工程科学与技术   2019, Vol. 51 Issue (1): 229-235

Vibration Analysis of Bearing-rotor System Based on Nonlinear Dynamic Characteristics
CHEN Dongju, HAN Jihong, GAO Xue, FAN Jinwei
Beijing Key Lab. of Advanced Manufacturing Technol., Beijing Univ. of Technol., Beijing 100124, China
Abstract: With the wide application of the aerostatic spindle in the ultra-precision machining process, the requirements for the kinematic precision of the spindle are increasing. It is necessary to predict and improve the precision of the spindle motion accurately. Based on the nonlinear dynamic characteristics of the aerostatic bearing, the vibration characteristics and prediction model of the aerostatic spindle were studied, and the influence of the nonlinear dynamic characteristic analysis on the spindle rotary precision was explored. Firstly, the dynamic flow model of the gas film of the aerostatic journal bearing was established, the nonlinear dynamic stiffness and damping coefficients were obtained by the perturbation method. The gas film was used as the spring-damping system to build the bearing-rotor system, and the dynamic vibration model of bearing-rotor system was established by dynamic analysis. Then nonlinear dynamic parameters were introduced into the vibration model, and the curve of the radial runout error, the deflection error and the total error of radial vibration were obtained by solving the model with MATLAB. The frequency domain analysis were conducted on the vibration signal. Finally, Rotation Error Measurement experimental of spindle was performed to inspect the results of vibration error analysis. From the dynamic analysis of the aerostatic journal bearings, the dynamic stiffness and dynamic damping of the bearing are all nonlinear, and the dynamic stiffness increases with the increase of eccentricity, and the dynamic damping decreases. From the vibration analysis of the bearing-rotor system, the following conclusions can be obtained. 1) The nonlinear analysis has an obvious influence on the deflection error, but the influence on the radial runout error is not obvious. It shows that the nonlinear analysis mainly affects the deflection error of the spindle and thus affects the total radial error. 2) The maximum amplitude of the deflection error is basically stable when the fixed value analysis is analyzed, while the maximum amplitude of the deflection error has an increase process and tends to be stable at the time of nonlinear analysis, and the maximum amplitude of the nonlinear analysis is obviously larger than the amplitude of the constant value analysis. 3) The total radial error of the nonlinear analysis and the constant value analysis is basically the same at the beginning of the gas supply, but with the increase of time, the maximum amplitude under the nonlinear analysis is larger than the maximum amplitude under the constant value analysis, which shows that the nonlinear analysis has no obvious effect on the error of radial runout and the deviation angle when the gas supply is started. When the gas supply is stable, the nonlinear dynamic stiffness and dynamic damping will obviously affect the vibration amplitude of the rotor. 4) From the frequency domain, the resonance frequency at the maximum amplitude of the nonlinear analysis is 964 Hz, and the resonance frequency at the maximum amplitude of the constant value analysis is 986 Hz, and the nonlinear analysis reduces the resonance frequency at the maximum amplitude. 5) When the frequency is higher than 1 500 Hz, the amplitude change of the rotor is very small, which shows that the vibration of the rotor is more stable when the frequency is greater than 1 500 Hz, and the vibration frequency of the gas film is not easy to resonate with the natural frequency. Experimental results indicated that the error of radial rotation error of the spindle based on nonlinear analysis is reduced by 1.43% to 6.54% compared to the constant value analysis. Therefore, the coupled vibration analysis of the bearing-rotor system can be achieved by applying the gas film as the spring-damping system to the rotor. The introduction of nonlinear dynamic characteristic parameters of bearing realizes the effect of bearing dynamic performance on spindle dynamic vibration. Based on nonlinear dynamic characteristics, the vibration analysis of the bearing-rotor system can more accurately analysis and predict the radial vibration error of the aerostatic spindle.
Key words: aerostatic journal bearing    bearing-rotor system    nonlinear analysis    vibration error

1 空气静压径向轴承结构及原理

 图1 空气静压径向轴承结构 Fig. 1 Structural of aerostatic journal bearing

2 空气静压轴承动态特性分析 2.1 气膜动态流动模型

 \begin{aligned}{b} \displaystyle \frac{\partial }{{\partial x}}\left(p{h^3}\frac{{\partial p}}{{\partial x}}\right) + \frac{\partial }{{\partial {\textit{z}}}}\left(p{h^3}\frac{{\partial p}}{{\partial {\textit{z}}}}\right) = 6\mu \displaystyle \frac{{\partial (Uph)}}{{\partial x}} + 12\mu \frac{{\partial (ph)}}{{\partial t}} \end{aligned} (1)

