Shock and Vibration

Volume 2016, Article ID 2403426, 8 pages

http://dx.doi.org/10.1155/2016/2403426

## Structural Modifications for Torsional Vibration Control of Shafting Systems Based on Torsional Receptances

^{1}College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China^{2}School of Engineering, University of Liverpool, Liverpool L69 3GH, UK^{3}State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116023, China

Received 28 June 2016; Revised 18 August 2016; Accepted 22 August 2016

Academic Editor: Jussi Sopanen

Copyright © 2016 Zihao Liu 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

Torsional vibration of shafts is a very important problem in engineering, in particular in ship engines and aeroengines. Due to their high levels of integration and complexity, it is hard to get their accurate structural data or accurate modal data. This lack of data is unhelpful to vibration control in the form of structural modifications. Besides, many parts in shaft systems are not allowed to be modified such as rotary inertia of a pump or an engine, which is designed for achieving certain functions. This paper presents a strategy for torsional vibration control of shaft systems in the form of structural modifications based on receptances, which does not need analytical or modal models of the systems under investigation. It only needs the torsional receptances of the system, which can be obtained by testing simple auxiliary structure attached to relevant locations of the shaft system and using the finite element model (FEM) of the simple structure. An optimization problem is constructed to determine the required structural modifications, based on the actual requirements of modal frequencies and mode shapes. A numerical experiment is set up and the influence of several system parameters is analysed. Several scenarios of constraints in practice are considered. The numerical simulation results demonstrate the effectiveness of this method and its feasibility in solving torsional vibration problems in practice.

#### 1. Introduction

Dynamic performance of structures plays an important role in engineering; however, there are always some circumstances in which structural dynamic performance does not meet the design requirements or actual situations in practice. Therefore, it is common that some existing structures need to be modified in order to acquire desired dynamic performance [1]. Many researchers have put forward many methods for the eigenstructure assignment problems [2–5]. One major way of doing that is to assign structure suitable natural frequencies and modes through structural modifications as a typical vibration control strategy, which usually requires knowledge of accurate structural parameters (e.g., mass, stiffness, and damping matrices [6, 7]) or modal data [8–10]. However, in most engineering problems, it is very difficult to gain such knowledge. Usually modal tests must be conducted [11] and model updating must be carried out [12], which is expensive and tedious for complicated structures. Moreover, the application of modal data in practice has a number of difficulties, which was discussed in [13]. On the other hand, some structural modification methods directly based on the measured system receptances or Frequency Response Functions (FRFs) [14–17] overcome those difficulties and provide effective solutions to this kind of problems, which belong to inverse structural dynamics. Structural modifications based on the measured receptance or FRFs were studied in forward analysis for prediction of receptances of the modified structure [18] and in inverse analysis for assigning natural frequencies and vibration nodes [19] and eigenstructures [13, 20].

For rotating machines, one of the most important problems is torsional vibration of shafts. The adverse impact caused by torsional vibration includes vibration of the whole machine, damage in the transmission system, excessive wear of bearings and gears, and even shaft fracture [21]. For shaft structures, numerical models are usually needed to evaluate torsional vibration characteristics in the engineering design stage, but some structural parameters (such as the rotary inertia of the motor and the actual torsional stiffness of gears) cannot be accurately obtained easily. Therefore, there are inevitably considerable discrepancies between the designs and the actual structures based on such imperfect theoretical models. So suppression of torsional vibration is a big challenge. If there is a method that does not require an accurate theoretical system model in solving the torsional vibration problems and can also achieve structural modifications to the system based on measured data, this method will bring many advantages in practice.

However, it should be noted that, perhaps partly owing to difficulties in accurately measuring torsional FRFs, inverse structural dynamics problems of rotating machineries based on measured data (especially for rotational receptance) nearly have never been studied before [22].

Although many researchers have put forward a number of methods for measuring transfer functions for rotational degrees of freedom (DoFs); for example, Mottershead et al. [23] proposed one indirect method based on T-block for obtaining rotational receptances, the progress in measuring torsional transfer functions in shaft systems is still very limited [23–27], and nearly none of these papers are about torsional vibration measurement of shaft structures. Recently, Lv et al. [22] put forward an indirect method to measure the torsional receptance. The method was implemented by using a T-shaped simple auxiliary structure attached to one end of the shaft system, and the torsional system receptances could be obtained accurately through combining the auxiliary structure’s finite element model (FEM) and test data of the whole structure.

This paper presents a theoretical strategy of structural modifications for suppression of torsional vibration of shaft systems, using “measured” torsional receptances and a structural optimization method. One main advantage of this method proposed in this paper is that it does not need any knowledge of mass and stiffness parameters or even analytical or modal models of the system under investigation; instead “measured” torsional receptance data are used, which can be obtained from measured translational vibration data obtained through an additional structure. In this paper, structural modifications for suppressing torsional vibration of a simplified model of a “real” rotating machine are studied. Several scenarios of practical constraints are considered. Theoretical results show the effectiveness of this method.

#### 2. Receptance-Based Method

A shaft system under a harmonic excitation, treated as a general linear discrete conservative dynamic system without damping, can be described bywhere is the mass (moment of inertia) matrix, is the acceleration vector, is the stiffness matrix, is the displacement vector, is the force amplitude vector, e is the Euler number, , is the angular velocity, and is the time variable.

Denote the change in mass matrix and change in stiffness matrix due to structural modifications as and , respectively. Then (1) becomesIt can be assumed that the response is harmonic in the form of ; then is the eigenvector. Substituting it into (2) yieldsThe original system FRF matrix is defined as . Equation (3) can then be rewritten asFor the eigenvalue problem, it is assumed that the desired natural frequency and mode are, respectively, and ; then the following equation is derived:In order to obtain the FRFs of the unknown shaft system, a simple auxiliary structure needs to be attached to the shaft at one end, as shown in Figure 1.