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Abstract A technique based on the Wiener path integral (WPI) is developed for determining the stochastic response of diverse nonlinear systems with fractional derivative elements. Specifically, a reduced-order WPI formulation is proposed, which can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the herein developed technique can determine, directly, any lower-dimensional joint response probability density function corresponding to a subset only of the response vector components. This is done by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in few only specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system. An indicative numerical example is considered pertaining to a stochastically excited tuned mass-damper-inerter nonlinear system with a fractional derivative element. Comparisons with relevant Monte Carlo simulation data demonstrate the accuracy and computational efficiency of the technique.
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Estimating the forcing function in a mechanical system by an inverse calibration method
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.
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- Award ID(s):
- 2031808
- PAR ID:
- 10381357
- Date Published:
- Journal Name:
- Journal of Vibration and Control
- Volume:
- 28
- Issue:
- 21-22
- ISSN:
- 1077-5463
- Page Range / eLocation ID:
- 3352 to 3363
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1177/10775463211031053
