Multiple-Frequency Force Estimation of Controlled Vibrating Systems with Generalized Nonlinear Stiffness

Francisco Beltran-Carbajal, Juan Eduardo Esquivel-Cruz, Hugo Yañez-Badillo, Ivan de Jesus Rivas-Cambero, David Sotelo () and Carlos Sotelo
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Francisco Beltran-Carbajal: Departamento de Energía, Unidad Azcapotzalco, Universidad Autónoma Metropolitana, Azcapotzalco, Mexico City 02200, Mexico
Juan Eduardo Esquivel-Cruz: Departamento de Posgrado, Unversidad Politécnica de Tulancingo, Tulancingo 43629, Mexico
Hugo Yañez-Badillo: Departamento de Investigación, TecNM: Tecnológico de Estudios Superiores de Tianguistenco, Tianguistenco 52650, Mexico
Ivan de Jesus Rivas-Cambero: Departamento de Posgrado, Unversidad Politécnica de Tulancingo, Tulancingo 43629, Mexico
David Sotelo: Tecnologico de Monterrey, School of Engineering and Sciences, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico
Carlos Sotelo: Tecnologico de Monterrey, School of Engineering and Sciences, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico

Mathematics, 2023, vol. 11, issue 13, 1-29

Abstract: An on-line estimation technique of multiple-frequency oscillatory forces combined with the Hilbert–Huang transform for an important class of actively controlled, forced vibrating mechanical systems with nonlinear stiffness forces is proposed. Polynomial parametric nonlinearities are incorporated in the significantly perturbed vibrating system dynamics. This class of nonlinear vibrating systems can exhibit harmful large-amplitude vibrations, which are inadmissible in many engineering applications. Disturbing oscillations can be also provoked due to interactions of the primary mechanical system to be actively protected against dangerous vibrations with other forced uncertain multidegree-of-freedom nonlinear vibrating systems. Taylor’s series expansion to dynamically model uncertain vibrating forces into a small time window for real-time estimation purposes is employed. Intrinsic mode functions of multiple-frequency vibrating forces can be then obtained by the Hilbert-Huang transform. Uncertain instantaneous frequencies and amplitudes of disturbing oscillations can be directly computed in temporal space. An active vibration control scheme for efficient and robust tracking of prescribed motion reference profiles based on multiple frequency force estimation is introduced as well. The presented closed-loop on-line estimation technique can be extended for other classes of nonlinear oscillatory systems. Analytical, experimental and numerical results to prove the estimation effectiveness are presented. Numerical results show reasonable estimation errors of less than 2%.

Keywords: mechanical vibrations; nonlinear stiffness; polynomial nonlinearity; active vibration control; harmonics estimation; Hilbert–Huang transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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