

2.4. Energy expressions

For the multi-span curved Timoshenko beams, the total strain energy (V) and kinetic energy (T) can be expressed as:

(29)
V = i = 1 N V b , i + i = 1 N - 1 V i , i + 1 s ,
(30)
T = i = 1 N T b , i ,

where Vb,i and Tb,i represent the strain energy and kinetic energy of the ith curved Timoshenko beams, and Vi,i+1s is the potential energy expression in the connective springs related to ith and i+1th beams. The detailed expression of the Vb,i, Vi,i+1s and Tb,i can be written as:

(31)
V b , i = 1 2 0 ϕ i E i A i u i R i θ + w i R i 2 + E i I i ψ i R i θ i 2 + k G i A i w i R i θ i - u i R i - ψ i 2 R i d θ i
            + 1 2 k θ 0 u i u i 2 + k θ 0 w i u i 2 + K θ 0 θ i ψ i 2 θ = 0 + 1 2 k θ 1 u i u i 2 + k θ 1 w i u i 2 + K θ 1 θ i ψ i 2 θ = ϕ ,
(32)
V i , i + 1 s = 1 2 k i , i + 1 c u u i θ = 0 u i + 1 θ = ϕ 2 + k i , i + 1 c w w i θ = 0 w i + 1 θ = ϕ 2
            + K i , i + 1 c θ ψ i θ = 0 ψ i + 1 θ = ϕ 2 ,
(33)
T b , i = 1 2 0 ϕ i ρ i A i u ¨ i 2 + ρ i A i w ¨ i 2 + ρ i I i ψ ¨ i 2 R i d θ i .

2.5. Solution procedure

Having established the admissible displacement functions and energy functions of the multi-span curved Timoshenko beams, next, the corresponding coefficients of the admissible functions should be determined. In a weak formulation such as the Rayleigh-Ritz technique, however, all the expansion coefficients are considered as the generalized coordinates independent of each other. The strong and weak solutions are mathematically equivalent if they are constructed with the same degree of smoothness over the solution domain. The Rayleigh-Ritz technique will be adopted in this study since the solution can be obtained more easily. More importantly, such a solution process is better suitable for the future modeling of built-up structures.

The Lagrangian for the multi-span curved Timoshenko beams can be generally expressed as:

(34)
L = T - V .

Substituting Eqs. (29) and (30) into the Lagrangian function Eq. (34), taking its derivatives with respect to each of the undetermined coefficients and making them equal to zero:

(35)
L Ξ = 0     and     Ξ = A m i , B m i , C m i , i = 1 ,   2 , , N , m = 1 ,   2 , , M ,             L ϑ = 0     and     ϑ = a l i , b l i , c l i , i = 1 ,   2 , , N , l = 1 ,   2 ,

a total of 3M+3N equations can be obtained and they can be summed up in a matrix form as:

(36)
K - ω 2 M G = 0 ,

where K and M represent the stiffness matrix and the mass matrix of the beam, respectively. They are defined as:

(37)
K = K 1,1 K 1,2 K 1 , i - 1 K 1 , i K 1 , i + 1 K 1 , N - 1 K 1 , N K 2,1 K 2,2 K 2 , i - 1 K 2 , i K 2 , i + 1 K 2 , N - 1 K 2 , N K i , 1 K i , 1 K i , i - 1 K i , i K i , i + 1 K i , N - 1 K i , N K N , 1 K N , 1 K N , i - 1 K N , i K N , i + 1 K N , N - 1 K N , N ,
(38)
M = M 1,1 M 1,2 M 1 , i - 1 M 1 , i M 1 , i + 1 M 1 , N - 1 M 1 , N M 2,1 M 2,2 M 2 , i - 1 M 2 , i M 2 , i + 1 M 2 , N - 1 M 2 , N M i , 1 M i , 1 M i , i - 1 M i , i M i , i + 1 M i , N - 1 M i , N M N , 1 M N , 1 M N , i - 1 M N , i M N , i + 1 M N , N - 1 M N , N ,
(39)
G = G 1 G 2 G i - 1 G i G i + 1 G N - 1 G N T .

