# 高阶非线性减振系统的主动控制的研究Study on the Active Control of Vertical Nonlinear Suspension System

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The vehicle suspension system can be simplified into a single-degree-of-freedom vibration model. Five nonlinear stiffness factors are considered in this model, and a time-delay damper is used to control its vertical vibration. Using multi-scale analysis, the theoretical approximate solution of the system is obtained. The effects of nonlinear stiffness coefficient, damping coefficient, feedback gain coefficient and time delay on vertical vibration reduction are studied, and the regularity is analyzed, which can provide theoretical guidance for vibration reduction projects in engineering applications.

1. 引言

2. 力学模型

$m\stackrel{¨}{x}\left(t\right)+kx\left(t\right)+\delta {x}^{5}\left(t\right)+c\stackrel{˙}{x}\left(t\right)+{f}_{sc}\stackrel{¨}{x}\left(t-\tau \right)={F}_{0}\mathrm{cos}\left(wt\right)$ (1)

Figure 1. Illustration of model

3. 摄动分析

$\frac{\text{d}x}{\text{d}t}=\frac{w{x}_{c}}{\Omega }\cdot \frac{\text{d}y}{\text{d}T}$$\frac{{\text{d}}^{2}x}{\text{d}{t}^{2}}=\frac{{w}^{2}{x}_{c}}{{\Omega }^{2}}\cdot \frac{{\text{d}}^{2}y}{{\text{d}}^{2}T}$$\frac{{\text{d}}^{2}x\left(t-\tau \right)}{\text{d}{t}^{2}}=\frac{{w}^{2}{x}_{c}}{{\Omega }^{2}}\cdot \frac{{\text{d}}^{2}y\left(T-{\tau }_{c}\right)}{\text{d}{T}^{2}}$ (2)

$y=\frac{x}{{x}_{c}}$$T=\frac{w}{\Omega }t$${\tau }_{c}=\frac{w}{\Omega }\tau$${w}_{0}=\sqrt{\frac{k{\Omega }^{2}}{m{w}^{2}}}$${\xi }_{1}=\frac{\delta {x}_{c}^{4}{\Omega }^{2}}{m{w}^{2}}$${\xi }_{2}=\frac{c\Omega }{mw}$${f}_{s}=\frac{{f}_{sc}}{{m}_{c}}$$F=\frac{{F}_{0}{\Omega }^{2}}{m{x}_{c}{w}^{2}}$ (3)

$\stackrel{¨}{y}\left(T\right)+{w}_{0}^{2}y\left(T\right)+{\xi }_{1}{y}^{5}\left(T\right)+{\xi }_{2}\stackrel{˙}{y}\left(T\right)+{f}_{s}\stackrel{¨}{y}\left(T-{\tau }_{c}\right)=F\mathrm{cos}\left(\Omega T\right)$ (4)

${\xi }_{1}=\epsilon {\stackrel{^}{\xi }}_{1}$${\xi }_{2}=\epsilon {\stackrel{^}{\xi }}_{2}$${f}_{s}=\epsilon {\stackrel{^}{f}}_{s}$$F=\epsilon \stackrel{^}{f}$$\tau ={\tau }_{c}=\stackrel{^}{\tau }$${y}_{\tau }=y\left(T-\stackrel{^}{\tau }\right)$$\Omega ={w}_{0}+\epsilon \sigma$

$\stackrel{¨}{y}\left(T\right)+{w}_{0}^{2}y\left(T\right)=\epsilon \left[-{\stackrel{^}{\xi }}_{1}{y}^{5}\left(T\right)-{\stackrel{^}{\xi }}_{2}\stackrel{˙}{y}\left(T\right)-{\stackrel{^}{f}}_{s}{\stackrel{˙}{y}}_{\stackrel{^}{\tau }}+\stackrel{^}{f}\mathrm{cos}\left(\Omega T\right)\right]$ (5)

$y\left(T,\epsilon \right)={y}_{0}\left({T}_{0},{T}_{1}\right)+\epsilon {y}_{1}\left({T}_{0},{T}_{1}\right)$${y}_{\stackrel{^}{\tau }}\left(T,\epsilon \right)={y}_{0\stackrel{^}{\tau }}\left({T}_{0},{T}_{1}\right)+\epsilon {y}_{1\stackrel{^}{\tau }}\left({T}_{0},{T}_{1}\right)$ (6)

