﻿ 一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学
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 力学学报  2016, Vol. 48 Issue (1): 163-172  DOI: 10.6052/0459-1879-15-144 0

### 引用本文 [复制中英文]

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Yue Yuan. LOCAL DYNAMICAL BEHAVIOR OF TWO-PARAMETER FAMILY NEAR THE NEIMARK-SACKER-PITCHFORK BIFURCATION POINT IN A VIBRO-IMPACT SYSTEM[J]. Chinese Journal of Ship Research, 2016, 48(1): 163-172. DOI: 10.6052/0459-1879-15-144.
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### 文章历史

2015-04-23 收稿
2015-07-14 录用
2015-07-23 网络版发表

1 力学模型以及对称周期运动

 图 1 具有对称刚性约束的三自由度碰撞振动系统 Fig. 1 Three-degree-of-freedom vibro-impact system with symmetric rigid constraints

 ${\pmb U}_m \ddot {\pmb x} + 2\zeta {\pmb U}_c \dot {\pmb x} + {\pmb U }_k {\pmb x} = {\pmb U}_f f\sin (\omega t + \tau )$ (1)

 $t= T\sqrt {\dfrac{K_3 }{M_3 }} ,\zeta = \dfrac{C_3 }{2\sqrt {K_3 M_3 } } ,\omega =\varOmega \sqrt {\dfrac{M_3 }{K_3 }} \\ f = \dfrac{P_3 }{P_0 } ,u_{m_i } = \dfrac{M_i }{M_3 } ,u_{k_i } = \dfrac{K_i }{K_3 } \\ u_{c_i } = \dfrac{C_i }{C_3 } ,u_{f_i } = \dfrac{P_i }{P_3 } ,x_i = \dfrac{X_i K_3 }{P_0 }$

 $y_{2 + } = \delta _{11} y_{2 - } + \delta _{12} y_{3 - } ,y_{3 + } = \delta _{21} y_{2 - } + \delta _{22} y_{3 - }$ (2)

 $\delta _{11} = \dfrac{u_{m_2 } - R}{1 + u_{m_2 } } ,\delta _{12} =\dfrac{1 + R}{1 + u_{m_2 } } \\ \delta _{21} = \dfrac{u_{m_2 } (1 + R)}{1 + u_{m_2 } } ,\delta _{22} = \dfrac{1 - u_{m_2 } R}{1 + u_{m_2 } }$

$y_{i - } = \dot {x}_{i - }$，$y_{i + } = \dot {x}_{i + }$分别表示$M_2$和$M_3$在碰撞之前和碰撞之后的无量纲化瞬时速度.

 ${\pmb I}\ddot {\xi } + {\pmb C}\dot {\xi } + {\pmb \varLambda}\xi = {\pmb \psi }^{\rm T}{\pmb U}_f f \sin(\omega t + \tau )$ (3)

 $x_i (t) = \sum\limits_i^3 { } \phi _{ij} \Big \{ {\rm e}^{ - \eta _j t}[a_j \cos(\omega _{dj} t) + b_j \sin (\omega _{dj} t)] + \\ \qquad A_j \sin (\omega t + \tau ) + B_j \cos (\omega t + \tau ) \Big\}$ (4)

 $x_i (t) = \left\{ \!\!\begin{array}{l} \sum\limits_i^3 { } \phi _{ij} [{\rm e}^{ - \eta _j t}(a_{ji} \cos(\omega _{dj} t) + \\ \qquad b_{ji} \sin (\omega _{dj} t)) + A_j \sin (\omega t + \tau _0 ) + \\ \qquad B_j \cos (\omega t + \tau _0 )] ,t \in [0,t_1] \\ \sum\limits_i^3 { } \phi _{ij} \Big\{ {\rm e}^{ - \eta _j (t - t_1 )} [a_{ji} \cos (\omega _{dj} (t - t_1 )) + \\ \qquad b_{ji} \sin (\omega _{dj}(t - t_1 ))] + A_j \sin (\omega t + \tau _0 ) + \\ \qquad B_j \cos (\omega t + \tau _0 )\Big\} ,t \in [t_1 ,t_2] \end{array} \!\!\right. (i = 1,2,3)$ (5)

2 庞加莱映射以及对称不动点的稳定性及分岔

 $\dot{\pmb X} = {\pmb F}({\pmb X},t)$ (6)