The detail expressions for the sub-stiffness and sub-mass matrices are not shown here since they are easy to gain. According to the above formula, the general vibration characteristics of the multi-span curved Timoshenko beam will be obtained. Specifically, the frequencies (or eigenvalues) can be obtained directly by solving the Eq. (36), and the mode shapes will be acquired by substituting the corresponding eigenvectors into the series representations of displacement and rotation components. It should also be noted that the current method is particularly advantageous in obtaining other variables of interest such as power flows. Since the displacements are constructed sufficiently smoothly as required in a strong formulation, post-processing the solution can be done easily through appropriate mathematical operations, including term-by-term differentiations.

3. Numerical results and discussion

In this section, a comprehensive investigation concerning the in-plane free vibration of multi-span curved Timoshenko beams with various boundary and coupling conditions is given to demonstrate the accuracy and reliability of the present method. Throughout these examples, unless otherwise stated, the non-dimensional Ω=ωL12/12ρ1/E1h121/2 is used in the presentation, and the material and geometry properties of all the curved beams under consideration are: ρi= 7800 kg/m3 (i=1, 2,…, N), μi= 0.3 (i=1, 2,…, N), Ei= 2.1×1011 Pa (i= 1, 2,…, N), ϕi= 120° (i= 1, 2,…, N), Ri= 1 m (i= 1, 2,…, N), and ai×bi= 0.005 m×0.005 m (i= 1, 2,…, N).

3.1. Determination of the boundary and coupling spring stiffness

In the present work, the general boundary and coupling conditions are implemented by the artificial stiffness-like spring technique introduced to simulate the boundary forces and displacements, with the help of which, the general boundary and coupling conditions of the multi-span curved Timoshenko beams can be achieved by assigning the proper stiffness to the boundary and coupling springs. Taking a clamped end boundary (C) and rigid coupling (R) conditions for example, it can be realized by simply setting the stiffness of the entire springs to be “infinitely large” which is instead of a sufficiently large number in the actual calculation. So, it’s of great significance to investigate the effects of the spring stiffness of the boundary and coupling spring on the modal characteristics.

Effects of elastic boundary and coupling stiffness parameters on the frequency parameters Ω of two-span curved Timoshenko beams are studied. In Fig. 2, variation of the lowest three frequency parameters ΔΩ versus the elastic boundary and coupling restraint parameters Γλ for two-span curved Timoshenko beams is shown. The elastic boundary restraint parameter Γλ refers to the situation that the beam is completely free at the left end boundary, rigid coupling restraint at the junction and elastically supported at the right end boundary, which is realized by only one group of spring component with the stiffness varying from 10-2 to 1014. According to Fig. 2, we can see that the frequency parameters exist the large change as the stiffness parameters increase in the certain range. In conclusion, the “infinitely large” in the actual calculations can be equal to 1×1014.

Fig. 2. Variation of the frequency parameters ΔΩ versus the elastic restraint and coupling parameters for Timoshenko beams

 Variation of the frequency parameters ΔΩ versus the elastic restraint  and coupling parameters for Timoshenko beams
 Variation of the frequency parameters ΔΩ versus the elastic restraint  and coupling parameters for Timoshenko beams
 Variation of the frequency parameters ΔΩ versus the elastic restraint  and coupling parameters for Timoshenko beams
 Variation of the frequency parameters ΔΩ versus the elastic restraint  and coupling parameters for Timoshenko beams
 Variation of the frequency parameters ΔΩ versus the elastic restraint  and coupling parameters for Timoshenko beams
 Variation of the frequency parameters ΔΩ versus the elastic restraint  and coupling parameters for Timoshenko beams

Then the vibration analysis will be conducted and the frequencies and modal shapes of multi-span curved Timoshenko beams with different boundary conditions including the arbitrary classical, general elastic, general elastic coupling and their combinations will be presented. Here the left and first joint of the beam is taken as the example, considering three types of classical boundary conditions, three types of elastic boundary conditions and four types of coupling boundary conditions which are commonly encountered in engineering practices, the corresponding spring stiffness parameters are given as follows respectively:

Boundary conditions:

C : k θ 0 u 1 = 1 0 14   N / m ,       k θ 0 w 1 = 1 0 14   N / m ,       K θ 0 θ 1 = 1 0 14   N m / r a d , S : k θ 0 u 1 = 1 0 14   N / m ,       k θ 0 w 1 = 1 0 14   N / m ,         K θ 0 θ 1 = 0   N m / r a d , F : k θ 0 u 1 = 0   N / m ,         k θ 0 w 1 = 0   N / m ,         K θ 0 θ 1 = 0   N m / r a d , E 1 : k θ 0 u 1 = 1 0 6   N / m ,         k θ 0 w 1 = 1 0 6   N / m ,         K θ 0 θ 1 = 0   N m / r a d , E 2 : k θ 0 u 1 = 0   N / m ,         k θ 0 w 1 = 0   N / m ,       K θ 0 θ 1 = 1 0 6   N m / r a d , E 3 : k θ 0 u 1 = 1 0 6   N / m ,       k θ 0 w 1 = 1 0 6   N / m ,       K θ 0 θ 1 = 1 0 6   N m / r a d .

