﻿ 一种含负刚度元件的新型动力吸振器的参数优化
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 力学学报  2015, Vol. 47 Issue (2): 320-327  DOI: 10.6052/0459-1879-14-275 0

### 引用本文 [复制中英文]

[复制中文]
Peng Haibo, Shen Yongjun, Yang Shaopu. PARAMETERS OPTIMIZATION OF A NEW TYPE OF DYNAMIC VIBRATION ABSORBER WITH NEGATIVE STIFFNESS[J]. Chinese Journal of Ship Research, 2015, 47(2): 320-327. DOI: 10.6052/0459-1879-14-275.
[复制英文]

### 文章历史

2014-09-11 收稿
2014-11-13 录用
2014–12–12 网络版发表

1. 石家庄铁道大学工程力学系, 石家庄050043;
2. 石家庄铁道大学机械工程学院, 石家庄050043

1 动力吸振器模型及参数优化

 图 1 动力吸振器模型 Fig. 1 The dynamical model of dynamic vibration absorber

 $\left.\!\! m_1 \ddot {x}_1 + k_1 x_1 + k_2 (x_1 - x_2 ) + \\ \qquad c(\dot {x}_1 - \dot {x}_2 ) = F\cos (\omega t) \\ m_2 \ddot {x}_2 + k_2 (x_2 - x_1 ) + c(\dot {x}_2 - \dot {x}_1 ) + kx_2 = 0 \!\!\right\}$ (1)

 $\mu = \dfrac{m_2 }{m_1 }\,,\quad \omega _1 = \sqrt {\dfrac{k_1 }{m_1 }} \,,\quad \omega _2 = \sqrt {\dfrac{k_2 }{m_2 }} \\ \xi = \dfrac{c}{2m_2 \omega _2 }\,,\quad \alpha = \dfrac{k}{k_2 }\,,\quad f = \dfrac{F}{m_1 }$

 $\left. \begin{array}{l} {{\ddot x}_1} + 2\mu {\omega _2}\xi {{\dot x}_1} + (\omega _1^2 + \mu \omega _2^2){x_1} - \\ \qquad 2\mu {\omega _2}\xi {{\dot x}_2} - \mu \omega _2^2{x_2} = f\cos (\omega t)\\ {{\ddot x}_2} + 2{\omega _2}\xi {{\dot x}_2} + (\alpha \omega _2^2 + \omega _2^2){x_2} - \\ \qquad 2{\omega _2}\xi {{\dot x}_1} - \omega _2^2{x_1} = 0 \end{array} \right\}$ (2)

1.1 解析解

 $\left. \begin{array}{l} \left[ {{s^2} + 2\mu {\omega _2}\xi s + (\omega _1^2 + \mu \omega _2^2)} \right]{X_1} - \\ \qquad 2\mu ({\omega _2}\xi s + \omega _2^2){X_2} = f{{\rm{e}}^{{\rm{j}}\omega t}}\\ ({s^2} + 2{\omega _2}\xi s + \alpha \omega _2^2 + \omega _2^2){X_2} - \\ \qquad (2{\omega _2}\xi s + \omega _2^2){X_1} = 0 \end{array} \right\}$ (3)

 $\left. \begin{array}{l} {B_1}H{({\rm{j}}\omega )_{{x_1}}} - \mu {B_2}H{({\rm{j}}\omega )_{{x_2}}} = f\\ {B_3}H{({\rm{j}}\omega )_{{x_2}}} - {B_2}H{({\rm{j}}\omega )_{{x_1}}} = 0 \end{array} \right\}$ (4)

 $B_1 = - \omega ^2 + 2{\rm j}\mu \omega _2 \xi \omega + \omega _1^2 + \mu \omega _2^2 \\ B_2 = = 2{\rm j}\omega _2 \xi \omega + \omega _2^2 \\ B_3 = - \omega ^2 + 2{\rm j}\omega _2 \xi \omega + \alpha \omega _2^2 + \omega _2^2$

 $\left.\!\! H({\rm j}\omega )_{x_1 } = \dfrac{B_3 f}{B_1 B_3 - \mu B_2^2 } \\ H({\rm j}\omega )_{x_2 } = \dfrac{B_1 f}{B_1 B_3 - \mu B_1 B_2 } \!\!\right\}$ (5)

