Damped Oscillation (general solution)
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Exercise:
The general solution for a damped oscillator with linear damping i.e. a damping term proportional to v_yt can be written as yt left C_ cosomega t + C_ sinomega t right e^-delta t Find the coefficients C_ and C_ for abcliste abc an oscillation starting from rest with an initial amplitude A; abc an oscillation starting from the equlibrium position with an initial velocity v_. abcliste
Solution:
The displacement for t is y left C_ cos + C_ sin right e^ left C_ + C_ right C_ The first derivative of yt is dot yt left -C_ sinomega t omega + C_ cosomega t omega right e^-delta t & quad - leftC_ cosomega t + C_ sinomega t right e^-delta t delta left - sinomega t C_ omega + C_ delta+ cosomega t C_ omega - C_ delta right e^-delta t For t only the cosine term remains: dot y C_ omega - C_ delta abcliste abc The initial conditions are yA and dot y. It follows that C_ A C_ omega - C_ delta v_ Longrightarrow C_ v_ + C_ delta v_ + A delta The solution can be written as yt left A cosomega t + v_ + Adelta sinomega t right e^-delta t abc The initial conditions are C_ C_ omega - C_ delta C_ omega v_ Longrightarrow C_ fracv_omega The solution can be written as yt fracv_omega sinomega t e^-delta t abcliste
The general solution for a damped oscillator with linear damping i.e. a damping term proportional to v_yt can be written as yt left C_ cosomega t + C_ sinomega t right e^-delta t Find the coefficients C_ and C_ for abcliste abc an oscillation starting from rest with an initial amplitude A; abc an oscillation starting from the equlibrium position with an initial velocity v_. abcliste
Solution:
The displacement for t is y left C_ cos + C_ sin right e^ left C_ + C_ right C_ The first derivative of yt is dot yt left -C_ sinomega t omega + C_ cosomega t omega right e^-delta t & quad - leftC_ cosomega t + C_ sinomega t right e^-delta t delta left - sinomega t C_ omega + C_ delta+ cosomega t C_ omega - C_ delta right e^-delta t For t only the cosine term remains: dot y C_ omega - C_ delta abcliste abc The initial conditions are yA and dot y. It follows that C_ A C_ omega - C_ delta v_ Longrightarrow C_ v_ + C_ delta v_ + A delta The solution can be written as yt left A cosomega t + v_ + Adelta sinomega t right e^-delta t abc The initial conditions are C_ C_ omega - C_ delta C_ omega v_ Longrightarrow C_ fracv_omega The solution can be written as yt fracv_omega sinomega t e^-delta t abcliste