 $\begin{array}{l} \displaystyle\frac{\partial }{{\partial \theta }}\left(P{H^3}\frac{{\partial P}}{{\partial \theta }}\right) + \frac{\partial }{{\partial {\textit{Z}}}}\left(P{H^3}\frac{{\partial P}}{{\partial {\textit{Z}}}}\right) = {\rm{ }}\varLambda \cdot \displaystyle \frac{{\partial (PH)}}{{\partial \theta }} + \varOmega \cdot \frac{{\partial (PH)}}{{\partial t}} \end{array}$ (2)
2.2 动刚度和动阻尼

 \left\{ {\begin{aligned} &{H = {H_0} + H'{\rm{ = }}{H_0} + \varepsilon {H_0}\sin\,\,wt}, \\ &{P = {P_0} + P'{\rm{ = }}{P_0} + \varepsilon {\rm{(}}{P_1}\sin \,\,wt + {P_2}\cos\,\,wt{\rm{)}}} \end{aligned}} \right. (3)

 \left\{ {\begin{aligned} \begin{aligned} &\!\!\displaystyle\frac{\partial }{{\partial \theta }}\left(P{H^3}\frac{{\partial P}}{{\partial \theta }}\right) + \frac{\partial }{{\partial {\textit{Z}}}}\left(P{H^3}\frac{{\partial P}}{{\partial {\textit{Z}}}}\right) = \varLambda \cdot \frac{{\partial (PH)}}{{\partial \theta }},\\ &\!\!{P_{\rm{0}}}{H_{\rm{0}}^{\rm{2}}}\displaystyle\frac{\partial }{{\partial \theta }}\left(\frac{{\partial {P_{\rm A}}}}{{\partial \theta }}\right)\!+\!{P_{\rm{0}}}{H_{\rm{0}}^{\rm{2}}}\frac{\partial }{{\partial {\textit{Z}}}}\left(\frac{{\partial {P_{\rm A}}}}{{\partial {\textit{Z}}}}\right)\!-\!\varLambda \cdot \frac{{\partial {P_{\rm A}}}}{{\partial \theta }} =\!-\!\varOmega \! \cdot\!w \cdot {P_{\rm B}}, \\ &\!\!{{P_{\rm{0}}}{H_{\rm{0}}^{\rm{2}}}\displaystyle\frac{\partial }{{\partial \theta }}\left(\frac{{\partial {P_{\rm B}}}}{{\partial \theta }}\right) \!+\! {P_{\rm{0}}}{H_{\rm{0}}^{\rm{2}}}\frac{\partial }{{\partial {\textit{Z}}}}\left(\frac{{\partial {P_{\rm B}}}}{{\partial {\textit{Z}}}}\right) \!-\! \varLambda \! \cdot\! \frac{{\partial {P_{\rm B}}}}{{\partial \theta }} = \varOmega w(P_0^2\!+\!{P_{\rm A}})} \end{aligned}\end{aligned}} \right. (4)

 \left\{ {\begin{aligned} & {{K_{\rm n}} = \displaystyle \frac{{\iint {{P_1} \cdot \cos \,\,\theta {\rm d}s}}}{h}}, \\ & {{C_{\rm n}} = \displaystyle \frac{{\iint {{P_2} \cdot \cos \,\, \theta {\rm d}s}}}{{\omega h}}} \end{aligned}} \right. (5)

3 空气静压主轴径向振动分析 3.1 空气静压主轴径向振动误差分析

 图2 空气静压主轴径向振动误差分析 Fig. 2 Error analysis of radial vibration of aerostatic spindle

 $\Delta x = {x_1} + {x_2} + x'$ (6)
3.2 轴承–转子动力学模型

 图3 轴承–转子系统 Fig. 3 Bearing-rotor system of aerostatic spindle

 \left\{ \begin{aligned} &m\ddot x{\rm{ + }}{F_1}{\rm{ + }}{F_2}{\rm{ + }}{F_3}{\rm{ + }}{F_4}{\rm{ = }}{F_{\rm r}} + mg + {F_x},\\ &{\rm{(}}{J_0}{\rm{)}}\ddot \theta {\rm{ + }}{M_1}{\rm{ + }}{M_2} - {M_3} - {M_4} = {F_{\rm r}} \cdot 4a \end{aligned} \right. (7)

