斜碰撞振动系统动力学研究
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摘要
斜碰撞振动系统不但具有正碰撞振动系统力学状态的非光滑特性，还由于相对切向速度和摩擦引起碰撞物体的切向变形，因而具有更大的研究难度。迄今，斜碰撞振动分析仍是一个开放的、富有挑战的难题。
本文以含斜碰撞振动的约束力学系统为对象，采用理论分析、计算和实验相结合的方法，通过对斜碰撞瞬间力学过程的描述、斜碰撞振动系统的连续分析、碰撞振动的映射和分叉、含斜碰撞约束力学系统与固定面的碰撞振动、两自由度斜碰撞振动系统的周期运动及其稳定性、斜碰撞振动系统的实验等研究，较全面地揭示了单自由度和两自由度斜碰撞振动系统中存在的复杂非线性动力学行为。论文共计八章，其主要研究内容及学术贡献如下：
(1)在瞬间碰撞假设和考虑碰撞面切向摩擦的情况下，斜碰撞前后的系统状态关系不能用任何简单的碰撞关系描述。为解决该问题，在第二章中提出了通过冲量步进来求解斜碰撞问题的思路，即在碰撞过程中逐次施加法向冲量和相应的切向摩擦冲量，由冲量-动量(矩)定理求得系统在每一步时的状态，直到碰撞物体脱离。这样可以在每一步计算时判断切向速度是否发生了反向。通过一个两自由度斜碰撞振动系统的碰撞过程分析了该方法的实施步骤，并与详细的解析分析结果进行了对比。和仅适应于特定斜碰撞系统的解析形式的碰撞关系相比，该算法为不同的力学模型提供了一种普适性的求取斜碰撞规律的思路。
(2)在第三章中建立了斜碰撞振动系统的Poincaré映射，研究了通过该映射方法确定系统周期运动及其稳定性，并具体计算了给定周期运动附近的局部碰撞映射，分析了斜碰撞振动系统可能存在的各种分叉行为。
(3)针对能归入单自由度斜碰撞类型的几种不同的斜碰撞振动系统，如简谐激励下含集中质量悬臂梁、简谐激励下的单摆或双摆与固定约束平面之间的单点斜碰撞振动问题，在第四章中建立了各自具体的斜碰撞关系，并研究了系统稳态响应随激励参数或系统物理参数变化的演变情况，观察到诸如周期碰撞振动的分叉、跳跃和混沌等复杂的非线性现象。
(4)第五章介绍了由弹簧摆和弹簧振子组成的两自由度斜碰撞振动系统周期碰撞振动的存在性及稳定性条件，针对系统稳态特征以及激励参数、阻尼和碰撞面切向摩擦对稳态特征的影响给出了数值仿真。结果表明，在无阻尼、无摩擦的简化情况下，周期斜碰撞振动运动具有严格的存在条件，并随激励参数的变化出现复杂的分叉和稳定性的反复切换；运动阻尼可在较大参数范围内对非稳定运
斜碰撞振动系统动力学研究
(5)
(6)
动镇定;相对而言，切向摩擦对系统稳态特征的影响不大。
为了从理论上检验瞬间碰撞模型，在第六章中采用并改进了计入局部弹塑性影
响的刚体碰撞过程法向力一位移关系，并利用阻尼函数和Coulomb摩擦力模型，
分析了两自由度斜碰撞振动系统在接触过程中法向和切向力的变化细节，给出
了碰撞接触时间随碰撞角度和碰撞速度变化的关系。对计入碰撞细节后的系统
稳态行为进行了数值仿真。讨论了连续模型和瞬间碰撞模型的应用领域。通过
连续模型的分析结果确定了瞬间碰撞假设在斜碰撞下的适用范围，即该假设适
用于碰撞角度不太大，碰撞物体相对法向接触速度不太小的斜碰撞情况。
作为理论和数值分析的延续，第七章介绍了一套两自由度斜碰撞振动系统动力
学实验装置及相应的实验研究。在该实验中观察到了不同的周期碰撞振动、混
沌碰撞运动以及一些典型的分叉过程，定性验证了由冲量步进算法得到的理论
分析结果和从连续分析结果得到的瞬间碰撞假设在斜碰撞情况下的适用范围，
从而部分验证了本文的分析和计算结果。
Compared with any vibrating systems involving normal impacts, an oblique-impact vibrating system features not only the non-smooth characteristic of impacts, but also complicated dynamics owing to the relative tangential velocity and corresponding friction between two colliding objects. The study of an oblique-impact process, thus, is an open, challenging problem when the tangential deformation and friction of two impact objects are taken into account.
