Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7314280, 12 pages

http://dx.doi.org/10.1155/2016/7314280

## Flexural Free Vibrations of Multistep Nonuniform Beams

Department of Road and Bridge, College of Transportation, Jilin University, Changchun 130022, China

Received 28 June 2016; Revised 20 September 2016; Accepted 11 October 2016

Academic Editor: Salvatore Caddemi

Copyright © 2016 Guojin Tan 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

This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as and was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.

#### 1. Introduction

Beams with nonuniform cross section are widely used in various engineering fields, such as bridges, tall buildings, and helicopter rotor blades. A large number of studies can be found in literature about the free vibrations of nonuniform beams. Vibration problems of beams with nonuniform cross section are often described by partial differential equations and in most cases it is extremely difficult to find their closed form solutions. Consequently, a wide range of approximate and numerical solutions such as Rayleigh-Ritz, Galerkin, finite difference, finite element, and spectral finite element methods have been used to obtain the natural vibration characteristics of variable-section beams [1–4].

Huang and Li investigated the free vibration of axially functionally graded beams with variable flexural rigidity and mass density [5]. A novel and simple approach was presented to solve modal shapes and corresponding frequencies through transforming traditional fourth-order governing differential equation into Fredholm integral equation. Koplow et al. proposed an analytical solution for the dynamic response of Euler-Bernoulli beams with step changes in cross section, which was verified by experimental tests and receptance coupling methods [6]. Firouz-Abadi et al. studied the transverse free vibrations of a class of variable-cross section beams using Wentzel-Kramers-Brillouin (WKB) approximation [7]. The governing equation of motion for the Euler-Bernoulli beam including axial force distribution was utilized to obtain a singular differential equation in terms of the natural frequency of vibration and a WKB expansion series was applied to find the solution. Inaudi and Matusevich investigated longitudinal vibration problems of variable-cross section rods using an improved power series method [8]. This method introduced domain partition implementation in matrix formulation, as an alternative to other power series techniques in vibration analysis. Therefore, the method solved linear differential equations efficiently up to a desired degree of accuracy and remedies two limitations of the conventional power series method. The Adomian decomposition method (ADM) is employed to investigate the free vibrations of tapered Euler-Bernoulli beams with a continuously exponential variation of width and a constant thickness along the length under various boundary conditions [9]. Duan and Wang demonstrated the free vibration of beams with multiple step changes using the modified discrete singular convolution (DSC) [10]. The jump conditions at the steps were used to overcome the difficulty in using ordinary DSC for dealing with ill-posed problems. A transfer matrix method and the Frobenius method were adopted by J. W. Lee and J. Y. Lee to solve the free vibration characteristics of a tapered Bernoulli-Euler beam and obtain the power series solution for bending vibrations [11].

Nevertheless, besides all advantages of such numerical methods, exact solutions can provide adequate insight into the physics of the problems and convenience for parametric studies. The other advantage of exact solutions is their significance in the field of inverse problems. An exact solution can be more useful than numerical solutions to design the characteristics and damage identification of a structure.

Wang derived the closed form solutions for free vibration of a flexural bar with variably distributed stiffness but uniform mass [12]. Using a systematic approach, Abrate obtained a closed form solution of longitudinal vibration for rods whose cross section varies as [13]. Kumar and Sujith found the exact solutions for longitudinal vibration of nonuniform rods whose cross section varies as and [14]. Li presented an exact approach for free longitudinal vibrations of one-step nonuniform rod with classical and nonclassical boundary conditions [15]. The approach assumed that the distribution of mass is arbitrary, and distribution of longitudinal stiffness is expressed as a functional relation with mass distribution and vice versa. Li et al. obtained exact solutions of flexural vibration for beam-like structures whose moment of inertia and mass per unit length vary as , and , , respectively [16, 17].

The components whose moment of inertia and mass per unit length satisfy and are widely used in civil engineering, such as the bridge with cross section height varying as . This kind of beam-like structure is usually solved by dividing it into several segments whose and satisfy different distributions. The paper derived the exact general solution of one-step nonuniform beam firstly and then obtained the exact solution of multistep nonuniform beam by combining the transfer matrix method and exact solutions of one-step beam. The exact solutions of present method not only can provide convenience for parametric studies, but also are very useful for inverse problems such as damage identification.

#### 2. Modal Shape Function of One-Step Beam

##### 2.1. Beams with Uniform Cross Section

The governing differential equation for undamped free flexural vibration of beam with uniform cross section can be written as [18]where is Young’s modulus, is moment of inertia, is mass per unit length, and is transverse displacement at position and time .

Assuming the beam performs a harmonic free vibration at equilibrium position, that is,where is mode shape function of uniform beam, is corresponding natural frequency.

Inserting (2) into (1) obtains

Equation (3) leads to the modal shape function of beam with uniform cross section:where , , , , are () are integration constants, .

##### 2.2. Beams with Variable Cross Section

The governing differential equation for undamped free flexural vibration of beam with variable cross section can be written as [19]where is Young’s modulus, is bending moment of inertia at position , is mass per unit length at position , and is transverse displacement at position and time .

Assuming the beam performs a harmonic free vibration at equilibrium position, that is,where is mode shape function of beam, is natural frequency.

Substituting (6) into (5) arrives at

The moment of inertia and mass per unit length of beam are assumed to vary as where , , and are arbitrary constants but not equal to 0 and is a positive integer*.*

Substituting (8) into (7) yields

LetIt can be derived that

Introducing (11) into (9), one arrives at

Equation (12) can be simplified as

Let

Equation (13) can be written as

The characteristic equation of (15) is

The four roots of (16) arewhere

The modal shape function of one-step beam with variable cross section can be written as where , , , and are four undetermined coefficients.

Since , then , and , it can be found that and are two real roots in the first equation of (17). From the second equation in (17), it can be found that when , and are also two real roots and when , and are two conjugate complex roots.

Modal shapes of one-step beam are determined by the roots as follows:

(1) When , , , and are all real roots, substituting into (19), one obtains

(2) When and are real roots, while and are two conjugate complex roots, letthe roots and can be expressed aswhere .

Substituting and (22) into (19) yieldsIn (20) and (23), () have the same meanings as those in (4) and they are integration constants. The solutions of (20) and (23) are the exact general solutions only at the condition of . This is because will be equal to zero in (10) if for . And the derivation of constant term for the second equation in (10) has no mathematical meaning. Meanwhile, from the numerical point of view, small value of () is adopted and the uniform beam is approached; then, it could be solved by (20) or (23). However, compared to solutions of uniform beam by (4), the solutions of approximate uniform beam are not exact general solutions but numerical solutions.

Equations (20) and (23) can be expressed with a uniform style as when the modal shape function is (20), for , and when it is (23), for , and , .

#### 3. Transfer Relationship for Undetermined Coefficients of Multistep Beam Modal Shapes

As shown in Figure 1, the multistep beam is divided into segments. For an arbitrary beam segment () with length , a local Cartesian coordinated system is established with the origin locating at the left end of segment, is bending moment of inertia for the th segment, and is mode shape function for the th segment, where is defined as belonging to the th segment in the local Cartesian coordinated system.