Stationary response of stochastic viscoelastic system with the right unilateral nonzero offset barrier impacts
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摘要
The stationary response of viscoelastic dynamical system with the right unilateral nonzero offset barrier impacts subjected to stochastic excitations is investigated. First, the viscoelastic force is approximately treated as equivalent terms associated with effects. Then, the free vibro-impact(VI) system is absorbed to describe the periodic motion without impacts and quasi-periodic motion with impacts based upon the level of system energy. The stochastic averaging of energy envelope(SAEE) is adopted to seek the stationary probability density functions(PDFs). The detailed theoretical results for Van der Pol viscoelastic VI system with the right unilateral nonzero offset barrier are solved to demonstrate the important effects of the viscoelastic damping and nonzero rigid barrier impacts condition. Monte Carlo(MC) simulation is also performed to verify the reliability of the suggested approach. The stochastic P-bifurcation caused by certain system parameters is further explored. The variation of elastic modulus from negative to zero and then to positive witnesses the evolution process of stochastic P-bifurcation. From the vicinity of the common value to a wider range, the relaxation time induces the stochastic P-bifurcation in the two interval schemes.
The stationary response of viscoelastic dynamical system with the right unilateral nonzero offset barrier impacts subjected to stochastic excitations is investigated. First, the viscoelastic force is approximately treated as equivalent terms associated with effects. Then, the free vibro-impact(VI) system is absorbed to describe the periodic motion without impacts and quasi-periodic motion with impacts based upon the level of system energy. The stochastic averaging of energy envelope(SAEE) is adopted to seek the stationary probability density functions(PDFs). The detailed theoretical results for Van der Pol viscoelastic VI system with the right unilateral nonzero offset barrier are solved to demonstrate the important effects of the viscoelastic damping and nonzero rigid barrier impacts condition. Monte Carlo(MC) simulation is also performed to verify the reliability of the suggested approach. The stochastic P-bifurcation caused by certain system parameters is further explored. The variation of elastic modulus from negative to zero and then to positive witnesses the evolution process of stochastic P-bifurcation. From the vicinity of the common value to a wider range, the relaxation time induces the stochastic P-bifurcation in the two interval schemes.
The stationary response of viscoelastic dynamical system with the right unilateral nonzero offset barrier impacts subjected to stochastic excitations is investigated. First, the viscoelastic force is approximately treated as equivalent terms associated with effects. Then, the free vibro-impact(VI) system is absorbed to describe the periodic motion without impacts and quasi-periodic motion with impacts based upon the level of system energy. The stochastic averaging of energy envelope(SAEE) is adopted to seek the stationary probability density functions(PDFs). The detailed theoretical results for Van der Pol viscoelastic VI system with the right unilateral nonzero offset barrier are solved to demonstrate the important effects of the viscoelastic damping and nonzero rigid barrier impacts condition. Monte Carlo(MC) simulation is also performed to verify the reliability of the suggested approach. The stochastic P-bifurcation caused by certain system parameters is further explored. The variation of elastic modulus from negative to zero and then to positive witnesses the evolution process of stochastic P-bifurcation. From the vicinity of the common value to a wider range, the relaxation time induces the stochastic P-bifurcation in the two interval schemes.
