齿轮—转子—滑动轴承系统非线性动力学特性的理论和试验研究
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摘要
齿轮-转子系统是动力传输系统振动问题的最主要来源,齿轮系统工作时产生的强烈振动,对系统的安全性和隐蔽性造成了严重的影响。目前,工程实践中复杂齿轮-转子-轴承系统的振动机理尚未清楚,针对此问题,从齿轮动力学和转子动力学的角度出发,考虑系统的齿侧间隙、时变啮合刚度、静态传动误差、输入输出扭矩激励、弹性轴和轴承的影响,建立了齿轮-转子系统的非线性动力学模型。对动力学模型进行了数值仿真,研究了转速对动态啮合力和系统的振动特性的影响。在判断系统的振动是否为混沌的过程中,采用混沌时间序列分析理论计算了齿轮传动系统的高维非线性方程组的最大Lyapunov指数,数值积分的
结果表明:动力学模型能合理地体现齿轮啮合的三种冲击状态;混沌时间序列分析理论能有效地计算出齿轮传动系统高维非线性方程组的最大Lyapunov指数。齿轮副扭转振动系统是最基础的齿轮系统,为了从近似解析解的角度研究齿侧间隙的大小对齿轮副扭转振动系统的幅频响应曲线的影响规律,将多项谐波平衡法与求解非线性方程组的最小二乘解的广义逆法相结合求解近似解析解,研究结果表明:齿侧间隙对系统幅频响应曲线的影响除了与齿侧间隙的大小密切相关之外,还与阻尼、时变啮合刚度谐波项幅值及预紧力紧密相关。当阻尼相对较小时,系统的幅频响应曲线受到齿侧间隙的影响比较明显,在不同的齿侧间隙下,主共振区和超谐共振区都会出现振幅跳跃现象;阻尼比增大到一定值后,随着齿侧间隙逐渐增大,主共振区始终出现振幅跳跃,但在振幅相对较小的超谐共振区振幅跳跃现象变得不明显;当阻尼比增大到较大值时,不同的齿侧间隙下,系统的幅频响应曲线都接近于线性系统的响应曲线。随着啮合刚度一次谐波项幅值的逐渐增大,齿侧间隙比相对较小时,幅频响应曲线在主共振区会体现出“硬化曲线”的特征;当齿轮间隙比增大到一定值时,不论啮合刚度一次谐波项幅值的大小为多少,系统的幅频响应曲线均体现出“软化曲线”的特征。在预紧力比较小的条件下,当齿侧间隙比相对较小时,幅频响应曲线只在局部转速范围内体现出“硬化曲线”;当齿侧间隙比相对较大时,幅频响应曲线体现为“软化曲线”。在预紧力比较大的条件下,齿侧间隙越小,“硬化曲线”的特征越明显;齿侧间隙需要增大到较大值才能使主共振区的幅频响应曲线变成“软化曲线”。
由于齿轮副扭转振动系统的模型假设转轴和轴承是刚性不变形的,与实际的齿轮-转子系统存在一定的差别,为了研究不同齿侧间隙对齿轮-转子系统的动力学特性的影响,采用数值仿真研究了不同齿侧间隙对系统的分岔和混沌的影响,研究结果表明:齿侧间隙对系统的第一阶弯曲临界转速处的振动状态的影响比较大,齿侧间隙相对较小时,系统的第一阶弯曲临界转速处的振动状态相对较好。除了齿侧间隙之外,支承刚度对齿轮-转子系统的分岔和混沌也具有重要的影响,采用数值仿真研究了支承刚度对系统的分岔和混沌的影响,研究结果表明:随着支承刚度的逐渐增大,系统的弯扭耦合临界转速均相应地增大,系统的分岔和混沌区域也相应地发生改变。
在考虑非线性啮合力的基础上,进一步考虑非线性油膜力的影响,建立了齿轮-转子-滑动轴承系统的动力学模型,采用数值仿真研究了系统的动态响应,研究结果表明:随着转速的逐渐升高,非线性啮合力和非线性油膜力分别在不同的转速范围内影响着齿轮-转子-滑动轴承系统的非线性振动特性,当转速相对较低时,系统的振动特性主要受到非线性啮合力的影响,随着转速的逐渐升高,非线性油膜力对系统振动特性的影响逐渐增大;当转速逐渐增大到接近系统的第一阶临界转速(可以是齿轮啮合引起的弯扭耦合临界转速而不一定是转子的纯弯曲临界转速)的二倍时,逐渐出现非线性油膜力引起的“半频涡动”;通过对比线性八参数油膜力和非线性油膜力对啮合力的影响,发现转速逐渐增大到接近系统的第一阶临界转速(可为弯扭耦合临界转速)的二倍时,非线性油膜力逐渐对非线性啮合力起作用,并使啮合力的频谱中出现“半频涡动”频率成份;随着转速的进一步增大,非线性油膜力对非线性啮合力的影响也越来越大,甚至超过不平衡质量对啮合力的影响;然而传统的线性油膜力对非线性啮合力则基本没有影响。
为了验证数值仿真结果的正确性,设计了齿轮-转子-滚动轴承系统振动试验台;提出了一种可以调整齿侧间隙的装置;在试验台上研究了齿侧间隙的大小对齿轮-转子-轴承系统的振动特性的影响规律,试验结果表明:在非共振转速下,存在较差齿侧间隙范围使齿轮系统的振幅相对较大,并存在较好齿侧间隙范围使齿轮系统的振幅相对较小;当齿侧间隙增大到一定值后,系统通常将保持单边冲击状态,系统的振幅将维持在一定的范围内,继续增大齿侧间隙不会再对系统的振动产生大的影响。在齿轮-转子-滚动轴承系统振动试验台的基础上,将主动齿轮-转子的滚动轴承改造成滑动轴承,并测量了若干转速下滑动轴承座的振动,试验结果表明:转速低于一阶弯曲临界转速之前,系统的振动主要受到非线性啮合力的影响,试验的结果基本验证了数值仿真的结果。
Geared rotor system is the most primary origin of the vibration problem of power transmission system, the security and concealment of system are affected by the severe vibration significantly. At present, the vibration mechanism of geared rotor bearing system is still not clear. Aiming at this problem, on the basis of gear dynamic and rotor dynamic, the effect of clearance, time-varying mesh stiffness, static transmission error, flexible shaft, bearing, input and output torque excitation are considered, and the nonlinear dynamic model of geared rotor system is derived. Numerical integration method is used to study the dynamic model, the effects of speed on the dynamic mesh force and vibration characteristic