Shock and Vibration

Volume 2018, Article ID 8178274, 18 pages

https://doi.org/10.1155/2018/8178274

## Periodic Responses of a Rotating Hub-Beam System with a Tip Mass under Gravity Loads by the Incremental Harmonic Balance Method

^{1}Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, China^{2}Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA

Correspondence should be addressed to J. L. Huang; nc.ude.usys.liam@ljgnauh

Received 8 August 2017; Revised 22 November 2017; Accepted 9 January 2018; Published 13 March 2018

Academic Editor: Francesco Pellicano

Copyright © 2018 D. W. Zhang 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

Dynamic characteristics of a flexible hub-beam system with a tip mass under gravity loads are investigated. The slope angle of the centroid line of the beam is utilized to describe its motion. Hamilton’s principle is used to derive the equations of motion and their boundary conditions. By using Lagrange’s equations, spatially discretized equations based on assumed mode method are derived, and the equations of motion are expressed in nondimensional matrix form. The incremental harmonic balance (IHB) method is used to solve for periodic responses of a high-dimensional model of the rotating hub-beam system with a tip mass for which convergence is reached. A frequency equation is derived giving the relationship between the nondimensional natural frequencies and three nondimensional parameters, that is, the rotating angular velocity, the tip mass, and the hub radius ratio. A comparative study is performed for nonlinear frequency responses of the system with a tip mass under different values of tip masses and damping ratios.

#### 1. Introduction

A number of systems in the fields of navigation and mechanical engineering can be modeled as rotating hub-beam system with a tip mass. For example, flexible manipulators, spacecraft structures, and cranes carrying moving loads can be studied in this way. In these systems, the beams carrying a tip mass are rotating in a horizontal plane with the whole systems mounted on a rotating hub. In order to study their dynamic characteristics, the rotating beam systems are simplified as a rotating hub-beam system with a tip mass model. In recent years much attention has been placed on linear and nonlinear dynamic characteristics of rotating beam with a tip mass, and earlier work has been done with the linear analysis. For instance, Conrad and Morgül [1] used linearized feedback law to study the stabilization of a flexible beam with a tip mass. Rao [2] derived the time-dependent equations of motion that governs the vibration of an Euler-Bernoulli beam; it was used to obtain the linear dynamic response of the beam under moving load mass. Demetriou [3] presented a method for construction of observer for linear second-order lumped and distributed parameter systems using parameter-dependent Lyapunov functions. Furta [4] proved that the attached point mass on a thin elastic beam plays a destabilizing role for any values of the problem parameters and studied the dynamical stability of the rectilinear shape of the beam by means of the direct Lyapunov method. In [5], the extended Hamilton principle was employed to derive the equations of motion of a rotating beam with a tip mass undergoing coupled torsional-bending vibrations and analyzing the exact frequencies leading to a better control of the system.

Recently, the nonlinear vibration of a rotating beam with a tip mass has been studied by numerous researchers. Yang et al. [6] presented a finite element model for a flexible hub-beam system with a tip mass where viscous damping of the hub and the air drag force were considered. They showed that the traditional linear model cannot account for the dynamic stiffening and it may lead to erroneous results in high-speed systems. Simulation results on the vibration measurements of the rotating flexible beam carrying a tip mass for different combinations of shaft and root flexibilities and arm properties were reported by Ismail et al. [7]. Gregory et al. [8] have pointed out the shortcomings of the sequential single-axis vibration method. They have also reported evidence of differences in failure modes and fatigue life for multiaxial loadings versus single-axis inputs by utilizing multiaxial electrodynamic shakers. Sunar and Al-Bedoor [9] tested the suitability of a piezoelectric (PZT) sensor in measuring vibrations of the rotating beams. A comparison between the results of finite element and experimental was reported and indicated that the root-embedded PZT sensor can be effectively used in measurements of blade vibration. In [10], Sinha et al. investigated a simultaneously precessing and nutating beam with a tip mass using a variant of Hill’s method and found that the stability of an only precessing beam depends on the inclination of the beam-centerline with the axis of precession. Patil and Gandhi [11] demonstrated chaotic behavior for an inverted flexible pendulum with a tip mass on a cart system; the dynamic model was validated with experimental data for a couple of cases of beam excitation. Li et al. [12] established the dynamic equations of the rigid-flexible coupling system in a noninertial coordinate system; the numerical simulation results showed that dynamic stiffening is produced by the coupling effect of the centrifugal inertial load distributed on the flexible beam and the transverse vibration deformation of the beam. Yong et al. [13] studied the rigid-flexible coupling system with a hub and concentrated mass; they utilized the second Lagrange equation and the assumed mode method to establish the dynamic equations of the system. The results showed that the concentrated mass mainly suppresses the bending vibration of the beam and exhibits damping characteristics.

