Structural Engineering and Mechanics

  • 제73권3호
  • /
  • Pages.271-286
  • /
  • 2020
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Generalized complex mode superposition approach for non-classically damped systems

  • Chen, Huating (Guangdong Provincial Key Laboratory of Earthquake Engineering and Applied Technology, Guangzhou University) ;
  • Liu, Yanhui (Guangdong Provincial Key Laboratory of Earthquake Engineering and Applied Technology, Guangzhou University) ;
  • Tan, Ping (Guangdong Provincial Key Laboratory of Earthquake Engineering and Applied Technology, Guangzhou University)
  • 투고 : 2019.03.03
  • 심사 : 2019.10.05
  • 발행 : 2020.02.10
⟨ 이전 논문 다음 논문 ⟩

초록

Passive control technologies are commonly used in several areas to suppress structural vibrations by the addition of supplementary damping, and some modal damping may be heavy beyond critical damping even for regular structures with energy dissipation devices. The design of passive control structures is typically based on (complex) mode superposition approaches. However, the conventional mode superposition approach is predominantly applied to cases of under-critical damping. Moreover, when any modal damping ratio is equal or close to 1.0, the system becomes defective, i.e., a complete set of eigenvectors cannot be obtained such that some well-known algorithms for the quadratic eigenvalue problem are invalid. In this paper, a generalized complex mode superposition method that is suitable for under-critical, critical and over-critical damping is proposed and expressed in a unified form for structural displacement, velocity and acceleration responses. In the new method, the conventional algorithm for the eigenvalue problem is still valid, even though the system becomes defective due to critical modal damping. Based on the modal truncation error analysis, modal corrected methods for displacement and acceleration responses are developed to approximately consider the contribution of the truncated higher modes. Finally, the implementation of the proposed methods is presented through two numerical examples, and the effectiveness is investigated. The results also show that over-critically damped modes have a significant impact on structural responses. This study is a development of the original complex mode superposition method and can be applied well to dynamic analyses of non-classically damped systems.

키워드

과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China, Guangzhou University