of system are also studied. On the process of judge the vibration is chaos or not, the maximum Lyapunov exponent of high dimensional nonlinear equation is calculated by chaotic time series analysis theory. The results of numerical integration show that the dynamic model can reveal three kind of impact status of gear mesh in reason. The chaotic time series analysis theory can calculate the maximum Lyapunov exponent of high dimensional nonlinear equation effectively.
The gear pair torsional vibration system is the most basical gear system, in order to study the effect of clearance’s size on the gear pair torsional vibration system’s amplitude-frequency response curve from approximate analytical solution method, the multi-term harmonic balance method is connected with the generalized inverse method which used to solve the minimum least-square solutions of nonlinear equation group. The results show that the effect of clearance on the amplitude-frequency response curve of gear pair torsional vibration system relates with the size of clearance, the damping, time-varying mesh stiffness and preload. When the damping is relatively small, the amplitude-frequency response curve of system affected by clearance obviously, the amplitude jump phenomena appear in primary resonance and super harmonic resonance under different clearance. When damping increases to certain value, with the increase of clearance, the amplitude jump phenomena always appear in primary resonance, but it is not obvious in the super harmonic resonance whose amplitude is relative small. When damping increases to large value, the amplitude-frequency response curves of system approach to the response curves of linear system under different clearance. Whit the increase of the first harmonic term of mesh stiffness’s value, when the clearance is relatively small, the amplitude-frequency response curve of system will reflect the character of hardening type curve. When clearance increase to certain value, no matter the first harmonic term of mesh stiffness’s value is small or large, the amplitude-frequency response curve of system just reflects the character of softening type curve. On the condition of preload is relatively small, when clearance is also relatively small, the amplitude-frequency response curve just reflect the character of hardening type curve in local speed range. When clearance is relatively large, the amplitude-frequency response curve reflects the character of softening type curve. On the condition of preload is relatively large, when the learance is small, with the preload becomes larger and larger, and then the character of hardening type curve also becomes more obvious, the clearance need to increase to larger value to make the amplitude-frequency response curve becomes softening type curve in the primary resonance.
