扭摆法转动惯量测量中的非线性问题研究
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摘要
转动惯量是表征物体在转动过程中惯性大小的物理量，在航天航空、汽车、机械、仪表等工业领域经常需要测量物体的转动惯量，因此，转动惯量的准确测量具有重要的实际意义。现有的转动惯量测量技术主要是针对线性测量系统。但是，随着测量对象的大型化、外形复杂化，在转动惯量测量过程中会出现非线性因素；随着合金材料，复合材料的大量使用，材料的非线性特性也会使转动惯量测量过程中出现非线性因素。这些非线性因素则会影响转动惯量的测量精度。
为了提高转动惯量的测量精度，本文考虑了非线性因素对扭摆法转动惯量测量的影响。建立了基于非线性动力学系统的时不变、时变转动惯量测量模型，分析各种非线性因素对转动惯量测量的影响。提出了适用于非线性系统的转动惯量计算方法。搭建基于扭摆法的转动惯量测量实验平台，验证了转动惯量测量模型和计算方法的有效性。本文主要的研究内容如下：
考虑了非线性因素对转动惯量测量的影响，利用分析动力学原理建立了基于非线性动力学系统的时不变和时变转动惯量测量模型。为了定量分析转动惯量测量模型，利用非线性微分方程的求解方法推导出测量模型的近似解析解，并利用数值解进行验证。利用数值仿真的方法分析了各种非线性因素对转动惯量测量的影响，从而为转动惯量的准确测量提供理论基础。
利用基于Hilbert变换的时频分析方法确定了物体转动惯量与瞬时无阻尼固有频率之间的函数关系，从而提出了基于Hilbert变换的转动惯量计算方法。首先，对物体角位移信号进行Hilbert变换并构造解析信号；然后，根据解析信号求出扭摆运动的瞬时无阻尼固有频率；最后，根据瞬时无阻尼固有频率计算物体的转动惯量。该方法适合非线性系统条件下的时不变、时变转动惯量的计算。针对不同的转动惯量测量模型，通过数值仿真的方法验证了基于Hilbert变换的转动惯量计算方法的正确性。
为了保证转动惯量的测量精度，分析了Hilbert变换中的端点效应对转动惯量测量的影响，并利用信号延拓的方法对端点效应进行抑制。为了减小噪声对转动惯量测量的影响，设计了基于Kaiser窗的有限冲击响应(FIR)滤波器。在进行转动惯量计算之前，需要利用滤波器对实验信号进行滤波。
为了验证转动惯量测量模型和计算方法的有效性，建立了基于扭摆法的转动惯量测量系统，分别设计了时不变转动惯量实验样件和时变转动惯量实验样件。该测量系统通过气浮转台实现实验样件的扭摆运动，利用光栅传感器测量扭摆运动的角位移信号。根据转动惯量测量非线性模型，并利用基于Hilbert变换的转动惯量计算方法计算实验样件的转动惯量。实验结果表明：基于非线性模型的转动惯量测量结果优于基于线性模型的转动惯量测量结果，从而验证了非线性测量模型的有效性；本文转动惯量计算方法可以准确计算时变和时不变转动惯量，从而验证了该计算方法的有效性。
The moment of inertia (MOI) expresses the inertia of the rigid body duringrotation. The MOI measurement has important practical significance because theMOI is measured in the industrial fields such as aircraft, automobile, machinery,instrumentation. The existing MOI measurement technique is suitable for linearmeasurement systems. Due to the complicated shape and large size of measurementobjects, there exist nonlinear factors in the MOI measurement. With the wideapplication of composite material and alloy material, nonlinear factors appear in theMOI measurement because of the nonlinear characteristic of the material. Thenonlinear factors affect the accuracy of MOI measurement.