This dissertation presents a systematic study on the dynamics of a few types of oblique-impact vibrating systems of either single degree of freedom or two degrees of freedom. The study, both computationally and experimentally oriented, focuses on the description of an instantaneous oblique-impact and the continuous analysis of an oblique-impact, the mapping and the local bifurcations of the oblique-impact vibrating systems, the characteristics of the oblique-impact vibration between a mechanical system and the fixed rigid stops, and the periodic vibro-impacts and their stability in an oblique-impact vibrating system. The dissertation, including 8 chapters, begins with a brief survey on the recent studies on the modeling and dynamic analysis of vibro-impacting systems in Chapter 1. There follows the main body of the dissertation, from Chapters 2 to 7. It terminates with a few concluding remarks in Chapter 8. The results and the main contributions of the dissertation are as following.
When the instantaneous impact is assumed and the friction between the contact surfaces is taken into account, the relations between the pre-impact state and the post-impact state cannot be described by using any simple impact laws. Hence, a numerical method, referred to as the incremental impulse method in the dissertation, is developed in Chapter 2. The method is illustrated and verified through a vibro-impacting system composed of a spring-pendulum and an oscillator, for which all the possible cases can be well analyzed in terms of analytic expressions. This method enables one to judge the direction of tangential micro-slip and to determine the state of the system at each incremental step of both normal impulse and tangential frictional impulse so that the possible reverse of relative micro-slip due to impact friction can be properly determined. Compared with the analytical expressions applicable to a few of special vibro-impacting systems only, the method provides a widely feasible way to solve the obliqu
e-impact problems of various dynamic systems.
In Chapter 3, the Poincare mapping is introduced to the study on oblique-impact vibrating systems. The periodic vibro-impacts and their stability are analyzed by using Shaw's method and the local impact mapping near a periodic vibro-impacting motion is analyzed by using the discontinuity bypass mapping. Furthermore, several possible bifurcations in the oblique-impact vibrating system are illustrated through a numerical example.
Chapter 4 is devoted to the study of 3 typical oblique-impact vibrating systems. That is, a slender cantilever beam with a lumped tip mass between two rigid stops of a basement subject to a horizontal harmonic excitation, a single pendulum impacting a fixed rigid stop under a harmonic excitation, and a double compound pendulum under the harmonic moment excitation with the end displacement limit. Based on the uniform method, the specific relations between the pre-impact state and the post-impact state are presented for each case. Detailed numerical studies are presented for the transitional dynamics of the steady-state motions of those systems with the variation of the excitation and the system parameters, with the illustrations given for rich nonlinear phenomena, such as the bifurcations of periodic vibro-impact motions and the chaotic vibro-impact motions.
