Shock and Vibration

Volume 2017 (2017), Article ID 8186976, 23 pages

https://doi.org/10.1155/2017/8186976

## A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods

Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat-Elzaton, Cairo, Egypt

Correspondence should be addressed to Tamer Ahmed El-Sayed

Received 2 June 2017; Revised 22 August 2017; Accepted 15 October 2017; Published 28 November 2017

Academic Editor: Toshiaki Natsuki

Copyright © 2017 Tamer Ahmed El-Sayed and Said Hamed Farghaly. 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 exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions. This set of solutions is derived to normalize the solution at the origin of the coordinates. The start, end, and intermediate boundary conditions involve concentrated masses and linear and rotational elastic supports. The beam start, end, and intermediate equations are assembled using the present normalized transfer matrix (NTM). The advantage of this method is that it is quicker than the standard method because the size of the complete system coefficient matrix is 4 × 4. In addition, during the assembly of this matrix, there are no inverse matrix steps required. The validity of this method is tested by comparing the results of the current method with the literature. Then the validity of the exact stepped analysis is checked using experimental and FE(3D) methods. The experimental results for stepped beams with single step and two steps, for sixteen different test samples, are in excellent agreement with those of the three-dimensional finite element FE(3D). The comparison between the NTM method and the finite element method results shows that the modal percentage deviation is increased when a beam step location coincides with a peak point in the mode shape. Meanwhile, the deviation decreases when a beam step location coincides with a straight portion in the mode shape.

#### 1. Introduction

Stepped beam-like structure plays an important role in the construction of mechanical and civil engineering systems. Flexural vibrations were first investigated by Euler-Bernoulli in the eighteenth century. The rotary inertia effect was considered by Rayleigh [1]. Almost 95 years ago, Timoshenko introduced a correction for the beam theory to include the shear deformation effect [2]. The effect of rotary inertia and shear deformations on the beam natural frequencies is small at the lower normal modes and large at the higher normal modes. Cowper [3] derived formulae for the precise evaluation of the shear coefficient for rectangular and round cross sections as a function of Poisson’s ratio .

The free vibrations of beams with discontinuities can be solved using either exact or approximate solution. The exact methods include the derivation of the transcendental eigenvalue equations in order to evaluate the beam natural frequencies. In the case of relatively simple problems closed form solutions for the eigenvalue problem were obtained [4, 5].

For complicated problems, the beam eigenvalues are obtained by the decomposition of the domain. Numerical assembly technique (NAT) is one of the common methods used for the evaluation of the eigenvalue problem for beams with multiple discontinuity [6–8]. In this method, the size of the frequency equation determinant is for beam with segments. Dynamic stiffness matrix is one of methods that is similar to the finite element method in assembling of the elements but with exact element rather than approximate element [9, 10]. The frequency equation determinant size for segments is . The Laplace transformation method is used to obtain a solution for a Timoshenko beam mounted on elastic foundation with several combinations of discrete in-span attachments and with several combinations of attachments at the boundaries [11]. The attachments include translation and rotational springs, masses, and undamped single degree of freedom system. The characteristics equation for beam with segments is . The transfer matrix method (TMM) is one of the favorable methods in the analysis of multispan beams. Many researchers used it to derive the frequency equation of a complicated beam system [12–14]. The advantage of this method is that, for segments beam, the size of the frequency equation determinant is which reduces the computational time. On the other hand, during the formulation of the beam frequency equation inverse matrix steps are required to form the final system transcendental eigenvalue problem. The use of linearly independent fundamental set of solution in solving the buckling and free vibrations of nonuniform rods was introduced by Li [15] and Li et al. [16]. This method enables obtaining the closed form solution of a multispan beam. A set of fundamental solutions which suit the analysis of single-span Timoshenko beams was introduced [17].