참고문헌

  1. Akgu, M.A. (1993), "A new family of mode-superposition methods for response calculations", J Sound Vib., 167(2), 289-302. https://doi.org/10.1006/jsvi.1993.1336.
  2. Amin, M. and Ang, A. (1968), "Nonstationary stochastic model of earthquake motions", J Eng Mech ASCE, 94(2), 559-583.
  3. Benfratello, S. and Muscolino, G. (2001), "Mode-superposition correction method for deterministic and stochastic analysis of structural systems", Comput Struct., 79(26-28), 2471-2480. https://doi.org/10.1016/S0045-7949(01)00084-0.
  4. Besselink, B., Tabak, U., Lutowska, A., Wouw, N., Nijmeijer, H., Rixen, D.J., Hochstenbach, M.E. and Schilders, W.H.A. (2013), "A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control", J Sound Vib., 332(19), 4403-4422. https://doi.org/10.1016/j.jsv.2013.03.025.
  5. Cacciola, P., Maugeri, N. and Muscolino, G. (2007), "A modal correction method for non-stationary random vibrations of linear systems", Probab. Eng. Mech., 22(2), 170-180. https://doi.org/10.1016/j.probengmech.2006.10.002.
  6. Camarda, C.J., Haftka, R.T. and Riley, M.F. (1987), "An evaluation of higher-order modal methods for calculating transient structural response", Comput Struct., 27(1), 89-101. https://doi.org/10.1016/00457949(87)90184-2.
  7. Chen, H.T., Tan, P. and Zhou, F.L. (2017a), "An improved response spectrum method for non-classically damped systems", Bull. Earthq. Eng., 15(10), 4375-4397. https://doi.org/10.1007/s10518-017-0144-4.
  8. Chen, H.T., Tan, P., Ma, H.T. and Zhou, F.L. (2017b), "Response spectrum analysis considering non-classical damping in the base-isolated benchmark building", Struct. Eng. Mech., 64(4), 473-485. https://doi.org/10.12989/sem.2017.64.4.473.
  9. Chen, J., Xu, R. and Zhang, M. (2014), "Acceleration response spectrum for predicting floor vibration due to occupant walking", J Sound Vib. 333(15), 3564-3579. https://doi.org/10.1016/j.jsv.2014.03.023.
  10. Chu, Y.L., Song, J.W. and Lee, G.C. (2009), "Modal analysis of arbitrarily damped three-dimensional linear structures subjected to seismic excitations", Technical Report, MCEER-09-0001, University of Buffalo, State University of New York.
  11. Clough, R. and Penzien, J. (1995), Dynamics of Structures, (3rd Edition), Computers and Structures Inc., California.
  12. Cornwell, R.E., Craig, R.R. and Johnston, C.P. (1983), "On the application of the mode acceleration method to structural dynamics problems", Earthq Eng Struct Dyn., 11(5), 679-688. https://doi.org/10.1002/eqe.4290110507.
  13. Der Kiureghian, A. and Nakamura, Y. (1993), "CQC modal combination rule for high-frequency modes", Earthq Eng Struct Dyn., 22(11), 943-956. https://doi.org/10.1002/eqe.4290221103.
  14. Dhileep, M. and Bose, P.R. (2008), "A comparative study of "missing mass" correction methods for response spectrum method of seismic analysis", Comput Struct., 86(21-22), 2087-2094. https://doi.org/10.1016/j.compstruc.2008.04.003.
  15. Dickens, J.M., Nakagawa, J.M. and Wittbrodt, M.J. (1997), "A critique of mode acceleration and modal truncation augmentation methods for modal response analysis", Comput Struct., 62(6), 985-998. https://doi.org/10.1016/s0045-7949(96)00315-x.
  16. Dickens, J.M. and Pool, K.V. (1992), "Modal truncation vectors and periodic time domain analysis applied to a cyclic symmetry structure", Comput Struct., 45(4), 685-696. https://doi.org/10.1016/0045-7949(92)90487-k.
  17. FEMA 273 (1997), NEHRP guidelines for the seismic rehabilitation of buildings. The Federal Emergency Management Agency; Washington, USA.
  18. Friswell, M., Prells, U. and Garvey, S. (2005), "Low-rank damping modifications and defective systems", J Sound Vib., 279(3-5), 757-774. https://doi.org/10.1016/j.jsv.2003.11.042.
  19. Hansteen, O.E. and Bell K. (1979), "On the accuracy of mode superposition analysis in structural dynamics", Earthq Eng Struct Dyn. 7(5), 405-411. https://doi.org/10.1002/eqe.4290070502.
  20. Huang, C., Liu, Z.S. and Chen, S.H. (1997), "An accurate modal method for computing response to periodic excitation", Comput Struct., 63(3), 625-631. https://doi.org/10.1016/s0045-7949(96)00367-7.
  21. Liu, G.H., Lian, J.J. and Liang, C. (2016) "Completeness verification of complex response spectrum method for underdamped and overdamped systems regarding the decoupled damping as mathematical parameter without physical meaning", J. Earthq. Eng., 20(7), 1104-1125. https://doi.org/10.1080/13632469. 2016.113 8165.
  22. Li, L., Hu, Y. and Wang, X. (2013), "Improved approximate methods for calculating frequency response function matrix and response of MDOF systems with viscoelastic hereditary terms", J Sound Vib., 332(15), 3945-3956. https://doi.org/10.1016/j.jsv.2013.01.043.
  23. Li, L., Hu, Y., Wang, X. and Lu, L. (2014), "A hybrid expansion method for frequency response functions of non-proportionally damped systems", Mech Syst Signal Pr., 42(1-2), 31-41. https://doi.org/10.1016/j.ymssp.2013.07.020.
  24. Liu, W., Tong, M. and Lee, G.C. (2005), "Optimization methodology for damper configuration based on building performance indices", J. Struct. Eng., 131(11), 1746-1756. https://doi.org/10.1061/(ASCE)0733-9445(2005)131:11(1746).
  25. Liu, Z.S., Wang, K., Huang, C. and Chen, S.H. (1996), "An extended hybrid method for contribution due to truncate lower- and higher-frequency modes in modal summation", Eng Struct., 18(7), 558-563. https://doi.org/10.1016/0141-0296(95)00123-9.
