A new method to estimate time-varying external forces acting upon a vibrating structure is presented in this paper. The method is developed for a force-calibrating device which is designed to be rigid in the operating frequency band and, therefore, it was believed that simple force-gauges should recover the applied forces. It has been observed that the inertia of the vibrating calibration device distorts the estimated force, even when the excitation is only one-third of the first natural frequency. Unlike traditional methods, the frequency response need not be inverted, alternatively a smoothed, Lagrange multiplier based estimation method is formed. With the proposed method, an electromagnetic excitation device can be calibrated such that the electromagnetic forces can be compared with theoretical estimates. The unique features of the proposed method are: (a) Compensation for inertia forces; (b) incorporation of measured reaction forces in a mixed analytical and experimentally obtained model; (c) possible independence on elastic stiffness forces; and (d) closed form, integral equations. New equations relating the externally applied forces and various measurable physical parameters are established, for example: The dependency upon the air-gap, current, and magnetic flux can be found.

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