On the other hand, there are numerous approximate methods to approximate the eigenvalue problem for the transverse vibrations of beams [21–23]. Finite element method is one of the most dominant methods for solving the free vibration of beams. The effect of step ratios and eccentricity on the free vibration of arbitrarily beam was investigated by Ju et al. [21]. The lowest three natural frequencies of a multistep up and down cantilever beam using a global Rayleigh-Ritz formulation, component modal analyses (CMA), ANSYS®, and experimental are evaluated [24]. Adomian decomposition method (ADM) was used to obtain the effect of step ratio and step location on the beam natural frequencies [25, 26]. The free and forced vibrations of beams with either single- or multiple-step changes using the composite element method (CEM) were introduced by Lu et al. [27]. The accuracy and convergence of CEM were compared with existing theoretical and experimental results. Differential transformation method (DTM) was applied in order to analyze the natural frequencies for different geometrically and material parameters stepped Bernoulli-Euler beam [28]. The differential quadrature element method (DQEM) was proposed to analyze the free vibration problem of beams with any discontinuities in cross-section [20]. Discrete singular convolution (DSC) was proposed for solving the free vibration analysis of stepped beams [29]. The solution for multistep Timoshenko beam using both numerical assembly technique and differential transformation method is proposed by Yesilce [19]. Experimental measurements were considered as a good tool for the validation of the analytical results.

The exact free vibration of two-span Timoshenko stepped beams has been investigated by Gutierrez et al. [18] and Rossi et al. [32]. Farghaly [33] derived the exact solution for four-span Timoshenko beam with attachments. Yesilce [19] investigated the free vibration of stepped beams using exact numerical assembly technique (NAT) and using approximate differential transformation method. Recently, Farghaly and El-Sayed [6] drive the exact solution for the lateral vibration of Timoshenko beam with generalized start, end, and intermediate conditions using numerical assembly technique (NAT).

The exact free vibration of a mechanical system composed of two elastic Timoshenko segments carried on an intermediate eccentric rigid body or on elastic supports was introduced by Farghaly and El-Sayed [34]. Their analysis was based on both analytical and experimental methods. They claimed a good agreement between the analytical and experimental results. Experimental setup using electromagnetic-acoustic transducer (EMAT) was introduced by Díaz-De-Anda et al. [35]. They compared the experimental results with those obtained theoretically using Timoshenko beam theory (TBT) with one and two shear coefficients. The flexural frequencies and amplitudes for cylindrical and rectangular Timoshenko beams were examined experimentally [35]. They found that the experimental results coincide very well with theoretical predictions. The transverse vibration of Bernoulli-Euler beams with discontinuous geometry and elastic support was investigated experimentally and analytically [36].

During the last decades, many literatures were focused on the problem of free and forced vibration analysis of Timoshenko beam and the accuracy of the natural frequency predictions. To the authors’ knowledge, there is not enough research that has tackled the experimental modal frequencies of stepped thick beams, computationally and experimentally. Therefore, the main aim of this work is to investigate the results of the modal frequencies for such beams using analytical, experimental, and the three-dimensional finite element FE(3D). An analytical analysis is proposed which is based on the derivation of a set of fundamental solutions that suits the analysis of Timoshenko beams. This set of solutions is used to modify the TMM to include no inverse matrix procedure which may be called normalized transfer matrix method (NTM). The comparison between the experimental NTM and FE(3D) is done for selected single-step and two-step application models. The percentage deviations between NTM and FE(3D) are investigated. The results show that the finite element results are very close to the experimental results. The study includes the effect of increasing the step ratio , step location parameter , and the length ratio. Finally, the capability of the present analysis to solve the free vibration of tapered beams has been investigated.

#### 2. Mathematical Model

The mathematical model for beam with multiple-stepped sections is shown in Figure 1. The total length of the beam is . The beam model is divided into segments. The beam has stations as shown in Figure 1. The station numbering corresponding to the start, intermediate, and end location is represented by , respectively. At each station, there are linear and rotational elastic supports and concentrated mass with mass moment of inertia. As shown in Figure 1, the beam segments are described by their material and cross-sectional properties and , which are the density, Young’s modulus of elasticity, rigidity modulus, cross-sectional area, and second moment of inertia, respectively. In this section, the frequency equation of the model is driven using the proposed normalized transfer matrix method. Since the current analysis is based on Timoshenko beam theory, the rotary inertia and shear deformations effects are considered. In the current analysis, the analytical solution is subject to the assumptions that the shear strain is assumed constant over the cross-section; therefore, a shear coefficient is used to compensate this assumption. In addition, the effect of stress concentration at the beam steps is neglected.