Accurate method for harmonic responses of non-classically damped systems in the middle frequency range

Abstract

This paper is aimed at eliminating the influence of the problem of the harmonic response analysis of non-classically damped systems with lower-higher-modal truncation. Based on the Neumann expansion theorem and the frequency shifting technique, the relationships satisfied by eigensolutions and system matrices are established and an explicit expression on the lower-higher-modal truncation error of harmonic responses can be expressed as a sum of available modes and system matrices. A correction method, which only involves the available modes and system matrices, is then presented to calculate the harmonic responses. The method maintains original-space without having to involve the state-space equation of motion such that it is efficient in computational effort and storage capacity. The method is convergent if and only if all the modes whose resonant frequencies lie within the range of excitation frequencies are available. Finally, a two-stage floating raft isolation system and a sandwich plate are used to to validate the effectiveness of the presented method.

Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

“…The finite element method (FEM) was used by Chen (2014) to analyze the impacts of the twist angle on inherent frequency, where the K-V damping effect was considered, whereas the explicit modal shapes and corresponding dynamic responses were not worked out. Liu et al. (2016) considered a nonclassically damped linear system with symmetric governing matrices, and because of the modal nonorthogonality, complex mode theory was introduced for equation decoupling.…”

“…(2003) also conducted an eigenvalue analysis of a nonproportional damped beam, and an expansion in state form was generated in conjunction with a transfer matrix technique. Such analyses were also performed in (Qu and Selvam, 2002; Li et al., 2016; Lzaro, 2016), wherein the governing matrices were uniformly symmetric, although the effects of nonproportional damping were included. Through this specific literature review, the following conclusions are drawn at this stage.…”

An Algebraic Method for Harmonic Responses based on Extended Hybrid Expansion Method

“…Often, a lot of rolling machine parts such as blowers, engines and boosters may cause oscillatory excitation [1]. Therefore, harmonic response analysis is playing an increasingly important role in many areas.…”

Sensitivity analysis of modal assurance criteria of damped systems

“…The indirect method to calculate the sensitivity of MAC values is based on the existing sensitivity method of mode shape. Several methods have been developed for the calculation of the sensitivity of mode shape, including the model method (Adhikari, 1999, 2002), the modified modal method (Wang, 1991; Li et al., 2014d; Zhang, 2014), Nelson’s method (Nelson, 1976; Friswell and Adhikari, 2000; Adhikari and Friswell, 2006; Li et al., 2012a, 2013a, 2013b; Weng et al., 2013), the algebraic method (Lee and Jung, 1997; Lee et al., 1999; Li et al., 2012b, 2014c), and the iterative method (Xie, 2012, 2013; Wu et al., 2015). Due to the fact that many design parameters are involved in practice and the diagonal elements of the MAC sensitivity matrix are usually considered, a direct method (Li et al., 2014a) was developed to efficiently and accurately calculate the sensitivity of MAC values.…”

“…The inclusion of the damping influence in structural analysis is extremely important if a model is to be applied in predicting vibration levels, transmissibility, transient responses, and design problems dominated by energy dissipation. It was shown that complex modal analysis can be a good choice to accurately calculate the frequency response (Adhikari, 2013; Li et al., 2014b, 2014e) and can be used to transform any viscously damped system into independent second-order equations (Ma et al., 2010; Morzfeld et al., 2011; Kawano et al., 2013). The damped system produces a set of normal real mode shapes only when it satisfies the mathematical conditions (Caughey and O’Kelly, 1965).…”