 ${\pmb F} \Big({\pmb X},t + \dfrac{2n\pi }{\omega }\Big) = {\pmb F}({\pmb X},t)$ (7)

 ${\pmb F}( - {\pmb X},t + \dfrac{n\pi }{\omega }) = - {\pmb F}({\pmb X},t)$ (8)

 ${\pmb R}^6\times {\pmb S}^1 = \{ x_1 ,y_1 ,x_2 ,y_2 ,x_3 ,y_3 , \\ \qquad t |(x_1 ,y_1 ,x_2 ,y_2,x_3 ,y_3 ) \in {\pmb R}^6 ,\ t \in {\pmb S}^1 \}$ (9)

 ${\pmb \varPi }_0 = \{ (x_1 ,y_1 ,x_2 ,y_2 ,x_3 ,y_3 ,t) \in \\ \qquad {\pmb R}^6\times {\pmb S}^1\big | x_2 - x_3 = h,y_i = \dot {x}_{i + } \}$ (10)

 ${\pmb \varPi }_1 = \{ (x_1 ,y_1 ,x_2 ,y_2 ,x_3 ,y_3 ,t) \in \\ \qquad {\pmb R}^6\times {\pmb S }^1\big |x_2 - x_1 = - h,y_i = \dot {x}_{i + } \}$ (11)

 ${\pmb R}:(x_1 ,y_1 ,x_2 ,y_2 ,x_3 ,y_3 ,t) \mapsto \\ \qquad \Big( - x_1 ,- y_1 ,- x_2 ,- y_2 ,- x_3 ,- y_3 ,t + \dfrac{n\pi }{\omega } \Big)$ (12)

 ${\pmb R}^2={\pmb I}$ (13)

 ${\pmb R \pmb F}({\pmb X}) = {\pmb F}({\pmb R \pmb X})$ (14)

 ${\pmb P } = {\pmb P }_4 \circ {\pmb P }_3 \circ {\pmb P }_2\circ {\pmb P }_1$ (15)

 ${\pmb D\pmb P} = {\pmb D\pmb P }_4 ({\pmb P }_3 \circ {\pmb P }_2 \circ {\pmb P }_1({\pmb X }_0 )) \cdot {\pmb D\pmb P }_3 ({\pmb P }_2 \circ {\pmb P }_1 ({\pmb X }_0 )) \cdot \qquad {\pmb D\pmb P }_2 ({\pmb P }_1 ({\pmb X }_0 )) \cdot {\pmb D\pmb P }_1 ({\pmb X }_0 )$ (16)

 ${\pmb Q } = {\pmb R }^{ - 1} \circ {\pmb Q }_u$ (17)

 ${\pmb P } = {\pmb Q }^2$ (18)

 ${\pmb X }_0 = {\pmb Q }({\pmb X }_0 )$ (19)

 ${\pmb Q }({\pmb X }_\alpha ) = {\pmb X }_\beta \ne {\pmb X }_\alpha$ (20)

 $x_i \Big(\dfrac{n\pi }{\omega }\Big) = - x_i (0) ,\quad\dot {x}_{i + }\Big (\dfrac{n\pi }{\omega } \Big) = - \dot {x}_{i + } (0)$ (21a)

 $x_2 (0) - x_1 (0) = h ,x_2 \Big (\dfrac{n\pi }{\omega }\Big) - x_1 \Big (\dfrac{n\pi }{\omega }\Big) = - h$ (21b)

 $\tau _0 = 2\tan ^{ - 1} \Big (\dfrac{V_c \pm \sqrt {V_c^2 + U_c^2 - h^2} }{U_c + h} \Big )$ (22)

 $a_j = E_{aj} \cos \tau _0 + F_{aj} \sin \tau _0$ (23)

 $b_j = I_{bj} a_1 + J_{bj} a_2 + K_{bj} a_3$ (24)

3 根据映射${\pmb Q}$确定映射 ${\pmb P}$在内伊马克沙克-音叉分岔点附近的范式表达

C1. ${\pmb D\pmb Q}({\pmb \mu })$有一对共轭复特征值以及一个$-1$的实特征值同时位于单位圆上：$\lambda _{1,2}= \tilde {\lambda },\bar {\tilde {\lambda }} = {\rm e}^{\pm {\rm i}\tilde {\theta}}$，$\lambda _3 =-1$；其余特征值在单位圆内.

