Universal Journal of Mechanical Engineering Vol. 4(2), pp. 50 - 55
DOI: 10.13189/ujme.2016.040205
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Closed-form Inverse Determination of Force Excitations for Undamped and Damped Linear-systems Using a Least-squares Approach

A.E. Segall ^*
Department of Engineering Science and Mechanics, The Pennsylvania State University, USA

ABSTRACT

An inverse vibration solution was derived for single DOF systems (undamped and damped) using a method based on the least-squares determination of polynomial coefficients representing the unknown force excitation. Initially, a closed-form and generalized direct-solution to the Vibration Equation was derived using standard polynomials to represent the excitation; while simple polynomials and a single DOF system were chosen to demonstrate the method analytically, more sophisticated functions such as Splines, B-Splines, Basis Functions, and/or Finite-Element analysis can be used with the advocated method. In order to solve the inverse problem, the generalized solutions were then used to enforce measured displacement data and the Levenberg-Marquardt algorithm employed to determine the unknown polynomial coefficients representing the base excitation. Results including random errors added to the data (up to ±5%) indicated that the method could be used to determine a time varying excitation; indeed, good agreement was seen between exact and polynomial solution (undamped), as well as an inversely predicted excitation for both damped and undamped systems subjected to a half-sine force history. Given the results, the method appears well suited for complicated scenarios provided the response can be approximated by a polynomial (or more sophisticated functions) and inverse calculations are restricted to the time interval bounding the measured data. In addition, the least-squares inverse approach should be adaptable to more complicated system and excitations by using general purpose finite-element codes. For the current analysis, the least-squares smoothing inherent in the process appears to have helped minimize the influence of measurement errors.

KEYWORDS
Vibration, Damped, Undamped, Inverse, Excitation

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] A.E. Segall , "Closed-form Inverse Determination of Force Excitations for Undamped and Damped Linear-systems Using a Least-squares Approach," Universal Journal of Mechanical Engineering, Vol. 4, No. 2, pp. 50 - 55, 2016. DOI: 10.13189/ujme.2016.040205.

(b). APA Format:
A.E. Segall (2016). Closed-form Inverse Determination of Force Excitations for Undamped and Damped Linear-systems Using a Least-squares Approach. Universal Journal of Mechanical Engineering, 4(2), 50 - 55. DOI: 10.13189/ujme.2016.040205.