Abstract：Here, probability response and bifurcation phenomenon of a non-smooth shape memory alloy (SMA) beam under external and parametric excitations were studied.Firstly, the approximate system equivalent to the original system was obtained with the non-smooth transformation.Then, the corresponding average Fokker-Planck-Kolmogorov equation was derived using the stochastic averaging method and the stochastic differential equation theory.Finally, the analytical results of the approximate system were obtained using the non-smooth inverse transformation, and those were verified with numerical simulation.In addition, using the extreme value condition of steady-state response, the critical condition of the system having stochastic bifurcation was further obtained, and its correctness was verified to make the prediction of stochastic bifurcation more intuitive.The study results showed that both damping coefficient and coefficient of restitution can induce the system’s stochastic bifurcation; critical values of bifurcation parameters are obtained.
李玉婷，冯进钤，岳晓乐. 外激与参激作用下非光滑形状记忆合金梁的概率响应[J]. 振动与冲击, 2021, 40(17): 1-6.
LI Yuting, FENG Jinqian, YUE Xiaole. Probabilistic response of non-smooth SMA beams under external and parametric excitations. JOURNAL OF VIBRATION AND SHOCK, 2021, 40(17): 1-6.
［1］XU M, WANG Y, JIN X L, et al.Incorporating dissipated impact into random vibration analyses through the modified hertzian contact model ［J］.Journal of Engineering Mechanics, 2013, 139(12): 1736-1743.
［2］XU M, WANG Y, JIN X L, et al.Random vibration with inelastic impact: equivalent non-linearization technique ［J］.Journal of Sound and Vibration, 2014, 333(1): 189-199.
［3］李超.一类碰撞振动系统的响应研究［D］.西安: 西北工业大学, 2015.
［4］RONG H W, WANG X D, XU W, et al.Resonant response of a non-linear vibro- impact system to combined deterministic harmonic and random excitations ［J］. International Journal of Non-Linear Mechanics, 2010, 45(5): 474-481.
［5］ZHU H T.Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises ［J］.Journal of Sound and Vibration, 2014, 333(3): 954-961.
［6］ZHU H T.Stochastic response of a vibro-impact Duffing system under external Poisson impulses ［J］.Nonlinear Dynamics, 2015, 82(1/2): 1001-1013.
［7］ZHU H T.Response of a vibro-impact Duffing system with a randomly varying damping term ［J］.International Journal of Non-Linear Mechanics, 2014, 65: 53-62.
［8］ZHU H T.Probabilistic solution of vibro-impact stochastic Duffing systems with a unilateral non-zero offset barrier ［J］.Physical A: Statistical Mechanics and its Applications, 2014, 410: 335-344.
［9］ZHU H T.Stochastic response of a parametrically excited vibro-impact system with a non-zero offset constraint ［J］.International Journal of Dynamics and Control, 2015, 4(2): 180-194.
［10］ZHU H T.A solution procedure for a vibro-impact problem under fully correlated Gaussian white noises ［J］.CMES: Computer Modeling in Engineering & Sciences, 2014, 97(3): 281-298.
［11］聂旭涛, 黄科.随机激励下导引头伺服机构动力学特性的研究［J］.振动与冲击, 2010, 29(2): 128-130.
NIE Xutao, HUANG Ke.Study on the dynamics of the seeker servo mechanism affected by random excitation ［J］.Journal of Vibration and Shock, 2010, 29(2): 128-130.
［12］张波, 曾京, 刘伟渭.Gauss白噪声参激下悬挂轮对系统的随机稳定性研究［J］.振动与冲击, 2015, 34(19): 49-56.
ZHANG Bo, ZENG Jing, LIU Weiwei.Research on stochastic stability of suspension wheelset system under Gauss white noise ［J］.Journal of Vibration and Shock, 2015, 34(19): 49-56.
［13］徐明, 金华斌.非弹性碰撞振动系统的首次穿越分析［J］.振动与冲击, 2016, 35(17): 197-200.
XU Ming, JIN Huabin. First-passage failure of the non-inelastic vibro-impact system ［J］.Journal of Vibration and Shock, 2016, 35(17): 197-200.
［14］张雷, 吴勇军.五自由度强非线性随机振动系统的首次穿越研究［J］.振动与冲击，2012, 31(12): 1-4.
ZHANG Lei, WU Yongjun.First-passage of 5-DOF strongly nonlinear random vibration systems ［J］.Journal of Vibration and Shock, 2012, 31(12): 1-4.
