Identifying time-dependent damping and stiffness functions by a simple and yet accurate method

The inverse vibration problem is a mathematical process to determine unknown mechanical parameters from measured vibration data. In this study the data of displacement are chosen in order to identify a time-dependent function of damping or stiffness. However, when both functions are to be identified we require both the data of displacement and velocity. This is the first time that a closed-form estimation method for the inverse vibration problems of estimating time-dependent parameters has been constructed. We are able to transform the inverse vibration problem into an identification problem governed by a parabolic-type partial differential equation (PDE). Then, a one-step group-preserving scheme (GPS) for the semi-discretization of PDE is established, which can be used to derive a closed-form solution for estimating parameters. The new Lie-group estimation method has three further advantages: it does not require any prior information on the functional forms of unknown functions; no initial guesses are required; and no iterations are required. Numerical examples were examined to show that the present approach is highly accurate and efficient even for identifying discontinuous and oscillatory parameters. Against the noise is good when only one function is estimated; however, the present approach is slightly weak against the noise when both functions are identified.