An Accurate Complex Modal Superposition Method

Civil-Comp Press, 2014 Proceedings of the Twelfth International Conference on Computational Structures Technology, B.H.V. Topping and P. Iványi, (Editors), Civil-Comp Press, Stirlingshire, Scotland.

Paper 163

Frequency Response Sensitivity: An Accurate Complex Modal Superposition Method L. Li, Y.J. Hu and X.L. Wang School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan, P.R. China

Abstract This paper is aimed at including the influence of the higher-order modes to the frequency response sensitivities of non-classically viscously damped systems. An accurate modal superposition method (AMSM), which only involves the available modes and system matrices, is presented to accurately calculate the frequency response sensitivities of non-classically damped systems. The AMSM is maintained in the original-space without having to involve the state-space equations of motion. The convergence condition only requires that all the modes whose resonant frequencies lie within the range of the excitation frequencies must be used for mode superposition. So it is easy to satisfy and the AMSM easily converges to the exact results. The computational complexity of the AMSM and the direct frequency response method (DFRM) is evaluated and compared. It is shown that the errors of the sensitivities calculated using the mode displacement method (MDM) are significant. The AMSM can reduce the modal-truncation error very rapidly. That is, the AMSM can give almost the same accuracy with the exact results of the DFRM, but save much computational cost. Therefore, the AMSM yields good trade-off between the accuracy and the computational complexity. Keywords: modal truncation error, residual flexibility, nonclassical damping, frequency response analysis, modal superposition method, sensitivity, harmonic response, design sensitivity analysis.

1

Introduction

Sensitivity analysis deals with the calculation of the rate of performance measures from changes in design parameters. The frequency response sensitivity of structural systems is of interest in dynamic problems subjected to harmonic loading that may be caused by reciprocating or rotating machine parts. Generally speaking, the frequency response sensitivity can be used to predict the change of frequency

1

response from changes in design parameters, select a search direction and obtain an approximation model for an optimization process. Frequency response sensitivity is of fundamental importance and plays a very important role in many areas, such as model updating, structural damage detection, vibration and noise control, system identification and optimization. Often there are two kinds of methods [direct frequency response method (DFRM) and modal superposition method], which are used to evaluate the frequency response sensitivity of a linear mechanical system. The DFRM is based on the direct frequency results in an exact calculation by using the matrix decomposition technique. The DFRM evaluate the frequency response sensitivity in a manner similar to a static-force problem at each excitation frequency. That is, in this method, the decomposition of the system dynamic matrix is involved for each excitation frequency. Therefore, the method is time-consuming when the number of degrees of freedom (DOF) and excitation frequencies is large. The modal superposition method calculates the frequency response sensitivity by expressing it as the summation of the contributions of modes. However, it is difficult, or even impossible, to obtain all the modes of large-scaled structures. Therefore, the mode displacement method (MDM), which only uses the lower modes to approximately calculate dynamic responses and their sensitivities, are usually used in engineering applications. It means that the modal-truncation scheme is generally used and the modal-truncation error is therefore involved. It was shown [1] that the errors of MDM may be significant if only a few lower modes are retained and the MDM may be not a good approach to perform the frequency response analysis. Many modal correction approaches [2-5] (such as the modal acceleration method, dynamic correction method, and force derivative method) has been presented to eliminate the modaltruncation error to dynamic responses. However, the studies are only restricted to the case of undamped or classically damped systems. In general, classical damping means that energy dissipation is almost uniformly distributed throughout the mechanical system. Of course, there is no reason why the condition must be satisfied. In practical, structural systems with two or more parts with significantly different levels of energy dissipation are frequently encountered. It was shown by experimental data [6] that no physical system is strictly classically damped. In the case of non-classically damped systems, the modal superposition method for evaluating the response needs complex modes. To avoid the complex modes, several approximation techniques [7] are developed to efficiently calculate the responses. Among these approximation methods, the most common method is so-called the proportional approximation method (PAM) or uncoupled mode superposition method, which is simply to ignore the off-diagonal (coupling) elements of the transformed damping matrix. Many authors [8-11] studied the accuracy of the PAM or investigated some conditions that the errors of the PAM may be small. It was shown that light damping, diagonal dominance of the transformed damping matrix and good separation of the normal modes (these conditions are once believed to produce small errors) are not sufficient conditions for the accuracy of the PAM. Although the PAM is a powerful approximate method, the results of the PAM are not always with acceptable accuracy. The accuracy results may be obtained by using the complex modal superposition method. It is

