一类干摩擦系统的周期解及粘滑分岔研究
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摘要
摩擦振子由于摩擦的非光滑性而蕴含着复杂的粘滑运动,这是摩擦振子的基本运动类型.本学位论文首先针对一单自由度干摩擦动力系统,讨论了其周期解的存在性及其应满足的条件,然后针对另一耦合的两自由度干摩擦动力系统,讨论了其在粘滑分界处应满足的必要条件.论文的主要工作如下：
1.针对一单自由度带干摩擦的振动系统,在摩擦类型为库仑摩擦时,讨论了其粘滑周期运动,解析地得到了粘滑周期运动的存在性条件,并在此基础之上,半解析地得到了系统单个粘滑周期运动内,粘、滑运动分界处的时间及其运动状态,最后通过数值仿真更加直观地说明了所得到的理论结果,并且通过数值仿真,得到了在分岔曲线附近,周期运动的周期T与2π的比值T/2π与参数ζ的关系.由于系统关于速度具有对称性,当v0>0时可以完全类似的讨论,并且能得到和v,o<0时类似的结果.
2.讨论了一类两自由度干摩擦系统的非光滑分岔和混沌现象,通过分析得到系统轨线在粘滑分界处应满足的必要条件,以及在切换流形上可能存在滑动现象的区域.通过数值仿真发现,当参数μm,较大时,系统均发生稳定的周期行为,然而当μm。减小到一定程度时,开始出现大片的混沌现象.在混沌参数区域中,系统夹杂着周期现象,除一般的简单周期运动外,其中还存在着极为复杂的周期运动.
The stick-slip behavior in friction oscillators is very complicated due to the non-smooth of the dry friction, which is the basic form of motion of dynamical systems with friction. In this thesis, the stick-slip periodic solution in a single-degree-of-freedom oscillator with dry friction is investigated in detail, and for a two-degree-of-freedom system with dry friction, the necessary condition for the stick-slip boundary is obtained. The main results of the research involve:
1. In a single-degree-of-freedom dynamical system with dry friction, under the assumption of kinetic friction being the Coulomb friction, the existence condition of the stick-slip periodic solution is analytically derived which gives out a new result in a class of friction systems. Moreover, the time and states of motion on the boundary of the stick and slip motions are semi-analytically obtained in a single stick-slip period. Finally, the theoretical results are validated by numerical simulation. At the same time, the relationship between the ratio T/2πwith period T as numerator and the parameterξnearby the bifurcation curve is obtained. Since the system is symmetrical with respect to the velocity, the case for v0>0 can be discussed in the same way as that of the case v0<0 which has been investigated in this paper, and it is clear that the similar conclusions hold.
2. In a two-degree-of-freedom dynamical system with dry friction, the non-smooth bifurcation and chaos phenomena are investigated. By the theoretical analysis, the necessary condition which must be satisfied for the stick-slip orbits in the boundary of stick and slip motions is obtained, and in the switching manifold, we find out the possible area where the slip motion will happen. Using the numerical simulation, it is found that the system possesses steady periodic behaviors when the parameterμm takes rather large values. However, a large number of chaos phenomena appear when the parameterμm is descended to a certain extent. In the parameter region of chaos, there are also lots of periodic phenomena which include complicated periodic motions besides the simple ones.
1.针对一单自由度带干摩擦的振动系统,在摩擦类型为库仑摩擦时,讨论了其粘滑周期运动,解析地得到了粘滑周期运动的存在性条件,并在此基础之上,半解析地得到了系统单个粘滑周期运动内,粘、滑运动分界处的时间及其运动状态,最后通过数值仿真更加直观地说明了所得到的理论结果,并且通过数值仿真,得到了在分岔曲线附近,周期运动的周期T与2π的比值T/2π与参数ζ的关系.由于系统关于速度具有对称性,当v0>0时可以完全类似的讨论,并且能得到和v,o<0时类似的结果.
