23.11: Solution to the Forced Damped Oscillator Equation
- Page ID
- 25900
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We shall now use complex numbers to solve the differential equation
\[F_{0} \cos (\omega t)=m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x \nonumber \]
We begin by assuming a solution of the form
\[x(t)=x_{0} \cos (\omega t+\phi) \nonumber \]
where the amplitude \(x_{0}\) and the phase constant \(\phi\) need to be determined. We begin by defining the complex function
\[z(t)=x_{0} e^{i(\omega t+\phi)} \nonumber \]
Our desired solution can be found by taking the real projection
\[x(t)=\operatorname{Re}(z(t))=x_{0} \cos (\omega t+\phi) \nonumber \]
Our differential equation can now be written as
\[F_{0} e^{i \omega t}=m \frac{d^{2} z}{d t^{2}}+b \frac{d z}{d t}+k z \nonumber \]
We take the first and second derivatives of Equation (23.D.3),
\[\frac{d z}{d t}(t)=i \omega x_{0} e^{i(\omega t+\phi)}=i \omega z \nonumber \]
\[\frac{d^{2} z}{d t^{2}}(t)=-\omega^{2} x_{0} e^{i(\omega t+\phi)}=-\omega^{2} z \nonumber \]
We substitute Equations (23.D.3), (23.D.6), and (23.D.7) into Equation (23.D.5) yielding
\[F_{0} e^{i \omega t}=\left(-\omega^{2} m+b i \omega+k\right) z=\left(-\omega^{2} m+b i \omega+k\right) x_{0} e^{i(\omega t+\phi)} \nonumber \]
We divide Equation (23.D.8) through by \(e^{i \omega t}\) and collect terms using yielding
\[x_{0} e^{i \phi}=\frac{F_{0} / m}{\left(\left(\omega_{0}^{2}-\omega^{2}\right)+i(b / m) \omega\right)} \nonumber \]
where we have used \(\omega_{0}^{2}=k / m\). Introduce the complex number
\[z_{1}=\left(\omega_{0}^{2}-\omega^{2}\right)+i(b / m) \omega \nonumber \]
Then Equation (23.D.9) can be written as
\[x_{0} e^{i \phi}=\frac{F_{0}}{m y} \nonumber \]
Multiply the numerator and denominator of Equation (23.D.11) by the complex conjugate \(\bar{z}_{1}=\left(\omega_{0}^{2}-\omega^{2}\right)-i(b / m) \omega\) yeilding
\[x_{0} e^{i \phi}=\frac{F_{0} \bar{z}_{1}}{m z_{1} \bar{z}_{1}}=\frac{F_{0}}{m} \frac{\left(\left(\omega_{0}^{2}-\omega^{2}\right)-i(b / m) \omega\right)}{\left(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+(b / m)^{2} \omega^{2}\right)} \equiv u+i v \nonumber \]
where
\[u=\frac{F_{0}}{m} \frac{\left(\omega_{0}^{2}-\omega^{2}\right)}{\left(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+(b / m)^{2} \omega^{2}\right)} \nonumber \]
\[v=-\frac{F_{0}}{m} \frac{(b / m) \omega}{\left(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+(b / m)^{2} \omega^{2}\right)} \nonumber \]
Therefore the modulus \(x_{0}\) is given by
\[x_{0}=\left(u^{2}+v^{2}\right)^{1 / 2}=\frac{F_{0} / m}{\left(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+(b / m)^{2} \omega^{2}\right)} \nonumber \]
and the phase is given by
\[\phi=\tan ^{-1}(v / u)=\frac{-(b / m) \omega}{\left(\omega_{0}^{2}-\omega^{2}\right)} \nonumber \]