TS-87
A Generalized Inverse Nonlinear Force Vibration Problem for Simultaneously Estimating the Time- Dependent External Forces
Cheng-Hung HUANG*
ABSTRACT
The Conjugate Gradient Method (CGM) is used to a generalized inverse nonlinear force vibration problem, (i.e. system parameters are function of time), in estimating the unknown time-dependent external forces simultaneously for a multiple-degree-of-freedom damped system by using the measured displacements. It is assumed that no prior information is available on the functional form of the unknown external forces in the present study, thus, it is classified as the function estimation in inverse calculation. The accuracy of the inverse analysis for a two-degree-of-freedom problem is examined by using the simulated exact and inexact displacement measurements in the numerical experiments. Results show that an excellent estimation on the external forces can be obtained with any arbitrary initial guesses within a couple of second's CPU time at Pentium III-500 MHz PC.
Key Words: Nonlinear Vibration, Inverse Problem, Force Estimation
1. INTRODUCTION
The direct solutions for a nonlinear damped force vibration problem in a multiple-degree-of-freedom system are concerned with the determination of the system displacement, velocity and acceleration at time t when the initial conditions, external forces and time-dependent system parameters are specified. In contrast, the inverse vibration solution for a damped system that we are going to discuss here involves the determination of the time-dependent external forces simultaneously from the knowledge of the displacement measurements at different time t.
The techniques of inverse problems were applied in many different area of engineering research. Many difficult but practical inverse heat transfer problems in thermal sciences were solved by using a very powerful algorithm, i.e. the Conjugate Gradient Method (CGM). For instant, Huang and Chen [1] used boundary element method and conjugate gradient method to estimate the growth of boundary thickness of a multiple region domain. Huang et al, [2] used the CGM to estimate the contact conductance for the plat-finned tube heat exchangers . Huang and Hsiung [3] used same technique in estimating the optimal shape of cooling passages in turbine blades. Moreover it has also been used in engineering fracture mechanics. For example, Huang and Shih [4] used CGM to estimate the Interfacial Cracks in Bimaterials, etc.
For the inverse vibration problems, the textbook by Gladwell [5] contains a general presentation of the inverse problem for undamped vibrating system, Bateman et al. [6] presented two force reconstruction techniques, i.e., the sum of weighted acceleration and the deconvolution techniques to evaluate the impact test. Michaels and Pao [7] presented an iterative method of deconvolution, which determined the inverse source problem for an oblique force on an elastic plate. Ma et al. [8] used the Kalman filter with a recursive estimator to determine the impulsive loads in a single-degree-of-freedom (SDOF) as well as for a multiple-degree-of-freedom (MDOF) lumped-mass systems.
In all the above references, the system parameters are all assumed constants. The discussions of the inverse nonlinear force vibration problems (i.e. the system parameters are function of time) in estimating the external forces simultaneously using the conjugate gradient method have never seen, to the author's best knowledge, in the literature.
The CGM is also called an iterative regularization method, which means the regularization procedure is performed during the iterative processes. The CGM derives basis from the perturbational principles (Alifanov [9]) and transforms the inverse problem to the solution of three problems, namely, the direct problem, the sensitivity problem and the adjoint problem, which will be discussed in detail in the text.