  26. Lu, L., Chen, P. and Pong, K. (2016), "Theory and experiment of an inertia-type vertical isolation system for seismic protection of equipment", J Sound Vib., 366(31), 44-61. https://doi.org/10.1016/j.jsv.2015.12.009.
  27. Lu, L.Y., Hung, J.H. and Lin, P.Y. (2014), "Seismic simulation test of equipment protection by using a fuzzy-controlled smart isolation system", J Vib Control., 20(16), 2501-2520. https://doi.org/10.1177/1077546313479989.
  28. Maddox, N.R. (1975), "On the number of modes necessary for accurate response and resulting forces in dynamic analyses", J Appl Mech ASME, 42(2), 516-517. https://doi.org/10.1115/1.3423622.
  29. Mentrasti, L. (2012), "Exact deflation in the complex modal analysis of low-rank non-classically damped structures", Eng. Struct., 45, 496-508. https://doi.org/10.1016/j.engstruct.2012.06.023.
  30. Moler, C.B. and Stewart, G.W. (1973), "An algorithm for generalized matrix eigenvalue problems", SIAM J. Num. Anal., 10(2), 241-256. https://doi.org/10.1137/0710024.
  31. Occhiuzzi, A. (2009), "Additional viscous dampers for civil structures: analysis of design methods based on effective evaluation of modal damping ratios", Eng. Struct., 31(5), 1093-1101. https://doi.org/10.1016/j.engstruct.2009.01.006
  32. Ohtori, Y., Christenson, R.E., Spencer Jr., B.F. and Dyke, S.J. (2004) "Benchmark control problems for seismically excited nonlinear buildings", J. Eng. Mech., 130(4), 366-385. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:4(366).
  33. Pozzi1, M. and Der Kiureghian, A. (2015), "Response spectrum analysis for floor acceleration", Earthq Eng Struct. Dyn., 44(12), 2111-2127. https://doi.org/10.1002/eqe.2583.
  34. Qu, Z.-Q. (2000), "Hybrid expansion method for frequency responses and their sensitivities, Part I: Undamped systems", J Sound Vib., 231(1), 175-193. https://doi.org/10.1006/jsvi.1999.2672.
  35. Qu, Z.-Q. and Selvam, R.P. (2000), "Hybrid expansion method for frequency responses and their sensitivities, Part II: Viscously damped systems", J Sound Vib., 238(3), 369-388. https://doi.org/10.1006/jsvi.2000.3085.
  36. Ruiz, P. and Penzien, J. (1968), "PSEAGN-artificial generation of earthquake accelerograms", Report of Earthquake Engineering Research Center, No. UCB/EERC69-3, University of California, Berkeley, California.
  37. Ruiz, P. and Penzien, J. (1969), Probabilistic study of behavior of structures during earthquakes. Report of Earthquake Engineering Research Center, No. EERC 69-3, University of California, Berkeley, California.
  38. Singh, M.P. (1980), "Seismic response by SRSS for non-proportional damping", J. Eng. Mech. ASCE, 106(6), 1405-1419. https://doi.org/10.1243/03093247V154235.
  39. Singh, M.P. and McCown, B.E. (1986), "Mode acceleration-based response spectrum approach for non-classically damped structures", Soil Dyn Earthq Eng., 5(4), 226-233. https://doi.org/10.1016/0267-7261(86)90008-4.
  40. Sivandi-Pour, A., Gerami, M. and Khodayarnezhad, D. (2014), "Equivalent modal damping ratios for non-classically damped hybrid steel concrete buildings with transitional storey", Struct. Eng. Mech., 50(3), 383-401. https://doi.org/10.12989/sem.2014.50.3.383.
  41. Suarez, L.E. and Gaviria, C.A. (2015), "Dynamic properties of a building with viscous dampers in non-proportional arrangement", Struct. Eng. Mech., 55(6), 1241-1260. https://doi.org/10.12989/sem.2015.55.6.1241.
  42. Takewaki, I. (2004), "Frequency domain modal analysis of earthquake input energy to highly damped passive control structures", Earthq Eng Struct Dyn., 33(5), 575-590. https://doi.org/10.1002/eqe.361.
  43. Traill-Nash, R.W. (1981), "Modal methods in the dynamics of systems with non-classical damping", Earthq Eng Struct Dyn., 9(2), 153-169. https://doi.org/10.1002/eqe.4290090205.
  44. Uriz, P. and Whittaker, A. (2001), "Retrofit of pre-Northridge steel moment-resisting frames using fluid viscous dampers", Struct. Design Tall Build., 10(3), 71-90. https://doi.org/10.1002/tal.199.
  45. Veletsos, A.S. and Ventura, C.E. (1986), "Modal analysis of non-classically damped linear systems", Earthq. Eng. Struct. Dyn., 14(2), 217-243. https://doi.org/10.1002/eqe.4290140205.
  46. Whittle, J., Williams, M., Karavasilis, T. and Blakeborough, A. (2012), "A comparison of viscous damper placement methods for improving seismic building design", J. Earthq. Eng., 16(4), 540-560. https://doi.org/10.1080/13632469.2011.653864.
  47. Xiao, W.W., Li, L. and Lei, S. (2017), "Accurate modal superposition method for harmonic frequency response sensitivity of non-classically damped systems with lower-higher-modal truncation", Mech. Syst. Signal Pr., 85, 204-217. http://dx.doi.org/10.1016/j.ymssp.2016.08.017.
  48. Yu, R.F., Wang, L., Liu, H. and Zhou, X.Y. (2005), "The CCQC3 Method for Multi-component Seismic Response of Non-classically Damped Irregular Structures", Structures Congress, New York, April.
  49. Yu, R.F. and Zhou, X.Y. (2008), "Response spectrum analysis for non-classically damped linear system with multiple-support excitations", Bull. Earthq. Eng., 6(2), 261-284. https://doi.org/10.1007/s10518-007-9048-z.
  50. Yu, R.F., Zhou, X.Y. and Yuan M.Q. (2012), "Dynamic response analysis of generally damped linear system with repeated eigenvalues", Struct. Eng. Mech., 42(4), 449-469. https://doi.org/10.12989/sem.2012.42.4.449.
  51. Zhou, X.Y., Yu, R.F and Dong, D. (2004), "Complex mode superposition algorithm for seismic responses of non-classically damped linear MDOF system", J. Earthq. Eng., 8(4), 597-641. https://doi.org/10.1142/S1363246904001602.