1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China；
2. College of Mechanical Engineering, Chongqing University, Chongqing 400044, China；
3. Engineering Laboratory for Detection, Control and Integrated System, Chongqing Technology and Business University, Chongqing 400044, China
The energy dissipation of the hysteretic damping that is described by the normalized Bouc-Wen model is studied. An equation for calculating the energy dissipation with arbitrary deformation amplitude is established along with a fast evaluation approach. On the basis of the equation, the equivalent linearization analysis method for the dynamic response of systems with stranded wire helical springs is proposed. Numeric simulations are carried out to verify the proposed method. The results obtained by the proposed method coincide with those obtained by the numerical method while the proposed method is much more efficient than the numerical method. The equivalent linearization method is able to improve the efficiency of the designing of systems with stranded wire helical springs significantly; therefore, it is of practical value for the designing of many practical systems which have approximate linear responses.
[1] 王时龙,雷松,周杰,等. 两端并圈多股弹簧的冲击响应研究[J]. 振动与冲击,2011,30(3):64-68.
WANG Shi-long，LEI Song，ZHOU Jie,et al. Impact response of stranded wires helical springs with closed ends[J]. Journal of Vibration and Shock,2011,30(3):64-68.
[2] Zhao Y，Wang S L，Zhou J，et al. Modeling and identification of the dynamic behavior of stranded wire helical springs[J]. Journal of Vibroengineering,2013, 15(1): 326-339.
[3] Peng Y X，Wang S L，Zhou J，et al. Structural design, numerical simulation and control system of a machine tool for stranded wire helical springs[J]. Journal of Manufacturing Systems, 2012, 31(1SI): 34-41.
[4] 王时龙,张明明,周杰,等. 振动状态下螺旋弹簧运动状态模型[J]. 机械工程学报,2012,48(1):78-83.
WANG Shi-long，ZHANG Ming-ming，ZHOU Jie，et al. Motion model of helical springs under vibrational condition[J]. Journal of Mechanical Engineering,2012,48(1):78-83.
[5] Bouc R. Forced vibration of mechanical systems with hysteresis[C].// Djadkov S. Proceedings of the 4th Conference on Nonlinear Oscillations. Prague：Academia,1967: 315.
[6] Wen Y K. Method for random vibration of hysteretic systems[J]. Journal of the Engineering Mechanics Division, 1976, 102(2): 249-263.
[7] 相恒波,方秦,王玮,等. 磁流变阻尼器抗爆隔震性能的数值模拟[J]. 振动与冲击,2007,26(11):106-111.
XIANG Heng-bo，FANG Qin，WANG We,et al. Numerical simulation on shock isolation of blast-resistant structure with magnetorheological dampers[J]. Journal of Vibration and Shock,2007,26(11):106-111.
[8] Ni Y Q，Ko J M，Wong C W，et al. Modelling and identification of a wire-cable vibration isolator via a cyclic loading test part 1: experiments and model development[J]. Proceedings of the Institution of Mechanical Engineers Part I-Journal of Systems and Control Engineering,1999, 213(I3): 163-171.
[9] Okuizumi N，Kimura K. Multiple time scale analysis of hysteretic systems subjected to harmonic excitation[J]. Journal of Sound and Vibration,2004, 272(3): 675-701.
[10] Wong C W，Ni Y Q，Lau S L. Steady-state oscillation of hysteretic differential model .1. Response analysis[J]. Journal of Engineering Mechanics-Asce,1994, 120(11): 2271-2298.
[11] Charalampakis A E，Koumousis V K. On the response and dissipated energy of bouc-wen hysteretic model[J]. Journal of Sound and Vibration. 2008, 309(3/5): 887-895.
[12] Ikhouane F, Manosa V, Rodellar J. Dynamic properties of the hysteretic bouc-wen model[J]. Systems & Control Letters, 2007, 56(3): 197-205.
[13] Olver F W. NIST handbook of mathematical functions[M]. Cambridge: Cambridge University Press, 2010:318.
[14] Braden B. The surveyor's area formula[J]. The College Mathematics Journal,1986, 17(4): 326-337.
[15] Burden R L，Faires J D. Numerical analysis[M]. Boston: Cengage Learning, 2010.