In a rotating hub-beam system with a tip mass, the effect of the gravity load of beam was not accounted for in the previous literature [14]. As the size and height of rotating machineries such as manipulators and cranes increase, it is necessary to consider the effect of the gravity load of the beam during the rotation. Park and Kim [15] derived the equations of motion of the rotating beam which include all dynamic effects and mainly studied the effects of curvature and tip mass that can change the dynamic response of the beam. Flatness based controller design techniques for a rotating hub-beam system with a payload attached to the tip of the beam in a gravitational field were utilized in [16]. Cai et al. [17] made a comparison between the first-order approximation coupling model and the zeroth-order approximation coupling model when prescribed torque drives the beam with a tip mass, which takes into account the influence of the gravitational force filed of the beam in the analysis of the dynamic characteristics of the system. As for a flexible hub geometrical nonlinearity beam with a tip mass, Emam [18] employed a flexural model to study the dynamic responses of a flexible hub geometrical beam with a tip mass of which the hub is restrained by a translational and a rotational spring; such a model accounts for the geometrical coupling between the axial and lateral deformations. If gravity of the beam is much less than the gravity of tip mass, it is not necessary to consider the effect of the gravity load of beam [19].

The main objective of this work is to use the IHB method to analyze nonlinear dynamic responses of a rotating hub-beam system with a tip mass for which convergence is reached. The vibration analysis is performed under the slope angle model with five included trial functions, and the model considers the effects of gravity loads of the beam and tip mass during the rotation. To deal with the rotating hub-beam system with a tip mass, the IHB method is applied, which was first introduced by Lau and Cheung [20, 21]. The IHB method was successfully applied to the analysis of nonlinear structural vibrations problems. Xu and Zhu [22] used the IHB method to determine parametric instability boundaries of a parametrically excited system. In order to improve calculation efficiency of the IHB method, the fast Fourier transform (FFT) was used to efficiently calculate coefficients of a discrete Fourier transform in practice [23]. Wang and Zhu [24] developed a modified IHB method that incorporated FFT and Broyden’s method to obtain the dynamic responses of belt-drive system with a noncircular sprocket. The configuration of the rotating beam is described by the slope angle of its centroid line; the slope angle model has been widely used in studying the nonlinear dynamics [25]. The nonlinear dynamic equation of the rotating hub-beam system with a tip mass is derived in Section 2 using Hamilton’s principle. Spatially discretized equations of the slope angle model neglecting the fourth- and higher-order nonlinear terms are derived by Lagrange’s equations in Section 3. The IHB method for calculating the periodic responses of the rotating hub-beam system with a tip mass is formulated in Section 4. The critical buckling load of the rotating beam under an axial tip mass is calculated in Section 5. Natural frequencies of the rotating beam with a constant rotating angular velocity are calculated in Section 6.1 using the slope angle model, and compared with those from the dynamical model in [14]. Nonlinear frequency responses of the system with a tip mass for different numbers of trial functions are calculated in Section 6.2. The influences of tip mass and damping ratio on nonlinear frequency responses are studied in Sections 6.3 and 6.4, respectively.

#### 2. Equations of Motion

A schematic diagram of a rotating beam with a tip mass is shown in Figure 1, a rotating planar beam is attached to a rigid hub of radius , and the beam rotates in a vertical plane around the central axis in the inertial coordinates . The effects of gravity loads of the beam and tip mass are considered. The beam has a slender shape that the shear deformation effect and rotary inertia of the beam are ignored. The position of a cross section of the beam is described by the arc-length coordinate along its centroid line. For the whole beam, the bending stiffness , cross-section area , and linear mass density can be approximated as constant. is the length of the beam, and is the tip mass at the end of the beam. The hub rotates with a given rotating angular velocity , where is time; the gravitational acceleration is . represents the space attitude motion of hub, andThe slope angle of the beam at the position is expressed as , then .