Abstract: To address the time-domain inversion problem of vibration responses of a simply supported rectangular plate, this paper first expresses the time-domain dynamic load at each time instant as a series of two-dimensional Legendre orthogonal polynomials, thereby transforming the complex load field into an identification problem of a finite number of coefficients. The vibration response of the thin plate is then decomposed into a linear superposition of modal displacements, and a time-domain transfer relationship between modal forces and modal displacements is established based on the modal superposition principle, thereby achieving a coupled mapping between the displacement response at any spatial point and the dynamic load at each time instant. On this basis, an iterative regularization method based on the least squares QR (LSQR) algorithm is adopted to solve the inversion problem, effectively overcoming the numerical instability caused by the ill-conditioned nature of the system. In this paper, spherical and planar distributed dynamic loads are separately inverted. The spatial and temporal distribution characteristics of the obtained dynamic loads are highly consistent with those of the original input loads, verifying the effectiveness of the proposed method. By introducing the signal-to-noise ratio (SNR) parameter, the spherical dynamic load is inverted under Gaussian white noise with an SNR of 30 dB, and the method is found to have good noise resistance and identification accuracy. In summary, the combination of two-dimensional Legendre orthogonal polynomials and the iterative regularization method has high feasibility and engineering application potential in time-domain dynamic load identification.

Key words: two-dimensional Legendre orthogonal polynomials, iterative regularization, time-domain inversion, Gaussian white noise

摘要： 针对简支矩形板振动响应的时域反演问题,该文首先将每一时刻的时域动载荷表示为二维勒让德正交多项式的级数形式,从而将复杂的载荷场转化为有限个系数的识别问题;然后将薄板的振动响应分解为模态位移的线性叠加,并依据模态叠加原理建立模态力与模态位移之间的时域传递关系,进而实现空间任意点的位移响应与各时刻动载荷之间的耦合映射;在此基础上,采用基于最小二乘QR（least squares QR,LSQR）算法的迭代正则化方法对反演问题进行求解,有效克服了系统病态性带来的数值不稳定问题。文章通过对球面分布和平面分布动载荷分别进行反演,所得动载荷在空间和时间上的分布特征与原始输入载荷高度一致,从而验证了方法的有效性;通过引入信噪比参数,并在信噪比为30 dB的高斯白噪声条件下对球面动载荷进行反演计算,验证了该方法具有良好的抗噪能力和识别精度。综上所述,二维勒让德正交多项式结合迭代正则化方法在时域动态载荷识别中具有较高的可行性与工程应用潜力。

关键词: 二维勒让德正交多项式, 迭代正则化, 时域反演, 高斯白噪声