引文
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[2]Arenas J P and Crocker M J 2010 Sound Vib.44 12
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[15]Potapov V 1997 J.Appl.Math.Mech.61 287
[16]Zhu W Q and Cai G Q 2011 Internat.J.Non-Linear Mech.46 720
[17]Huang Q H and Xie W C 2008 J.Appl.Mech.75 021012
[18]Di Paola M,Failla G and Pirrotta A 2012 Probabilist.Eng.Mech.2885
[19]Soize C and Poloskov I E 2012 Comput.Math.Appl.64 3594
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[21]Brogliato B 1996 Nonsmooth Impact Mechanics:Models,Dynamics and Control,LNCIS 220(Berlin:Springer Verlag)
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[25]Feng Q 2003 Comput.Methods Appl.Mech.Eng.192 2339
[26]Feng Q and He H 2003 Eur.J.Mech.A Solids 22 267
[27]Namachchivaya N S and Park J H 2005 J.Appl.Mech.72 862
[28]Park J H and Namachchivaya N S 2004 ASME 2004 International Mechanical Engineering Congress and Exposition November 13-19,2004Anaheim,USA,pp.189-200
[29]Zhuravlev V F 1976 Mech.Solids 11 23
[30]Wu Y and Zhu W Q 2008 Phys.Lett.A 372 623
[31]Feng J,Xu W and Wang R 2008 J.Sound Vib.309 730
[32]Dimentberg M,Gaidai O and Naess A 2009 Internat.J.Non-Linear Mech.44 791
[33]Wang D L,Xu W and Zhao X R 2016 Chaos 26 033103
[34]Feng J Q,Xu W and Niu Y J 2010 Acta Phys.Sin.59 157(in Chinese)
[35]Kumar P,Narayanan S and Gupta S 2014 Probabilist.Eng.Mech.38143
[36]Kumar P,Narayanan S and Gupta S 2016 Nonlinear Dynam.85 439
[37]Kumar P,Narayanan S and Gupta S 2017 Int.J.Mech.Sci.127 103
[38]Su M B and Rong H W 2011 Chin.Phys.B 20 060501
[39]Rong H W,Wang X D,Xu W and Fang T 2005 Acta Phys.Sin.54 4610(in Chinese)
[40]Ma S J,Xu W and Li W 2006 Acta Phys.Sin.55 4013(in Chinese)
[41]Yang L H,Ge Y and Ma X K 2017 Acta Phys.Sin.66 190501(in Chinese)
[42]Li W,Zhang M T and Zhao J F 2017 Chin.Phys.B 26 090501
[43]Kumar P,Narayanan S and Gupta S 2016 Probabilist.Eng.Mech.4570
[44]Kumar P,Narayanan S and Gupta S 2016 Procedia Eng.144 998
[45]Zhu W Q and Lin Y K 1991 J.Eng.Mech.117 1890
[46]Xu W,He Q,Rong H W and Fang T 2003 Proceedings of the Fifth International Conference on Stochastic Structural Dynamics-SSD03(Zhu W Q Ed.)(Boca Raton:CRC Press)pp.509-515
[47]Dtchetgnia Djeundam S,Yamapi R,Kofane T and Aziz-Alaoui M 2013Chaos 23 033125
[48]Arnold L 2013 Random Dynamical Systems(Berlin:Springer-Verlag)
[49]Zakharova A,Feoktistov A,Vadivasova T and Scholl E 2013 Eur.Phys.J.Spec.TOP.222 2481
[2]Arenas J P and Crocker M J 2010 Sound Vib.44 12
[3]Lu L M,Wen J H,Zhao H G and Wen X S 2014 Acta Phys.Sin.63154301(in Chinese)
[4]Kiik J C,Kurasov P and Usman M 2015 Phys.Lett.A 379 1871
[5]Khan Z,El Naggar M and Cascante G 2011 J.Franklin I.348 1363
[6]Findley W N and Davis F A 2013 Creep and Relaxation of Nonlinear Viscoelastic Materials(Amsterdam:North-Holland)
[7]Del Nobile M,Chillo S,Mentana A and Baiano A 2007 J.Food Eng.78 978
