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.

本文提出并演示了一种基于校准的积分公式，用于在给定位移或加速度数据的情况下解析质量-弹簧-阻尼器系统中的力函数。所提出的方法在振动的背景下是新颖的，在传热领域得到了彻底的研究。当由具有恒定属性、线性、时不变常微分方程的系统进行数学描述时，该方法可以进一步扩展和推广到与机器部件、车辆悬架、平移和旋转系统、齿轮系统等相关的多变量系统。强制函数的解析方法和随后的数值重建基于求解运动方程的无参数逆公式。校准方法是在频域中制定的，并利用降维产生的几个观察结果，导致代数化系统涉及输入-输出关系和具有所有系统参数的传递函数。从校准工作中消除传递函数代替实验数据，从而导致系统误差的减少。这些无参数、减少系统误差的方面是所提出方法的独特和新颖的优点。形成了仅包含未知强迫函数和实验数据的第一类沃尔泰拉积分方程。与所有不适定问题一样，必须引入正则化以稳定系统。建立了一种未来时间技术，用于根据所选的正则化参数形成一系列预测。使用相平面分析和互相关原理的组合来估计最佳正则化参数。最后，进行数值模拟以验证所提出的方法。