Shock and Vibration

Volume 2015, Article ID 507581, 11 pages

http://dx.doi.org/10.1155/2015/507581

## Transverse Free Vibration of Axially Moving Stepped Beam with Different Length and Tip Mass

^{1}State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China^{2}Department of Mechanics, Shanghai University, Shanghai 200444, China

Received 10 April 2015; Accepted 26 May 2015

Academic Editor: Rafał Burdzik

Copyright © 2015 Guoliang Ma 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

Axially moving stepped beam (AMSB) with different length and tip mass is represented by adopting Euler-Bernoulli beam theory, and its characteristics and displacements of transverse free vibration are calculated by using semianalytical method. Firstly, the governing equation of the transverse free vibration is established based on Hamilton’s principle. The equation is cast into eigenvalue equation through the complex modal analysis. Then, a scheme is proposed to derive the continuous condition accordingly as the displacement, rotation, bending moment, and shear force are all equal at the connections of any two segments. Another scheme is to derive frequency equation from the given boundary conditions which contain a tip mass in the last segment. Finally, the natural frequency and modal function are calculated by using numerical method according to the eigenvalue equation and frequency equation. Due to the introduction of modal truncation, displacement and, the free vibration solution can be obtained by adopting modal superposition after Hilbert transform. The numerical examples illustrate that length, velocity, mass, and geometry affect characteristics and displacements significantly; the series of methods are effective and accurate to investigate the vibration of the AMSB with different length and tip mass after comparing several results.

#### 1. Introduction

Axially moving structures, simply modeled as axially moving beams (strings) in some cases, are applied in various industrial applications such as magnetic tapes, textile fibers, band saw blades, conveyor belts, transmission chains, aerial cableways, pipes conveying fluid, and steel strips in a thin steel sheet production line. Furthermore, the stepped beam has excellent mechanical properties, constant strength, and optimized distribution of quality, which is also widely used as basic component of many mechanical devices. Moreover, the axially moving stepped beams (AMSB), with different length and tip mass, are made of same or different parts traveling at different speeds; they draw more and more attentions of researchers. There are a wide application and prospect in various fields such as extended arm-type mechanism and air refuelling pipe. This kind of beam actually is multibody system; multibody system analysis (MSA) is an analytical tool used to solve the dynamics problems of complex mechanical systems [1].

There are a number of studies on axially moving systems that have been carried out by scholars in their literatures. Generally, the most important rule is that the inherent frequency and modal function change as the axially moving speed changes. Scholars develop analytical, semianalytical, and numerical methods to investigate the axially moving beams (strings) under different boundary conditions and calculate the natural frequencies, modal functions, responses, and stability at constant or variable speed, which are the major dynamic contents. Mote Jr. [2] firstly calculated the natural frequencies after solving the frequency equation derived from the governing equation of an axially moving beam under pinned-pinned boundary conditions. Simpson [3] presented the frequencies and modal functions under pinned-pinned and clamped-clamped boundary conditions, while the tension effects were not considered. Wickert and Mote Jr. [4] proposed a complex modal analysis procedure for axially moving materials, and the procedure could be used to determine the frequencies and the modal functions of moving string and beam. Stylianou and Tabarrok [5] used finite element method (FEM) to solve the dynamics problems of an axially moving uniform beam with variable length; they divided uniform beam into finite uniform elements. Öz and Pakdemirli [6] applied the method of multiple scales to solve the stability boundaries of an axially accelerating beam under pinned-pinned, clamped-clamped; and fixed supported conditions; respectively, where the time-dependent velocity is assumed to vary harmonically near a constant mean speed . Ponomareva, Sandilo, and van Horssen [7, 8] concentrated on studying the transversal vibrations of continuum with time-varying velocity and variable length: transient from string to beam behaviour where the velocity is given by . In their papers, it is explicitly shown how to work with a combined model that is a string model at the low frequencies and a tensioned beam model at the higher frequencies. Ghayesh and Amabili [9] solved the steady-state transverse response of an axially moving beam with time-dependent axial speed . Park et al. [10] investigated vibrations of an axially moving beam with deployment or retraction considering the actual application, where the speed is . Recently, Malookani and van Horssen investigated the resonances and the applicability of Galerkin’s truncation method for an axially moving string with time-varying velocity [11]. Kazemirad et al. concentrated on thermal effects on nonlinear vibrations of an axially moving beam with an intermediate spring-mass support [12]. Furthermore, there are a lot of studies on axially moving Timoshenko and Viscoelastic beams [3, 13–17] and axially moving plates [18].

These beams above are almost homogeneous beams (strings); however, there are many nonuniform structures in engineering. In recent years, some scholars used analytical and numerical methods to investigate the nonuniform beam and made some progress in the research of stepped or variable cross section, heterogeneous beam. Gupta [19] used the FEM to calculate the frequency of variable cross section beam. This general method rapidly develops, while it has limitation such as complex computing process and long computing time. Lee and Ke [20] adopted analytical method to study the free vibration of nonuniform beam with general elastically retrained boundary conditions. Ece et al. [21] calculated the vibration characteristics of variable cross section rectangular beam. Leigh and Kunz [1] studied structural dynamics problems of multibody beam applying a mixed space time finite element scheme. Mao and Pietrzko [22] used Adomian decomposition method to investigate the vibration of stepped beam. Nevertheless, these objects above are static beams, and the dynamics problem of nonuniform beam has been analyzed without considering the axially moving effect. Some methods are not universally applicable, and the analytical method is complex and difficult to solve the nonuniform axially moving beam. It is well known that the axially moving nonuniform systems usually are modeled as stepped or segmental beams by some ways. If the transverse vibration continuously exists, this will obviously affect the stability and position accuracy of the slender members, which even can lead to the structure instability. Therefore, vibration analysis of stepped beam is important to consider the influence of moving axially for the dynamic design of the industrial transmission.

The remaining part of the paper is organized as follows. In Section 2, in consideration of the mature theory for uniform beam, stepped beam is divided into many segments with a view to obtain uniform beam. Every segment can be investigated as a uniform axially moving beam; then the frequency equation and the eigenvalue equation are derived by using analytical method. In Section 3, the characteristic and displacement of transverse free vibration can be calculated by using numerical method. This paper shows two examples and presents analysis and comparison to illustrate effect of velocity and mass on the vibration characteristics and displacement of AMSB. Finally, some concluding remarks are made in Section 4.

#### 2. Axially Moving Stepped Beam

##### 2.1. Free Vibration Equation

The cantilever structure which consisted of an AMSB with different length and tip mass is investigated as shown in Figure 1. The variable cross section beam (nonuniform beam) can be simplified as stepped beam. Considering a multibody stepped beam composed of several uniform segments, Euler-Bernoulli beam theories are used to model slender structures assuming that every beam is relatively thin compared to its length. There is segment () with mass per length , moment of inertial , and the material stiffness . The beam undergoes the bending vibration described by the transverse displacement ; is the axial coordinate, where and are the time. Every beam travels at axial transport speed representing the variable speed of any segment , where the length of one segment is . The tip constant mass lies in the right end of the variable length AMSB.