［15］DIMENTBERG M F, IOURTCHENKO D V.Stochastic and/or chaotic response of a vibration system to imperfectly periodic sinusoidal excitation ［J］.International Journal of Bifurcation and Chaos, 2005, 15(6): 2057-2061.
［16］YANG G D, XU W, FENG J Q, et al.Response analysis of Rayleigh-Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations ［J］.Nonlinear Dynamics, 2015, 82(4): 1797-1810.
［17］FENG J Q, XU W, RONG H W, et al.Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations ［J］.International Journal of Non-Linear Mechanics, 2009, 44(1): 51-57.
［18］FENG J Q, XU W, WANG R.Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise ［J］.Journal of Sound and Vibration, 2008, 309(3/4/5): 730-738.
［19］HUANG Z L, LIU Z H, ZHU W Q.Stationary response of multi degree-of-freedom vibro-impact systems under white noise excitations ［J］.Journal of Sound and Vibration, 2004, 275(1/2): 223-240.
［20］GU X D, ZHU W Q.A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations ［J］.Journal of Sound and Vibration, 2014, 333(9): 2632-2642.
［21］郑丽文, 金肖玲, 黄志龙.改进的基于雅可比椭圆函数的随机平均法［J］.力学季刊, 2014, 35(4): 531-540.
ZHENG Liwen, JIN Xiaoling, HUANG Zhilong.Improved random average method based on Jacobian elliptic function ［J］.Chinese Quarterly of Mechanics, 2014, 35(4): 531-540.
［22］郑丽文.基于雅可比椭圆函数的随机平均法及其应用［D］.杭州: 浙江大学, 2015.
［23］徐文俊, 郑丽文, 马品奎.一种改进的基于Jacobi椭圆函数的随机平均法［J］.振动工程学报, 2019(3): 444-451.
XU Wenjun, ZHENG Liwen, MA Pinkui.An improved random average method based on Jacobi elliptic function ［J］.Journal of Vibration Engineering, 2019(3): 444-451.
［24］刘先斌, 陈虬, 陈大鹏.非线性随机动力系统的稳定性和分岔研究［J］.力学进展, 1996, 26(4): 437-452.
LIU Xianbin, CHEN Qiu, CHEN Dapeng.Research on stability and bifurcation of nonlinear stochastic dynamic system ［J］.Advances in Mechanics, 1996, 26(4): 437-452.
［25］郝颖, 吴志强.三稳态Van der Pol-Duffing振子的随机P-分岔［J］.力学学报, 2013, 45(2): 257-264.
HAO Ying, WU Zhiqiang.Random P-bifurcation of tri-stable Van der Pol-Duffing oscillator ［J］.Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(2): 257-264.
［26］吴志强, 郝颖.乘性色噪声激励下三稳态Van der Pol-Duffing振子随机P-分岔［J］.物理学报, 2015, 64(6): 57-62.
WU Zhiqiang, HAO Ying.Random P-bifurcation of tri-stable Van der Pol-Duffing oscillator excited by multiplicative colored noise ［J］.Acta Physica Sinica, 2015, 64(6): 57-62.
［27］吴志强, 郝颖.随机激励Van der Pol-Duffing方程三峰P-分岔［J］.中国科学: 物理学 力学 天文学, 2013, 43(4): 524-529.
WU Zhiqiang, HAO Ying.Three-peak P-bifurcation of Van der Pol-Duffing equation with random excitation ［J］.Scientia Sinica Pysica,Mechanica & Astronomica, 2013, 43(4): 524-529.
［28］徐伟, 贺群, 戎海武,等.Duffing-Van der Pol振子随机分岔的全局分析［J］.物理学报, 2003, 52(6): 1365-1371.
XU Wei, HE Qun, RONG Haiwu, et al.Global analysis of random bifurcation of Duffing-Van der Pol oscillator ［J］.Acta Physica Sinica, 2003, 52(6): 1365-1371.
［29］葛根, 竺致文, 许佳.形状记忆合金梁在简谐和白噪声激励下的混沌及安全盆侵蚀现象［J］.振动与冲击, 2012, 31(23): 2-11.
GE Gen, ZHU Zhiwen, XU Jia.Chaos and safe basin erosion of shape memory alloy beams excited by simple harmony and white noise ［J］.Journal of Vibration and Shock, 2012, 31(23): 2-11.
［30］冯进钤.典型碰撞振动系统的非线性动力学［M］.北京: 科学出版社, 2018.
［31］朱位秋.非线性随机动力学与控制［M］.北京: 科学出版社, 2003.
