E=K+U=0.5m(dtdx)2+0.5kx2
dtdx=−aoexp(−bt)[bcos(wdt+ϕ)+wdsin(wdt+ϕ)] Thus
K=0.5ma02e−2bt[b2cos2(wdt+ϕ)+wd2sin2(wdt+ϕ)+bwdsin2(wdt+ϕ)] We have:
k=mwd2
U=0.5ma02wd2e−2btcos2(wdt+ϕ)⟨E⟩=0.5ma02e−2bt⟨A(t)⟩
A(t)=(b2+wd2)cos2(wdt+ϕ)+wd2sin2(wdt+ϕ)+bwdsin2(wdt+ϕ)
⟨A(t)⟩=0.5((b2+wd2)+wd2) As for weakly damped harmonic oscillator
b≪wd
⟨A(t)⟩=wd2
⟨E⟩=0.5ma02wd2e−2bt=E0e−2bt
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