Coupling conditions:

R : k 1,2 c u = 1 0 14   N / m ,         k 1,2 c w = 1 0 14   N / m ,         K 1,2 c θ = 1 0 14   N m / r a d , C 1 : k 1,2 c u = 1 0 6   N / m ,         k 1,2 c w = 1 0 6   N / m ,         K 1,2 c θ = 0   N m / r a d , C 2 : k 1,2 c u = 0   N / m ,         k 1,2 c w = 0   N / m ,         K 1,2 c θ = 1 0 6   N m / r a d , C 3 : k 1,2 c u = 1 0 6   N / m ,         k 1,2 c w = 1 0 6   N / m ,         K 1,2 c θ = 1 0 6   N m / r a d .

As previously, the classical boundary conditions are defined in terms of the boundary spring parameter, the appropriateness of which deserves great attention and will be discussed and proved in later sub-sections. Notably, in this paper the boundary conditions of the multi-span curved Timoshenko beam are represented by several simple letter strings introduced to make the expression succinct (seen in the Fig. 3), i.e., CRC1S identifies the three-span curved Timoshenko beams with C, S boundary conditions at the left and right ends boundary of beam, and R, C1 coupling conditions at joint 1 and joint 2, respectively.

3.2. Convergence study

Theoretically, there are infinite terms in the modified Fourier series solution. However, the series is numerically truncated and only finite terms are counted in actual calculations. The excellent convergence of the proposed method will be proved firstly. Considering the single curved Timoshenko beam as an element of the multi-span curved Timoshenko beams, thus the convergence can be studied by just checking the single curved Timoshenko beam’s. In the Table 1, the first six frequency parameters Ω for CC and FF curved Timoshenko beams with eleven truncation schemes (i.e. M= 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) are presented. The frequency parameters of the beams are calculated by MATAB on a notebook. The configuration of the computer is: Inter Core(TM) i7-4970 CPU and 8 GB RAM. It is obvious that the present method has an excellent convergence, and is sufficiently accurate even when only a small number of terms are included in the series expressions. The maximum difference between the M= 12 and M= 18 is less than 0.051 % for the worst case. Besides, from the table, we can see that although the series are truncated as much as 50, the computing time is less 0.09 s. Unless otherwise stated, the truncated number of the displacement expressions will be uniformly selected as M= 12 in the following discussions.

Then the accuracy and reliability of the current formulation will be validated further by some more numerical examples. In each case, the convergence study is performed and for brevity purpose, only the converged results are presented here.

Table 1. Convergence of the first six frequency parameters Ω for a single curved Timoshenko beam with C-C and F-F boundary conditions

M
C C
F F
1
2
3
4
5
6
1
2
3
4
5
6
8
51.48
100.3
183.1
245.8
317.8
381.1
19.87
56.80
114.4
190.8
285.2
397.0
9
51.48
100.2
183.1
245.6
317.5
381.1
19.78
56.80
114.4
190.7
284.9
396.4
10
51.44
100.2
183.0
245.6
317.4
380.8
19.78
56.80
114.4
190.7
284.8
395.9
11
51.44
100.1
183.0
245.6
317.3
380.8
19.76
56.80
114.4
190.7
284.7
395.7
12
51.43
100.1
182.9
245.6
317.3
380.7
19.76
56.80
114.4
190.6
284.7
395.6
13
51.43
100.1
182.9
245.5
317.3
380.7
19.75
56.80
114.4
190.6
284.7
395.5
14
51.42
100.1
182.9
245.5
317.3
380.7
19.75
56.80
114.4
190.6
284.7
395.5
15
51.42
100.1
182.9
245.5
317.2
380.7
19.74
56.80
114.4
190.6
284.6
395.4
16
51.42
100.1
182.9
245.5
317.2
380.7
19.74
56.80
114.4
190.6
284.6
395.4
17
51.42
100.1
182.9
245.5
317.2
380.7
19.74
56.80
114.4
190.6
284.6
395.4
18
51.42
100.1
182.9
245.5
317.2
380.7
19.74
56.80
114.4
190.6
284.6
395.4

