﻿ 一类分段光滑隔振系统的非线性动力学设计方法
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 力学学报  2016, Vol. 48 Issue (1): 192-200  DOI: 10.6052/0459-1879-15-099 0

### 引用本文 [复制中英文]

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Gao Xue, Chen Qian, Liu Xianbin. NONLINEAR DYNAMICS DESIGN FOR PIECEWISE SMOOTH VIBRATION ISOLATION SYSTEM[J]. Chinese Journal of Ship Research, 2016, 48(1): 192-200. DOI: 10.6052/0459-1879-15-099.
[复制英文]

### 文章历史

2015-03-25 收稿
2015-11-03 录用
2015-11-04 网络版发表

1 主共振区的动力学设计 1.1 主共振分岔分析

 $M\ddot {x} + C\dot {x} + F_{\rm k} (x) = F\cos \omega t$ (1)

 ${F_{\rm{k}}}(x) = \left\{ \begin{array}{l} {K_1}x{\mkern 1mu} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \ge {a_{\rm{c}}}{\kern 1pt} \\ {K_2}x + ({K_1} - {K_2}){a_{\rm{c}}}{\mkern 1mu} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x < {a_{\rm{c}}} \end{array} \right.$ (2)

 $\begin{array}{l} {[ - (\sin {\varphi _0}\cos {\varphi _0} + {\varphi _0} + \frac{2}{\beta }\sin {\varphi _0} - \pi )\frac{\varepsilon }{\pi } + (\lambda _\omega ^2 - 1)]^2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {(2{\xi _1}{\lambda _\omega })^2} = \frac{{F_1^2}}{{{\beta ^2}a_{\rm{c}}^2\omega _0^4}} \end{array}$ (3)

(3)其中,λω=ω/ω0,β=|a/ac|.需要注意的是,以上只是针对稳定振幅会超过临界位移时的频响方程;而对于稳定振幅不会超过临界位移的情况,系统则体现出典型的线性频响特性.

$\begin{array}{l} Y = 1/\beta {\mkern 1mu} ,\;\;\cos {\varphi _0} = - 1/\beta = - Y\\ \sin {\varphi _0} = \sqrt {1 - 1/{\beta ^2}} = 1 - \frac{1}{2}{Y^2} + O({Y^3})\\ {\varphi _0} = \arccos ( - 1/\beta ) = \pi /2 + Y + {Y^3}/6 + O({Y^4}) \end{array}$

 $G(Y ,\mu; \alpha _1 ,\alpha _2 ) = \left[ { \Big (\dfrac{1}{3}Y^3 - 2Y \Big) + \mu } \right]^2 + \alpha _1 - \alpha _2 Y^2 = 0$ (4)

$\alpha _2 = - \dfrac{(Y^2 - 2)^3}{Y^2 + 2}\,, \ \ \alpha _1 = \alpha _2 Y^2 - \left( {\dfrac{\alpha _2 Y}{Y^2 - 2}} \right)^2\,;$

 图 2 典型的主共振曲线 Fig.2 Different frequency responses of primary resonance

 $\mu = - (\frac{1}{3}{Y^3} - 2Y) \pm \sqrt { - ({\alpha _1} - {\alpha _2}{Y^2})} {\rm{ }}$ (5)

 $\frac{{d\mu }}{{dY}} = - ({Y^2} - 2) \pm ({\alpha _2}Y){[ - ({\alpha _1} - {\alpha _2}{Y^2})]^{ - 1/2}}$ (6)

 $\alpha _1 = \alpha _2 - 4\alpha _2^2$ (7)

 图 1 主共振的参数划分 Fig.1 Parameter zone of primary resonance response with different topology characteristics

1.2 避免跳跃发生的条件

 $\begin{array}{l} [ - (\sin {\varphi _0}\cos {\varphi _0} + {\varphi _0} + \frac{2}{\beta }\sin {\varphi _0} - \frac{{2{d_\lambda }}}{\beta }\sin {\varphi _0} - \pi )\frac{\varepsilon }{\pi } + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\lambda _\omega ^2 - 1){]^2} + {(2{\xi _1}{\lambda _\omega })^2} = \frac{{F_1^2}}{{{\beta ^2}a_{\rm{c}}^2\omega _0^4}} \end{array}$ (8)
<

 $\lambda _\omega ^2 = - (Z_1 + 2\xi _1^2 )\pm \sqrt {4Z_1 \xi _1^2 + 4\xi _1^4 + Z_2 }$ (9)

$\begin{array}{l} {Z_1} = [ - (1 - 2{d_\lambda })\frac{1}{\beta }\sqrt {\frac{{{\beta ^2} - 1}}{{{\beta ^2}}}} - \arccos ( - \frac{1}{\beta }) + \pi ]\frac{\varepsilon }{\pi } - 1\\ {Z_2} = \frac{{F_1^2}}{{{\beta ^2}a_{\rm{c}}^2\omega _0^4}} \end{array}$