2.讨论了一类两自由度干摩擦系统的非光滑分岔和混沌现象,通过分析得到系统轨线在粘滑分界处应满足的必要条件,以及在切换流形上可能存在滑动现象的区域.通过数值仿真发现,当参数μm,较大时,系统均发生稳定的周期行为,然而当μm。减小到一定程度时,开始出现大片的混沌现象.在混沌参数区域中,系统夹杂着周期现象,除一般的简单周期运动外,其中还存在着极为复杂的周期运动.
The stick-slip behavior in friction oscillators is very complicated due to the non-smooth of the dry friction, which is the basic form of motion of dynamical systems with friction. In this thesis, the stick-slip periodic solution in a single-degree-of-freedom oscillator with dry friction is investigated in detail, and for a two-degree-of-freedom system with dry friction, the necessary condition for the stick-slip boundary is obtained. The main results of the research involve:
1. In a single-degree-of-freedom dynamical system with dry friction, under the assumption of kinetic friction being the Coulomb friction, the existence condition of the stick-slip periodic solution is analytically derived which gives out a new result in a class of friction systems. Moreover, the time and states of motion on the boundary of the stick and slip motions are semi-analytically obtained in a single stick-slip period. Finally, the theoretical results are validated by numerical simulation. At the same time, the relationship between the ratio T/2πwith period T as numerator and the parameterξnearby the bifurcation curve is obtained. Since the system is symmetrical with respect to the velocity, the case for v0>0 can be discussed in the same way as that of the case v0<0 which has been investigated in this paper, and it is clear that the similar conclusions hold.
2. In a two-degree-of-freedom dynamical system with dry friction, the non-smooth bifurcation and chaos phenomena are investigated. By the theoretical analysis, the necessary condition which must be satisfied for the stick-slip orbits in the boundary of stick and slip motions is obtained, and in the switching manifold, we find out the possible area where the slip motion will happen. Using the numerical simulation, it is found that the system possesses steady periodic behaviors when the parameterμm takes rather large values. However, a large number of chaos phenomena appear when the parameterμm is descended to a certain extent. In the parameter region of chaos, there are also lots of periodic phenomena which include complicated periodic motions besides the simple ones.
引文
[1]Leine R I. Bifurcation in discontinuous mechanical systems of Filippov-type[M]. Ph. d. thesis, Technische Universiteit Eindhoven, the Netherlands,2000.
[2]Li Q H, Chen Y M, Wei Y H, Lu Q S. The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system[J]. International Journal of Non-Linear Mechanics,2011,46(1):197-203.
[3]Nordmark A B. Non-periodic motion caused by grazing incidence in an impact oscillator[J]. Journal of Sound and Vibration,1991,145(2):279-297.
[4]Feigin M I. On the structure of C-bifurcation boundaries of piecewise continuous systems[J]. J Appl Math Mech,1978,42(1):820-829.
[5]Di Bernardo M, Feigin M I, Hogan S J, Homer M E. Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems[J]. Chaos, Solitons & Fractals,1999, 10(28):1881-1908.
[6]Bernardo M D, Budd C J, Champneys AR. Unified framework for the analysis of grazing and border-collisions in piecewise-smooth systems[J]. Phys Rev Lett,2001,86(1): 2553-2556.
[7]Popp K, Stelter P. Stick-slip vibration and chaos[J]. Philos Trans Royal Soc Lond A,1990, 332(1624):89-105.
[8]Galvanetto U, Knudsen C. Event maps in a stick-slip system[J]. Nonlinear Dyn,1997, 13(2):99-115.
[9]Galvanetto U. Some discontinuous bifurcations in a two-block stick-slip system [J]. Journal of Sound and Vibration,2001,248(4):653-669.
[10]Galvanetto U. Sliding bifurcation in the dynamics of mechanical systems with dry friction-remarks for engineers and applied scientists[J]. Journal of Sound and Vibration, 2004,276(12):121-139.
[11]Awrejcewicz J, Holicke M M. Melnikov's method and stick-slip chaotic oscillations in very weakly forced mechanical systems[J]. International Journal of Bifurcation and Chaos, 1999,9(3):505-518.
[12]Awrejcewicz J, Sendkowski D. How to predict stick-slip chaos in R4[J]. Phys Lett A, 2004,330(5):371-376.
