On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Applied Mechanics |
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| Online Access: | https://www.mdpi.com/2673-3161/4/1/9 |
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| author | Luiz M. B. C. Campos Manuel J. S. Silva |
| author_facet | Luiz M. B. C. Campos Manuel J. S. Silva |
| author_sort | Luiz M. B. C. Campos |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>16</mn></mrow></semantics></math></inline-formula>th of the energy of the free oscillation. |
| first_indexed | 2024-03-11T06:59:57Z |
| format | Article |
| id | doaj.art-ce2e9fa087cf41a9894f1d5ea0117a9a |
| institution | Directory Open Access Journal |
| issn | 2673-3161 |
| language | English |
| last_indexed | 2024-03-11T06:59:57Z |
| publishDate | 2023-01-01 |
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| series | Applied Mechanics |
| spelling | doaj.art-ce2e9fa087cf41a9894f1d5ea0117a9a2023-11-17T09:20:16ZengMDPI AGApplied Mechanics2673-31612023-01-014114117810.3390/applmech4010009On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying ForcingLuiz M. B. C. Campos0Manuel J. S. Silva1CCTAE, IDMEC, LAETA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, PortugalCCTAE, IDMEC, LAETA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, PortugalThe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>16</mn></mrow></semantics></math></inline-formula>th of the energy of the free oscillation.https://www.mdpi.com/2673-3161/4/1/9damped oscillationsfree oscillationsforced oscillationsresonanceactive vibration suppressionforcing |
| spellingShingle | Luiz M. B. C. Campos Manuel J. S. Silva On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing Applied Mechanics damped oscillations free oscillations forced oscillations resonance active vibration suppression forcing |
| title | On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing |
| title_full | On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing |
| title_fullStr | On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing |
| title_full_unstemmed | On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing |
| title_short | On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing |
| title_sort | on the countering of free vibrations by forcing part ii damped oscillations and decaying forcing |
| topic | damped oscillations free oscillations forced oscillations resonance active vibration suppression forcing |
| url | https://www.mdpi.com/2673-3161/4/1/9 |
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