 $\bar {X}_1 = \left| {H({\rm j}\omega )_{x_1 } } \right|\,, \quad \bar {X}_2 = \left| {H({\rm j}\omega )_{x_2 } } \right|$ (6)

1.2 最优参数

 $A_1 = \left| {\dfrac{X_1 }{X_{st} }} \right| = \sqrt {\dfrac{A^2 + B^2\xi ^2}{C^2 + D^2\xi ^2}}$ (7)

 $A = \lambda ^2 - (1 + \alpha )\nu ^2\,,\ \ B = 2\lambda \nu \\ C = \lambda ^4 - (1 + \nu ^2 + \alpha \nu ^2 + \mu \nu ^2)\lambda ^2 + \\ \qquad (1 + \alpha + \alpha \mu \nu ^2)\nu ^2 \\ D = 2\lambda \nu \left[{1 + \alpha \mu \nu ^2 - (1 + \mu )\lambda ^2} \right] \\ \nu = \dfrac{\omega _2 }{\omega _1 }\,,\ \ \lambda = \dfrac{\omega }{\omega _1 }\,,\ \ X_{st} = F / k_1$

 图 2 不同阻尼比下归一化幅频响应曲线 Fig. 2 The normalized amplitude-frequency curves for different damping ratios

 $[\lambda ^2 - \nu ^2(1 + \alpha )]\big / \big[\lambda ^4 - \lambda ^2(1 + \nu ^2 + \alpha \nu ^2 + \mu \nu ^2) + \\ \qquad \nu ^2(1 + \alpha + \alpha \mu \nu ^2)\big] = \\ \qquad \big[1 + \alpha \mu \nu ^2 - \lambda ^2(1 + \mu )\big]^{-1}$ (8)

 $(2 + \mu )\lambda ^4 - 2\lambda ^2\left[{1 + \nu ^2(1 + \alpha )(1 + \mu )} \right] + \\ \qquad 2\nu ^2(1 + \alpha ) + \alpha \mu \nu ^4(2 + \alpha ) = 0$ (9)

 $\lambda _P^2 + \lambda _Q^2 = \dfrac{2\left[{1 + \nu ^2(1 + \alpha )(1 + \mu )} \right] }{2 + \mu }$ (10)

 $\dfrac{1}{1 + \alpha \mu \nu^2 - \lambda _P^2 (1 + \mu )} = \\ \qquad \dfrac{ - 1}{1 + \alpha \mu \nu ^2 - \lambda _Q^2 (1 + \mu )}$ (11)

 $\lambda _P^2 + \lambda _Q^2 = \dfrac{2 + 2\alpha \mu \nu ^2}{1 + \mu }$ (12)

 $\left[{\alpha + (1 + \mu )^2} \right] \nu ^2 - 1 = 0$ (13)

 $\nu _{\rm opt} = \sqrt {\dfrac{1}{\alpha + (1 + \mu )^2}}$ (14)

 $\lambda _P^2 = \dfrac{1 + \alpha + \mu - (1 + \mu )\sqrt {\dfrac{\mu }{2 + \mu }} }{\alpha + (1 + \mu )^2}$ (15a)

 $\lambda _Q^2 = \dfrac{1 + \alpha + \mu + (1 + \mu )\sqrt {\dfrac{\mu }{2 + \mu }} }{\alpha + (1 + \mu )^2}$ (15b)

 $\left. {A_1 } \right|_{(\lambda _P ,\lambda _Q )} = \dfrac{\alpha + (1 + \mu )^2}{(1 + \mu )^2}\sqrt {\dfrac{2 + \mu }{\mu }}$ (16)

 $\dfrac{\partial A_1^2 }{\partial \lambda ^2} = 0$ (17)

 $A_1^2 = \frac{p}{q} = \frac{A^2 + B^2\xi ^2}{C^2 + D^2\xi ^2}$ (18)

 $\dfrac{\partial A_1^2 }{\partial \lambda ^2} = \dfrac{\partial }{\partial \lambda ^2}\left( {\dfrac{p}{q}} \right) = \dfrac{{p}'q - p{q}'}{q^2} = 0$ (19)

 ${p}' = 2\lambda ^2 - 2\nu ^2(1 + \alpha - 2\xi ^2)$ (20a)