 \left\{ \begin{aligned} &{F\!\!_i} = {K_i}{x_i} + {C_i}{{\dot x}_i}{\rm{ , }}i = 1 \sim {\rm{4}};\\ &{M_i} = {F\!\!_i}{l_i}{\kern 1pt} {\rm{, }}i = 1 \sim {\rm{4}};\\ &{F\!\!_x} = me{\omega ^2}\cos \,\,\omega t;\\ &{x_1} = x + 3a\theta ;\\ & {\rm{ }}{x_2} = x + a\theta ;\\ &{x_3} = x - a\theta ;\\ &{\rm{ }}{x_4}{\kern 1pt} = x - 3a\theta ;\\ &{l_1} = {l_4} = 3a,{\rm{ }}{l_2} = {l_3} = a{\rm{ }} \end{aligned} \right. (8)

 \begin{aligned} \quad &\left[ {\begin{aligned} &m \;\;\; 0\\ &0 \;\; {{J_0}} \end{aligned}} \right]\left[ {\begin{aligned} &{\ddot x}\\ &{\ddot \theta } \end{aligned}} \right] + \left[ {\begin{aligned} &P \;\;\; Q\\ &{P'} \;\; {Q'} \end{aligned}} \right] + \left[ {\begin{aligned} &S \;\;\; I\\ &{S'} \;\; {I'} \end{aligned}} \right]\left[ {\begin{aligned} &x\\ &\theta \end{aligned}} \right]{\rm{ = }}\\ & \quad \quad \left[ {\begin{aligned} &{{F_{\rm r}} + mg + me{\omega ^2}\cos \,\,\omega t - S \cdot e}\\ &{4a{F_{\rm r}} - S' \cdot e} \end{aligned}} \right] \end{aligned} (9)

 \left\{ \begin{aligned} &P = {C_1} + {C_2} + {C_3} + {C_4}{\rm{ }},\\ &P' = Q = 3a{C_1} + a{C_2} - a{C_3} - 3a{C_4},\\ &Q' = 9{a^2}{C_1} + {a^2}{C_2} + {a^2}{C_3} + 9{a^2}{C_4},\\ &S = {K_1} + {K_2} + {K_3} + {K_4},\\ &S' = I = 3a{K_1} + a{K_2} - a{K_3} - 3a{K_4},\\ &I' = 9{a^2}{K_1} + {a^2}{K_2} - {a^2}{K_3} - 9{a^2}{K_4} \end{aligned} \right. (10)

 $\Delta x = {x_1} + {x_2} + x' = x + 4a \cdot \theta$ (11)
4 仿真结果及分析 4.1 仿真动态特性分析

 图4 动刚度与动阻尼仿真结果 Fig. 4 Simulation results of dynamic stiffness and dynamic damping

 \left\{ {\begin{aligned} &{{K_i} = {K_{\rm n}}({{{x_i}} / {{h_0}}}),{\rm{ }}i = {\rm{1}} \sim {\rm{4}}}; \\ & {{C_i} = {C_{\rm n}}({{{x_i}} / {{h_0}}}),{\rm{ }}i = {\rm{1}} \sim {\rm{4}}} \end{aligned}} \right. (12)
4.2 仿真径向振动误差分析

 图5 空气静压主轴径向振动误差曲线 Fig. 5 Error curves of radial vibration of aerostatic spindle

1）径向跳动误差包括两种形式的振动，在以频率为主轴回转频率的振动过程中同时进行一定幅度的上下振动，这是由主轴转子偏心运动和外载荷 ${F_{\rm r}}$ 造成的。

2）由图5（c）可看出，在供气瞬间转子有一个明显的振动冲击，转子的振幅从零迅速上升至某一值，这体现了主轴从开始供气至稳定供气的过程中转子不稳定的振动特性。

3）非线性分析对径向跳动误差的影响不明显，但对偏角振动误差有明显影响；定值分析时偏角误差的最大振幅基本稳定，但非线性分析时偏角误差的最大振幅存在一个增加过程并最终趋于稳定，其最大振幅明显大于定值分析时振幅。因此，非线性分析主要对主轴的偏角误差造成影响，从而影响径向总误差。

4）在供气开始一段时间内，非线性分析与定值分析下的径向总误差基本一致；随着时间的增加，非线性分析下的最大振幅大于定值分析下的最大振幅。这说明开始供气时非线性分析对径向跳动误差和偏角误差没有造成明显影响，当供气稳定时非线性的动刚度与动阻尼会对主轴转子振动幅度产生明显影响。

 图6 振动总误差的频域分析 Fig. 6 Frequency domain analysis of total error

5 空气静压主轴径向回转误差试验

 图7 回转误差测试装置 Fig. 7 Rotary error testing device

 图8 回转误差对比 Fig. 8 Comparison of rotary errors

6 结　论

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