[1] Chen Yong, Donald M. McFarland, Billie F. Spencer, et al. Improved complex modal superposition methodwith inclusion of overdamped modes [J]. Journal of Southeast University (English Edition), 2023, 39 (3): 213-224. [doi:10.3969/j.issn.1003-7985.2023.03.001]

Copy

Improved complex modal superposition methodwith inclusion of overdamped modes()

Share：

Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

39

Issue:

2023 3

Page:

213-224

Research Field:

Civil Engineering

Publishing date:

2023-09-20

Info

Title:

Improved complex modal superposition methodwith inclusion of overdamped modes

Author(s):

Chen Yong¹;  2;  Donald M. McFarland³;  4;  Billie F. Spencer;  Jr.²;  Lawrence A. Bergman³

¹College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310027, China
²Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana 61801, USA
³Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana 61801, USA
⁴College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China

Keywords:

nonclassically damped system;  continuous system;  forced vibration;  complex mode superposition method;  overdamped mode;  dynamic response

PACS:

TU311.3

DOI:

10.3969/j.issn.1003-7985.2023.03.001

Abstract:

To address the involvement of overdamped complex modes that appear in a nonproportionally damped system, an improved complex mode superposition(ICMS)theory is proposed for forced vibration analysis, which suggests the use of exact modes in pairs for CMS-based dynamic analysis of whether the modes are overdamped or not. A typical nonproportionally damped system, namely, a cantilever beam with attached multiple arbitrarily placed external dampers, is considered an example because the first mode of the system is likely to be overdamped with the increase in damping. First, the relationship between the response in a complex modal space and the actual dynamic response is elucidated, based on which complete theoretical ICMS approaches for attaining time-domain response, transfer functions, and variances are expounded in detail. By decomposing the governing equation into real and imaginary parts, the original equation of motion in the complex domain is represented by an augmented state-space equation with real-valued matrices, which considerably reduces the difficulties observed in computing the time-varying response using complex-valued matrices. Additionally, for external excitations that can be regarded as filtered white noise, an efficient method for evaluating the variance response is proposed, which effectively reduces the computational cost. The results from the application of the proposed CMS-based methods are compared with those obtained by an assumed-mode(AM)method and finite element analysis(FEA). It can be found that the current results are closer to those obtained by FEA than those by the AM method. Finally, the optimal damping and optimal position of the dampers are investigated using an enumeration method, which reveals that the use of multiple dampers with small damping demonstrates a better effect than that of a single damper with large damping.

References:

[1] Main J A, Krenk S. Efficiency and tuning of viscous dampers on discrete systems[J].Journal of Sound and Vibration, 2005, 286(1/2): 97-122. DOI: 10.1016/j.jsv.2004.09.022.
[2] Caughey T K, O’Kelly M E J. Classical normal modes in damped linear dynamic systems[J].Journal of Applied Mechanics, 1965, 32(3): 583-588. DOI: 10.1115/1.3627262.
[3] Srikantha P A. On the necessary and sufficient conditions for the existence of classical normal modes in damped linear dynamic systems[J].Journal of Sound and Vibration, 2003, 264(3): 741-745. DOI: 10.1016/S0022-460X(02)01506-7.
[4] Hassanpour P A, Esmailzadeh E, Cleghorn W L, et al. Generalized orthogonality condition for beams with intermediate lumped masses subjected to axial force[J].Journal of Vibration and Control, 2010, 16(5): 665-683. DOI: 10.1177/1077546309106526.
[5] Warburton G B. Soil-structure interaction for tower structures[J].Earthquake Engineering & Structural Dynamics, 1978, 6(6): 535-556. DOI: 10.1002/eqe.4290060603.
[6] Wu J S, Lin T L. Free vibration analysis of a uniform cantilever beam with point masses by an analytical-and-numerical-combined method[J].Journal of Sound and Vibration, 1990, 136(2): 201-213. DOI: 10.1016/0022-460X(90)90851-P.
[7] Pacheco B M, Fujino Y, Sulekh A. Estimation curve for modal damping in stay cables with viscous damper[J].Journal of Structural Engineering, 1993, 119(6): 1961-1979. DOI: 10.1061/(asce)0733-9445(1993)119: 6(1961).
[8] Gürg�F6;ze M. On the eigenvalues of viscously damped beams, carrying heavy masses and restrained by linear and torsional springs[J].Journal of Sound and Vibration, 1997, 208(1): 153-158. DOI: 10.1006/jsvi.1997.1165.
[9] Wu J S, Chen D W. Dynamic analysis of a uniform cantilever beam carrying a number of elastically mounted point masses with dampers[J].Journal of Sound and Vibration, 2000, 229(3): 549-578. DOI: 10.1006/jsvi.1999.2504.
[10] Wu J J. Use of equivalent-damper method for free vibration analysis of a beam carrying multiple two degree-of-freedom spring-damper-mass systems[J].Journal of Sound and Vibration, 2005, 281(1/2): 275-293. DOI: 10.1016/j.jsv.2004.01.013.
[11] Chang T P. Forced vibration of a mass-loaded beam with a heavy tip body[J].Journal of Sound and Vibration, 1993, 164(3): 471-484. DOI: 10.1006/jsvi.1993.1229.
[12] Yang B. Exact receptances of nonproportionally damped dynamic systems[J].Journal of Vibration and Acoustics, 1993, 115(1): 47-52. DOI: 10.1115/1.2930313.
[13] Gürg�F6;ze M, Erol H. On the frequency response function of a damped cantilever simply supported in-span and carrying a tip mass[J].Journal of Sound and Vibration, 2002, 255(3): 489-500. DOI: 10.1006/jsvi.2001.4118.
[14] Gürg�F6;ze M, Erol H. Dynamic response of a viscously damped cantilever with a viscous end condition[J].Journal of Sound and Vibration, 2006, 298(1/2): 132-153. DOI: 10.1016/j.jsv.2006.04.042.
[15] Veletsos A S, Ventura C E. Modal analysis of non-classically damped linear systems[J].Earthquake Engineering & Structural Dynamics, 1986, 14(2): 217-243. DOI: 10.1002/eqe.4290140205.
[16] Foss K A. Co-ordinates which uncouple the equations of motion of damped linear dynamic systems[J].Journal of Applied Mechanics, 1958, 25(3): 361-364. DOI: 10.1115/1.4011828.
[17] Zhao Y P, Zhang Y M. Improved complex mode theory and truncation and acceleration of complex mode superposition[J].Advances in Mechanical Engineering, 2016, 8(10): 168781401667151. DOI: 10.1177/1687814016671510.
[18] de Domenico D, Ricciardi G. Dynamic response of non-classically damped structures via reduced-order complex modal analysis: Two novel truncation measures[J]. Journal of Sound and Vibration, 2019, 452: 169-190. DOI: 10.1016/j.jsv.2019.04.010.
[19] Oliveto G, Santini A, Tripodi E. Complex modal analysis of a flexural vibrating beam with viscous end conditions[J].Journal of Sound and Vibration, 1997, 200(3): 327-345. DOI: 10.1006/jsvi.1996.0717.
[20] Oliveti G, Santini A. Complex modal analysis of a continuous model with radiation damping[J].Journal of Sound and Vibration, 1996, 192(1): 15-33. DOI: 10.1006/jsvi.1996.0173.
[21] Fan Z J, Lee J H, Kang K H, et al. The forced vibration of a beam with viscoelastic boundary supports[J].Journal of Sound and Vibration, 1998, 210(5): 673-682. DOI: 10.1006/jsvi.1997.1353.
[22] Krenk S. Complex modes and frequencies in damped structural vibrations[J].Journal of Sound and Vibration, 2004, 270(4/5): 981-996. DOI: 10.1016/S0022-460X(03)00768-5.
[23] Impollonia N, Ricciardi G, Saitta F. Dynamic behavior of stay cables with rotational dampers[J].Journal of Engineering Mechanics, 2010, 136(6): 697-709. DOI: 10.1061/(asce)em.1943-7889.0000115.
[24] Alati N, Failla G, Santini A. Complex modal analysis of rods with viscous damping devices[J].Journal of Sound and Vibration, 2014, 333(7): 2130-2163. DOI: 10.1016/j.jsv.2013.11.030.
[25] Chen Y, McFarland D M, Spencer B F Jr, et al. A beam with arbitrarily placed lateral dampers: Evolution of complex modes with damping[J].Journal of Vibration and Control, 2018, 24(2): 379-392. DOI: 10.1177/1077546316641592.
[26] Jacquot R G. Random vibration of damped modified beam systems[J].Journal of Sound and Vibration, 2000, 234(3): 441-454. DOI: 10.1006/jsvi.1999.2894.
[27] Jacquot R G. The spatial average mean square motion as an objective function for optimizing damping in damped modified systems[J].Journal of Sound and Vibration, 2003, 259(4): 955-965. DOI: 10.1006/jsvi.2002.5209.

Memo

Memo:

Biographies: Chen Yong(1974—), male, Ph.D., professor; Donald M. McFarland(corresponding author), male, doctor, professor, dmmcf@dmmcf.net.
Foundation items: National Science Foundation through TeraGrid Resources provided by the National Center for Supercomputing Applications(No. TG-MSS100016).
Citation: Chen Yong, Donald M. McFarland, Billie F. Spencer, Jr., et al. Improved complex modal superposition method with inclusion of overdamped modes[J].Journal of Southeast University(English Edition), 2023, 39(3):213-224.DOI:10.3969/j.issn.1003-7985.2023.03.001.

Common functions