The model of gear pair torsional vibration system assumes rotor and bearing are rigid, it has certain difference comparing with practical geared rotor system, in order to study the effect of clearance on geared rotor system, numerical integration method is used to study the effect of clearance on the bifurcation and chaos of system, the results show that the first lateral critical speed of system is affected by clearance greatly, when the clearance is relatively small, the vibration state is relatively preferable. Bearing stiffness also has important effect on the bifurcation and chaos of system, except for clearance. The effect of bearing stiffness on the bifurcation and chaos of system is studied by numerical integration. The results show that the lateral-torsion critical speed will increase with the increase of bearing stiffness, and the region of bifurcation and chaos also change accordingly.
On the basis of considering nonlinear dynamic force, the effect of nonlinear oil film force is considered further, dynamic model of geared rotor-oil journal bearing system is proposed. The nonlinear dynamic responses of system are investigated by numerical integration method. The results show that with the increase of rotational speed, the nonlinear vibration characteristic of system is affected by nonlinear mesh force and nonlinear oil film force respectively in different speed range. The vibration characteristic of system is mainly affected by nonlinear mesh force when speed is relatively low. With the increase of speed, the influence of nonlinear oil film force on the vibration also increases gradually, and the sub-synchronous forward precession phenomena appear when the speed approaches the twice of first order critical speed (it can be lateral-torsional critical speed caused by gear mesh, it is not need to be pure lateral critical speed of rotor). By comparing the effect of linear eight parameter oil film fore and nonlinear oil film fore on mesh force, the results show that when rotational speed increases to twice of the first order critical speed of system (it can be lateral-torsional critical speed), the nonlinear mesh force is affected by nonlinear oil film force gradually, the sub-synchronous forward precession frequency appears in spectrum plot of mesh force. With the increase of speed, the effect of nonlinear oil film fore on nonlinear mesh force also increase, it may exceed the effect of unbalance on mesh force. However, the conventional linear oil film force does not affect the mesh force.
In order to validate the correctness of numerical integration’s results, a geared rotor bearing system vibration test-bed is designed, and a kind of equipment which is used to adjust clearance is innovated independently, the effect of clearance on the vibration characteristic of geared rotor bearing system is studied on the test-bed by this new equipment. The experimental data show that when the speed is not resonance speed, there is the undesirable range of clearance which causes the vibration of system relatively large, there is also the preferable range of clearance which cause the vibration of system relatively small. When clearance increase to certain value, the meshing state of system commonly retains the single side impact, the vibration amplitude of system will retain in certain range, continue to increase clearance will not affect the vibration of system greatly. The rolling bearing of driving geared rotor system is transformed to oil journal bearing on the basis of geared rotor rolling bearing system test-bed, the vibration of system is measured under several speed, the experimental data show that when the rotational speed is low and far from the first order lateral critical speed, the vibration of system is affected by nonlinear mesh force mainly, the results of numerical integration are validated by the experimental results basically.
