This paper considers the calculation of the sensitivity of modal assurance criteria (MAC) values of viscously damped systems. A direct method is proposed by constructing a Lagrange function whose Lagrange multiplier only needs to be calculated once for different design parameters. It is efficient in computational time and storage capacity. An extensive numerical analysis is used to show the effectiveness of the derived results. Several applications concerning the sensitivity of MAC values are discussed on the basis of the experimental data and the numerical analysis of a long beam structure with bolted joints.
AdhikariS (1999) Rates of change of eigenvalues and eigenvectors in damped dynamic system. AIAA Journal37: 1452–1458.
2.
AdhikariS (2002) Derivative of eigensolutions of nonviscously damped linear systems. AIAA Journal40: 2061–2069.
3.
AdhikariS (2013) Structural Dynamic Analysis with Generalized Damping Models: Analysis, John Wiley & Sons.
4.
AdhikariSFriswellMI (2006) Calculation of eigensolution derivatives for nonviscously damped systems using Nelson's method. AIAA Journal44: 1799–1806.
5.
AllemangRJ (2003) The modal assurance criterion – twenty years of use and abuse. Sound and Vibration37: 14–23.
6.
BohleKFritzenCP (2003) Results obtained by minimising natural frequency and MAC-value errors of a plate model. Mechanical Systems and Signal Processing17: 55–64.
7.
CaugheyTKO'KellyMEJ (1965) Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics32: 583–588.
8.
FriswellMIAdhikariS (2000) Derivatives of complex eigenvectors using Nelson's method. AIAA Journal38: 2355–2357.
9.
FriswellMIMottersheadJE (1995) Finite Element Model Updating in Structural Dynamics, Dordrecht: Kluwer Academic Publishers.
10.
HjørungnesA (2011) Complex-Valued Matrix Derivatives, Cambridge: Cambridge University Press.
11.
KawanoDTMorzfeldMMaF (2013) The decoupling of second-order linear systems with a singular mass matrix. Journal of Sound and Vibration332: 6829–6846.
12.
KohBHDykeSJ (2007) Structural health monitoring for flexible bridge structures using correlation and sensitivity of modal data. Computers & Structures85: 117–130.
13.
LeeIWJungGH (1997) An efficient algebraic method for the computation of natural frequency and mode shape sensitivities. Part I. Distinct natural frequencies. Computers & Structures62: 429–435.
14.
LeeIWKimDOJungGH (1999) Natural frequency and mode shape sensitivities of damped systems. Part I. Distinct natural frequencies. Journal of Sound and Vibration223: 399–412.
15.
LiLHuYJWangXL (2014a) Direct way of computing the variability of modal assurance criteria. Mechanics Research Communications55: 53–58.
16.
LiLHuYJWangXL (2012a) A parallel way for computing eigenvector sensitivity of asymmetric damped systems with distinct and repeated eigenvalues. Mechanical Systems and Signal Processing30: 61–77.
17.
LiLHuYJWangXL (2013a) A study on design sensitivity analysis for general nonlinear eigenproblems. Mechanical Systems and Signal Processing34: 88–105.
18.
LiLHuYJWangXL (2014b) Accurate method for harmonic responses of non-classically damped systems in the middle frequency range. Journal of Vibration and Control. DOI: 10.1177/1077546314533579.
19.
LiLHuYJWangXL (2014c) Design sensitivity and Hessian matrix of generalized eigenproblems. Mechanical Systems and Signal Processing43: 272–294.
20.
LiLHuYJWangXL (2014d) Inclusion of higher modes in the eigensensitivity of nonviscously damped systems. AIAA Journal52: 1316–1322.
21.
LiLHuYJWangXL (2015) Efficient and accurate calculation of sensitivity of damped eigensystems. Computers & Structures146: 163–175.
22.
LiLHuYJWangXL (2012b) Computation of eigensolution derivatives for nonviscously damped systems using the algebraic method. AIAA Journal50: 2282–2284.
23.
LiLHuYJWangXL (2013b) Eigensensitivity analysis of damped systems with distinct and repeated eigenvalues. Finite Elements in Analysis and Design72: 21–34.
24.
LiLHuYJWangXL (2014e) A hybrid expansion method for frequency response functions of non-proportionally damped systems. Mechanical Systems and Signal Processing42: 31–41.
25.
MaFMorzfeldMImamA (2010) The decoupling of damped linear systems in free or forced vibration. Journal of Sound and Vibration329: 3182–3202.
26.
MaoKLiBWuJ (2010) Stiffness influential factors-based dynamic modeling and its parameter identification method of fixed joints in machine tools. International Journal of Machine Tools and Manufacture50: 156–164.
27.
MassaFTisonTLallemandB (2011) Structural modal reanalysis methods using homotopy perturbation and projection techniques. Computer Methods in Applied Mechanics and Engineering200: 2971–2982.
ModakSKundraTNakraB (2000) Model updating using constrained optimization. Mechanics Research Communications27: 543–551.
30.
MorzfeldMMaFParlettBN (2011) The transformation of second-order linear systems into independent equations. SIAM Journal on Applied Mathematics71: 1026–1043.
31.
MottersheadJEMichaelLFriswellMI (2011) The sensitivity method in finite element model updating: a tutorial. Mechanical Systems and Signal Processing25: 2275–2296.
32.
NelsonRB (1976) Simplified calculation of eigenvector derivatives. AIAA Journal14: 1201–1205.
33.
OuisseMCoganS (2010) Robust design of spot welds in automotive structures: a decision-making methodology. Mechanical Systems and Signal Processing24: 1172–1190.
34.
RibeiroDCalçadaRDelgadoR (2012) Finite element model updating of a bowstring-arch railway bridge based on experimental modal parameters. Engineering Structures40: 413–435.
35.
SestieriAIbrahimSR (1994) Analysis of errors and approximations in the use of modal coordinates. Journal of Sound and Vibration177: 145–157.
36.
WangBP (1991) Improved approximate methods for computing eigenvector derivatives in structural dynamics. AIAA Journal29: 1018–1020.
37.
WengSZhuH-PXiaY (2013) Substructuring approach to the calculation of higher-order eigensensitivity. Computers & Structures117: 23–33.
38.
WuBYangSLiZ (2015) A preconditioned conjugate gradient method for computing eigenvector derivatives with distinct and repeated eigenvalues. Mechanical Systems and Signal Processing50–51: 249–259.
39.
XieHQ (2012) An iterative method for partial derivatives of eigenvectors of quadratic eigenvalue problems. BIT Numerical Mathematics52: 525–536.
40.
XieHQ (2013) Simultaneous iterative method for eigenpair derivatives of damped systems. AIAA Journal51: 236–239.
41.
ZhangJ-TZhuHZhouS-Q (2012) Optimal design of a rod shape ultrasonic motor using sequential quadratic programming and finite element method. Finite Elements in Analysis and Design59: 11–17.
42.
ZhangZ-Y (2014) Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues. Applied Mathematics and Mechanics35: 437–452.
