Research On Separation Method For Identify The First Derivative Of Structural Stiffness Function

Damage detection, fault diagnosis of the engineering can prevent catastrophicfunctional failure occurs effectively, the research has important theoretical andpractical significance. Identifying the physical parameters for fault diagnosis is themost common and effective method. This paper analyzes why the diagnosis is hardto detection in the existence of error, that is changes in the physical parameters ofthe fault is small compared to the value is before the fault. The changes of parameteris small before and after the failure, however, the first derivative of the parametershas changed significantly. Based on the above facts, by identifying the shape of thefirst derivative of structural stiffness function, we can accurately identify the faultcaused by cracks and fatigue, etc. In this paper, we discussed the identification ofthe first derivative of the problem of the structural stiffness function.In this paper we firstly performed Fourier transform on the wave equation ofaxial vibration differential equation of the bar, by using the separation method weestablished the first derivative differential inversion model for identification of thestructural stiffness function, and transformed the parameter identification probleminto the first kind operator equation. Parameter identification is a mathematicalphysics inverse problem, inverse problem is generally nonlinear and ill-posed. Thispaper discusses how to solve the "ill-posed" operator equation of the first kind andanalyses of the theory of regularization and smoothing for solving ill-posedproblems. In this paper, for nonlinear problems we use an iterative method, in eachiteration, the nonlinear problem becomes linear problems. It is difficult to give agood initial value the first derivative of structure stiffness function in each iteration,so this paper uses the homotopy regularization method for it. The homotopyregularization not only can achieve the minimum of objective functional (theoriginal purpose of regularization method), and avoid falling into local minimum,but also can accelerate the convergence speed of the iteration method. Finally, thepaper introduced morbid equations with the Hilbert matrices and Pascal matrix ofthe coefficient matrix, examples containing bar structures of a crack and two cracks,demonstrates the feasibility of regularization, smoothing and the homotopyregularization method for solving operator equations of the first kind and theeffectiveness of the method proposed for identify the first derivative of the structuralstiffness function to fault diagnosis.In summary, this paper presents a creative way to identify the first derivative ofthe structural stiffness function for fault diagnosis for the first time. For the ill-posedproblems encountered in identifying problems, we analysis the rational choice of regularization parameters and frequencies, zero smooth factor, the homotopyregularization method for improving speed and accuracy for solving the problem. Inthe theoretical analysis and numerical simulation, it fully demonstrates validity andsuperiority of the method proposed.