结果表明:动力学模型能合理地体现齿轮啮合的三种冲击状态;混沌时间序列分析理论能有效地计算出齿轮传动系统高维非线性方程组的最大Lyapunov指数。齿轮副扭转振动系统是最基础的齿轮系统,为了从近似解析解的角度研究齿侧间隙的大小对齿轮副扭转振动系统的幅频响应曲线的影响规律,将多项谐波平衡法与求解非线性方程组的最小二乘解的广义逆法相结合求解近似解析解,研究结果表明:齿侧间隙对系统幅频响应曲线的影响除了与齿侧间隙的大小密切相关之外,还与阻尼、时变啮合刚度谐波项幅值及预紧力紧密相关。当阻尼相对较小时,系统的幅频响应曲线受到齿侧间隙的影响比较明显,在不同的齿侧间隙下,主共振区和超谐共振区都会出现振幅跳跃现象;阻尼比增大到一定值后,随着齿侧间隙逐渐增大,主共振区始终出现振幅跳跃,但在振幅相对较小的超谐共振区振幅跳跃现象变得不明显;当阻尼比增大到较大值时,不同的齿侧间隙下,系统的幅频响应曲线都接近于线性系统的响应曲线。随着啮合刚度一次谐波项幅值的逐渐增大,齿侧间隙比相对较小时,幅频响应曲线在主共振区会体现出“硬化曲线”的特征;当齿轮间隙比增大到一定值时,不论啮合刚度一次谐波项幅值的大小为多少,系统的幅频响应曲线均体现出“软化曲线”的特征。在预紧力比较小的条件下,当齿侧间隙比相对较小时,幅频响应曲线只在局部转速范围内体现出“硬化曲线”;当齿侧间隙比相对较大时,幅频响应曲线体现为“软化曲线”。在预紧力比较大的条件下,齿侧间隙越小,“硬化曲线”的特征越明显;齿侧间隙需要增大到较大值才能使主共振区的幅频响应曲线变成“软化曲线”。
由于齿轮副扭转振动系统的模型假设转轴和轴承是刚性不变形的,与实际的齿轮-转子系统存在一定的差别,为了研究不同齿侧间隙对齿轮-转子系统的动力学特性的影响,采用数值仿真研究了不同齿侧间隙对系统的分岔和混沌的影响,研究结果表明:齿侧间隙对系统的第一阶弯曲临界转速处的振动状态的影响比较大,齿侧间隙相对较小时,系统的第一阶弯曲临界转速处的振动状态相对较好。除了齿侧间隙之外,支承刚度对齿轮-转子系统的分岔和混沌也具有重要的影响,采用数值仿真研究了支承刚度对系统的分岔和混沌的影响,研究结果表明:随着支承刚度的逐渐增大,系统的弯扭耦合临界转速均相应地增大,系统的分岔和混沌区域也相应地发生改变。
在考虑非线性啮合力的基础上,进一步考虑非线性油膜力的影响,建立了齿轮-转子-滑动轴承系统的动力学模型,采用数值仿真研究了系统的动态响应,研究结果表明:随着转速的逐渐升高,非线性啮合力和非线性油膜力分别在不同的转速范围内影响着齿轮-转子-滑动轴承系统的非线性振动特性,当转速相对较低时,系统的振动特性主要受到非线性啮合力的影响,随着转速的逐渐升高,非线性油膜力对系统振动特性的影响逐渐增大;当转速逐渐增大到接近系统的第一阶临界转速(可以是齿轮啮合引起的弯扭耦合临界转速而不一定是转子的纯弯曲临界转速)的二倍时,逐渐出现非线性油膜力引起的“半频涡动”;通过对比线性八参数油膜力和非线性油膜力对啮合力的影响,发现转速逐渐增大到接近系统的第一阶临界转速(可为弯扭耦合临界转速)的二倍时,非线性油膜力逐渐对非线性啮合力起作用,并使啮合力的频谱中出现“半频涡动”频率成份;随着转速的进一步增大,非线性油膜力对非线性啮合力的影响也越来越大,甚至超过不平衡质量对啮合力的影响;然而传统的线性油膜力对非线性啮合力则基本没有影响。
为了验证数值仿真结果的正确性,设计了齿轮-转子-滚动轴承系统振动试验台;提出了一种可以调整齿侧间隙的装置;在试验台上研究了齿侧间隙的大小对齿轮-转子-轴承系统的振动特性的影响规律,试验结果表明:在非共振转速下,存在较差齿侧间隙范围使齿轮系统的振幅相对较大,并存在较好齿侧间隙范围使齿轮系统的振幅相对较小;当齿侧间隙增大到一定值后,系统通常将保持单边冲击状态,系统的振幅将维持在一定的范围内,继续增大齿侧间隙不会再对系统的振动产生大的影响。在齿轮-转子-滚动轴承系统振动试验台的基础上,将主动齿轮-转子的滚动轴承改造成滑动轴承,并测量了若干转速下滑动轴承座的振动,试验结果表明:转速低于一阶弯曲临界转速之前,系统的振动主要受到非线性啮合力的影响,试验的结果基本验证了数值仿真的结果。
Geared rotor system is the most primary origin of the vibration problem of power transmission system, the security and concealment of system are affected by the severe vibration significantly. At present, the vibration mechanism of geared rotor bearing system is still not clear. Aiming at this problem, on the basis of gear dynamic and rotor dynamic, the effect of clearance, time-varying mesh stiffness, static transmission error, flexible shaft, bearing, input and output torque excitation are considered, and the nonlinear dynamic model of geared rotor system is derived. Numerical integration method is used to study the dynamic model, the effects of speed on the dynamic mesh force and vibration characteristic of system are also studied. On the process of judge the vibration is chaos or not, the maximum Lyapunov exponent of high dimensional nonlinear equation is calculated by chaotic time series analysis theory. The results of numerical integration show that the dynamic model can reveal three kind of impact status of gear mesh in reason. The chaotic time series analysis theory can calculate the maximum Lyapunov exponent of high dimensional nonlinear equation effectively.