[13]Awrejcewicz J, Sendkowski D. stick-slip chaos detection in coupled oscillators with friction[J]. International Journal of Solids and Structures,2005,42(21-22):5669-5682.
[14]Halse C, Homer M, Bernardo M D. C-bifurcations and period-adding in one-dimensional piecewise-smooth maps[J]. Chaos, Solitons & Fractals,2003,18(5):953-976.
[15]Wagg D J. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[J]. Chaos, Solitons & Fractals,2004,22(3):541-548.
[16]Iwan W D. The dynamic Response of Bilinear Hysteretic Systems[M]. California: California Institute of Technology,1961.
[17]Menq C H, Bielak J, Griffin J H. The influence of microslip on vibratory response(Part I, Part Ⅱ) [J]. Journal of Sound and vibration,1986,107(2):279-293.
[18]Wang J H, Shish W L. The influence of a variable friction coefficient on the dynamic behavior of a blade with a friction damper [J]. Journal of Sound and Vibration,1991, 149(1):137-145.
[19]Bouc R. Forced vibration of mechanical system with hysteresis [M]. Proeeedings of 4th Conference on Nonlinear oscillation. Prague, Czechoslovakia,1967.
[20]Wen Y K. Method for random vibration of hysteretic systems [J]. Journal of Engineering Mechanics, ASCE, NewYork,1976,102(2):249-263.
[21]Wen Y K. Equivalent linearization for hysteretic system under random excitation[J]. Journal of Applied Mechanics,1980,47(1):150-154.
[22]Bernardo M D, Johansson K H, Vasca F. Self-oscillations in relay feedback systems: symmetry and bifurcations [J]. International Journal of Bifurcation and Chaos,2011, 11(1):1121-1140.
[23]Feigin M I. Forced Oscillations in Systems with Discontinuous Nonlinearities[M]. Nauka, Moscow,1994.
[24]Kowalczyk P S, Bernardo M D. Existence of stable asymmetric limit cycles and chaos in unforced symmetric relay feedback systems[J]. Proceedings of the European Control Conference, Porto,2001,1999-2004.
[25]Nordmark A B. Non-periodic motion caused by grazing incidence in impact oscillators[J]. Journal of Sound and Vibration,1991,145(2):279-297.
[26]Li Q H, Chen Y M, Qin Z Y. Existence of stick-slip periodic solutions in a dry friction oscillator[J]. Chinese Physics Letters,2011,28(3):030502.
[27]Den Hartog J P. Forced vibrations with combined Coulomb and viscous friction[J]. Transaction of the American Society of Mechanical Engineer,1931,53(9):107-115.
[28]Hong H K, Liu C S. Coulomb friction oscillator:modeling and responses to harmonic loads and base excitations[J]. Journal of Sound and Vibration,2000,229(5):1171-1192.
[29]Hong H K, Liu C S. Non-Sticking oscillation formulae for coulomb friction under harmonic loading[J]. Journal of Sound and Vibration,2001,224(5):883-898.
[30]Feeny B, Guran A, Hinrichs N, etal. A historical review on dry friction and stick-slip phenomena[J]. Americal Society of Mechanical Engineers,1998,51(5):321-341.
[31]Galvanetto U. Some discontinuous bifurcations in a two-block stick-slip system[J]. Journal of Sound and Vibration,2001,248(4):653-669.
[32]Bernardo M D, Kowalczyk P, Nordmark A B. Bifurcations of dynamical systems with sliding:derivation of normal form mapping[J]. Physica D,2002,170(3-4):175-205.
[33]Kowalcyk P, Bernardo M D. Two-parameter degenerate sliding bifurcations in filippov systems[J]. Physica D,2005,204(3-4):204-229.
[34]郭树起,杨绍普.三分之一固有频率谐激励下干摩擦振子的双Stop粘滑运动[J].振动与冲击,2009,28(4)：81-85.
[35]杨绍普,郭树起.一类摩擦振子的纯滑动运动分析[J].振动与冲击,2009,28(10)：43-48.
[36]Galvanetto U. Bifurcations and chaos in a four-dimensional mechanical system with dry friction[J]. Journal of Sound and Vibration,1997,204(4):690-695.
