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.
AdhikariS (1999) Modal analysis of linear asymmetric nonconservative systems. Journal of Engineering Mechanics125: 1372–1379.
2.
AdhikariS (2011) An iterative approach for nonproportionally damped systems. Mechanics Research Communications38: 226–230.
3.
AdhikariS (2013a) Structural Dynamic Analysis with Generalized Damping Models: Analysis, Chichester, UK: John Wiley & Sons.
4.
AdhikariS (2013b) Structural Dynamic Analysis with Generalized Damping Models: Identification, Chicheter, UK: John Wiley & Sons.
5.
AdhikariSPascualB (2009) Eigenvalues of linear viscoelastic systems. Journal of Sound and Vibration325: 1000–1011.
6.
AdhikariSPascualB (2011) Iterative methods for eigenvalues of viscoelastic systems. Journal of Vibration and Acoustics133: 021002.
7.
AkgunMA (1993) A new family of mode-superposition methods for response calculations. Journal of Sound and Vibration167: 289–302.
8.
BatheK-J (1996) Finite Element Procedures, New Jersey: Prentice Hall.
9.
BenfratelloSMuscolinoG (2001) Mode-superposition correction method for deterministic and stochastic analysis of structural systems. Computers & Structures79: 2471–2480.
10.
BilelloCDi PaolaMSalamoneS (2005) A correction method for dynamic analysis of linear continuous systems. Computers & Structures83: 662–670.
11.
BilelloCDi PaolaMSalamoneS (2008) A correction method for the analysis of continuous linear one-dimensional systems under moving loads. Journal of Sound and Vibration315: 226–238.
12.
BorinoGMuscolinoG (1986) Mode-superposition methods in dynamic analysis of classically and non-classically damped linear systems. Earthquake Engineering & Structural Dynamics14: 705–717.
13.
CacciolaPMaugeriNMuscolinoG (2007) A modal correction method for non-stationary random vibrations of linear systems. Probabilistic Engineering Mechanics22: 170–180.
14.
CamardaCJHaftkaRTRileyMF (1987) An evaluation of higher-order modal methods for calculating transient structural response. Computers & Structures27: 89–101.
15.
CaugheyTKO’KellyMEJ (1965) Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics – ASME32: 583–588.
16.
ChenSH (2007) Matrix Perturbation Theory in Structural Dynamic Design, Beijing: Science Press.
CortésFElejabarrietaMJ (2006) Computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments. Computer Methods in Applied Mechanics and Engineering195: 6448–6462.
19.
D’AveniAMuscolinoG (2001) Improved dynamic correction method in seismic analysis of both classically and non-classically damped structures. Earthquake Engineering and Structural Dynamics30: 501–517.
20.
Di PaolaMFaillaG (2004) A correction method for dynamic analysis of linear systems. Computers & Structures82: 1217–1226.
21.
DickensJMNakagawaJMWittbrodtMJ (1997) A critique of mode acceleration and modal augmentation methods for modal response analysis. Computers & Structures62: 985–998.
22.
Fawzy I and Bishop RED (1976) On the dynamics of linear non-conservative systems. In Proceedings of the Royal Society of London, Series A 352: 25–40.
23.
FischerP (2000) Eigensolution of nonclassically damped structures by complex subspace iteration. Computer Methods in Applied Mechanics and Engineering189: 149–166.
24.
GawronskiWSawickiJT (1997) Response errors of non-proportionally lightly damped structures. Journal of Sound and Vibration200: 543–550.
25.
HansteenOEBellK (1979) On the accuracy of mode superposition analysis in structural dynamics. Earthquake Engineering & Structural Dynamics7: 405–411.
HolzUBGolubGHLawKH (2004) A subspace approximation method for the quadratic eigenvalue problem. SIAM Journal on Matrix Analysis and Applications26: 498–521.
28.
HuangCLiuZ-SChenS-H (1997a) An accurate modal method for computing response to periodic excitation. Computers & Structures63: 625–631.
29.
HuangCLiuZSChenSH (1997b) An accurate modal method for computing response to periodic excitation. Computers & Structures63: 625–631.
30.
KlerkDDRixenDVoormeerenS (2008) General framework for dynamic substructuring: history, review and classification of techniques. AIAA Journal46: 1169–1181.
KulkarniSHNgSF (1992) Inclusion of higher modes in the analysis of non-classically damped systems. Earthquake Engineering & Structural Dynamics21: 543–549.
33.
KwakM (1993) Perturbation method for the eigenvalue problem of lightly damped systems. Journal of Sound and Vibration160: 351–357.
34.
