Shock and Vibration

Volume 2017 (2017), Article ID 3924921, 26 pages

https://doi.org/10.1155/2017/3924921

## Forced Vibration of a Timoshenko Beam Subjected to Stationary and Moving Loads Using the Modal Analysis Method

Department of Mechanical Engineering, Inha University, 100 Inha-ro, Nam-gu, Incheon 402-751, Republic of Korea

Correspondence should be addressed to Usik Lee

Received 30 September 2016; Accepted 7 November 2016; Published 4 January 2017

Academic Editor: Tony Murmu

Copyright © 2017 Taehyun Kim et al. 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.

#### Abstract

The modal analysis method (MAM) is very useful for obtaining the dynamic responses of a structure in analytical closed forms. In order to use the MAM, accurate information is needed on the natural frequencies, mode shapes, and orthogonality of the mode shapes a priori. A thorough literature survey reveals that the necessary information reported in the existing literature is sometimes very limited or incomplete, even for simple beam models such as Timoshenko beams. Thus, we present complete information on the natural frequencies, three types of mode shapes, and the orthogonality of the mode shapes for simply supported Timoshenko beams. Based on this information, we use the MAM to derive the forced vibration responses of a simply supported Timoshenko beam subjected to arbitrary initial conditions and to stationary or moving loads (a point transverse force and a point bending moment) in analytical closed form. We then conduct numerical studies to investigate the effects of each type of mode shape on the long-term dynamic responses (vibrations), the short-term dynamic responses (waves), and the deformed shapes of an example Timoshenko beam subjected to stationary or moving point loads.

#### 1. Introduction

The dynamic analysis of elastic structures subjected to moving loads (or masses) has been an interesting research topic in structural engineering. When moving loads are applied to a structure, dynamic deflections and stresses may become considerably higher than those induced by static loads. Because of these characteristics of moving load problems, various structures subjected to moving loads have been investigated including beams, bridges, railroads, highway structures, pavement, and overhead cranes. The discussion in this study will be limited to the flexural one-dimensional (1D) beam structures.

To examine the transverse vibrations of a 1D beam structure, the Timoshenko beam model has been widely adopted to take into account the effects of shear deformation and rotatory inertia on the dynamic responses. In transverse vibration analysis, various solution techniques have been described in the literature including MAM or eigenfunction expansion methods [1, 2], mode summation methods or assumed mode methods [3–5], semianalytical methods [6–11], integral transform methods (Laplace-Carson transform and Fourier transform) [12–14], transfer matrix method [15], Lagrange multiplier methods [16, 17], Galerkin methods [18, 19], finite element methods [20, 21], finite difference method [22], time-domain spectral element method [23], and frequency-domain spectral element method [24].

In order to obtain analytical closed-form solutions for a moving load problem by using the MAM, information is needed regarding the eigensolutions (natural frequencies and mode shapes) and the orthogonality properties of the mode shapes. To obtain the eigensolutions for a Timoshenko beam subjected to specific boundary conditions, we begin by obtaining general solutions for the corresponding free vibration problem. Many researchers have developed general solutions of the transverse vibrations of a Timoshenko beam including Traill-Nash and Collar [25], Huang [26], and Han et al. [27]. In [25–27], the general solutions are obtained for two frequency ranges, and , excluding the cutoff frequency . van Rensburg and van der Merwe [28] seemed to be the first to present general solutions for three frequency ranges, , , and . Leissa and Qatu [29] also presented the same general solutions for three frequency ranges. However, in this study, we develop a new expression of general solutions for two frequency ranges, and , including the cutoff frequency .

By imposing the boundary conditions for a specific problem on the general solutions, we obtain eigensolutions for the specific problem. In this study, we limited our consideration to simply supported (hinged-hinged or pinned-pinned) boundary conditions. For simply supported Timoshenko beams, Traill-Nash and Collar [25] first reported the appearance of a “second frequency spectrum” when the vibration frequency is larger than a specific frequency known as the cutoff frequency . They suggested that the pure shearing oscillation may occur at . However, they did not present the natural frequencies in explicit analytical form. Dolph [30] presented the natural frequencies and mode shapes for both bending and shear vibrations in explicit analytical form. However, he did not present the mode shapes at . Though Huang [26] presented the mode shapes for a simply supported Timoshenko beam, the mode shapes fail to satisfy the boundary conditions for bending moment as criticized by van Rensburg and van der Merwe [28]. Han et al. [27] derived the natural frequencies and mode shapes for bending and transverse shear vibrations in explicit analytical form and discussed the MAM used to obtain forced vibration responses. However, they did not investigate whether there are mode shapes at or not. van Rensburg and van der Merwe [28] derived the mode shapes for the bending and transverse shear vibrations by determining the coefficients of assumed mode shapes needed to satisfy governing equations. They reported that itself is a natural frequency and presented the mode shape at , which has been recognized as the “pure shear mode” [31]. However, they did not present natural frequencies in explicit forms. Thus, in this study, we presented a complete set of natural frequencies and mode shapes for all frequency ranges in explicit forms.