C2. $\dfrac{\partial \tilde {\lambda }({\pmb \mu }_c )}{\partial \mu _1 } \ne 0$，以及 $\dfrac{\partial \tilde {\lambda }({\pmb \mu }_c )}{\partial \mu _2 } \ne 0$，这是映射${\pmb Q }$的两参数簇的穿越条件；

C3. 非共振条件：$\tilde {\lambda }_{1,2}^n \ne 1$，$n = 1$，2，$\cdots$，6；且$\tilde {\lambda }_{1,2}^n \ne - 1$，$n = 4$，5.

 ${u}' = - u + \tilde {a}_1 u^3 + \tilde {a}_2 uz\bar {z} + O((\left| u\right| + \left| z \right|)^5)$ (25a)

 ${z}' = {\lambda }'_0 z + \tilde {b}_1 u^2z + \tilde {b}_2 z^2\bar {z} +O((\left| u \right| + \left| z \right|)^5)$ (25b)

 ${u}' = u + a_1 u^3 + a_2 uz\bar {z} + O((\left| u \right| + \left| z\right|)^5)$ (26a)

 ${z}' = \lambda _0 z + b_1 u^2z + b_2 z^2\bar {z} + O((\left| u \right| +\left| z \right|)^5)$ (26b)

4 在内伊马克沙克-音叉分岔点附近的两参数范式开折

 ${\pmb F }:{\pmb R }\times {\pmb C } \to {\pmb R }\times {\pmb C } \\ \left( \!\! \begin{array}{c} {u}' \\ {z}' \end{array}\!\! \right) = \left[\!\! \begin{array}{cc} { - 1} & 0 \\ 0 & \tilde {\lambda } \end{array}\!\! \right]\left(\!\! \begin{array}{c} u \\ z \end{array}\!\! \right) + \\ \qquad \left(\!\! \begin{array}{c} { - \varepsilon _1 u + \tilde {a}_1 u^3 + \tilde {a}_2 u\left| z \right|^2+ O\left[{(\left| u \right| + \left| z \right|)^5} \right]} \\ {\tilde {\lambda }(\bar {\varepsilon }_2 z + \bar {b}_1 u^2z + \bar {b}_2z\left| z \right|^2 + O\left[{(\left| u \right| + \left| z \right|)^5}\right]} \end{array}\!\! \right)$ (27)

 $\varepsilon _1 = \varepsilon _1 ({\pmb \mu }) = - \lambda _3 ({\pmb \mu }) - 1 = \\ \qquad - \dfrac{\partial \tilde {\lambda }_3 (\mu _1 ,\mu _2)}{\partial \mu _1 }\mu _1 - \dfrac{\partial \tilde {\lambda }_3 (\mu _1 ,\mu_2 )}{\partial \mu _2 }\mu _2 \\ \bar {\varepsilon }_2 = \bar {\varepsilon }_2 ({\pmb \mu }) = \tilde{\lambda }({\pmb \mu })\bar {\tilde {\lambda }}({\pmb \mu }) - 1 = \\ \qquad\bar {\tilde {\lambda }}\dfrac{\partial \tilde {\lambda }(\mu _1 ,\mu _2)}{\partial \mu _1 }\mu _1 + \bar {\tilde {\lambda }}\dfrac{\partial \tilde{\lambda }(\mu _1 ,\mu _2 )}{\partial u_2 }\mu _2 \\ \bar {b}_1 \approx \dfrac{b_1 }{\tilde {\lambda }} ,\quad\bar {b}_2 \approx \dfrac{b_2 }{\tilde {\lambda }}$

 $\left( \!\! \begin{array}{c} {x}' \\ {r}' \\ {\tilde {\theta }}'\end{array} \!\! \right) = \left(\!\! \begin{array}{c} {X(x,r,\tilde {\theta },\varepsilon )} \\ {R(x,y,\tilde {\theta },\varepsilon )} \\ {\varTheta (x,y,\tilde {\theta },\varepsilon )}\end{array} \!\! \right) = \\ \qquad \left(\!\! \begin{array}{c} { - x - \varepsilon _1 x + a_1 x^3 + a_2 xr^2 + h.o.t.} \\ {r + \varepsilon _2 r + \beta _1 x^2r + \beta _2 r^3 + h.o.t.} \\ {\tilde {\theta } + \arg \tilde {\lambda }_0 + \varepsilon _3 + \gamma _1x^2 + \gamma _2 r^2 + h.o.t.} \end{array} \!\! \right)$ (28)