The gear pair torsional vibration system is the most basical gear system, in order to study the effect of clearance’s size on the gear pair torsional vibration system’s amplitude-frequency response curve from approximate analytical solution method, the multi-term harmonic balance method is connected with the generalized inverse method which used to solve the minimum least-square solutions of nonlinear equation group. The results show that the effect of clearance on the amplitude-frequency response curve of gear pair torsional vibration system relates with the size of clearance, the damping, time-varying mesh stiffness and preload. When the damping is relatively small, the amplitude-frequency response curve of system affected by clearance obviously, the amplitude jump phenomena appear in primary resonance and super harmonic resonance under different clearance. When damping increases to certain value, with the increase of clearance, the amplitude jump phenomena always appear in primary resonance, but it is not obvious in the super harmonic resonance whose amplitude is relative small. When damping increases to large value, the amplitude-frequency response curves of system approach to the response curves of linear system under different clearance. Whit the increase of the first harmonic term of mesh stiffness’s value, when the clearance is relatively small, the amplitude-frequency response curve of system will reflect the character of hardening type curve. When clearance increase to certain value, no matter the first harmonic term of mesh stiffness’s value is small or large, the amplitude-frequency response curve of system just reflects the character of softening type curve. On the condition of preload is relatively small, when clearance is also relatively small, the amplitude-frequency response curve just reflect the character of hardening type curve in local speed range. When clearance is relatively large, the amplitude-frequency response curve reflects the character of softening type curve. On the condition of preload is relatively large, when the learance is small, with the preload becomes larger and larger, and then the character of hardening type curve also becomes more obvious, the clearance need to increase to larger value to make the amplitude-frequency response curve becomes softening type curve in the primary resonance.
The model of gear pair torsional vibration system assumes rotor and bearing are rigid, it has certain difference comparing with practical geared rotor system, in order to study the effect of clearance on geared rotor system, numerical integration method is used to study the effect of clearance on the bifurcation and chaos of system, the results show that the first lateral critical speed of system is affected by clearance greatly, when the clearance is relatively small, the vibration state is relatively preferable. Bearing stiffness also has important effect on the bifurcation and chaos of system, except for clearance. The effect of bearing stiffness on the bifurcation and chaos of system is studied by numerical integration. The results show that the lateral-torsion critical speed will increase with the increase of bearing stiffness, and the region of bifurcation and chaos also change accordingly.
On the basis of considering nonlinear dynamic force, the effect of nonlinear oil film force is considered further, dynamic model of geared rotor-oil journal bearing system is proposed. The nonlinear dynamic responses of system are investigated by numerical integration method. The results show that with the increase of rotational speed, the nonlinear vibration characteristic of system is affected by nonlinear mesh force and nonlinear oil film force respectively in different speed range. The vibration characteristic of system is mainly affected by nonlinear mesh force when speed is relatively low. With the increase of speed, the influence of nonlinear oil film force on the vibration also increases gradually, and the sub-synchronous forward precession phenomena appear when the speed approaches the twice of first order critical speed (it can be lateral-torsional critical speed caused by gear mesh, it is not need to be pure lateral critical speed of rotor). By comparing the effect of linear eight parameter oil film fore and nonlinear oil film fore on mesh force, the results show that when rotational speed increases to twice of the first order critical speed of system (it can be lateral-torsional critical speed), the nonlinear mesh force is affected by nonlinear oil film force gradually, the sub-synchronous forward precession frequency appears in spectrum plot of mesh force. With the increase of speed, the effect of nonlinear oil film fore on nonlinear mesh force also increase, it may exceed the effect of unbalance on mesh force. However, the conventional linear oil film force does not affect the mesh force.
In order to validate the correctness of numerical integration’s results, a geared rotor bearing system vibration test-bed is designed, and a kind of equipment which is used to adjust clearance is innovated independently, the effect of clearance on the vibration characteristic of geared rotor bearing system is studied on the test-bed by this new equipment. The experimental data show that when the speed is not resonance speed, there is the undesirable range of clearance which causes the vibration of system relatively large, there is also the preferable range of clearance which cause the vibration of system relatively small. When clearance increase to certain value, the meshing state of system commonly retains the single side impact, the vibration amplitude of system will retain in certain range, continue to increase clearance will not affect the vibration of system greatly. The rolling bearing of driving geared rotor system is transformed to oil journal bearing on the basis of geared rotor rolling bearing system test-bed, the vibration of system is measured under several speed, the experimental data show that when the rotational speed is low and far from the first order lateral critical speed, the vibration of system is affected by nonlinear mesh force mainly, the results of numerical integration are validated by the experimental results basically.
引文
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