To apply the MAM to a forced vibration analysis of a Timoshenko beam, the orthogonality properties of mode shapes are essential. For simply supported Timoshenko beams, Dolph [30] derived the orthogonality properties of mode shapes and other researchers [1, 32, 33] used the orthogonality properties derived by Dolph [30] for the modal analysis of forced vibration problems. However, Dolph [30] and other researchers [1, 32, 33] did not consider the orthogonality of the mode shapes at . Although van Rensburg et al. [34] mentioned the existence of a mode shape at , they did not include it in their free vibration analysis of a simply supported Timoshenko beam. Roux et al. [31] included the pure shear mode shape at in a series solution of the free vibration of a simply supported Timoshenko beam, but they did not apply the orthogonality properties of pure shear mode shape at to determine the coefficients of the series solution. Based on our literature survey of the modal analysis of forced vibrations of simply supported Timoshenko beams, we find that there have been no reports in which the pure shear mode shape at is considered in themodal analysis of the forced vibrations of simply supported Timoshenko beams. We also find that there have been no reports in which the vibrations of a simply supported Timoshenko beam induced by a stationary or moving bending moment are considered by using the MAM. Thus, in this study, we present the closed-form solutions of a simply supported Timoshenko beam subjected to stationary or moving bending moment, including the pure shear mode shape at .

In this study, we discuss the mathematical formulation of the general solutions of the free vibration of a Timoshenko beam subjected to arbitrary boundary conditions in Section 2. The general solutions are presented for frequency ranges and . We then derive natural frequencies and mode shapes in explicit forms for the case of simply supported boundary conditions. Finally, we present the orthogonality properties of the mode shapes. In Section 3, we describe the MAM for the forced vibration of a simply supported Timoshenko beam subjected to arbitrary initial conditions and to stationary or moving loads (a point transverse force and a point bending moment). In Section 4, we describe our numerical results. Lastly, in Section 5, we present concluding remarks.

#### 2. Mathematical Theory

##### 2.1. Mathematical Model of a Timoshenko Beam

The governing equations for a Timoshenko beam of length can be written in a matrix form as [35]where is the transverse displacement, is the rotation of the cross section due to bending, is the external transverse force, is the external bending moment, is Young’s modulus, is the shear modulus, is the mass density, is the shear coefficient factor, is the cross-sectional area, and is the area moment of inertia. The natural and geometric boundary conditions relevant to (1) are given bywhere and are the resultant transverse shear force and bending moment, respectively, defined byFinally, the initial conditions are given by

##### 2.2. General Solutions

To derive the eigenfunctions (natural modes) for a Timoshenko beam, we must first obtain the general solutions for the free vibration problem. Thus, we consider the homogeneous governing equation reduced from (1) as follows:The solution of (7) is assumed to be in the following form:where is the imaginary unit and is the angular frequency. By substituting (8) into (7), an eigenvalue problem is obtained as follows:

We assume that the solutions of (7) are in the following form:where denotes the wavenumber. Substituting (10) into (9) gives algebraic equations as follows:For the existence of nontrivial solutions, the determinant of the two-by-two matrix in (11) must vanish at certain values of , that is, at eigenvalues. From this condition, a dispersion equation is obtained as follows:In order to obtain the four eigenvalues, the above quartic equation can be reduced to a quadratic equation by replacing with (where ). By solving this quadratic equation, we can obtain four eigenvalues as follows:where , , and are always real numbers and is the cutoff frequency defined by

By using the four eigenvalues given by (13), the general solutions of (9) can be written as follows:(i)When (ii)When

By substituting each eigenvalue into (11), we obtain the ratios and . By using the results, (16) and (17) can be rewritten in terms of sinusoidal and hyperbolic functions as follows:(i)When (ii)When where is the radius of gyration and

The present expression of general solutions given by (18) and (19) is equivalent to the expression for three frequency ranges, , , and , by van Rensburg and van der Merwe [28].

##### 2.3. Natural Frequencies and Mode Shapes

To obtain the natural frequencies and mode shapes in analytical closed forms for specific boundary conditions, we considered three frequency ranges separately: (a) , (b) , and (c) . Our study was limited to the simply supported boundary conditions represented by

###### 2.3.1. When

Substituting (18) into (21) gives a matrix equation as follows:where

From the first and second relations in (22), we obtain . Then, from the third and fourth relations of (22), we obtainFor the existence of nontrivial solutions of , we obtain a characteristic equation from (24) as follows:

Since , , and , if , then the following condition can be obtained from (25):From (26), we obtainApplying (27) to (15a), (15b), and (15c) yields natural frequencies as follows:whereNote that is the maximum value of integer satisfying , which can be determined from (28) in closed form as follows:

To obtain the mode shapes corresponding to the natural frequencies , we can determine and by substituting (27) into (24) as follows:The th mode shape corresponding to can then be obtained from (18) in the following form:where

###### 2.3.2. When

Substituting (19) into (21) gives the following matrix equation:where

From the first and second relations in (34), we find . Then, the third and fourth relations in (34) can be written asFor the existence of nontrivial solutions of , we can obtain a characteristic equation from (36) as Since and , if , the following two conditions can be obtained from (37):or

In Section 2.3.1, we derived the natural frequencies and mode shapes for from the same condition given by (38). Thus, the natural frequencies and mode shapes for can be obtained as follows:

From the second condition (39), we obtainBy substituting (41) into (15), we can obtain the natural frequencies as follows:where and are defined in (29). To derive the mode shapes corresponding to , we can determine and by substituting (41) into (36) as follows:The mode shapes corresponding to are then obtained from (19) as follows: where

###### 2.3.3. When

The general solution at can be readily obtained from either (18) or (19), by allowing to approach , as follows:whereTo obtain the first two terms in (46) from either (18) or (19), L’Hospital’s rule is applied.

Applying the simply supported boundary conditions given by (21) to (46) yields the following eigenvalue problem:The necessary condition for the existence of a nontrivial solution of (48) (i.e., the determinant of the matrix of eigenvalue problem must vanish at the eigenvalue) is self-satisfied. Thus, we conclude that the cutoff frequency is also a natural frequency of a simply supported Timoshenko beam, which was described by van Rensburg and van der Merwe [28]. Now we must determine the mode shape corresponding to the natural frequency .

From (47), note that and . Thus, from (48), it can be shown that the following should be satisfied: andTo satisfy (49), we consider the following two cases.

*Case 1*. If is not an integer, then . In this case, the corresponding mode shape can be derived directly from (46) as follows:This mode shape is identical to the pure shear mode shape presented by van Rensburg and van der Merwe [28]. Accordingly, the subscript is adopted in (50) to emphasize the pure shear mode shape.

*Case 2*. If is an integer (i.e., ), then . In this case, the natural frequency happens to be equal to the natural frequency of a bending mode shape and they become a double natural frequency. The mode shapes at this double natural frequency are given by

Thus, for the modal analysis of the transverse vibrations of a simply supported Timoshenko beam subjected to a stationary or moving load, we need to consider the following three types of mode shapes:where .

##### 2.4. Orthogonality of Mode Shapes

For the modal analysis, we must derive the orthogonality properties of the mode shapes given by (52). Because any set of natural frequencies and mode shapes are the eigensolutions of the eigenvalue problem represented by (9), the th and th sets of eigensolutions must satisfy the following two equations separately as follows:

From (54), we obtainBy applying the simply supported boundary conditions, the left-hand side of (54) vanishes. Then, by using the definition of in (3), the right-hand side of (54) can be rewritten as

From (55), the orthogonality property of mode shapes with respect to can be derived as follows:where represent the Kronecker delta symbol [36] and is the modal mass. By using (56), we can derive the orthogonality property of mode shapes with respect to , from (53), as follows:

By substituting each mode shape given by (52) into (56), it can be shown that the following orthogonality properties are satisfied:where the modal masses are defined by

To derive normalized mode shapes (i.e., the normal modes) from (52), all modal masses given by (64) are set to unit value as follows: . Then, from (64), the coefficients of each normal mode shape are determined as follows:

#### 3. Modal Analysis of Forced Vibration

The forced vibration responses of (1) can be represented by using the normal mode summation method [37] as follows:where , , and are generalized coordinates to be determined in order to satisfy initial conditions.

Substituting (61) into (1) and applying the orthogonality conditions of the normal mode shapes yield the modal equations as follows:where the generalized forces are defined by

The initial conditions for (62) can be derived from (6) by using the orthogonality properties of normal mode shapes as follows:where

By using (3), (6), and (52), we can write the initial conditions for as

From (63) and (65), we note that the pure shear mode shape must be included in the modal analysis when a Timoshenko beam is subjected to external bending moment , initial rotation , and initial angular velocity . However, there have been no reports in the literature in which the external transverse force , bending moment , and arbitrary initial conditions were fully considered in the modal analysis of forced vibrations by taking into account the pure shear mode shape . In this study, we derived the vibration responses of a Timoshenko beam for two cases:(1)Case 1: when the beam is subjected to a stationary impulsive point transverse force and a stationary impulsive point bending moment.(2)Case 2: when the beam is subjected to a moving point transverse force and a moving point bending moment.

##### 3.1. Case 1: Stationary Impulsive Point Transverse Force and Bending Moment

As shown in Figure 1(a), a stationary impulsive point transverse force and a stationary bending moment acting on the arbitrary positions and of the beam can be expressed by employing Dirac delta functions and [36] as follows:where is the magnitude of the transverse impulsive point force and is the magnitude of the impulsive point bending moment.