[37]Galvanetto U, Bishop S R. Computational techniques for nonlinear dynamics in multiple friction oscillators[J]. Computer methods in applied mechanics and engineering,1998, 163(1-4):373-382.
[38]Dankowicz H, Nordmark A B. On the origin and bifurcations of stick-slip oscillations[J]. Physica D,2000,136(3-4):280-302.
[39]Bernardo M D, Johansson K H. Self-oscillations in relay feedback systems:symmetry and bifurcations[J]. International Journal of Bifurcation and Chaos,2001,11(4): 1121-1140.
[40]Galuanentto U. Some discontinous bifurcations in a two-block stick-slip system[J]. Journal of Sound and Vibration,2001,248(4):653-669.
[41]Bernardo M D, Kowalczyk P, Nordmark A B. Bifurcations of dynamical systems with sliding derivation of normal-form mappings[J]. Physica D,2002,170(3-4):175-205.
[42]Shin K, Brennan M J, Oh J E, Harris C J. Analysis of disc brake noise using a two degree of freedom model [J]. Journal of Sound and Vibration,2002,254(5):837-848.
[43]Wagg D J. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[J]. Chaos, Solitons & Fractals,2004,22(3):541-548.
[44]Domenico, Lucio D. Dry friction influence on the stability of a mechanical system with two degree of freedom[J]. Non-linear analysis, Non-linear systems and Chaos,2005, 90-93.
[45]Kowalczyk P, Bernardo M D. Two-parameter degenerate sliding bifurcations in filippov systems[J]. Physica D,2005,204(3-4):204-229.
[46]Kowalczyk P, Piiroinen P T. Two-parameter sliding bifurcations of periodic solution in a dry-friction oscillator[J]. Physica D,2008,237(8):1053-1073.
[47]Tomoda A, Watanabe T, Tanaka k. Seismic response of a two-degree-of-freedom system with friction based on the mass ratio[J]. Journal of System Design and Dynamics,2009, 3(2):227-236.
[48]Nordmark A B, Dankowicz H, Champneys A. Discontinuity induced bifucations in systems with impacts and friction-Discontinuities in the impact law[J]. International Journal of Non-linear Mechanics,2009,44(10):1011-1023.
[49]Leu D K. A simple dry friction model for metal forming process[J]. Journal of Materials Processing Technology,2009,209(5):2361-2368.
[50]Yang F H, Zhang W, Wang J. Sliding bifurcations and chaos induced by dry friction in a braking system[J]. Chaos, Solitons & Fractals,2009,40(3):1060-1075.
[51]Jiang J. Deternimation of the global responses characteristics of a piecewise smooth dynamical system with contact[J]. Nonlinear Dyn,2009,57(3):351-361.
[52]Li Z C, Wang Q, Gao H P. Control of friction oscillator by Lyapunov redesign based on delayed state feedback[J]. Acta Mech Sin,2009,25(2):257-264.
[53]Wen G L, Xu H D, Xie J H. Controlling Hopf-Hopf interaction bifurcations of a two-degree-of-freedom self-excited system with dry friction[J]. Nonlinear Dyn,2011, 64(1-2):49-57.
[2]Li Q H, Chen Y M, Wei Y H, Lu Q S. The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system[J]. International Journal of Non-Linear Mechanics,2011,46(1):197-203.
[3]Nordmark A B. Non-periodic motion caused by grazing incidence in an impact oscillator[J]. Journal of Sound and Vibration,1991,145(2):279-297.
[4]Feigin M I. On the structure of C-bifurcation boundaries of piecewise continuous systems[J]. J Appl Math Mech,1978,42(1):820-829.
[5]Di Bernardo M, Feigin M I, Hogan S J, Homer M E. Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems[J]. Chaos, Solitons & Fractals,1999, 10(28):1881-1908.
[6]Bernardo M D, Budd C J, Champneys AR. Unified framework for the analysis of grazing and border-collisions in piecewise-smooth systems[J]. Phys Rev Lett,2001,86(1): 2553-2556.
[7]Popp K, Stelter P. Stick-slip vibration and chaos[J]. Philos Trans Royal Soc Lond A,1990, 332(1624):89-105.