 $\begin{array}{l} q' = 4{\nu ^2}{\xi ^2}\left[{1 + \alpha \mu {\nu ^2} - 3\lambda (1 + \mu )} \right] \cdot \\ \;\;\;\;\;\;\left[{1 + \alpha \mu {\nu ^2} - \lambda (1 + \mu )} \right] + \\ \;\;\;\;\;\;2\left[{2\lambda - 1 - {\nu ^2}(1 + \alpha + \mu )} \right] \cdot \\ \;\;\;\;\;\;\left\{ {{\lambda ^2} - \lambda + \alpha \mu {\nu ^4} - {\nu ^2}\left[{(1 + \alpha )(\lambda - 1) + \mu \lambda } \right]} \right\} \end{array}$ (20b)

 $\dfrac{p}{q} = \dfrac{1}{[1 + \alpha \mu \nu ^2 - \lambda ^2(1 + \mu )]^2}$ (21)

 $\xi _P^2 = \dfrac{\mu \left[{2\mu + 3 - \sqrt {\mu (2 + \mu )} } \right] }{4[(2 + \mu )(1 + \mu + \alpha ) - (1 + \mu )\sqrt {\mu (2 + \mu )}]}$ (22a)

 $\xi _Q^2 = \frac{{\mu \left[{2\mu + 3 + \sqrt {\mu (2 + \mu )} } \right]}}{{4[(2 + \mu )(1 + \mu + \alpha ) + (1 + \mu )\sqrt {\mu (2 + \mu )}]}}$ (22b)

 $\xi _{\rm opt} = \sqrt {\dfrac{\mu (3 + 3\alpha + 3\mu + 2\alpha \mu )}{8(1 + \mu )^2 + 4(2 + \mu )[\alpha ^2 + 2\alpha (1 + \mu )]}}$ (23)

 $\left. {A_1 } \right|_{\lambda = 0} = \left. {A_1 } \right|_{(\lambda _P ,\lambda _Q )}$ (24)

 $\sqrt {\dfrac{(1 + \alpha )^2\left[{\alpha + (1 + \mu )^2} \right]^2}{\left[{\alpha ^2 + (1 + \mu )^2 + \alpha (1 + \mu )(2 + \mu )} \right]^2}} = \\ \qquad \sqrt {\dfrac{(2 + \mu )\left[{\alpha + (1 + \mu )^2} \right]^2}{\mu (1 + \mu )^4}}$ (25)

 $\alpha _1 = - (1 + \mu )^2$ (26a)

 $\alpha _{2,\;3} = - 1 - \mu \pm (1 + \mu )\sqrt { {\mu }/(2 + \mu)}$ (26b)

 $\alpha _{4,\;5} = - 1 - \mu (2 + \mu )\pm (1 + \mu )\sqrt {\mu (2 + \mu )}$ (26c)

 $\alpha _{\rm opt} = - 1 - \mu (2 + \mu ) + (1 + \mu )\sqrt {\mu (2 + \mu )}$ (27)

2 数值仿真和模型对比 2.1 数值仿真

 图 3 数值解和解析解比较 Fig. 3 The comparison between the amplitude-frequency curves of the analytical and numerical solution
2.2 与其他形式动力吸振器模型的对比 2.2.1 简谐激励下响应比较

 图 4 与其他形式动力吸振器模型的对比 Fig. 4 The comparison with other DVAs
2.2.2 随机激励下响应比较

 $S_N (\omega ) = \left| {H_{Nx_1 } ({\rm j}\omega )} \right|^2S_0$ (28a)

 $S_V (\omega ) = \left| {H_{Vx_1 } ({\rm j}\omega )} \right|^2S_0$ (28b)

 $S_R (\omega ) = \left| {H_{Rx_1 } ({\rm j}\omega )} \right|^2S_0$ (28c)

