{"title":"On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing","authors":"L. Campos, Manuel J. S. Silva","doi":"10.3390/applmech4010009","DOIUrl":null,"url":null,"abstract":"The present two-part paper concerns the active vibration suppression for the simplest damped continuous system, namely the transverse oscillations of an elastic string, with constant tension and mass density per unit length and friction force proportional to the velocity, described by the telegraph or wave-diffusion equation, in two complementary parts. The initial part I considers non-resonant and resonant forcing, by concentrated point forces or continuous force distributions independent of time, with phase shift between the forced and free oscillations, in the absence of damping, in which case the forced telegraph equation reduces to the forced classical wave equation. The present and final part II uses the forced wave-diffusion equation to model the effect of damping, both as amplitude decay and phase shift in time, for non-resonant and resonant forcing by a single point force, with constant magnitude or magnitude decaying exponentially in time at an arbitrary rate. Assuming a finite elastic string fixed at both ends, the free oscillations are (i) sinusoidal modes in space-time with exponential decay in time due to damping. The non-resonant forced oscillations at an applied frequency distinct from a natural frequency are also (ii) sinusoidal in space-time, with constant amplitude and a phase shift such that the work of the applied force balances the dissipation. For resonant forcing at an applied frequency equal to a natural frequency, the sinusoidal oscillations in space-time have (iii) a constant amplitude and a phase shift of π/2. In both cases, the (ii) non-resonant or (iii) resonant forcing dominates the decaying free oscillations after some time. Even by optimizing the forcing to minimize the total energy of oscillation, it remains below the energy of the free oscillation alone, but only for a short time—generally a fraction of the period. A more effective method of countering the damped free oscillations is to use forcing with amplitude decaying exponentially in time; by suitable choice of the forcing decay relative to the free damping, the total energy of oscillation over all time can be reduced to no more than 1/16th of the energy of the free oscillation.","PeriodicalId":8048,"journal":{"name":"Applied Mechanics Reviews","volume":"196 1","pages":""},"PeriodicalIF":12.2000,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mechanics Reviews","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.3390/applmech4010009","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
Abstract
The present two-part paper concerns the active vibration suppression for the simplest damped continuous system, namely the transverse oscillations of an elastic string, with constant tension and mass density per unit length and friction force proportional to the velocity, described by the telegraph or wave-diffusion equation, in two complementary parts. The initial part I considers non-resonant and resonant forcing, by concentrated point forces or continuous force distributions independent of time, with phase shift between the forced and free oscillations, in the absence of damping, in which case the forced telegraph equation reduces to the forced classical wave equation. The present and final part II uses the forced wave-diffusion equation to model the effect of damping, both as amplitude decay and phase shift in time, for non-resonant and resonant forcing by a single point force, with constant magnitude or magnitude decaying exponentially in time at an arbitrary rate. Assuming a finite elastic string fixed at both ends, the free oscillations are (i) sinusoidal modes in space-time with exponential decay in time due to damping. The non-resonant forced oscillations at an applied frequency distinct from a natural frequency are also (ii) sinusoidal in space-time, with constant amplitude and a phase shift such that the work of the applied force balances the dissipation. For resonant forcing at an applied frequency equal to a natural frequency, the sinusoidal oscillations in space-time have (iii) a constant amplitude and a phase shift of π/2. In both cases, the (ii) non-resonant or (iii) resonant forcing dominates the decaying free oscillations after some time. Even by optimizing the forcing to minimize the total energy of oscillation, it remains below the energy of the free oscillation alone, but only for a short time—generally a fraction of the period. A more effective method of countering the damped free oscillations is to use forcing with amplitude decaying exponentially in time; by suitable choice of the forcing decay relative to the free damping, the total energy of oscillation over all time can be reduced to no more than 1/16th of the energy of the free oscillation.
期刊介绍:
Applied Mechanics Reviews (AMR) is an international review journal that serves as a premier venue for dissemination of material across all subdisciplines of applied mechanics and engineering science, including fluid and solid mechanics, heat transfer, dynamics and vibration, and applications.AMR provides an archival repository for state-of-the-art and retrospective survey articles and reviews of research areas and curricular developments. The journal invites commentary on research and education policy in different countries. The journal also invites original tutorial and educational material in applied mechanics targeting non-specialist audiences, including undergraduate and K-12 students.