 $\frac{{d{\lambda _\omega }}}{{d\beta }} = \frac{1}{{2{\lambda _\omega }}}( - \frac{{d{Z_1}}}{{d\beta }} - \frac{{4\xi _1^2\frac{{d{Z_1}}}{{d\beta }} + \frac{{d{Z_2}}}{{d\beta }}}}{{2\sqrt {4{Z_1}\xi _1^2 + 4\xi _1^4 + {Z_2}} }})$ (10)

 图 3 跳跃发生的临界条件 Fig.3 Critical condition for jump avoidance
 图 4 临界频响函数 Fig.4 Critical frequency response
2 隔振有效区的动力学设计 2.1 倍周期分岔

 $\left. \begin{array}{l} \dot x = {f_1}(t,x){\mkern 1mu} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {S_1}\\ \dot x = {f_2}(t,x){\mkern 1mu} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {S_2}{\kern 1pt} {\kern 1pt} \end{array} \right\}$ (11)

 $\{ \Sigma |h(x) = x - {a_0} = 0\}$ (12)

 ${\Sigma _{x,1}} = \{ (x,\dot x,t)|x = {a_{\rm{c}}},\dot x > 0\}$ (13)

 ${\Sigma }_{x,2} = \{ (x,\dot x, t) | x = a_{\rm c} , \dot {x} < 0\}$ (14)

 $P = {P_2}^\circ {P_1}$ (15)

 $DP = D{P_2}^\circ D{P_1}$ (16)

 $D{P_1} = \frac{{\partial ({v_1}({v_0},{t_0}),{t_1}({v_0},{t_0}))}}{{\partial ({v_0},{t_0})}} = \left[ \begin{array}{l} \frac{{\partial {v_1}}}{{\partial {v_0}}}\\ \frac{{\partial {t_1}}}{{\partial {v_0}}} \end{array} \right.\left. \begin{array}{l} \frac{{\partial {v_1}}}{{\partial {t_0}}}\\ \frac{{\partial {t_1}}}{{\partial {t_0}}} \end{array} \right]$ (17)

 $a_{\rm c} = x_1 (t_0 ) \ \ \hbox{和} \ \ v_0 = \dot {x}_1 (t_0 )$ (18)

t1时刻,位移和速度分别为

 $a_{\rm c} = x_1 (t_1 (v_0 ,t_0 );v_0 ,t_0 ,A_1 (v_0 ,t_0 ),B_1 (v_0 ,t_0 ))$ (19)
 $v_1 (v_0 ,t_0 ) = \dot {x}_1 (t_1 (v_0 ,t_0 );v_0 ,t_0 ,A_1 (v_0 ,t_0 ), B_1 (v_0 ,t_0 ))$ (20)

 $0 \equiv \dfrac{\partial a_{\rm c} }{\partial t_0 } = \dfrac{\partial x_1 }{\partial A_1 } \dfrac{\partial A_1 }{\partial t_0 } + \dfrac{\partial x_1 }{\partial B_1 }\dfrac{\partial B_1 } {\partial t_0 } + \dfrac{\partial x_1 }{\partial t_1 }\dfrac{\partial t_1 }{\partial t_0 } + \dfrac{\partial x_1 }{\partial t_0 }$ (21)
 $\dfrac{\partial v_1 }{\partial t_0 } = \dfrac{\partial \dot {x}_1 }{\partial A_1 }\dfrac{\partial A_1 }{\partial t_0 } + \dfrac{\partial \dot {x}_1 }{\partial B_1 }\dfrac{\partial B_1 }{\partial t_0 } + \dfrac{\partial \dot {x}_1 }{\partial t_1 }\dfrac{\partial t_1 }{\partial t_0 } + \dfrac{\partial \dot {x}_1 }{\partial t_0 }$ (22)

 $0 \equiv \dfrac{\partial a_{\rm c} }{\partial v_0 } = \dfrac{\partial x_1 }{\partial A_1 }\dfrac{\partial A_1 }{\partial v_0 } + \dfrac{\partial x_1 }{\partial B_1 }\dfrac{\partial B_1 }{\partial v_0 } + \dfrac{\partial x_1 }{\partial t_1 }\dfrac{\partial t_1 }{\partial v_0 } + \dfrac{\partial x_1 }{\partial v_0 }$ (23)
 $\dfrac{\partial v_1 }{\partial v_0 } = \dfrac{\partial \dot {x}_1 }{\partial A_1 }\dfrac{\partial A_1 }{\partial v_0 } + \dfrac{\partial \dot {x}_1 }{\partial B_1 }\dfrac{\partial B_1 }{\partial v_0 } + \dfrac{\partial \dot {x}_1 }{\partial t_1 }\dfrac{\partial t_1 }{\partial v_0 } + \dfrac{\partial \dot {x}_1 }{\partial v_0 }$ (24)