2

obvious that the modal correction approaches used in undamped systems can be extended to non-classically damped systems based on the state-space formula (see, e.g., [12]). However, the state-space equation based approaches may not be only computationally expensive, but also lack the physical insight provided by the superposition of the complex modes of the equation of motion with original space. Some works [1, 13, 14] were developed to eliminate the modal truncation error of the frequency responses of damped systems without having to involve the statespace formula. Li et al. [1] presented an explicit expression on the contribution of the higher-order modes to the frequency responses of non-classically viscously damped systems by expressing it as a power-series expansion. And the method maintains original-space. It should be noted that the errors of frequency response sensitivities may be larger due to the fact that the frequency response sensitivities need the mode-superposition of the lower modes for two times. This paper is aimed at including the influence of the higher-order modes to the frequency response sensitivities of non-classically viscously damped systems. Based on the explicit expression [1], an accurate modal superposition method (AMSM) is presented to accurately calculate the sensitivities of the frequency responses of non-classically damped systems. The AMSM 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 convergence condition only requires that all the modes whose resonant frequencies lie within the range of excitation frequencies must be used for mode superposition. So it is easy to satisfy and the AMSM can be easily converged to exact results. It will be shown that the errors of frequency response sensitivities calculated using MDM may be significant if only a few lower modes are retained and the MDM may be not a good approach to perform the frequency response sensitivity analysis. The AMSM can reduce the modal-truncation error very rapidly. Also, the AMSM can give almost the same accuracy with the DFRM, but save much computational cost.

2

Frequency response sensitivity analysis

The equations of motion for a multiple DOF linear system subjected to external dynamic forces can be written in the form of

(t )  Cq (t )  Kq(t )  f (t ) Mq

(1)

where M   N  N , C   N  N and K   N  N are, respectively, the mass, viscous damping and stiffness matrices (here we only consider symmetric system matrices). Here q(t) and f(t) are the displacement and force vectors, respectively. The governing dynamic responses of the damped system subjected to harmonic excitations, i.e., the force vector takes the form, f(t)=Fhexp(iωt), can be modelled using the following matrix problem:

  M+iC  K  X(i )  F 2

h

3

(2)

or

D(i ) X(i )  Fh

(3)

D(i )   2M +iC  K

(4)

with the dynamic stiffness matrix

Since the forcing function is harmonic, the resultant steady-state response is also harmonic, i.e., q(t)=X(iω)exp(iωt) where X(iω) is the complex response vector. The viscously damped system cannot be simultaneously decoupled using classical normal modes unless it satisfies the conditions given in [15], in which case the system is said to be a proportionally (classically) damped system. Increasing the use of special energy-dissipating devices in vibration and noise control, the need to consider the complex modal analysis of non-classically damping systems is more than ever before. To this end, the concern of this study is the non-classically damped system.

2.1 Direct frequency response method The direct frequency response method which is an exact method and calculates the frequency response in a manner similar to a static-force problem at each excitation frequency. The response calculated by the DFRM can be given by

X(i )  D1 (i )Fh

(5)

By differentiating Equation (2) with respect to a design parameter p, the derivative of the response satisfies

X(i )  D1 (i )PF (i ) p

(6)

where the pseudo-force vector is given by

PF (i ) 

Fh D(i )  X(i ) p p

(7)

with

D(i ) M C K   2  i  p p p p

4

(8)

As can be seen, once the pseudo-force vector is calculated, the response sensitivity can be also calculated in a manner similar to a static-force problem at each excitation frequency.

2.2 Real modal superposition analysis The corresponding undamped modes (normal modes) can be obtained by solving the following eigenproblem

Ku j   2j Mu j  j  1, 2, , N

(9)

where ωj is the jth frequency and in order of ascent; uj denotes the normal mode corresponding to the jth frequency ωj. These normal modes satisfy the orthogonality relationship over the mass and stiffness matrices and can be normalized such that

uTj Mu k   jk , uTj Ku k   2j  jk  j  1, 2, , N

(10)

By ignoring the mode coupling, the response calculated by the PAM can be given by N

u j uTj Fh

j 1

 2  i2 j j   2j

XPAM (i )  

(11)

where the conventional modal damping ratio can be expressed as

j 

1 2 j

uTj Cu j

(12)

And the response sensitivity can be expressed as N u j uTj  X(i )   Fh D(i )  XPAM (i )       2 2  p  p PAM j 1   i2 j j   j  p 

(13)

The PAM is simply to ignore the off-diagonal (coupling) elements of the transformed damping matrix. The PAM only requires the real (undamped) modes, so it is efficient. However, the results of the PAM are not always with acceptable accuracy.

2.3 Complex modal superposition analysis The eigenvalue problem for the viscously damped system can be expressed as

 M   C  K  φ 2 j

j

j

 0  j  1, 2, , 2 N

5

(14)

Here λj and φj denote the jth eigenvalue and eigenvector. In this study, we assume that the eigenvalues of the system are distinct and the system is underdamped such that it has N complex conjugate pairs of eigenvalues. The eigenvalues with positive imaginary part are in the order as follows

Im(1 )