基于小波分析的线性时变结构参数识别方法研究
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摘要
结构的动力学参数决定着结构的动力学特性，其在结构的有限元建模、模型修正、灵敏度分析、振动控制、损伤识别和结构健康监测等方面都起着重要的作用。所以结构参数识别研究历来很受重视，但以往的动力学参数识别都只局限于针对时不变结构，时变结构的动力学问题一直是学术界研究的难点，尤其是线性时变结构的参数识别问题，理论研究一直没有太大的突破。可是在实际工程中，大型的柔性航天结构、高速（超高速）飞行器以及高速列车等动力学问题又亟待解决，为此本文以小波分析为研究基础，主要进行了线性时变结构的参数识别方法研究。
本文的主要研究内容和研究成果包括：
（1）推导了函数积分运算的连续小波变换计算方法，基于此法，在小波域内，仅利用线性时变结构的加速度响应信号，就可以计算出速度响应信号和位移响应信号的连续小波变换值。并由此把二阶振动微分方程转变为线性代数方程组，再基于短时时不变的假设，构造最小二乘问题，识别出每个时刻点时变结构的质量，刚度和阻尼系数。本文所提出的这一新识别方法，仅从时变结构的加速度响应信号出发就可以进行时变参数辨识，有较强的工程实用性。
（2）研究了小波尺度函数积分问题，提出了一种将小波分析和状态空间技术相结合的时变结构参数识别方法：通过引入状态向量，把时变结构的二阶振动微分方程改写为一阶状态空间方程，对状态方程的状态向量进行Daubechies小波尺度函数投影计算，借助尺度函数的正交性，把一阶状态方程解耦为线性代数方程组，基于短时时不变假设，求解方程组，识别出每个时刻点的状态转移矩阵，再将状态转移矩阵与时变结构的物理空间模型作对照，假设时变结构的质量系数为已知常数，或是其质量的时变特性已被规律性掌握，便可以得到每个时刻的系统刚度矩阵和阻尼矩阵。这种小波状态空间识别方法是小波-伽辽金法在时变参数识别应用中的重要补充，在识别过程中，本方法可以成功避免计算Daubechies小波二阶连接系数的数学难点，提高了参数的识别效率。
（3）时变自回归模型的自回归系数是随时间变化的，可以用一组正交基进行展开，本文将区间B样条小波引入到时变自回归模型中，其时变的自回归系数用区间B样条小波基函数进行展开，仅仅利用时变结构的加速度响应信号，就可以准确、快速地识别出时变结构的瞬时频率，文中的这一识别技术对于工程实际中时变结构的瞬时频率快速识别是一个非常好的选择。
（4）文中对上述三种时变参数识别方法，基于多种动力学模型，作了大量、细致的数值仿真研究，更设计完成了具有质量时变特性的悬臂梁振动测试实验。实验中利用采集的时变结构加速度响应信号，快速准确地识别了悬臂梁时变结构的前三阶瞬时频率。实验也进一步验证了区间B样条小波基函数时变自回归识别方法的可行性、有效性以及抗噪声能力。
Structural dynamic parameters are the main parameters that are able to determine thedynamic characteristics of the mechanical structures, which could offer important reference tostructure finite element modeling, dynamic model updating, sensitivity analysis, vibration controland structural health monitoring, etc. The identification of dynamic parameters is always of primeimportance in vibration analysis. However by far studies on dynamic parameters identificationhave been mainly limited to linear time invariant structures and it has been being very difficult tomake progress in research on dynamic of time varying structure, because theoretically it has notgeneral solution up to now, especially the time varying parameters identification issue. But thereare a number of thus problems in engineering to solve urgently, for example, the stretchingdynamics of large spacecraft flexible structure, varying mass problem of supersonic rocket, bridgevibration caused by the high speed train, etc. Therefore the parameters identification methodsissue for time varying structures based on wavelet analysis are studied in this dissertation.