[8]Renaud F,Dion J L,Chevallier G,Tawfiq I and Lemaire R 2011 Mech.Syst.Signal.Pr.25 991
[9]Christensen R 2012 Theory of Viscoelasticity:an Introduction(New York:Academic Press)
[10]Roscoe R 1950 Br.J.Appl.Phys.1 171
[11]Drozdov A D 1998 Viscoelastic Structures:Mechanics of Growth and Aging(San Diego:Academic Press)
[12]de Esp?ndola J J,Bavastri C A and Lopes E M 2010 J.Franklin I.347102
[13]Han X and Wang M 2009 Math.Methods Appl.Sci.32 346
[14]Zhu W Q 1992 Random Vibration(Beijing:Science Press)(in Chinese)
[15]Potapov V 1997 J.Appl.Math.Mech.61 287
[16]Zhu W Q and Cai G Q 2011 Internat.J.Non-Linear Mech.46 720
[17]Huang Q H and Xie W C 2008 J.Appl.Mech.75 021012
[18]Di Paola M,Failla G and Pirrotta A 2012 Probabilist.Eng.Mech.2885
[19]Soize C and Poloskov I E 2012 Comput.Math.Appl.64 3594
[20]Zhu W Q 1996 Appl.Mech.Rev.49 S72
[21]Brogliato B 1996 Nonsmooth Impact Mechanics:Models,Dynamics and Control,LNCIS 220(Berlin:Springer Verlag)
[22]Shaw S and Holmes P 1983 J.Appl.Mech.50 849
[23]Dankowicz H and Nordmark A B 2000 Phys.D 136 280
[24]Dimentberg M and Iourtchenko D 2004 Nonlinear Dynam.36 229
[25]Feng Q 2003 Comput.Methods Appl.Mech.Eng.192 2339
[26]Feng Q and He H 2003 Eur.J.Mech.A Solids 22 267
[27]Namachchivaya N S and Park J H 2005 J.Appl.Mech.72 862
[28]Park J H and Namachchivaya N S 2004 ASME 2004 International Mechanical Engineering Congress and Exposition November 13-19,2004Anaheim,USA,pp.189-200
[29]Zhuravlev V F 1976 Mech.Solids 11 23
[30]Wu Y and Zhu W Q 2008 Phys.Lett.A 372 623
[31]Feng J,Xu W and Wang R 2008 J.Sound Vib.309 730
[32]Dimentberg M,Gaidai O and Naess A 2009 Internat.J.Non-Linear Mech.44 791
[33]Wang D L,Xu W and Zhao X R 2016 Chaos 26 033103
[34]Feng J Q,Xu W and Niu Y J 2010 Acta Phys.Sin.59 157(in Chinese)
[35]Kumar P,Narayanan S and Gupta S 2014 Probabilist.Eng.Mech.38143
[36]Kumar P,Narayanan S and Gupta S 2016 Nonlinear Dynam.85 439
[37]Kumar P,Narayanan S and Gupta S 2017 Int.J.Mech.Sci.127 103
[38]Su M B and Rong H W 2011 Chin.Phys.B 20 060501
[39]Rong H W,Wang X D,Xu W and Fang T 2005 Acta Phys.Sin.54 4610(in Chinese)
[40]Ma S J,Xu W and Li W 2006 Acta Phys.Sin.55 4013(in Chinese)
[41]Yang L H,Ge Y and Ma X K 2017 Acta Phys.Sin.66 190501(in Chinese)
[42]Li W,Zhang M T and Zhao J F 2017 Chin.Phys.B 26 090501
[43]Kumar P,Narayanan S and Gupta S 2016 Probabilist.Eng.Mech.4570
[44]Kumar P,Narayanan S and Gupta S 2016 Procedia Eng.144 998
[45]Zhu W Q and Lin Y K 1991 J.Eng.Mech.117 1890
[46]Xu W,He Q,Rong H W and Fang T 2003 Proceedings of the Fifth International Conference on Stochastic Structural Dynamics-SSD03(Zhu W Q Ed.)(Boca Raton:CRC Press)pp.509-515
[47]Dtchetgnia Djeundam S,Yamapi R,Kofane T and Aziz-Alaoui M 2013Chaos 23 033125
[48]Arnold L 2013 Random Dynamical Systems(Berlin:Springer-Verlag)
[49]Zakharova A,Feoktistov A,Vadivasova T and Scholl E 2013 Eur.Phys.J.Spec.TOP.222 2481