Estimating the forcing function in a mechanical system by an inverse calibration method

Abstract

This article proposes and demonstrates a calibration-based integral formulation for resolving the forcing function in a mass–spring–damper system, given either displacement or acceleration data. The proposed method is novel in the context of vibrations, being thoroughly studied in the field of heat transfer. The approach can be expanded and generalized further to multi-variable systems associated with machine parts, vehicle suspensions, translational and rotational systems, gear systems, etc. when mathematically described by a system of constant property, linear, time-invariant ordinary differential equations. The analytic approach and subsequent numerical reconstruction of the forcing function is based on resolving a parameter-free inverse formulation for the equation(s) of motion. The calibration approach is formulated in the frequency domain and takes advantage of several observations produced by the dimensionality reduction leading to an algebratized system involving an input–output relationship and a transfer function possessing all the system parameters. The transfer function is eliminated in lieu of experimental data, from a calibration effort, thus leading to a reduction of systematic errors. These parameter-free, reduced systematic error aspects are the distinct and novel advantages of the proposed method. A first-kind Volterra integral equation is formed containing only the unknown forcing function and experimental data. As with all ill-posed problems, regularization must be introduced for system stabilization. A future-time technique is instituted for forming a family of predictions based on the chosen regularization parameter. The optimal regularization parameter is estimated using a combination of phase–plane analysis and cross-correlation principles. Finally, a numerical simulation is performed verifying the proposed approach.

A concept of computerized accelerometers

“…Accelerometers are widely applied in science, engineering and are of profound impact on our daily life, for example, in seismological studies [4, [7][8][9], vibration and shock test of automobiles and trains [1,2,10-13], the dynamical testing of aircraft structures [14], consumer electronics products [15,16], manned or unmanned navigation [15], precision livestock farming [17], biomedical and health monitoring applications [18] and a variety of military applications [15,16]. The acceleration responses with accelerometers installed on structures are also important to reconstruct loads for design and health monitoring of structures such as buildings and bridges [19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Recovering the Forcing Function in Systems with One Degree of Freedom Using ANN and Physics Information

“…Liu et al [15] solved the non-linear vibration problem by transforming the non-linear ordinary differential equations into parabolic ordinary differential equations due to their robustness against large noise. Recently, Rice et al [16] proposed a calibration-based integral formulation for estimating the forcing function in the spring mass damper system from response data. For a detailed review of past and present literature on dynamic load identification techniques, interested readers are advised to refer to [17].…”

Development and Validation of a Double-Sensor Hump Calibration Method for Articulated Vehicle Model Identification

“…An alternative to the conventional method is the inverse method. Theoretically, the vehicle calibration process can be considered as an inverse problem, which is generally positioned as the problem of determining the parameters of a system from its input–output correspondence [ 24 ]. In practice, Rozyn et al [ 25 ] used measurements of the response of the sprung mass when a vehicle was driven over an unknown road to determine the parameters of the vehicle, but the applicability of their method was limited to estimating the sprung mass parameters.…”