[8]Galvanetto U, Knudsen C. Event maps in a stick-slip system[J]. Nonlinear Dyn,1997, 13(2):99-115.
[9]Galvanetto U. Some discontinuous bifurcations in a two-block stick-slip system [J]. Journal of Sound and Vibration,2001,248(4):653-669.
[10]Galvanetto U. Sliding bifurcation in the dynamics of mechanical systems with dry friction-remarks for engineers and applied scientists[J]. Journal of Sound and Vibration, 2004,276(12):121-139.
[11]Awrejcewicz J, Holicke M M. Melnikov's method and stick-slip chaotic oscillations in very weakly forced mechanical systems[J]. International Journal of Bifurcation and Chaos, 1999,9(3):505-518.
[12]Awrejcewicz J, Sendkowski D. How to predict stick-slip chaos in R4[J]. Phys Lett A, 2004,330(5):371-376.
[13]Awrejcewicz J, Sendkowski D. stick-slip chaos detection in coupled oscillators with friction[J]. International Journal of Solids and Structures,2005,42(21-22):5669-5682.
[14]Halse C, Homer M, Bernardo M D. C-bifurcations and period-adding in one-dimensional piecewise-smooth maps[J]. Chaos, Solitons & Fractals,2003,18(5):953-976.
[15]Wagg D J. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[J]. Chaos, Solitons & Fractals,2004,22(3):541-548.
[16]Iwan W D. The dynamic Response of Bilinear Hysteretic Systems[M]. California: California Institute of Technology,1961.
[17]Menq C H, Bielak J, Griffin J H. The influence of microslip on vibratory response(Part I, Part Ⅱ) [J]. Journal of Sound and vibration,1986,107(2):279-293.
[18]Wang J H, Shish W L. The influence of a variable friction coefficient on the dynamic behavior of a blade with a friction damper [J]. Journal of Sound and Vibration,1991, 149(1):137-145.
[19]Bouc R. Forced vibration of mechanical system with hysteresis [M]. Proeeedings of 4th Conference on Nonlinear oscillation. Prague, Czechoslovakia,1967.
[20]Wen Y K. Method for random vibration of hysteretic systems [J]. Journal of Engineering Mechanics, ASCE, NewYork,1976,102(2):249-263.
[21]Wen Y K. Equivalent linearization for hysteretic system under random excitation[J]. Journal of Applied Mechanics,1980,47(1):150-154.
[22]Bernardo M D, Johansson K H, Vasca F. Self-oscillations in relay feedback systems: symmetry and bifurcations [J]. International Journal of Bifurcation and Chaos,2011, 11(1):1121-1140.
[23]Feigin M I. Forced Oscillations in Systems with Discontinuous Nonlinearities[M]. Nauka, Moscow,1994.
[24]Kowalczyk P S, Bernardo M D. Existence of stable asymmetric limit cycles and chaos in unforced symmetric relay feedback systems[J]. Proceedings of the European Control Conference, Porto,2001,1999-2004.
[25]Nordmark A B. Non-periodic motion caused by grazing incidence in impact oscillators[J]. Journal of Sound and Vibration,1991,145(2):279-297.
[26]Li Q H, Chen Y M, Qin Z Y. Existence of stick-slip periodic solutions in a dry friction oscillator[J]. Chinese Physics Letters,2011,28(3):030502.
[27]Den Hartog J P. Forced vibrations with combined Coulomb and viscous friction[J]. Transaction of the American Society of Mechanical Engineer,1931,53(9):107-115.
[28]Hong H K, Liu C S. Coulomb friction oscillator:modeling and responses to harmonic loads and base excitations[J]. Journal of Sound and Vibration,2000,229(5):1171-1192.
[29]Hong H K, Liu C S. Non-Sticking oscillation formulae for coulomb friction under harmonic loading[J]. Journal of Sound and Vibration,2001,224(5):883-898.
[30]Feeny B, Guran A, Hinrichs N, etal. A historical review on dry friction and stick-slip phenomena[J]. Americal Society of Mechanical Engineers,1998,51(5):321-341.
