In real physical systems there is always some damping (i. e. gamma > 0), from fr

Question

In real physical systems there is always some damping (i. e. gamma > 0), from frictional effects if nothing else. a Show that where xh(t) is the homogeneous solution of the equation m + y + kx = F0 cos(omega t). [Thus, xh(t) is a transient solution - one that dies out with the passage of time, leaving only the particular solution xp(t), called the steady state solution. ] Find the transient motion and steady state oscillations of a damped mass-spring system with m = 1, gamma = 2, and k = 12 under the influence of an external force F(t) = 18cos(2t) with x(0) = 3 and (0) = 0.

Explanation / Answer

for transient solution we have let solution x^pt p^2+2p+12=0 so x is found in the form x=Ae^pt +B e^-pt p=(-2+-2*11^.5)/2=-1+-11^.5 i x=e^(-t)[C cos 11^.5t + Dsin 11^.5 t] putting the initial conditions x(0)=3,gives C=3, x'(0)=0,gives D=0 so x=3e^(-t) cos 11^.5t for particular integral=18cos (2t)/(p^2+2p+12)=18/2(p+4)=9/(p+4)=9e^-4t finally x(t)=e^(-t)[C cos 11^.5t + Dsin 11^.5 t]+9e^-4t

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