In Chapter 5, the existence of the periodic vibro-impacts, as well as the corresponding stability is analyzed for an oblique-impact vibrating system composed of a spring-pendulum and an oscillator. When the system damping and the tangential friction
本文以含斜碰撞振动的约束力学系统为对象，采用理论分析、计算和实验相结合的方法，通过对斜碰撞瞬间力学过程的描述、斜碰撞振动系统的连续分析、碰撞振动的映射和分叉、含斜碰撞约束力学系统与固定面的碰撞振动、两自由度斜碰撞振动系统的周期运动及其稳定性、斜碰撞振动系统的实验等研究，较全面地揭示了单自由度和两自由度斜碰撞振动系统中存在的复杂非线性动力学行为。论文共计八章，其主要研究内容及学术贡献如下：
(1)在瞬间碰撞假设和考虑碰撞面切向摩擦的情况下，斜碰撞前后的系统状态关系不能用任何简单的碰撞关系描述。为解决该问题，在第二章中提出了通过冲量步进来求解斜碰撞问题的思路，即在碰撞过程中逐次施加法向冲量和相应的切向摩擦冲量，由冲量-动量(矩)定理求得系统在每一步时的状态，直到碰撞物体脱离。这样可以在每一步计算时判断切向速度是否发生了反向。通过一个两自由度斜碰撞振动系统的碰撞过程分析了该方法的实施步骤，并与详细的解析分析结果进行了对比。和仅适应于特定斜碰撞系统的解析形式的碰撞关系相比，该算法为不同的力学模型提供了一种普适性的求取斜碰撞规律的思路。
(2)在第三章中建立了斜碰撞振动系统的Poincaré映射，研究了通过该映射方法确定系统周期运动及其稳定性，并具体计算了给定周期运动附近的局部碰撞映射，分析了斜碰撞振动系统可能存在的各种分叉行为。
(3)针对能归入单自由度斜碰撞类型的几种不同的斜碰撞振动系统，如简谐激励下含集中质量悬臂梁、简谐激励下的单摆或双摆与固定约束平面之间的单点斜碰撞振动问题，在第四章中建立了各自具体的斜碰撞关系，并研究了系统稳态响应随激励参数或系统物理参数变化的演变情况，观察到诸如周期碰撞振动的分叉、跳跃和混沌等复杂的非线性现象。
(4)第五章介绍了由弹簧摆和弹簧振子组成的两自由度斜碰撞振动系统周期碰撞振动的存在性及稳定性条件，针对系统稳态特征以及激励参数、阻尼和碰撞面切向摩擦对稳态特征的影响给出了数值仿真。结果表明，在无阻尼、无摩擦的简化情况下，周期斜碰撞振动运动具有严格的存在条件，并随激励参数的变化出现复杂的分叉和稳定性的反复切换；运动阻尼可在较大参数范围内对非稳定运
斜碰撞振动系统动力学研究
(5)
(6)
动镇定;相对而言，切向摩擦对系统稳态特征的影响不大。
为了从理论上检验瞬间碰撞模型，在第六章中采用并改进了计入局部弹塑性影
响的刚体碰撞过程法向力一位移关系，并利用阻尼函数和Coulomb摩擦力模型，
分析了两自由度斜碰撞振动系统在接触过程中法向和切向力的变化细节，给出
了碰撞接触时间随碰撞角度和碰撞速度变化的关系。对计入碰撞细节后的系统
稳态行为进行了数值仿真。讨论了连续模型和瞬间碰撞模型的应用领域。通过
连续模型的分析结果确定了瞬间碰撞假设在斜碰撞下的适用范围，即该假设适
用于碰撞角度不太大，碰撞物体相对法向接触速度不太小的斜碰撞情况。
作为理论和数值分析的延续，第七章介绍了一套两自由度斜碰撞振动系统动力
学实验装置及相应的实验研究。在该实验中观察到了不同的周期碰撞振动、混
沌碰撞运动以及一些典型的分叉过程，定性验证了由冲量步进算法得到的理论
分析结果和从连续分析结果得到的瞬间碰撞假设在斜碰撞情况下的适用范围，
从而部分验证了本文的分析和计算结果。
Compared with any vibrating systems involving normal impacts, an oblique-impact vibrating system features not only the non-smooth characteristic of impacts, but also complicated dynamics owing to the relative tangential velocity and corresponding friction between two colliding objects. The study of an oblique-impact process, thus, is an open, challenging problem when the tangential deformation and friction of two impact objects are taken into account.
This dissertation presents a systematic study on the dynamics of a few types of oblique-impact vibrating systems of either single degree of freedom or two degrees of freedom. The study, both computationally and experimentally oriented, focuses on the description of an instantaneous oblique-impact and the continuous analysis of an oblique-impact, the mapping and the local bifurcations of the oblique-impact vibrating systems, the characteristics of the oblique-impact vibration between a mechanical system and the fixed rigid stops, and the periodic vibro-impacts and their stability in an oblique-impact vibrating system. The dissertation, including 8 chapters, begins with a brief survey on the recent studies on the modeling and dynamic analysis of vibro-impacting systems in Chapter 1. There follows the main body of the dissertation, from Chapters 2 to 7. It terminates with a few concluding remarks in Chapter 8. The results and the main contributions of the dissertation are as following.