LeeI-WKimM-CRobinsonA (1998) Efficient solution method of eigenproblems for damped structural systems using modified Newton-Raphson technique. Journal of Engineering Mechanics124: 576–580.
LiLHuYJWangXL (2013a) Design sensitivity analysis of dynamic response of nonviscously damped systems. Mechanical Systems and Signal Processing41: 613–638.
37.
LiLHuYJWangXL (2013b) Improved approximate methods for calculating frequency response function matrix and response of MDOF systems with viscoelastic hereditary terms. Journal of Sound and Vibration332: 3945–3956.
38.
LiLHuYJWangXL (2014a) Eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems. Journal of Sound and Vibration333: 1182–1192.
39.
LiLHuYJWangXL (2014b) Harmonic response calculation of viscoelastic structures using classical normal modes: an iterative method. Computers & Structures133: 39–50.
40.
Li L, Hu YJ and Wang XL (2014c) Inclusion of higher modes in the eigensensitivity of nonviscously damped systems. AIAA Journalhttp://dx.doi.org/10.2514/1.J052797.
41.
LiLHuYJWangXL (2014d) A hybrid expansion method for frequency response functions of non-proportionally damped systems. Mechanical Systems and Signal Processing42: 31–41.
42.
LiuGRQuekSS (2003) The Finite Element Method: A Practical Course, England: Elsevier Science Ltd.
43.
LiuZSChenSH (1994) Contribution of the truncated modes to eigenvector derivatives. AIAA Journal32: 1551–1553.
44.
LiuZSChenSHLiuWT (1994a) An accurate modal method for computing responses to harmonic excitation. International Journal of Analytical and Experimental Modal Analysis9: 1–13.
45.
LiuZSChenSHZhaoYQ (1994b) An accurate method for computing eigenvector derivatives for free-free structures. Computers & Structures52: 1135–1143.
46.
LiuZSWangKHuangC (1996) An extended hybrid method for contribution due to truncated lower- and higher-frequency modes in modal summation. Engineering Structures18: 558–563.
47.
MaddoxNR (1975) On the number of modes necessary for the accurate response and resulting forces in dynamic analyses. Journal of Applied Mechanics – ASME42: 516–517.
NingWWangJZhangJ-H (2010) An improved series expansion method of frequency response function under medium and high frequency excitations. International Journal of Mechanics and Materials in Design6: 11–15.
50.
PalmeriALombardoM (2011) A new modal correction method for linear structures subjected to deterministic and random loadings. Computers & Structures89: 844–854.
51.
QuZ-Q (2000) Hybrid expansion method for frequency responses and their sensitivities, Part I: undamped systems. Journal of Sound and Vibration231: 175–193.
52.
QuZ-Q (2007) Adaptive mode superposition and acceleration technique with application to frequency response function and its sensitivity. Mechanical Systems and Signal Processing21: 40–57.
53.
QuZ-QSelvamRP (2000) Hybrid expansion method for frequency responses and their sensitivities, Part II: viscously damped systems. Journal of Sound and Vibration238: 369–388.
54.
RajakumarC (1993) Lanczos algorithm for the quadratic eigenvalue problem in engineering applications. Computer Methods in Applied Mechanics and Engineering105: 1–22.
55.
RayleighL (1877) Theory of Sound (two vols), New York: Dover Publications.
56.
SestieriAIbrahimSR (1994) Analysis of errors and approximations in the use of modal coordinates. Journal of Sound and Vibration177: 145–157.
57.
TangJWangW (1995) Perturbation method for determining eigensolutions of weakly damped systems. Journal of Sound and Vibration187: 671–681.
58.
TisseurFMeerbergenK (2001) The quadratic eigenvalue problem. SIAM Review43: 235–286.
59.
TsaiHCKellyJM (1988) Non-classical damping in dynamic analysis of base-isolated structures with internal equipment. Earthquake Engineering & Structural Dynamics16: 29–43.
60.
UdwadiaFE (2009) A note on nonproportional damping. Journal of Engineering Mechanics135: 1248–1256.
61.
WarburtonGSoniS (1977) Errors in response calculations for non-classically damped structures. Earthquake Engineering & Structural Dynamics5: 365–376.
62.
WilkinsonJH (1965) The Algebraic Eigenvalue Problem, Oxford: Oxford University Press.
63.
ZengQH (1995) Highly accurate modal method for calculating eigenvector derivatives in viscous damping systems. AIAA Journal33: 746–751.
64.
ZhaoYQLiuZSChenSH (1999) An accurate modal truncation method for eigenvector derivatives. Computers & Structures73: 609–614.
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