[31]Galvanetto U. Some discontinuous bifurcations in a two-block stick-slip system[J]. Journal of Sound and Vibration,2001,248(4):653-669.
[32]Bernardo M D, Kowalczyk P, Nordmark A B. Bifurcations of dynamical systems with sliding:derivation of normal form mapping[J]. Physica D,2002,170(3-4):175-205.
[33]Kowalcyk P, Bernardo M D. Two-parameter degenerate sliding bifurcations in filippov systems[J]. Physica D,2005,204(3-4):204-229.
[34]郭树起,杨绍普.三分之一固有频率谐激励下干摩擦振子的双Stop粘滑运动[J].振动与冲击,2009,28(4)：81-85.
[35]杨绍普,郭树起.一类摩擦振子的纯滑动运动分析[J].振动与冲击,2009,28(10)：43-48.
[36]Galvanetto U. Bifurcations and chaos in a four-dimensional mechanical system with dry friction[J]. Journal of Sound and Vibration,1997,204(4):690-695.
[37]Galvanetto U, Bishop S R. Computational techniques for nonlinear dynamics in multiple friction oscillators[J]. Computer methods in applied mechanics and engineering,1998, 163(1-4):373-382.
[38]Dankowicz H, Nordmark A B. On the origin and bifurcations of stick-slip oscillations[J]. Physica D,2000,136(3-4):280-302.
[39]Bernardo M D, Johansson K H. Self-oscillations in relay feedback systems:symmetry and bifurcations[J]. International Journal of Bifurcation and Chaos,2001,11(4): 1121-1140.
[40]Galuanentto U. Some discontinous bifurcations in a two-block stick-slip system[J]. Journal of Sound and Vibration,2001,248(4):653-669.
[41]Bernardo M D, Kowalczyk P, Nordmark A B. Bifurcations of dynamical systems with sliding derivation of normal-form mappings[J]. Physica D,2002,170(3-4):175-205.
[42]Shin K, Brennan M J, Oh J E, Harris C J. Analysis of disc brake noise using a two degree of freedom model [J]. Journal of Sound and Vibration,2002,254(5):837-848.
[43]Wagg D J. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[J]. Chaos, Solitons & Fractals,2004,22(3):541-548.
[44]Domenico, Lucio D. Dry friction influence on the stability of a mechanical system with two degree of freedom[J]. Non-linear analysis, Non-linear systems and Chaos,2005, 90-93.
[45]Kowalczyk P, Bernardo M D. Two-parameter degenerate sliding bifurcations in filippov systems[J]. Physica D,2005,204(3-4):204-229.
[46]Kowalczyk P, Piiroinen P T. Two-parameter sliding bifurcations of periodic solution in a dry-friction oscillator[J]. Physica D,2008,237(8):1053-1073.
[47]Tomoda A, Watanabe T, Tanaka k. Seismic response of a two-degree-of-freedom system with friction based on the mass ratio[J]. Journal of System Design and Dynamics,2009, 3(2):227-236.
[48]Nordmark A B, Dankowicz H, Champneys A. Discontinuity induced bifucations in systems with impacts and friction-Discontinuities in the impact law[J]. International Journal of Non-linear Mechanics,2009,44(10):1011-1023.
[49]Leu D K. A simple dry friction model for metal forming process[J]. Journal of Materials Processing Technology,2009,209(5):2361-2368.
[50]Yang F H, Zhang W, Wang J. Sliding bifurcations and chaos induced by dry friction in a braking system[J]. Chaos, Solitons & Fractals,2009,40(3):1060-1075.
[51]Jiang J. Deternimation of the global responses characteristics of a piecewise smooth dynamical system with contact[J]. Nonlinear Dyn,2009,57(3):351-361.
[52]Li Z C, Wang Q, Gao H P. Control of friction oscillator by Lyapunov redesign based on delayed state feedback[J]. Acta Mech Sin,2009,25(2):257-264.
[53]Wen G L, Xu H D, Xie J H. Controlling Hopf-Hopf interaction bifurcations of a two-degree-of-freedom self-excited system with dry friction[J]. Nonlinear Dyn,2011, 64(1-2):49-57.