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Resonance is not a phenomenon reserved for undamped mechanisms. Consider the model: d²x dr +2. +5.x = cos (w-t) dt2 dt corresponding to a 1 kg mass attached to a spring which has a restoring constant of k = 5 N/m and a damping constant of c = 2 N/(m/s). (a) Determine the particular solution of the equation, keeping w undetermined. (Note: Do not include the complementary solution in your answer!) xp (t) = 2w sin (wt) + (5-w²) cos(wt) 4w²+(5-w²)² Note: To include w, type "omega". (b) Regardless of initial conditions, in the limit as t→∞ we have that x (t)→ xp (t). That is, the solution converges asymptotically to the particular solution. To better understand this response, determine the amplitude in the phase-shifted cosine form A (w) cos (w.t-a) of x₂ (t): 1 A (w) = √(46²)² + (5 − w²) ² X (c) Determine the forcing frequency w which maximize the amplitude A (w) and then determine the maximal amplitude of the response. (Hint: Consider taking the derivative of A (w)!) √3 maximum of A (w): 1 4

Resonance is not a phenomenon reserved for undamped mechanisms. Consider the model: d²x dr +2. +5.x = cos (w-t) dt2 dt corresponding to a 1 kg mass attached to a spring which has a restoring constant of k = 5 N/m and a damping constant of c = 2 N/(m/s). (a) Determine the particular solution of the equation, keeping w undetermined. (Note: Do not include the complementary solution in your answer!) xp (t) = 2w sin (wt) + (5-w²) cos(wt) 4w²+(5-w²)² Note: To include w, type "omega". (b) Regardless of initial conditions, in the limit as t→∞ we have that x (t)→ xp (t). That is, the solution converges asymptotically to the particular solution. To better understand this response, determine the amplitude in the phase-shifted cosine form A (w) cos (w.t-a) of x₂ (t): 1 A (w) = √(46²)² + (5 − w²) ² X (c) Determine the forcing frequency w which maximize the amplitude A (w) and then determine the maximal amplitude of the response. (Hint: Consider taking the derivative of A (w)!) √3 maximum of A (w): 1 4

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Resonance is not a phenomenon reserved for undamped mechanisms. Consider the model: d²x dr +2. +5.x = cos (w.t) dt² dt corresponding to a 1 kg mass attached to a spring which has a restoring constant of k = 5 N/m and a damping constant of c = 2 N/(m/s). (a) Determine the particular solution of the equation, keeping w undetermined. (Note: Do not include the complementary solution in your answer!) xp (t) = 2w sin (wt) + (5-w²) cos(wt) 4w²+(5-w²)² Note: To include w, type "omega". (b) Regardless of initial conditions, in the limit as t→ ∞ we have that x (t)→ xp (t). That is, the solution converges asymptotically to the particular solution. To better understand this response, determine the amplitude in the phase-shifted cosine form A (w) cos (w.t-a) of x₂ (t): 1 A (w) = √(46²)² + (5 − w²)² X (c) Determine the forcing frequency w which maximize the amplitude A (w) and then determine the maximal amplitude of the response. (Hint: Consider taking the derivative of A (w)!) √3 maximum of A (w): 144
Transcribed Image Text:Resonance is not a phenomenon reserved for undamped mechanisms. Consider the model: d²x dr +2. +5.x = cos (w.t) dt² dt corresponding to a 1 kg mass attached to a spring which has a restoring constant of k = 5 N/m and a damping constant of c = 2 N/(m/s). (a) Determine the particular solution of the equation, keeping w undetermined. (Note: Do not include the complementary solution in your answer!) xp (t) = 2w sin (wt) + (5-w²) cos(wt) 4w²+(5-w²)² Note: To include w, type "omega". (b) Regardless of initial conditions, in the limit as t→ ∞ we have that x (t)→ xp (t). That is, the solution converges asymptotically to the particular solution. To better understand this response, determine the amplitude in the phase-shifted cosine form A (w) cos (w.t-a) of x₂ (t): 1 A (w) = √(46²)² + (5 − w²)² X (c) Determine the forcing frequency w which maximize the amplitude A (w) and then determine the maximal amplitude of the response. (Hint: Consider taking the derivative of A (w)!) √3 maximum of A (w): 144
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