The main contents of this dissertation are as follows:
（1） in chapter two, the continuous wavelet transform based algorithm for functionalintegration is formulated and the continuous wavelet transform values of displacement andvelocity responses signals are calculated subsequently, only using the measurements ofacceleration responses signals. As a result, vibration differential equations of motion are betransformed into linear algebraic equations with wavelet-based coefficients. On the assumptionthat the time varying structural dynamic parameters are constant in a short period, the structurephysical parameters (mass, stiffness and damping coefficients) can be extracted by solving a leastsquare problem in every moment. Since the structural acceleration response data is just used toidentify the dynamic parameters, the proposed identification method has a strong practicability.
（2） in chapter three, a wavelet-based state space time varying parameters identificationmethod is proposed based on the research results of wavelet scaling functional integrationproblem. For an arbitrary linear time varying system, the second order vibration differentialequations are first rewritten as the first order state equations using the state space theory. The statevectors are projected by using the Daubechies wavelet scaling functions and the first order statespace equations are transformed into simple linear equations based on the orthogonality of thewavelet scaling functions. This allows the time varying equivalent state space system matrices ateach moment to be identified directly via solving the linear equations on the assumption that thetime varying structural dynamic parameters are constant in a short period. The stiffness and damping matrices are determined by comparing the identified equivalent system matrices with thephysical system matrices on the premise of the system time varying mass characteristics areknown in advance. The system modal parameters are extracted though eigenvalue decompositionof the state space system matrices. The proposed algorithm is a development of Wavelet-Gelerkinmethod for time varying parameters identification as there is no need to calculate the second orderconnection coefficients of Daubechies wavelet during the identification procedure. Thisimprovement can supply a higher calculation speed.
（3） in chapter four, the time varying auto regressive (T-AR) model has time dependent autoregressive coefficients which can be expanded by a set of orthogonal basis functions. Thisdissertation utilizes a time varying auto regressive model using B-spline wavelet on the interval asbasis function. The time varying auto regressive model’s coefficients are expanded with B-splinewavelet on the interval basis function first. Subsequently the system acceleration response signalsare utilized for structural instantaneous frequencies identification. Since the parameters are beextracted quickly and accurately, the proposed identification method is priority for the engineers.
（4） in this dissertation, three proposed time varying parameters identification algorithmsabove are investigated with a number of different simulation models. The performance of theB-spline wavelet on the interval based T-AR method is illustrated using a cantilever beam withtime varying mass characteristics. The first three orders instantaneous frequencies of thecantilever beam are quickly and accurately identified only by using the measurement accelerationresponse signals. The test results demonstrate the good performance (the feasibility, effectiveness,and the anti-noise ability) of the T-AR identification method.
本文的主要研究内容和研究成果包括：
（1）推导了函数积分运算的连续小波变换计算方法，基于此法，在小波域内，仅利用线性时变结构的加速度响应信号，就可以计算出速度响应信号和位移响应信号的连续小波变换值。并由此把二阶振动微分方程转变为线性代数方程组，再基于短时时不变的假设，构造最小二乘问题，识别出每个时刻点时变结构的质量，刚度和阻尼系数。本文所提出的这一新识别方法，仅从时变结构的加速度响应信号出发就可以进行时变参数辨识，有较强的工程实用性。
（2）研究了小波尺度函数积分问题，提出了一种将小波分析和状态空间技术相结合的时变结构参数识别方法：通过引入状态向量，把时变结构的二阶振动微分方程改写为一阶状态空间方程，对状态方程的状态向量进行Daubechies小波尺度函数投影计算，借助尺度函数的正交性，把一阶状态方程解耦为线性代数方程组，基于短时时不变假设，求解方程组，识别出每个时刻点的状态转移矩阵，再将状态转移矩阵与时变结构的物理空间模型作对照，假设时变结构的质量系数为已知常数，或是其质量的时变特性已被规律性掌握，便可以得到每个时刻的系统刚度矩阵和阻尼矩阵。这种小波状态空间识别方法是小波-伽辽金法在时变参数识别应用中的重要补充，在识别过程中，本方法可以成功避免计算Daubechies小波二阶连接系数的数学难点，提高了参数的识别效率。
（3）时变自回归模型的自回归系数是随时间变化的，可以用一组正交基进行展开，本文将区间B样条小波引入到时变自回归模型中，其时变的自回归系数用区间B样条小波基函数进行展开，仅仅利用时变结构的加速度响应信号，就可以准确、快速地识别出时变结构的瞬时频率，文中的这一识别技术对于工程实际中时变结构的瞬时频率快速识别是一个非常好的选择。
（4）文中对上述三种时变参数识别方法，基于多种动力学模型，作了大量、细致的数值仿真研究，更设计完成了具有质量时变特性的悬臂梁振动测试实验。实验中利用采集的时变结构加速度响应信号，快速准确地识别了悬臂梁时变结构的前三阶瞬时频率。实验也进一步验证了区间B样条小波基函数时变自回归识别方法的可行性、有效性以及抗噪声能力。