3.3. Multi-span curved Timoshenko beams with general boundary and coupling restraints

In this sub-section, multi-span curved Timoshenko beams with general boundary and coupling restraints are investigated. Firstly, the accuracy and reliability of the present method is validated by a verification study about the classical boundary conditions. In Tables 2-4, the first eight frequency parameters Ω with classical boundary and rigid coupling conditions for single curved Timoshenko beam, two-span curved Timoshenko beam and three-span curved Timoshenko beam are presented, respectively. The results obtained from the FEM (ABAQUS) are also listed in the table as the reference, and the two results match very well. The differences between the two results are very small, and do not exceed 0.92 % for the worst case. Next, the in-plane vibration of multi-span curved Timoshenko beams with general elastic restraints will be studied.

Table 2. Frequency parameters Ω for a single curved Timoshenko beam with different classical boundary conditions

Mode
C C
S S
F F
C F
C S
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
50.175
50.366
30.385
30.384
19.779
19.776
3.844
3.843
40.407
40.396
2
103.27
103.55
76.737
76.740
57.139
57.133
16.068
16.070
89.665
89.797
3
184.71
185.32
148.12
148.13
115.69
115.68
53.24
53.24
167.38
167.48
4
280.60
281.13
234.56
234.51
194.28
194.25
111.77
111.77
257.99
257.39
5
394.14
394.77
345.31
345.36
292.72
292.66
190.23
190.25
374.08
374.34
6
537.97
537.33
471.24
471.20
410.94
410.84
288.65
288.68
504.16
503.84
7
678.58
678.24
621.35
621.45
548.88
548.73
406.84
406.83
659.35
660.13
8
871.79
871.95
786.73
786.58
706.52
706.35
544.72
544.76
829.13
828.82
*The FEM results are form ABAQUS software

Table 3. Frequency parameters Ω for a two-span curved Timoshenko beam with different classical boundary conditions

Mode
C R C
S R S
F R F
C R F
C R S
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
6.0095
6.0268
2.6165
2.6790
5.9400
5.9393
0.9826
0.9917
4.8761
4.8775
2
21.271
21.355
15.224
15.228
10.775
10.774
5.583
5.605
17.572
17.619
3
39.781
40.096
30.361
30.384
26.717
26.728
7.471
7.414
35.513
35.452
4
63.996
63.565
54.465
54.546
42.689
42.715
24.178
24.054
60.169
59.783
5
89.462
89.797
76.729
76.740
69.369
69.550
41.879
41.761
83.537
83.629
6
128.51
128.56
112.13
112.27
96.454
96.480
67.750
67.816
119.86
119.86
7
164.83
164.48
148.16
148.13
133.07
133.16
93.253
93.330
157.84
157.63
8
206.60
207.20
786.73
786.58
170.17
170.16
130.83
130.83
201.78
201.98
*The FEM results are form ABAQUS software

Table 4. Frequency parameters Ω for a three-span curved Timoshenko beam with different classical boundary conditions

Mode
C R R C
S R R S
F R R F
C R R F
C R R S
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
3.7242
3.7150
1.2770
1.2829
2.4610
2.4609
0.5704
0.5730
2.0351
2.0198
2
6.6241
6.6359
3.7794
3.7831
7.1872
7.1861
2.0940
2.0782
5.7446
5.7586
3
12.917
12.963
11.519
11.524
7.9489
7.9423
5.0785
5.0696
12.107
12.078
4
25.576
25.561
20.204
20.192
17.296
17.304
7.2865
7.3351
22.485
22.495
5
37.095
37.055
30.397
30.385
28.778
28.758
15.235
15.241
34.004
33.939
6
53.729
53.758
46.921
46.902
39.009
39.023
27.269
27.243
50.196
50.189
7
71.283
71.390
62.487
62.495
54.833
54.815
38.089
38.092
66.462
66.550
8
85.488
85.646
76.729
76.751
73.932
73.921
54.281
54.296
81.550
81.701
*The FEM results are form ABAQUS software

Tables 5-7 show the first eight frequency parameters Ω of single curved Timoshenko beam, two-span curved Timoshenko beam and three-span curved Timoshenko beam subjected to classical-elastic restraints and elastic boundaries, respectively. Besides, due to the lack of the open reported reference results and to be used as the comparison, the contrast results obtained using an FEM (ABAQUS) model are also given in Tables 5-7. An excellent agreement is achieved between the current and the FEM solutions. Finally, the in-plane vibrations of multi-span curved Timoshenko beams with general elastic boundary and coupling restraints are presented.

