Shock and Vibration

Volume 2015 (2015), Article ID 690196, 9 pages

http://dx.doi.org/10.1155/2015/690196

## Sensor Placement Optimization of Vibration Test on Medium-Speed Mill

^{1}School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China^{2}Department of Civil, Structural and Environmental Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, USA

Received 8 September 2014; Revised 22 December 2014; Accepted 24 December 2014

Academic Editor: Stathis C. Stiros

Copyright © 2015 Lihua Zhu 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

Condition assessment and decision making are important tasks of vibration test on dynamic machines, and the accuracy of dynamic response can be achieved by the sensors placed on the structure reasonably. The common methods and evaluation criteria of optimal sensor placement (OSP) were summarized. In order to test the vibration characteristic of medium-speed mill in the thermal power plants, the optimal placement of 12 candidate measuring points in , , and directions on the mill was discussed for different targeted modal shapes, respectively. The OSP of medium-speed mill was conducted using the effective independence method (EfI) and QR decomposition algorithm. The results showed that the order of modal shapes had an important influence on the optimization results. The difference of these two methods on the sensor placement optimization became smaller with the decrease of the number of target modes. The final scheme of OSP was determined based on the optimal results and the actual test requirements. The field test results were basically consistent with the finite element analysis results, which indicated the sensor placement optimization for vibration test on the medium-speed mill was feasible.

#### 1. Introduction

The model updating, condition identification, condition assessment, and decision making of the concerned structure require enough efficient data to reflect structural properties. It is a common way to achieve the related data by sensors. However, restricted by testing technology and cost, a practical problem is how to select a set with a minimum number of sensor locations from all possibilities [1]. Optimal sensor placement (OSP) has been one of the more important tasks in dynamic test, especially for those high-rise buildings and complicated structures.

The basic principle of sensor placement optimization is that limited measuring points can get enough efficient data which can reflect the structural properties. Recently, most methods aim to achieve the best identification of structural characteristics, including the frequencies and mode shapes. Among these methods, effective independence (EfI) method is the most widely used method, which was proposed by Kammer [2]. For modal kinetic method (MKE), the target of OSP is to ensure the maximization of the measured kinetic mode energy that improves signal-to-noise ratio [3]. Besides, EfI is an iterated version of the MKE with reorthonormalized mode shapes [4]. The modal assurance criterion (MAC) method is based on QR decomposition with column pivoting, and these measuring points are chosen to minimize off-diagonal element of MAC matrix [5].

So far, a multitude of researches have been conducted on sensor placement optimization for large-span bridges and high-rise buildings in civil engineering. Yi et al. conducted sensor placement optimization for Dalian International Trade Mansion [1]. Chow et al. presented optimal sensor configuration of a typical transmission tower for the purpose of structural model updating [6]. Wang et al. established the wind and structural health monitoring system implemented on the Runyang Yangtze River Bridge [7]. However, there are few researches on the sensor placement optimization for those structures which are not very high or large, while the vibration modes are very complicated. The medium-speed mill is one of the most commonly used auxiliary power items of equipment in thermal power plants. It generates significant disturbing force when it is running, so it is very important to monitor its vibration condition. The study of Zhu et al. shows that vibration modes of medium-speed mill are very complicated [8].

In order to obtain the reliable dynamic response, reasonable condition assessment, and decision making of medium-speed mill, the measuring points should be placed on the locations that reflect abundant structural information of medium-speed mill. So the OSP for vibration test on medium-speed mill is extremely necessary. In this paper, a reliable finite element model (FEM) of medium-speed mill supported by spring vibration-isolated foundation is built, and two OSP methods are applied to the vibration test of medium-speed mill. Finally, considering the actual situation of vibration test, an OSP scheme is proposed in the vibration test on medium-speed mill.

#### 2. The Algorithm and Evaluation Criteria of OSP

##### 2.1. Effective Independence Method

The aim of the EfI method is to select measurement positions that make the mode shapes of interest as linearly independent as possible while containing sufficient information about the target modal responses in the measurements.

The measured structural response can be expressed as where is the matrix of FEM target mode shapes, is the coefficient response vector and also is modal coordinate, and is a sensor noise vector, assumed stationary random process with a mean value zero. There must be a deviation for corresponding real generalized coordinate . Assuming that this process is an effective unbiased estimation, the covariance matrix is obtained as where is the Fisher information matrix (FIM) by assuming the measured noise is independent and has the same statistical properties. The matrix can be written as Then, the maximization of is equivalent to the maximization of , and thus can be used to simplify FIM. Constructing matrix , where is the effective independence allocation matrix. The diagonal elements of the represent the contribution of the candidate points to the modal matrix linearly independent.

##### 2.2. The Algorithm for OSP Based on QR Decomposition

QR decomposition is derived from the maximization of the FIM. The linearly independent row vectors are extracted from modal matrix by QR decomposition that ensures a larger norm of matrix. These degrees of freedom (DOFs) corresponding with linearly independent row vectors can be used as an ideal placement for sensors. The decomposition can be expressed as where is the unit conversion matrix, descending order according to the value of the diagonal elements of . We choose the modal DOFs in accordance with the sequence of the elements in .

##### 2.3. The Evaluation Criteria of OSP

###### 2.3.1. Fisher Information Matrix Determinant

According to the principle of EfI method, the greater the trace or the value of FIM determinant is, the smaller the covariance is and the more effective the modal coordinate is. Meanwhile, FIM also measures the abundance modal information from the response of structures, and the greater the determinant value is, the more abundant modal information it contains [5].

###### 2.3.2. Modal Assurance Criterion

The quantity of measured DOFs is far less than the whole DOFs of structural model and the responses are affected by measurement accuracy and noise; thus, the modal vector measured cannot retain the original space property. MAC is a good method to evaluate the space angle of modal vectors [9]. The formula is expressed as where and are the th and th modal vectors, respectively. The off-diagonal elements of MAC matrix should be minimized for OSP, and thus the modal matrix measured can keep better orthogonality.

#### 3. OSP for Vibration Test on Medium-Speed Mill

##### 3.1. Description of the Medium-Speed Mill

Medium-speed mill is an important auxiliary dynamic machine in the thermal power plants. To mitigate the vibration, the spring vibration-isolated foundation is normally used to support the mill. A spring vibration-isolated foundation for medium-speed mill is typically comprised of upper and lower bedplates and spring isolators. An elevation view of a vibration-isolated foundation for medium-speed mill is shown in Figure 1. As shown, there are two buttresses over the lower bedplate, and the distance between the upper bedplate and buttresses is 330 mm. Spring isolators are placed between the two buttresses and the upper bedplate. Twelve sets of spring isolators are placed on each buttress; thus, a total of 24 sets of isolators are placed.