 $D{P_2} = \frac{{\partial ({v_2}({v_1},{t_1}),{t_2}({v_1},{t_1}))}}{{\partial ({v_1},{t_1})}} = \left[ \begin{array}{l} \frac{{\partial {v_2}}}{{\partial {v_1}}}\\ \frac{{\partial {t_2}}}{{\partial {v_1}}} \end{array} \right.\left. \begin{array}{l} \frac{{\partial {v_2}}}{{\partial {t_1}}}\\ \frac{{\partial {t_2}}}{{\partial {t_1}}} \end{array} \right]$ (25)

 $\lambda _{1,2} = \dfrac{1}{2}(T\pm \sqrt {T^2 - 4D} )$ (26)

$\begin{array}{l} D = {{\rm{e}}^{ - {\textstyle{{2\pi c} \over \omega }}}}{\mkern 1mu} ,\quad T = {{\rm{e}}^{ - {\textstyle{{\pi c} \over \omega }}}}\left[ { - {s_ + }{s_ - }(\frac{{\omega _2^2 + \omega _1^2}}{{{\omega _1}{\omega _2}}}) + 2{c_ + }{c_ - }} \right]\\ {\omega _2} = \sqrt {{k_2} - \xi _1^2} {\mkern 1mu} ,\;\;{\omega _1} = \sqrt {{k_1} - \xi _1^2} \\ {s_ + } = \sin [{\omega _2}({t_2} - {t_1})]{\mkern 1mu} ,\;{s_ - } = \sin [{\omega _1}({t_1} - {t_0})]\\ {c_ + } = \cos [{\omega _2}({t_2} - {t_1})]{\mkern 1mu} ,\;\;{c_ - } = \cos [{\omega _1}({t_1} - {t_0})] \end{array}$

 ${{\rm{e}}^{{\textstyle{{ - 4\pi {\xi _1}} \over \omega }}}} + 2{{\rm{e}}^{{\textstyle{{ - 2\pi {\xi _1}} \over \omega }}}}( - \frac{1}{2}\frac{{\omega _1^2 + \omega _2^2}}{{{\omega _1}{\omega _2}}}{s^2} + {c^2}) + 1 = 0$ (27)

 $\Big( - \dfrac{1}{2}\dfrac{\omega _1^2 + \omega _2^2 }{\omega _1 \omega _2 }s^2 + c^2) < - 1$ (28)

 $\begin{array}{l} \frac{{{\xi _1}}}{\omega } + \frac{1}{{2\pi }}\ln [(\frac{1}{2}\frac{{\omega _1^2 + \omega _2^2}}{{{\omega _1}{\omega _2}}}{s^2} - {c^2}) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{{(\frac{1}{2}\frac{{\omega _1^2 + \omega _2^2}}{{{\omega _1}{\omega _2}}}{s^2} - {c^2})}^2} - 1} {\mkern 1mu} ] = 0 \end{array}$ (29)

 图 5 亚谐分岔发生的临界条件( $\omega _0^2 = 1{\mkern 1mu} 833,\varepsilon = 0.6$) Fig.5 Critical condition for occurrence of period-doubling bifurcation ( $\omega _0^2 = 1{\mkern 1mu} 833,\varepsilon = 0.6$ )
 图 6 不同非线性因子ε所对应的倍周期分岔发生的临界条件 Fig.6 Critical conditions for occurrence of period-doubling bifurcation under different nonlinear factorsε

2.2 多稳态运动

 图 7 速度频率响应 Fig.7 Velocity frequency response
 图 8 定相位庞加莱截面上的不动点(12 Hz) Fig.8 Fixed points on fixed phase Poincar′e section (12 Hz)

 图 9 庞加莱截面上位移状态的时间序列(12 Hz) Fig.9 Time series on fixed phase Poincar′e section (12 Hz)
 图 10 庞加莱截面上位移状态的时间序列(14 Hz) Fig.10 Time series on fixed phase Poincar′e section (14 Hz)

3 总 结

(1)采用平均法研究了主共振的幅频特性曲线,并利用奇异性理论分析了主共振的分岔,进而基于修正后的频响方程给出了避免由鞍结分岔诱导的跳跃现象发生的动力学设计方法.

(2)隔振有效区内的倍周期分岔主要由周期1运动的特征值穿越-1(穿出)所导致,增大阻尼可以抑制倍周期分岔的发生,但在被动隔振技术中,增大线性阻尼却会削弱隔振有效区的隔振效果.

(3)数值仿真研究证实,在噪声影响下,分段光滑隔振系统的响应会在不同稳态间跃迁,非常不利于隔振.因此,在完成跳跃与倍周期分岔的防治设计后,应采用数值仿真校验系统是否存在多稳态运动.