Table 5. Frequency parameters Ω for a single curved Timoshenko beam with different elastic boundary conditions

Mode
C E 1
C E 2
E 1 E 1
E 2 E 2
E 3 E 3
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
40.422
40.392
6.0279
6.0268
30.379
30.379
10.411
10.412
51.944
51.944
2
89.646
89.757
21.352
21.355
76.685
76.689
34.648
34.649
103.37
103.38
3
167.37
167.43
66.572
66.565
148.01
148.03
83.464
83.464
188.04
188.05
4
257.27
257.10
128.58
128.56
234.05
234.08
151.77
151.78
279.93
279.95
5
373.97
374.04
213.19
213.20
344.79
344.84
240.53
240.56
403.43
403.48
6
502.77
502.62
315.79
315.76
469.28
469.36
348.80
348.84
532.41
534.42
7
658.81
659.25
439.47
439.62
619.61
619.72
477.05
477.08
695.95
696.19
8
824.37
824.33
581.93
581.93
778.92
779.14
624.80
624.91
849.83
850.06
*The FEM results are form ABAQUS software

Table 6. Frequency parameters Ω for a two-span curved Timoshenko beam with different elastic boundary conditions

Mode
C R E 1
C R E 2
E 1 R E 1
E 2 R E 2
E 3 R E 3
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
4.9212
4.9475
0.5005
0.8976
2.6791
2.6790
2.6791
2.6790
6.0279
6.0265
2
17.533
17.618
6.039
6.089
15.228
15.227
10.411
10.412
21.352
21.352
3
35.505
35.450
10.746
10.845
30.397
30.381
15.235
20.407
40.370
40.389
4
60.124
60.132
27.821
27.599
54.539
54.539
34.666
34.649
66.536
66.547
5
83.395
83.615
48.025
47.940
76.729
76.715
54.539
54.546
89.720
89.742
6
119.87
119.84
73.712
73.815
112.21
112.23
83.464
83.464
128.47
128.47
7
157.84
157.83
101.57
101.58
148.09
148.09
112.24
112.27
167.33
167.36
8
201.71
202.02
139.77
139.85
191.66
191.66
151.77
151.78
212.97
213.00
*The FEM results are form ABAQUS software

Table 7. Frequency parameters Ω for a three-span curved Timoshenko beam with different elastic boundary conditions

Mode
C R R E 1
C R R E 2
E 1 R R E 1
E 2 R R E 2
E 3 R R E 3
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
2.0251
2.0198
0.8059
0.8075
1.2843
1.2830
1.2843
1.2829
3.7169
3.7147
2
5.7442
5.7586
2.8042
2.7927
3.7831
3.7831
3.7831
3.7831
6.6352
6.6359
3
12.053
12.078
5.2625
5.2651
11.519
11.524
10.415
10.411
12.954
12.962
4
22.500
22.495
10.304
10.306
20.204
20.191
11.519
11.525
25.540
25.558
5
33.976
33.938
16.707
16.710
30.397
30.383
20.204
20.191
37.058
37.051
6
50.200
50.189
29.551
29.539
46.884
46.899
34.666
34.649
53.729
53.751
7
66.460
66.543
42.468
42.453
62.487
62.487
46.884
46.902
71.356
71.371
8
81.543
81.694
58.182
58.208
76.7 29
76.733
62.487
62.491
85.598
85.609
*The FEM results are form ABAQUS software

In Tables 8 and 9, the detailed comparisons between results obtained by the present method and those provided by FEM solutions (ABAQUS) are presented, in which two types of multi-span curved Timoshenko beams (two-span curved Timoshenko beam and three-span curved Timoshenko beam) are included. It’s very clear that the current results have a great agreement with the reference data. In order to improve our comprehension of the effects of elastic boundary and coupling restraints on vibration characteristic of multi-span curved Timoshenko beams. The first six mode shapes of the single curved Timoshenko beam, two-span curved Timoshenko beam and three-span curved Timoshenko beam with different boundary and coupling restraints are given in Figs. 4-6, respectively. It can be seen that the elastic boundary and coupling restrain have a quite significant effect on the vibration characteristics of the beam structures.

