The method of constructing any scale wavelet finite element (WFE) based on the one-dimensional or two-dimensional Daubechies scaling functions was presented, and the corresponding WFE adaptive lifting algorithm was given. In order to obtain the nested increasing approximate subspaces of multiscale finite element, the Daubechies scaling functions with the properties of multi-resolution analysis were employed as the finite element interpolating functions. Thus, the WFE could adaptively mesh the singularity domain caused by local cracks, which resulted in better approximate solutions than the traditional finite element methods. The calculations of natural frequencies of cracked beam were used to check the accuracy of given methods. In addition, the results of cracked cantilever beam and engineering application were satisfied. So, the current methods can provide effective tools in the numerical modeling of the fault prognosis of incipient crack.

This paper proposes a fault diagnosis method for plant machinery in an unsteady operating condition using instantaneous power spectrum (IPS) and genetic programming (GP). IPS is used to extract feature frequencies of each machine state from measured vibration signals for distinguishing faults by relative crossing information. Excellent symptom parameters for detecting faults are automatically generated by the GP. The excellent symptom parameters generated by GP can sensitively reflect the characteristics of signals for precise diagnosis. The method proposed in this paper is verified by applying it to the fault diagnosis of a rolling bearing.

An important property of wavelet multiresolution analysis is the capability to represent functions in a dynamic multiscale manner, so the solution in the wavelet domain enables a hierarchical approximation to the exact solution. The typical problem that arises when using Daubechies wavelets in numerical analysis, especially in finite element analysis, is how to calculate the connection coefficients, an integral of products of wavelet scaling functions or derivative operators associated with these. The method to calculate multiscale connection coefficients for stiffness matrices and load vectors is presented for the first time. And the algorithm of multiscale lifting computation is developed. The numerical examples are given to verify the effectiveness of such a method.

In this paper we present an experimental investigation of the identification of crack location and size. By providing the first three natural frequencies through vibration measurements, curves of crack equivalent stiffness versus crack location are plotted, and the intersection of the three curves predicts the crack location and size. In the experiments, the cracked specimens were made using a wire-cut electrical discharge machine, and the cantilever beams were excited next to the free end by means of an impulse force hammer. In order to obtain the accurate natural frequencies from the transient signal measured, the method of zoom fast Fourier transform is adopted to improve frequency resolution. From experimental results, it is observed that the identification errors of crack location and size are less than 2% and 4%, respectively. The effectiveness of crack identification through vibration measurements is verified.

小波变换是非平稳信号处理的有力工具，小波性质取决于它的基函数。在模态分析、机械监测诊断等领域,冲击响应信号十分普遍。一种单边衰减复指数型的Laplace小波基函数具有分析冲击响应信号的优势，采用Laplace小波相关滤波方法可提取振动信号中的冲击响应分量，成功地诊断出内燃机进气阀的磨损故障和识别出大型水轮机轴系的一阶固有频率。