When the instantaneous impact is assumed and the friction between the contact surfaces is taken into account, the relations between the pre-impact state and the post-impact state cannot be described by using any simple impact laws. Hence, a numerical method, referred to as the incremental impulse method in the dissertation, is developed in Chapter 2. The method is illustrated and verified through a vibro-impacting system composed of a spring-pendulum and an oscillator, for which all the possible cases can be well analyzed in terms of analytic expressions. This method enables one to judge the direction of tangential micro-slip and to determine the state of the system at each incremental step of both normal impulse and tangential frictional impulse so that the possible reverse of relative micro-slip due to impact friction can be properly determined. Compared with the analytical expressions applicable to a few of special vibro-impacting systems only, the method provides a widely feasible way to solve the obliqu
e-impact problems of various dynamic systems.
In Chapter 3, the Poincare mapping is introduced to the study on oblique-impact vibrating systems. The periodic vibro-impacts and their stability are analyzed by using Shaw's method and the local impact mapping near a periodic vibro-impacting motion is analyzed by using the discontinuity bypass mapping. Furthermore, several possible bifurcations in the oblique-impact vibrating system are illustrated through a numerical example.
Chapter 4 is devoted to the study of 3 typical oblique-impact vibrating systems. That is, a slender cantilever beam with a lumped tip mass between two rigid stops of a basement subject to a horizontal harmonic excitation, a single pendulum impacting a fixed rigid stop under a harmonic excitation, and a double compound pendulum under the harmonic moment excitation with the end displacement limit. Based on the uniform method, the specific relations between the pre-impact state and the post-impact state are presented for each case. Detailed numerical studies are presented for the transitional dynamics of the steady-state motions of those systems with the variation of the excitation and the system parameters, with the illustrations given for rich nonlinear phenomena, such as the bifurcations of periodic vibro-impact motions and the chaotic vibro-impact motions.
In Chapter 5, the existence of the periodic vibro-impacts, as well as the corresponding stability is analyzed for an oblique-impact vibrating system composed of a spring-pendulum and an oscillator. When the system damping and the tangential friction
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73 Pilipchuk V N, Ibrahim R A. Dynamics of a two-pendulum model with impact interaction and an elastic support. Nonlinear Dynamics, 2000, 21(3): 221-247
74 Aidanp J O, Gupta R B. Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system. Journal of Sound and Vibration, 1993, 165(2): 305-327
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55 Han W,Hu H Y,Jin D P.Vibro-impacts of a slender cantilever beam with a lumped tip mass between two rigid stops.Transactions of Nanjing University Aeronautics & Astronautics,2001,18(2):137-143
56 Masri S F.Steady-state response of a multi-degree system with an impact damper. Transactions of the ASME, Journal of Applied Mechanics,1973,40(1):127-132
57 Nigm M M.Shabana A A.Effect of an impact damper on a multi-degree of freedom system.Journal of Sound and Vibration,1983,89(4):541-547
58 Bishop S R,Thompson M G Foale S.Prediction of period-1 impacts in a driven beam.Proceedings of the Royal Society of London,Series A, 1996,452(1954): 2579-2592
59 金栋平，胡海岩，吴志强．基于Hertz接触模型的柔性梁碰撞振动分析．振动工程学报，1998，11(1)：46-51