${\epsilon }^{0}:{D}_{0}^{2}{y}_{0}+{\omega }_{0}^{2}{y}_{0}=0$ (7)

${\epsilon }^{1}:{D}_{0}^{2}{y}_{1}+{w}_{0}^{2}{y}_{1}=-2{D}_{0}{D}_{1}{y}_{0}+\left[-{\stackrel{^}{\xi }}_{1}{y}_{0}^{5}-{\stackrel{^}{\xi }}_{2}{D}_{0}{y}_{0}-{\stackrel{^}{f}}_{s}{\stackrel{˙}{y}}_{0\stackrel{^}{\tau }}+\stackrel{^}{f}\mathrm{cos}\left({w}_{0}{T}_{0}+\sigma {T}_{1}\right)\right]$ (8)

${y}_{0}=a\left({T}_{1}\right)\mathrm{cos}\left[{w}_{0}{T}_{0}+\phi \left({T}_{1}\right)\right]=A\left({T}_{1}\right){\text{e}}^{i{w}_{0}{T}_{0}}+cc$ (9)

$\stackrel{^}{f}\mathrm{cos}\left({w}_{0}{T}_{0}+\sigma {T}_{1}\right)=\frac{1}{2}\stackrel{^}{f}{\text{e}}^{i\left({w}_{0}{T}_{0}+\sigma {T}_{1}\right)}+cc$ (10)

${y}_{0\tau }={A}_{\tau }\left({T}_{1}\right){\text{e}}^{i{w}_{0}\left({T}_{0}-\tau \right)}+cc$ (11)

${A}_{\tau }=A\left({T}_{1}\right)-\epsilon \tau {A}^{\prime }\left({T}_{1}\right)+\frac{{\left(\epsilon \tau \right)}^{2}}{2}{A}^{″}\left({T}_{1}\right)+\cdots$ (12)

 (13)

$-i2{w}_{0}{D}_{1}A-10{\stackrel{^}{\xi }}_{1}{A}^{3}{\stackrel{¯}{A}}^{2}-i{w}_{0}{\stackrel{^}{\xi }}_{2}A-i{w}_{0}{\stackrel{^}{f}}_{s}A{\text{e}}^{-i{w}_{0}\stackrel{^}{\tau }}+\frac{\stackrel{^}{f}}{2}{\text{e}}^{i\sigma {T}_{1}}=0$ (14)

$-i{w}_{0}{D}_{1}a+a{w}_{0}{D}_{1}\phi -\frac{5}{16}{\stackrel{^}{\xi }}_{1}{a}^{5}-i{w}_{0}{\stackrel{^}{\xi }}_{2}\frac{a}{2}-i{w}_{0}{\stackrel{^}{f}}_{s}\frac{a}{2}{\text{e}}^{-i{w}_{0}\stackrel{^}{\tau }}+\frac{\stackrel{^}{f}}{2}{\text{e}}^{i\left(\sigma {T}_{1}-\phi \right)}=0$ (15)

${D}_{1}a=-\frac{a}{2}{\stackrel{^}{\xi }}_{2}-{\stackrel{^}{f}}_{s}\frac{a}{2}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)+\frac{\stackrel{^}{f}}{2{w}_{0}}\mathrm{sin}\left(\sigma {T}_{1}-\phi \right)$ (16)

$a{D}_{1}\phi =\frac{5}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{5}+{\stackrel{^}{f}}_{s}\frac{a}{2}\mathrm{sin}\left({w}_{0}\tau \right)-\frac{\stackrel{^}{f}}{2{w}_{0}}\mathrm{cos}\left(\sigma {T}_{1}-\phi \right)$ (17)

$\varphi =\sigma {T}_{1}-\phi$ ，则上式为：

 (18)

$a\sigma -a{D}_{1}\varphi =\frac{5}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{5}+{\stackrel{^}{f}}_{s}\frac{a}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)-\frac{\stackrel{^}{f}}{2{w}_{0}}\mathrm{cos}\varphi$ (19)

