含对称刚性约束振动系统的周期运动和分岔

研究了两类含对称刚性约束振动系统的周期运动和分岔。刚性约束导致两振动系统在简谐激振力作用下发生碰撞振动,并呈现不同的碰撞形式。对比两类系统的相关结果,讨论了间隙值和激振频率对两振动系统对称碰撞周期运动的稳定性和分岔的影响,分析了对称碰撞周期运动的分岔规律。对于较大的间隙值,激振频率的递减通常导致对称碰撞周期运动首先发生Neimark-Sacker分岔;对于较小的间隙值,激振频率的递减通常导致对称碰撞周期运动发生叉式分岔。研究了单周期对称碰撞运动、单周期反对称碰撞运动、单周期4-碰撞运动、倍周期4-碰撞运动和倍周期6-碰撞运动的Neimark-Sacker分岔。研究结果表明间隙值和激振频率的变化可能导致含对称刚性约束振动系统呈现复杂且形式多样的概周期碰撞运动。

PERIODIC-IMPACT MOTIONS AND BIFURCATIONS OF VIBRATORY SYSTEMS WITH SYMMETRICAL RIGID CONSTRAINTS

Two harmonically excited systems with symmetrical rigid constraints are considered. Repeated impacts usually occur in such systems due to the rigid constraints. The impact formats that occur in two systems are different, i.e., the components of one system collide with each other, and one of components of the other system collides with rigid obstacles. Dynamics of these systems are studied with special attention to pitchfork and Neimark-Sacker bifurcations associated with several periodic-impact motions. Two important parameters, the clearance and the excitation frequency, are considered, and the influence of these parameters on the symmetrical double-impact periodic motion and bifurcations is analyzed with emphases. With a decrease in the excitation frequency, symmetrical double-impact periodic motion usually undergoes Neimark-Sacker bifurcation for large values of clearances and pitchfork bifurcation for small values of clearances. Neimark-Sacker bifurcations, associated with symmetrical double-impact periodic motion, asymmetrical double-impact periodic motion, single-period four-impact symmetrical motion, double-period four-impact motion and double-period six-impact motion, etc., are analyzed in detail. The studies show that the vibratory systems with symmetrical rigid amplitude constraints may exhibit complex and rich quasi-periodic impact motions associated with several periodic-impact ones.