Fig. 3. A simple letter string of a multi-span curved Timoshenko beam

 A simple letter string of a multi-span curved Timoshenko beam

Fig. 4. The lowest six mode shapes for single curved Timoshenko beams with different boundary conditions

 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for single curved Timoshenko beams  with different boundary conditions

According to the above analysis, it can be seen that the current method is reliable to make correct predictions of the modal characteristics for the multi-span curved Timoshenko beam with the elastic restraint boundary and coupling conditions as well as the classical boundary and rigid coupling conditions. It should be noted that for sake of simplifying the research, only the three-span curved Timoshenko beam is studied in this paper, but it doesn’t mean the current method is restricted to the three-span. Through the theoretical formulations, it can be seen that when the number of the curved beams is added, which merely increases the dimensional of the stiffness matrix and mass stiffness, the corresponding analysis can be easily obtained.

Table 8. Frequency parameters Ω for a two-span curved Timoshenko beam with different elastic coupling conditions

Mode
C C 1 C
C C 2 C
E 1 C 1 E 1
E 2 C 2 E 2
E 3 C 3 E 3
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
3.7459
3.8427
3.7635
3.8427
0.0468
0.0471
0.0147
0.0152
6.0279
6.0261
2
16.066
16.068
6.0004
6.0261
10.775
10.773
0.0147
0.0160
21.352
21.351
3
40.316
40.392
16.149
16.068
30.379
30.380
0.0736
0.0573
40.370
40.385
4
53.236
53.221
21.392
21.355
42.709
42.700
0.0736
0.0581
66.536
66.547
5
89.610
89.786
53.002
53.221
76.699
76.707
5.9249
5.9382
89.720
89.727
6
111.74
111.75
66.666
66.565
96.46
96.43
10.415
10.411
128.47
128.48
7
167.54
167.48
111.78
111.75
148.05
148.07
26.717
26.724
167.33
167.35
8
190.40
190.20
128.02
128.56
170.09
170.04
34.666
34.648
212.97
213.02
*The FEM results are form ABAQUS software

Table 9. Frequency parameters Ω for a three-span curved Timoshenko beam with different elastic coupling conditions

Mode
C C 1 C 1 C
C C 2 C 2 C
E 1 C 1 C 1 E 1
E 2 C 2 C 2 E 2
E 3 C 3 C 3 E 3
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
Present
FEM*
1
2.9826
2.9699
3.5457
3.5304
7.1096
7.1025
3.9340
3.9321
3.7152
3.7147
2
3.0338
3.0255
4.8516
4.8533
14.4988
14.4950
7.9195
7.9195
6.6360
6.6359
3
9.8554
9.9093
7.5732
7.5835
30.380
30.382
10.415
10.411
12.962
12.962
4
17.469
17.459
16.892
16.881
37.192
37.191
23.442
23.421
25.557
25.558
5
36.253
36.227
19.151
19.127
48.616
48.599
30.250
30.253
37.045
37.047
6
46.308
46.244
28.191
28.247
76.703
76.715
34.666
34.649
53.745
53.747
7
54.380
54.373
54.312
54.480
87.846
87.843
63.628
63.588
71.362
71.367
8
84.173
84.292
60.592
60.555
105.051
105.007
75.809
75.795
85.574
85.591
*The FEM results are form ABAQUS software

Fig. 5. The lowest six mode shapes for two-span curved Timoshenko beams with different boundary conditions

 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for two-span curved Timoshenko beams  with different boundary conditions

Fig. 6. The lowest six mode shapes for three-span curved Timoshenko beams with different boundary conditions

 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions
 The lowest six mode shapes for three-span curved Timoshenko beams  with different boundary conditions

4. Conclusions

A unified method is presented for in-plane vibration analysis of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions. Each of the displacements and rotations of every curved Timoshenko beam, is expressed as a modified Fourier series, which is constructed as the linear superposition of a standard one-dimensional Fourier cosine series supplemented with auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the ends and accelerate the convergence of series representations. All the expansion coefficients are determined by the Rayleigh-Ritz technique as the generalized coordinates. The excellent accuracy and reliability of the current solutions are confirmed by comparing the present results with FEM solution, and numerous new results for multi-span curved Timoshenko beams with various classical cases, classical-elastic restraints, and elastic boundary and coupling conditions, are presented, which can be served as the benchmark solutions for other computational techniques in the future research.

Acknowledgements

The authors would like to thank the anonymous reviewers for their useful and constructive comments.

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