Question 33.5: Find the reponse function for the damped harmonic oscillator......
Find the reponse function for the damped harmonic oscillator (mass m, spring constant k, damping α) whose equation of motion is given by
m\ddot{x} + α\dot{x} + kx = f (33.74)
and show that eqn 33.73 holds.
\boxed{\widetilde{χ}^{\prime } (0) = \mathcal{P} \int_{-\infty }^{\infty }{\frac{dω^{\prime } }{\pi } }\frac{\widetilde{χ} ^{\prime \prime }(ω^{\prime) } }{ω^{\prime}} }. (33.73)
Writing the resonant frequency ω^{2} _{0} = k/m , and writing the damping γ = α/m, we have
\ddot{x} + γ\dot{x } + ω^{2}_{0} x = \frac{f}{m}, (33.75)
and Fourier transforming this gives immediately that
\widetilde{χ} (ω)= \frac{\widetilde{x} (ω) }{ \widetilde{f} (ω)} =\frac{1}{m}\left[\frac{1}{ω^{2}_{0} − ω^{2} −iωγ} \right]. (33.76)
Hence, the imaginary part of the response function is
\widetilde{χ}^{\prime \prime } (ω)=\frac{1}{m}\left[\frac{ωγ}{(ω^{2} − ω^{2}_{0} )^{2} +(ωγ)^{2} } \right], (33.77)
and the static susceptibility is
\widetilde{χ} ^{\prime } (0) = \frac{1}{ mω^{2}_{0} } = \frac{1}{k} . (33.78)
The real and imaginary parts of \widetilde{χ}(ω) are plotted in Fig. 33.2(a). The imaginary part shows a peak near ω_{0}. Equation 33.77 shows that \widetilde{χ} ^{\prime \prime } (ω)/ω =(γ/m)[(ω^{2} − ω^{2}_{0} )+(ωγ)^{2}] and straightforward integration shows that \int_{-\infty }^{\infty }{} ( \widetilde{χ} ^{\prime \prime } (ω)/ω)dω = π/(mω^{2}_{0} )=π\widetilde{χ} ^{\prime } (0) and hence that eqn 33.73 holds. This is illustrated in Fig. 33.2(b)