60 金栋平，胡海岩．两柔性梁碰撞类型的实验研究．实验力学，1999，14(2)：129-135
61 金栋平，胡海岩，吴志强．碰撞参数对其周期运动的影响．非线性动力学学报，1997，4(4)：312-316
62 Ivanov A P. Impact Oscillations: linear theory of stability and bifurcation. Journal of Sound and Vibration, 1994, 178(3): 361-378
63 Toulemonde C, Gontier C. Multiple degree of freedom impact oscillator. Europe Journal of Mechanics, A/Solids, 1997, 16(5): 879-904
64 Toulemonde C, Gontier C. Sticking motions of impact oscillator. Europe Journal of Mechanics, A/Solids, 1998, 18(2): 339-366
65 Fredriksson M H, Nordmark A B. Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators. Proceedings of the Royal Society of London, Series A, 1997, 453(1961): 1261-1276
66 Fredriksson M H. Grazing bifurcations in multibody systems. Nonlinear Analysis, Theory, Method & Applications, 1997, 30(7): 4475-4483
67 Fredriksson M H, Nordmark A B. On normal form calculations in impact oscillators. Proceedings of the Royal Society of London, Series A, 2000, 456(1994): 315-329
68 舒仲周．有碰撞存在的多体振动系统的周期运动和稳定条件．振动工程学报，1990，3(3)：42-51
69 曹登庆，舒仲周．存在间隙的多自由度系统的周期运动及Robust稳定性．力学学报，1997，29(1)：74-83
70 刘才山，陈滨．多柔体系统碰撞动力学研究综述．力学进展．2000，30(1)：7-14
71 刘才山，陈滨．结构柔性对系统碰撞动力学响应的影响．北京大学学报(自然科学版)，2000，36(1)：64-69
72 Zhuravlev V F. A method for analyzing vibration-impact systems by means of special functions, Izvestiya AN SSSR Mekhanika Tverdogo Tela (Mechanics of Solids), 1976, 11 (2): 30-34
73 Pilipchuk V N, Ibrahim R A. Dynamics of a two-pendulum model with impact interaction and an elastic support. Nonlinear Dynamics, 2000, 21(3): 221-247
74 Aidanp J O, Gupta R B. Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system. Journal of Sound and Vibration, 1993, 165(2): 305-327
75 Jin D P, Hu H Y. Forced periodic vibro-impacts and their stability of dual elastic components. Acta Mechanica Sinica, 1997, 13(4): 366-376
76 李群宏，陆启韶．一类双自由度碰撞振动系统运动分析．力学学报，2001，33(6)：776-786
77 Luo G W, Xie J H. Hopf bifurcations of a two degree-of-freedom vibro-impact system. Journal of Sound and Vibration, 1998, 213(3): 391-480
78 罗冠炜，谢建华，孙训方．具有单侧刚性约束的两自由度振动系统在强共振条件下的拟周期运动与混沌．固体力学学报，2000，21(2)：138-144
79 罗冠炜，谢建华，孙训方．两自由度塑性碰撞振动系统的动力学研究．力学学报，2000，32(5)：579-586
80 Luo G W, Xie J H. Bifurcations and chaos in a system with impacts. Physica D, 2001, 148(3-4): 183-200
81 Luo G W, Xie J H, Guo S H L. Periodic motions and global bifurcations of a two degree-of-freedom system with plastic vibro-impact. Journal of Sound and Vibration, 2001, 240(5): 837-858
82 谢建华，文桂林，肖建．两自由度碰撞振动系统分叉参数的确定．振动工程学报， 2001，14(3)：249-253
83 Galvanetto U, Bishop S R. Characterization of the dynamics of a four-dimensional stick-slip system by a scalar variable. Chaos, Solitons & Fractals, 1995, 5(11): 2171-2179
84 Hogan S J. Damping in rigid block dynamics contained between side-walls. Chaos, Solitons and Fractals, 2000, 11 (4): 495-506
85 Lenci S, Rega G. Optimal numerical control of single-well to cross-well chaos transition in mechanical systems. Chaos, Solitons and Fractals, 2003, 15(1): 173-186
86 Cundall P A, Hart R D. Numerical modelling of discontinua. Engineering Computations, 1992, 9(2): 101-113
87 Meijaard J P. Efficient numerical integration of the equations of motion of non-smooth mechanical systems. Zeitschrift fur Angewandte Mathematik und Mechanik, 1997, 77(6): 419-427
88 Janin O, Lamarque C H. Comparison of several numerical methods for mechanical systems with impacts. International Journal for Numerical Methods in Engineering, 2001, 51(9): 1101-1132
89 Moore D B, Shaw S W. The experimental response of an impacting pendulum system. International Journal of Non-Linear Mechanics, 1990, 25(1):1-16
90 Hinrichs N, Oestreich M, Popp K, Dynamics of oscillators with impact and friction. Chaos, Solitons and Fractals, 1997, 8(4): 535-558
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