$\frac{\stackrel{^}{f}}{2}\mathrm{sin}\varphi =\frac{a}{2}{w}_{0}{\stackrel{^}{\xi }}_{2}+{\stackrel{^}{f}}_{s}{w}_{0}\frac{a}{2}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)$ (20)

$\frac{\stackrel{^}{f}}{2}\mathrm{cos}\varphi =\frac{5}{16}{\stackrel{^}{\xi }}_{1}{a}^{5}+{\stackrel{^}{f}}_{s}{w}_{0}\frac{a}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)-a\sigma {w}_{0}$ (21)

$\frac{{\stackrel{^}{f}}^{2}}{4}={\left[\frac{a}{2}{w}_{0}{\stackrel{^}{\xi }}_{2}+{\stackrel{^}{f}}_{s}{w}_{0}\frac{a}{2}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)\right]}^{2}+{\left[\frac{5}{16}{\stackrel{^}{\xi }}_{1}{a}^{5}+{\stackrel{^}{f}}_{s}{w}_{0}\frac{a}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)-a\sigma {w}_{0}\right]}^{2}$ (22)

(23)

4. 系统参数对减振性能的影响

${F}^{2}={\left[a{w}_{0}{\xi }_{2}\right]}^{2}+{\left[\frac{5}{8}{\xi }_{1}{a}^{5}-2a{w}_{0}\Omega +2a{w}_{0}^{2}\right]}^{2}$ (24)

Figure 2. Stiffness versus amplitude

5. 系统的稳定性与分岔

${S}_{1}=\frac{\partial {f}_{1}}{\partial a}=-\frac{{\stackrel{^}{\xi }}_{2}}{2}-\frac{{\stackrel{^}{f}}_{s}}{2}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)$${S}_{2}=\frac{\partial {f}_{1}}{\partial \varphi }=\frac{\stackrel{^}{f}}{2{w}_{0}}\mathrm{cos}\varphi =\frac{5}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{5}+{\stackrel{^}{f}}_{s}\frac{a}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)-a\sigma$

${S}_{3}=\frac{\partial {f}_{2}}{\partial a}=\frac{1}{a}\left[\sigma -\frac{25}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{4}-{\stackrel{^}{f}}_{s}\frac{1}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)\right]$${S}_{4}=\frac{\partial {f}_{2}}{\partial \varphi }=\frac{\stackrel{^}{f}}{2{w}_{0}}\mathrm{sin}\varphi =-\frac{1}{2}{\stackrel{^}{\xi }}_{2}-{\stackrel{^}{f}}_{s}\frac{1}{2}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)$

$cr=\left[\frac{5}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{4}+{\stackrel{^}{f}}_{s}\frac{1}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)-\sigma \right]\left[\sigma -\frac{25}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{4}-{\stackrel{^}{f}}_{s}\frac{1}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)\right]$

$\mathrm{det}\left[\begin{array}{cc}{S}_{1}-\lambda & {S}_{2}\\ {S}_{3}& {S}_{4}-\lambda \end{array}\right]=0$ (25)

Figure 3. Damping versus amplitude

Figure 4. Time delay versus amplitude

Figure 5. Feedback gain versus amplitude

$B=\left[{\stackrel{^}{\xi }}_{2}+{\stackrel{^}{f}}_{s}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)\right]>0$

$A={\left[\frac{{\stackrel{^}{\xi }}_{2}}{2}+\frac{{\stackrel{^}{f}}_{s}}{2}\mathrm{cos}\left({w}_{0}\stackrel{^}{\tau }\right)\right]}^{2}-\left[\frac{5}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{4}+{\stackrel{^}{f}}_{s}\frac{1}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)-\sigma \right]\left[\sigma -\frac{25}{16{w}_{0}}{\stackrel{^}{\xi }}_{1}{a}^{4}-{\stackrel{^}{f}}_{s}\frac{1}{2}\mathrm{sin}\left({w}_{0}\stackrel{^}{\tau }\right)\right]>0$

6. 结论

Figure 6. Stability domain versus oscillating curve

NOTES

*通讯作者。

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