20.3: Example- Pendulum Driven at near Double the Natural Frequency
- Page ID
- 29535
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A simple pendulum of length \(ℓ\), mass \(m\) is attached to a point which oscillates vertically \(y=a \cos \Omega t\). Measuring \(y\) downwards, the pendulum position is
\begin{equation}
x=\ell \sin \phi, y=a \cos \Omega t+\ell \cos \phi
\end{equation}
The Lagrangian
\begin{equation}
\begin{array}{c}
L=\dfrac{1}{2} m\left(\dot{x}^{2}+\dot{y}^{2}\right)+m g \ell \cos \phi \\
=\dfrac{1}{2} m\left(\ell^{2} \cos ^{2} \phi\right) \dot{\phi}^{2}+\dfrac{1}{2} m(a \Omega \sin \Omega t+\ell \dot{\phi} \sin \phi)^{2}+m g \ell \cos \phi \\
=\dfrac{1}{2} m \ell^{2} \dot{\phi}^{2}-m a \ell \Omega \sin \Omega t \dfrac{d}{d t} \cos \phi+a^{2} \Omega^{2} \sin ^{2} \Omega t+m g \ell \cos \phi
\end{array}
\end{equation}
The purely time-dependent term will not affect the equations of motion, so we drop it, and since the equations are not affected by adding a total derivative to the Lagrangian, we can integrate the second term by parts (meaning we’re dropping a term \(\begin{equation}
\left.\dfrac{d}{d t}(\operatorname{mal} \Omega \sin \Omega t \cos \phi)\right)
\end{equation}\) to get
\begin{equation}
L=\dfrac{1}{2} m \ell^{2} \dot{\phi}^{2}+m a \ell \Omega^{2} \cos \Omega t \cos \phi+m g \ell \cos \phi
\end{equation}
(We’ve also dropped the term \(\begin{equation}
m g a \cos \Omega t
\end{equation}\) from the potential energy term—it has no \(\phi \text { or } \dot{\phi}\) dependence, so will not affect the equations of motion.)
The equation for small oscillations is
\begin{equation}
\ddot{\phi}+\omega_{0}^{2}\left[1+(4 a / \ell) \cos \left(2 \omega_{0}+\varepsilon\right) t\right] \phi=0, \quad \omega_{0}^{2}=g / \ell
\end{equation}
Comparing this with
\begin{equation}
\ddot{x}+\omega_{0}^{2}\left[1+h \cos \left(2 \omega_{0}+\varepsilon\right) t\right] x=0
\end{equation}
we see that \(4 a / \ell \equiv h\), so the parametric resonance range around \(2 \omega_{0}=2 \sqrt{g / \ell} \text { is of width } \dfrac{1}{2} h \omega_{0}=2 a \sqrt{g / \ell^{3}}\).