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# Simultaneously Estimating the Time-Dependent Damping and Stiffness Coefficients with the Aid of Vibrational Data

Chein-Shan Liu1, Jiang-Ren Chang2, Kai-Huey Chang2, Yung-Wei Chen2

Department of Mechanical and Mechatronic Engineering,Taiwan Ocean University, Keelung, Taiwan; Departmentof Harbor and River Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw
Department of Systems Engineering and Naval Architec-ture, Taiwan Ocean University, Keelung, Taiwan

Computers, Materials & Continua 2008, 7(2), 97-108. https://doi.org/10.3970/cmc.2008.007.097

## Abstract

For the inverse vibration problem a mathematical method is required to determine unknown parameters from the measurement of vibration data. When both damping and stiffness functions are identified, it is a rather difficult problem. In this paper we will propose a feasible method to simultaneously estimate both the time-dependent damping and stiffness coefficients through three mathematical transformations. First, the second-order equation of motion is transformed into a self-adjoint first-order system by using the concept of integrating factor. Then, we transform these two ODEs into two hyperbolic type PDEs. Finally, we apply a one-step group preserving scheme for the semi-discretizations of PDEs to obtain two uncoupled algebraic equations, of which the first one is used to estimate the damping coefficient while the second one is used to estimate the stiffness coefficient. The estimated results are acceptable for that used in vibrational engineering. We also discuss the use of velocity and acceleration data as inputs in the estimation. However, it leads to a bad result, and is not suggested for the use in estimation.

## Keywords

C. . Liu, J. . Chang, K. . Chang and Y. . Chen, "Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data," Computers, Materials & Continua, vol. 7, no.2, pp. 97–108, 2008.

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