As a warm-up, suppose we have a underdamped, unforced spring-mass system with mass 0.8 kg, spring constant 18 N/m, and damping coefficient 5 kg/s. We pull the mass 0.3 m from its rest position and let it go while imparting an initial velocity of 0.7 m/s. (a) Set up and solve the initial value problem for this spring-mass system. (b) Write your answer from part (a) in phase-amplitude form, i.e. as y(t) = Aeat sin(ßt – 4) and graph the result. Compare with a graph of your answer from (a) to check that you have the correct amplitude and phase shift. (c) Find the values of t at which y(t) attains a local maximum (there will be infinitely many, of course), and observe that the time distance between any two consecutive maxima is always the same. (d) Compute the ratio of the displacements at two consecutive maxima. (That is, if y has a maximum at tn, and the next one occurs at tn+1, compute y(tn)/y(tn+1).) Observe that this ratio is constant.

As a warm-up, suppose we have a underdamped, unforced spring-mass system with mass 0.8 kg, spring constant 18 N/m, and damping coefficient 5 kg/s. We pull the mass 0.3 m from its rest position and let it go while imparting an initial velocity of 0.7 m/s. (a) Set up and solve the initial value problem for this spring-mass system. (b) Write your answer from part (a) in phase-amplitude form, i.e. as y(t) = Aeat sin(ßt – 4) and graph the result. Compare with a graph of your answer from (a) to check that you have the correct amplitude and phase shift. (c) Find the values of t at which y(t) attains a local maximum (there will be infinitely many, of course), and observe that the time distance between any two consecutive maxima is always the same. (d) Compute the ratio of the displacements at two consecutive maxima. (That is, if y has a maximum at tn, and the next one occurs at tn+1, compute y(tn)/y(tn+1).) Observe that this ratio is constant.

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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