Shock and Vibration

Volume 2016 (2016), Article ID 8479721, 9 pages

http://dx.doi.org/10.1155/2016/8479721

## Understanding Power Spectrum Density Transmissibility

State Key Lab of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China

Received 3 August 2015; Revised 6 November 2015; Accepted 15 November 2015

Academic Editor: Salvatore Russo

Copyright © 2016 Yu 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

Power spectrum density transmissibility (PSDT) is a type of complex frequency domain function proposed recently. It describes the relation between cross-spectra of system outputs. Since PSDTs with same local-reference degree of freedom (DOF) combination but with different transferring output DOFs cross each other at the system’s poles under certain load condition, the functions have been used as the primary data in operational modal analysis (OMA) to extract modal parameters, and such technique is named as PSDT-based OMA (PSDTOMA). Because PSDT is a concept that appears recently, researches on which, especially, in-depth discussions aim at the essence and properties of PSDT are very rare. For appropriate application of PSDTOMA, it is necessary to perform further study on such problem obviously. In this paper, two paths to get PSDT, which, respectively, are referred to as PSDT estimator and PSDT syntheticism, are given firstly; some properties about PSDT are explored based on the PSDT syntheticism, and the relation between PSDT and single reference transmissibility function (STF) is analyzed. Finally, the above conclusions are verified with numerical values and experimental data.

#### 1. Introduction

Similar to more common-used transfer functions, transmissibility is a type of (complex) frequency domain functions, which describe the relation between physical quantities of the same type at different locations in a dynamical system. According to the type of physical quantities described, the transmissibility about system dynamic characteristics can be divided into force transmissibility and motion transmissibility.

Power spectrum density transmissibility (PSDT), which this paper studies, was first proposed by Yan and Ren [1] in 2012 as the third kind of motion transmissibility following the traditional single reference transmissibility function (STF) [2–4] and multi-/poly-reference transmissibility function (MTF/PTF) [5–7], and it is defined as the ratio of cross-spectra between responses at different locations in a system. Similar to STF, PSDT is a scalar (complex) frequency domain function as well, representing the relation between responses of local degree of freedom (DOF) and reference DOF. Because PSDT introduces the concept of transferring output DOF additionally and the PSDT, depending on different transferring output DOFs, intersects at the system poles under certain loading condition [1, 8], in theory, the STF, depending on different loading conditions, can be replaced with the PSDT, depending on different transferring output DOFs, which is used as the primary data of operational modal analysis (OMA) to lower identification method’s demand on the number of loading conditions. This OMA process is called PSDT-based OMA (PSDTOMA). Same as STF-based OMA (sTOMA), PSDTOMA has the characteristic that identification results are unaffected by harmonic component in excitation, compared to the traditional OMA methods.

Because PSDT was proposed recently, it has not been studied thoroughly and deeply. To the authors’ knowledge, all the current PSDT-related studies focus on the application layer [1, 8], and no discussion on its essence and property has been made. In this paper, two paths to get PSDT, which, respectively, are referred to as PSDT estimator and PSDT syntheticism, are given firstly; some properties about PSDT are explored based on the PSDT syntheticism, and the relation between PSDT and single reference transmissibility function (STF) which is similar to PSDT is analyzed. Finally, the above conclusions are verified with numerical values and experimental data.

#### 2. Power Spectrum Density Transmissibility

Under a certain load condition, PSDT is defined as the ratio of cross-spectral densities between response at DOF and responses at DOFs , , respectively [1, 8]. It is expressed aswhere and are cross-spectral densities between responses at DOFs , and responses at DOFs , , respectively; DOFs , , and are referred to as local DOF, reference DOF, and transferring output DOF, respectively.

In the frequency domain, the relation between inputs and outputs of a linear time invariant (LTI) system can be expressed aswhere is output power spectrum matrix; is input power spectrum matrix; is transfer function matrix; denote the complex conjugate and transpose of . According to (2), the cross-spectral density between response at two DOFs, such as and , which also is the entry in the th row and th column of , can be expressed aswhere and are row vectors consisting of entries in the th and th row of , respectively. Combining (1) and (3), we get another expression of PSDT:Equations (1) and (4) are two opposite paths to get the value of PSDT. Equation (1) is referred to as PSDT estimator, which is used for getting PSDT estimation from outputs, while (4) is referred to as PSDT syntheticism, which is used for getting PSDT synthesis from inputs and transfer function. The relation among PSDT syntheticism, PSDT estimator, PSDT synthesis, and PSDT estimation is shown in Figure 1. In theory, PSDT estimation is a consistent approach of PSDT synthesis, and this makes PSDT synthesis be regarded as an evaluation criterion for PSDT estimation.