In order to improve the accuracy of MOI measurement, the influence ofnonlinear factors on the MOI measurement based on torsion pendulum is considered.The time-varying and time invariant MOI measurement models based on thenonlinear dynamic systems are described. The influence of the nonlinear factors onthe MOI measurement is analyzed. The MOI calculation method for the nonlineardynamic system is presented. The MOI measurement system based on torsionpendulum is established. The validity of MOI measurement model and calculationmethod is verified by the experimental results. The main contents are as follows:
The influence of nonlinear factors on the MOI measurement is considered. Thetime-varying and time invariant MOI measurement models based on nonlineardynamic systems are established by the principal of analytical dynamics. For thequantitative analysis of the MOI measurement model, the approximate analyticalsolution is computed by the solving method of nonlinear differential equation. Thenumerical solution is used to validate the analytical solution. The influence of thenonlinear factors on the MOI measurement is analyzed by the numerical simulationmethod, which provides theoretical basis for the MOI accurate measurement.
The functional relationship between MOI and instantaneous undamped naturalfrequency is determined by the time-frequency analysis method based on Hilberttransformation. The MOI calculation method based on Hilbert transform ispresented. The analytical signal is formed by the Hilbert transform of the angulardisplacement signal. The MOI is calculated by the instantaneous undamped natural frequency which is computed by the analytical signal. This method is suitable forthe calculation of the time-varying and time invariant MOI under the condition ofnonlinear system. For different MOI measurement models，the correctness of theMOI calculation method based on Hilbert transform is verified by the numericalsimulation method.
In order to ensure the accuracy of MOI measurement, the influence of the endeffect in Hilbert transform on the MOI measurement is analyzed. The signalextension is used to inhibit the end effect. In order to reduce the effect of the noiseon the MOI measurement, the finite impact response (FIR) filter based on Kaiserwindow is designed. It is used to filter the experimental signal before the MOIcalculation.
In order to verify the validity of MOI measurement model and calculationmethod, the MOI measurement system based on torsion pendulum is established.The experimental samples with time invariant and time-varying MOI are designed.The experimental samples can make torsion pendulum movement by the air-hoveredturntable. The angular displacement signal is measured by the gating sensor.According to the nonlinear model of MOI measurement, the MOI of theexperimental samples is calculated by the calculation method based on the Hilberttransform. The experimental results show that the MOI measurement results basedon the nonlinear model are better than the results based on the linear model. So theexperimental results verify the validity of the nonlinear measurement model. Theexperimental results show that the accurate calculation of time invariant and thetime-varying MOI is achieved by the method presented by this paper. So theexperimental results verify the validity of the MOI calculation method presented bythis paper.
为了提高转动惯量的测量精度，本文考虑了非线性因素对扭摆法转动惯量测量的影响。建立了基于非线性动力学系统的时不变、时变转动惯量测量模型，分析各种非线性因素对转动惯量测量的影响。提出了适用于非线性系统的转动惯量计算方法。搭建基于扭摆法的转动惯量测量实验平台，验证了转动惯量测量模型和计算方法的有效性。本文主要的研究内容如下：
考虑了非线性因素对转动惯量测量的影响，利用分析动力学原理建立了基于非线性动力学系统的时不变和时变转动惯量测量模型。为了定量分析转动惯量测量模型，利用非线性微分方程的求解方法推导出测量模型的近似解析解，并利用数值解进行验证。利用数值仿真的方法分析了各种非线性因素对转动惯量测量的影响，从而为转动惯量的准确测量提供理论基础。
利用基于Hilbert变换的时频分析方法确定了物体转动惯量与瞬时无阻尼固有频率之间的函数关系，从而提出了基于Hilbert变换的转动惯量计算方法。首先，对物体角位移信号进行Hilbert变换并构造解析信号；然后，根据解析信号求出扭摆运动的瞬时无阻尼固有频率；最后，根据瞬时无阻尼固有频率计算物体的转动惯量。该方法适合非线性系统条件下的时不变、时变转动惯量的计算。针对不同的转动惯量测量模型，通过数值仿真的方法验证了基于Hilbert变换的转动惯量计算方法的正确性。
为了保证转动惯量的测量精度，分析了Hilbert变换中的端点效应对转动惯量测量的影响，并利用信号延拓的方法对端点效应进行抑制。为了减小噪声对转动惯量测量的影响，设计了基于Kaiser窗的有限冲击响应(FIR)滤波器。在进行转动惯量计算之前，需要利用滤波器对实验信号进行滤波。
为了验证转动惯量测量模型和计算方法的有效性，建立了基于扭摆法的转动惯量测量系统，分别设计了时不变转动惯量实验样件和时变转动惯量实验样件。该测量系统通过气浮转台实现实验样件的扭摆运动，利用光栅传感器测量扭摆运动的角位移信号。根据转动惯量测量非线性模型，并利用基于Hilbert变换的转动惯量计算方法计算实验样件的转动惯量。实验结果表明：基于非线性模型的转动惯量测量结果优于基于线性模型的转动惯量测量结果，从而验证了非线性测量模型的有效性；本文转动惯量计算方法可以准确计算时变和时不变转动惯量，从而验证了该计算方法的有效性。