 $\begin{array}{l} \sigma _N^2 = \int_{ - \infty }^\infty {{S_N}(\omega )d\omega = } {S_0}\int_{ - \infty }^\infty {|{H_{N{x_1}}}({\rm{j}}\omega ){|^2}{\rm{d}}\omega = } \\ \;\;\;\;\;\;\;\frac{{\pi {S_0}}}{{2\omega _1^3\mu \xi \nu (1 + \alpha + \mu \alpha {\nu ^2}){{(\alpha {\nu ^2} - 1)}^2}}} \cdot \\ \;\;\;\;\;\;\{ (1 + \alpha )\{ {[{\nu ^2}(1 + \alpha ) - 1]^2} + \\ \;\;\;\;\;\;4{\nu ^2}{\xi ^2}\} + \mu {\nu ^2}[4{\xi ^2}(1 + \alpha ) + \\ \;\;\;\;\;\;2{\nu ^2}(1 + \alpha + 2\alpha {\xi ^2}) - 1] + {\mu ^2}{\nu ^4}(1 + 4\alpha {\xi ^2})\} \end{array}$ (29a)

 $\begin{array}{l} \sigma _V^2 = \int_{ - \infty }^\infty {{S_V}(\omega ){\rm{d}}\omega = {S_0}} \int_{ - \infty }^\infty {|{H_{V{x_1}}}({\rm{j}}\omega ){|^2}{\rm{d}}\omega = } \\ \;\;\;\;\;\;\frac{{\pi {S_0}}}{{2\omega _1^3\mu \xi \nu }}[1 + {\nu ^4}{(1 + \mu )^2} + \\ \;\;\;\;\;\;{\nu ^2}(4\mu {\xi ^2} + 4{\xi ^2} - \mu - 2)] \end{array}$ (29b)

 $\begin{array}{l} \sigma _R^2 = \int_{ - \infty }^\infty {{S_R}(\omega ){\rm{d}}\omega = {S_0}} \int_{ - \infty }^\infty {|{H_{R{x_1}}}({\rm{j}}\omega ){|^2}{\rm{d}}\omega = } \\ \;\;\;\;\;\;\frac{{\pi {S_0}}}{{2\omega _1^3\mu \xi {\nu ^5}}} \cdot [1 + {\nu ^4} + (\mu + 4{\xi ^2} - 2)] \end{array}$ (29c)

 $\sigma _N^2 = \dfrac{3.302\pi S_0 }{\omega _1^2 }$ (30a)

 $\sigma _V^2 = \dfrac{6.401\pi S_0 }{\omega _1^2 }$ (30b)

 $\sigma _R^2 = \dfrac{5.780\pi S_0 }{\omega _1^2 }$ (30c)

 图 5 随机激励时间历程 Fig. 5 The time history of the random excitation
 图 6 不含吸振器的主系统时间历程 Fig. 6 The time history of the primary system without DVA
 图 7 含沃伊特型吸振器的主系统时间历程 Fig. 7 The time history of the primary system with Voigt DVA
 图 8 含接地型吸振器的主系统时间历程 Fig. 8 The time history of the primary system with the DVA by Ren
 图 9 含负刚度吸振器的主系统时间历程 Fig. 9 The time history of the primary system with the DVA in this paper

3 结 论

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PARAMETERS OPTIMIZATION OF A NEW TYPE OF DYNAMIC VIBRATION ABSORBER WITH NEGATIVE STIFFNESS
Peng Haibo, Shen Yongjun, Yang Shaopu
1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
2. Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
Fund: The project was supported by the National Natural Science Foundation of China (11372198), the Program for New Century Excellent Talents in University of Ministry of Education of China (NCET-11-0936), the Cultivation Plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province (LJRC018), the Program for Advanced Talent in the Universities of Hebei Province (GCC2014053), and the Program for Advanced Talent in Hebei (A201401001).
Abstract: A new type of dynamic vibration absorber with negative stiffness spring is presented and studied analytically in detail. At first the analytical solution is obtained based on the Laplace transform method, and it could be found that there exist two fixed points independent of the damping ratio in the normalized amplitude-frequency curves. The optimum tuning ratio and damping ratio are obtained based on the fixed-point theory. According to the characteristics of the negative stiffness element, the optimal negative stiffness ratio is obtained and it could keep the system stable. The comparison of the analytical solution with the numerical one verifies the correctness and satisfactory precision of the analytical solution. The comparison with other two traditional dynamic vibration absorbers under the harmonic and random excitation show that the presented dynamic vibration absorber performs better in vibration absorption. The result could provide theoretical basis for the optimal design of similar dynamic vibration absorber.
Key words: dynamic vibration absorber    negative stiffness    vibration control    fixed-point theory    parameter optimization