 $\left( \begin{array}{c} {x}' \\ {r}' \end{array} \right) = \left( \begin{array}{l} (1 + 2\varepsilon _1 )x - 2a_1 (1 + 2\varepsilon _1 )x^3 - \\ \qquad 2a_2 (1 +\varepsilon _1 + \varepsilon _2 )xr^2 \\ (1 + 2\varepsilon _2 )r + 2\beta _1 (1 + \varepsilon _1 + \varepsilon _2)x^2r + \\ \qquad 2\beta _2 (1 + 2\varepsilon _2 )r^3\end{array} \right)$ (29)

 $L_1 :a_2 \varepsilon _2 + \beta _2 \varepsilon _1 = 0 hbox{或} \ a_1 \varepsilon _2 + \beta _1\varepsilon _1 = 0 \\ L_2 :a_1 \varepsilon _2 + \beta _1 \varepsilon _1 = 0 hbox{或} \ a_2 \varepsilon_2 + \beta _2 \varepsilon _1 = 0$
 图 2 映射P 在内伊马克沙克-音叉分岔点附近的范式局部 两参数开折 Fig. 2 The local two-parameter unfolding near Neimark-Saker-pitchfork point the of the map P

5 在内伊马克沙克-音叉分岔点附近的局部两参数动力学数值分析

 图 3 庞加莱截面投影图：音叉分岔 Fig. 3 The projected Poncar′e section: pitchfork bifurcation

 图 4 庞加莱截面投影图：内伊马克沙克-分岔 Fig. 4 The projected Poncar′e section: Neimark-Saker bifurcation

 图 5 两个共轭不动点和一个对称的拟周期吸引子： 内伊马克沙克-音叉分岔 Fig. 5 Two conjugate fixed points and a single symmetric quasi-periodic attractor: Neimark-Saker-pitchfork bifurcation

 图 6 通过内伊马克沙克-音叉分岔产生两个共轭的拟周期吸引子： 一对共轭的T2 环面 Fig. 6 Two conjugated quasi-periodic attractors induced by Neimark-Saker-pitchfork bifurcation: a pair of conjugate T2 tori
6 结 论

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LOCAL DYNAMICAL BEHAVIOR OF TWO-PARAMETER FAMILY NEAR THE NEIMARK-SACKER-PITCHFORK BIFURCATION POINT IN A VIBRO-IMPACT SYSTEM
Yue Yuan
Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China
Abstract: A three-degree-of-freedom vibro-impact system with symmetry is considered. Due to the symmetry, the Poncar é map P is the second iteration of another virtual implicit map Q. It is shown that the symmetric period n-2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poncaré map. Then we can investigate bifurcations of the symmetric period n-2 motion by researching into bifurcations of the associated symmetric fixed point. Based on the symmetry of the system, it is shown that the Neimark-Saker-pitchfork bifurcation of the symmetric fixed point of the Poncaré map P corresponds to the Neimark-Saker-flip bifurcation of the map Q. By using the map Q, according to the two-parameter unfolding of the normal form, we reveal the possible local dynamical behaviors of the symmetric fixed point of the Poncaré map P near the Neimark-Saker-pitchfork bifurcation point in detail. Near this codimension two bifurcation point, the dynamic behaviors of the vibro-impact system can be expressed by a single symmetric fixed point, a pair of conjugate fixed points, a pair of conjugate quasi-periodic attractors or a single symmetric quasi-periodic attractor in the projected Poncaré section. The numerical simulation represents various possible cases near the Neimark- -Saker-pitchfork bifurcation point. It is shown that the interaction of the Neimark-Saker bifurcation and the pitchfork bifurcation may lead into the creation of some new results. The symmetric fixed point bifurcates into a pair of conjugate unstable fixed points firstly, and the two conjugate fixed points will bifurcate into the same symmetric quasi-periodic attractor finally.
Key words: vibro-impact system    symmetric fixed point    two-parameter unfolding    Neimark-Sacker-pitchfork bifurcation    conjugate quasi-periodic attractors