Structural dynamic parameters are the main parameters that are able to determine thedynamic characteristics of the mechanical structures, which could offer important reference tostructure finite element modeling, dynamic model updating, sensitivity analysis, vibration controland structural health monitoring, etc. The identification of dynamic parameters is always of primeimportance in vibration analysis. However by far studies on dynamic parameters identificationhave been mainly limited to linear time invariant structures and it has been being very difficult tomake progress in research on dynamic of time varying structure, because theoretically it has notgeneral solution up to now, especially the time varying parameters identification issue. But thereare a number of thus problems in engineering to solve urgently, for example, the stretchingdynamics of large spacecraft flexible structure, varying mass problem of supersonic rocket, bridgevibration caused by the high speed train, etc. Therefore the parameters identification methodsissue for time varying structures based on wavelet analysis are studied in this dissertation.
The main contents of this dissertation are as follows:
（1） in chapter two, the continuous wavelet transform based algorithm for functionalintegration is formulated and the continuous wavelet transform values of displacement andvelocity responses signals are calculated subsequently, only using the measurements ofacceleration responses signals. As a result, vibration differential equations of motion are betransformed into linear algebraic equations with wavelet-based coefficients. On the assumptionthat the time varying structural dynamic parameters are constant in a short period, the structurephysical parameters (mass, stiffness and damping coefficients) can be extracted by solving a leastsquare problem in every moment. Since the structural acceleration response data is just used toidentify the dynamic parameters, the proposed identification method has a strong practicability.
（2） in chapter three, a wavelet-based state space time varying parameters identificationmethod is proposed based on the research results of wavelet scaling functional integrationproblem. For an arbitrary linear time varying system, the second order vibration differentialequations are first rewritten as the first order state equations using the state space theory. The statevectors are projected by using the Daubechies wavelet scaling functions and the first order statespace equations are transformed into simple linear equations based on the orthogonality of thewavelet scaling functions. This allows the time varying equivalent state space system matrices ateach moment to be identified directly via solving the linear equations on the assumption that thetime varying structural dynamic parameters are constant in a short period. The stiffness and damping matrices are determined by comparing the identified equivalent system matrices with thephysical system matrices on the premise of the system time varying mass characteristics areknown in advance. The system modal parameters are extracted though eigenvalue decompositionof the state space system matrices. The proposed algorithm is a development of Wavelet-Gelerkinmethod for time varying parameters identification as there is no need to calculate the second orderconnection coefficients of Daubechies wavelet during the identification procedure. Thisimprovement can supply a higher calculation speed.
（3） in chapter four, the time varying auto regressive (T-AR) model has time dependent autoregressive coefficients which can be expanded by a set of orthogonal basis functions. Thisdissertation utilizes a time varying auto regressive model using B-spline wavelet on the interval asbasis function. The time varying auto regressive model’s coefficients are expanded with B-splinewavelet on the interval basis function first. Subsequently the system acceleration response signalsare utilized for structural instantaneous frequencies identification. Since the parameters are beextracted quickly and accurately, the proposed identification method is priority for the engineers.
（4） in this dissertation, three proposed time varying parameters identification algorithmsabove are investigated with a number of different simulation models. The performance of theB-spline wavelet on the interval based T-AR method is illustrated using a cantilever beam withtime varying mass characteristics. The first three orders instantaneous frequencies of thecantilever beam are quickly and accurately identified only by using the measurement accelerationresponse signals. The test results demonstrate the good performance (the feasibility, effectiveness,and the anti-noise ability) of the T-AR identification method.
引文
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