17.2: Two Coupled Pendulums
- Page ID
- 29506
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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’ll take two equal pendulums, coupled by a light spring. We take the spring restoring force to be directly proportional to the angular difference between the pendulums. (This turns out to be a good approximation.)
For small angles of oscillation, we take the Lagrangian to be
\begin{equation}
L=\dfrac{1}{2} m \ell^{2} \dot{\theta}_{1}^{2}+\dfrac{1}{2} m \ell^{2} \dot{\theta}_{2}^{2}-\dfrac{1}{2} m g \ell \theta_{1}^{2}-\dfrac{1}{2} m g \ell \theta_{2}^{2}-\dfrac{1}{2} C\left(\theta_{1}-\theta_{2}\right)^{2}
\end{equation}
Denoting the single pendulum frequency by \(\omega_{0}\), the equations of motion are (writing \(\omega_{0}^{2}=g / \ell, k=C / m \ell^{2}\), so \(\left.[k]=T^{-2}\right)\)
\begin{equation}
\begin{array}{l}
\ddot{\theta}_{1}=-\omega_{0}^{2} \theta_{1}-k\left(\theta_{1}-\theta_{2}\right) \\
\ddot{\theta}_{2}=-\omega_{0}^{2} \theta_{2}-k\left(\theta_{2}-\theta_{1}\right)
\end{array}
\end{equation}
We look for a periodic solution, writing
\begin{equation}
\theta_{1}(t)=A_{1} e^{i \omega t}, \quad \theta_{2}(t)=A_{2} e^{i \omega t}
\end{equation}
(The final physical angle solutions will be the real part.)
The equations become (in matrix notation):
\begin{equation}
\omega^{2}\left(\begin{array}{c}
A_{1} \\
A_{2}
\end{array}\right)=\left(\begin{array}{cc}
\omega_{0}^{2}+k & -k \\
-k & \omega_{0}^{2}+k
\end{array}\right)\left(\begin{array}{c}
A_{1} \\
A_{2}
\end{array}\right)
\end{equation}
Denoting the \(2 \times 2 \text { matrix by } \mathbf{M}\)
\begin{equation}
\mathbf{M} \vec{A}=\omega^{2} \vec{A}, \quad \vec{A}=\left(\begin{array}{l}
A_{1} \\
A_{2}
\end{array}\right)
\end{equation}
This is an eigenvector equation, with \(\omega^{2}\) the eigenvalue, found by the standard procedure:
\begin{equation}
\operatorname{det}\left(\mathbf{M}-\omega^{2} \mathbf{I}\right)=\left|\begin{array}{cc}
\omega_{0}^{2}+k-\omega^{2} & -k \\
-k & \omega_{0}^{2}+k-\omega^{2}
\end{array}\right|=0
\end{equation}
Solving, \(\omega^{2}=\omega_{0}^{2}+k \pm k\), that is
\begin{equation}
\omega^{2}=\omega_{0}^{2}, \quad \omega^{2}=\omega_{0}^{2}+2 k
\end{equation}
The corresponding eigenvectors are (1,1) and (1,−1).