The moment of inertia (MOI) expresses the inertia of the rigid body duringrotation. The MOI measurement has important practical significance because theMOI is measured in the industrial fields such as aircraft, automobile, machinery,instrumentation. The existing MOI measurement technique is suitable for linearmeasurement systems. Due to the complicated shape and large size of measurementobjects, there exist nonlinear factors in the MOI measurement. With the wideapplication of composite material and alloy material, nonlinear factors appear in theMOI measurement because of the nonlinear characteristic of the material. Thenonlinear factors affect the accuracy of MOI measurement.
In order to improve the accuracy of MOI measurement, the influence ofnonlinear factors on the MOI measurement based on torsion pendulum is considered.The time-varying and time invariant MOI measurement models based on thenonlinear dynamic systems are described. The influence of the nonlinear factors onthe MOI measurement is analyzed. The MOI calculation method for the nonlineardynamic system is presented. The MOI measurement system based on torsionpendulum is established. The validity of MOI measurement model and calculationmethod is verified by the experimental results. The main contents are as follows:
The influence of nonlinear factors on the MOI measurement is considered. Thetime-varying and time invariant MOI measurement models based on nonlineardynamic systems are established by the principal of analytical dynamics. For thequantitative analysis of the MOI measurement model, the approximate analyticalsolution is computed by the solving method of nonlinear differential equation. Thenumerical solution is used to validate the analytical solution. The influence of thenonlinear factors on the MOI measurement is analyzed by the numerical simulationmethod, which provides theoretical basis for the MOI accurate measurement.
The functional relationship between MOI and instantaneous undamped naturalfrequency is determined by the time-frequency analysis method based on Hilberttransformation. The MOI calculation method based on Hilbert transform ispresented. The analytical signal is formed by the Hilbert transform of the angulardisplacement signal. The MOI is calculated by the instantaneous undamped natural frequency which is computed by the analytical signal. This method is suitable forthe calculation of the time-varying and time invariant MOI under the condition ofnonlinear system. For different MOI measurement models，the correctness of theMOI calculation method based on Hilbert transform is verified by the numericalsimulation method.
In order to ensure the accuracy of MOI measurement, the influence of the endeffect in Hilbert transform on the MOI measurement is analyzed. The signalextension is used to inhibit the end effect. In order to reduce the effect of the noiseon the MOI measurement, the finite impact response (FIR) filter based on Kaiserwindow is designed. It is used to filter the experimental signal before the MOIcalculation.
In order to verify the validity of MOI measurement model and calculationmethod, the MOI measurement system based on torsion pendulum is established.The experimental samples with time invariant and time-varying MOI are designed.The experimental samples can make torsion pendulum movement by the air-hoveredturntable. The angular displacement signal is measured by the gating sensor.According to the nonlinear model of MOI measurement, the MOI of theexperimental samples is calculated by the calculation method based on the Hilberttransform. The experimental results show that the MOI measurement results basedon the nonlinear model are better than the results based on the linear model. So theexperimental results verify the validity of the nonlinear measurement model. Theexperimental results show that the accurate calculation of time invariant and thetime-varying MOI is achieved by the method presented by this paper. So theexperimental results verify the validity of the MOI calculation method presented bythis paper.
引文
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