8.9: Damped Resonance
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We now solve the dimensionless equations given by Equation \ref{8.34}, Equation \ref{8.35} and Equation \ref{8.36}, written here as
\[\ddot{x}+\alpha \dot{x}+x=\cos \beta t, \label{8.37}\]
where the physical constraints of our three applications requires that \(\alpha>0\) . The homogeneous equation has characteristic equation
\[r^{2}+\alpha r+1=0 \nonumber \]
so that the solutions are
\[r_{\pm}=-\frac{\alpha}{2} \pm \frac{1}{2} \sqrt{\alpha^{2}-4} \nonumber \]
When \(\alpha^{2}-4<0\) , the motion of the unforced oscillator is said to be underdamped; when \(\alpha^{2}-4>0\) , overdamped; and when \(\alpha^{2}-4=0\) , critically damped. For all three types of damping, the roots of the characteristic equation satisfy \(\operatorname{Re}\left(r_{\pm}\right)<0\) . Therefore, both linearly independent homogeneous solutions decay exponentially to zero, and the long-time asymptotic solution of Equation \ref{8.37} reduces to the non-decaying particular solution. Since the initial conditions are satisfied by the free constants multiplying the decaying homogeneous solutions, the long-time asymptotic solution is independent of the initial conditions.
If we are only interested in the long-time asymptotic solution of Equation \ref{8.37}, we can proceed directly to the determination of a particular solution. As before, we consider the complex ode
\[\ddot{z}+\alpha \dot{z}+z=e^{i \beta t}, \nonumber \]
with \(x_{p}=\operatorname{Re}\left(z_{p}\right)\) . With the ansatz \(z_{p}=A e^{i \beta t}\) , we have
\[-\beta^{2} A+i \alpha \beta A+A=1 \nonumber \]
or
\[\begin{align} A &=\frac{1}{\left(1-\beta^{2}\right)+i \alpha \beta} \\ &=\left(\frac{1}{\left(1-\beta^{2}\right)^{2}+\alpha^{2} \beta^{2}}\right)\left(\left(1-\beta^{2}\right)-i \alpha \beta\right) . \label{8.38} \end{align} \]
To determine \(x_{p}\) , we utilize the polar form of a complex number. The complex number \(z=x+i y\) can be written in polar form as \(z=r e^{i \phi}\) , where \(r=\sqrt{x^{2}+y^{2}}\) and \(\tan \phi=y / x\) . We therefore write
\[\left(1-\beta^{2}\right)-i \alpha \beta=r e^{i \phi}, \nonumber \]
with
\[r=\sqrt{\left(1-\beta^{2}\right)^{2}+\alpha^{2} \beta^{2}}, \quad \tan \phi=-\frac{\alpha \beta}{1-\beta^{2}} . \nonumber \]
Using the polar form, \(A\) in Equation \ref{8.38} becomes
\[A=\left(\frac{1}{\sqrt{\left(1-\beta^{2}\right)^{2}+\alpha^{2} \beta^{2}}}\right) e^{i \phi} \nonumber \]
and \(x_{p}=\operatorname{Re}\left(A e^{i \beta t}\right)\) becomes
\[\begin{align} x_{p} &=\left(\frac{1}{\sqrt{\left(1-\beta^{2}\right)^{2}+\alpha^{2} \beta^{2}}}\right) \operatorname{Re}\left\{e^{i(\beta t+\phi)}\right\} \\ &=\left(\frac{1}{\sqrt{\left(1-\beta^{2}\right)^{2}+\alpha^{2} \beta^{2}}}\right) \cos (\beta t+\phi) \label{8.39}\end{align} \]
We conclude with a couple of observations about Equation \ref{8.39}. First, if the forcing frequency \(\omega\) is equal to the natural frequency \(\omega_{0}\) of the undamped oscillator, then \(\beta=1\) and \(A=1 / i \alpha\) , and \(x_{p}=(1 / \alpha) \sin t\) . The oscillator position is observed to be \(\pi / 2\) out of phase with the external force, or in other words, the velocity of the oscillator, not the position, is in phase with the force. Second, the value of \(\beta\) that maximizes the amplitude of oscillation is the value of \(\beta\) that minimizes the denominator of Equation \ref{8.39}. To determine \(\beta_{m}\) we thus minimize the function \(g\left(\beta^{2}\right)\) with respect to \(\beta^{2}\) , where
\[g\left(\beta^{2}\right)=\left(1-\beta^{2}\right)^{2}+\alpha^{2} \beta^{2} . \nonumber \]
Taking the derivative of \(g\) with respect to \(\beta^{2}\) and setting this to zero to determine \(\beta_{m}\) yields
\[-2\left(1-\beta_{m}^{2}\right)+\alpha^{2}=0 \nonumber \]
or
\[\beta_{m}=\sqrt{1-\frac{\alpha^{2}}{2}} \approx 1-\frac{\alpha^{2}}{4}, \nonumber \]
the last approximation valid if \(\alpha<<1\) and the dimensionless damping coefficient is small. We can interpret this result by saying that small damping slightly lowers the "resonance" frequency of the undamped oscillator.