Chapter 9

Three coupled pendulums swing perpendicularly to the horizontal line containing their points of suspension, and the following equations of motion are satisfied: $$\begin{array}{rl}-m{\ddot{x}}_1& =cm{x}_1+d(x_1-x_2)\\ -M{\ddot{x}}_2& =cM{x}_2+d(x_2-x_1)+d(x_2-x_3)\\ -m{\ddot{x}}_3& =cm{x}_3+d(x_3-x_2)\end{array}$$ where $x_1,x_2$ and $x_3$ are measured from the equilibrium points, $m,M$ and $m$ are the masses of the pendulum bobs and $c$ and $d$ are positive constants. Find the normal frequencies of the system and sketch the corresponding patterns of oscillation. What happens as $d\to 0$ or $d\to \mathrm{\infty }$ ?

A double pendulum, smoothly pivoted at $A$, consists of two light rigid rods, $AB$ and $BC$, each of length $l$, which are smoothly jointed at $B$ and carry masses $m$ and $\alpha m$ at $B$ and $C$ respectively. The pendulum makes small oscillations in one plane under gravity; at time $t,AB$ and $BC$ make angles $\theta (t)$ and $\varphi (t)$ respectively with the downward vertical. Find quadratic expressions for the kinetic and potential energies of the system and hence show that the normal modes have angular frequencies given by $${\omega }^2=\frac{g}{l}[1+\alpha \pm \sqrt{\alpha (1+\alpha )}]$$ For $\alpha =1/3$, show that in one of the normal modes the mid-point of $BC$ does not move during the motion.

The simultaneous reduction to diagonal form of two real symmetric quadratic forms. Consider the two real symmetric quadratic forms $u^{T}Au$ and $u^{T}Bu$, where $u^T$ stands for the row matrix $(xyz)$, and denote by ${\mathrm{u}}^n$ those column matrices that satisfy $${\mathrm{Bu}}^n={\lambda }_n{\mathrm{Au}}^n$$ in which $n$ is a label and the ${\lambda }_n$ are real, non-zero and all different. (a) By multiplying (E9.1) on the left by ${\left({\mathrm{u}}^m\right)}^{\mathrm{T}}$ and the transpose of the corresponding equation for ${\mathrm{u}}^m$ on the right by ${\mathrm{u}}^n$, show that ${\left({\mathrm{u}}^m\right)}^{\mathrm{T}}{\mathrm{Au}}^n=0$ for $n\ne m$ (b) By noting that $A{u}^n={\left({\lambda }_n\right)}^{-1}B{u}^n$, deduce that ${\left({\mathrm{u}}^m\right)}^{\mathrm{T}}{\mathrm{Bu}}^n=0$ for $m\ne n$. It can be shown that the $u^n$ are linearly independent; the next step is to construct a matrix $\mathrm{P}$ whose columns are the vectors ${\mathrm{u}}^n$. (c) Make a change of variables $u=Pv$ such that $u^{\mathrm{T}}$ Au becomes $v^{\mathrm{T}}Cv$, and $u^{\mathrm{T}}Bu$ becomes $v^{\mathrm{T}}Dv$. Show that $C$ and $D$ are diagonal by showing that $c_{ij}=0$ if $i\ne j$ and similarly for $d_{ij}$ Thus $\mathrm{u}=\mathrm{Pv}$ or $\mathrm{v}={\mathrm{P}}^{-1}\mathrm{u}$ reduces both quadratics to diagonal form. To summarise, the method is as follows: (a) find the ${\lambda }_n$ that allow (E9.1) a non-zero solution, by solving $|\mathrm{B}-\lambda \mathrm{A}|=0$; (b) for each ${\lambda }_n$ construct ${\mathrm{u}}^n$; (c) construct the non-singular matrix $\mathrm{P}$ whose columns are the vectors ${\mathrm{u}}^n$; (d) make the change of variable $\mathrm{u}=\mathrm{Pv}$.

Three particles of mass $m$ are attached to a light horizontal string having fixed ends, the string being thus divided into four equal portions of length $a$ each under a tension $T$. Show that for small transverse vibrations the amplitudes ${\mathrm{x}}^i$ of the normal modes satisfy $\mathrm{Bx}=\left({\mathrm{ma}\omega }^2/T\right)\mathrm{x}$, where $\mathrm{B}$ is the matrix $$\left(\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 2\end{array}\right)$$ Estimate the lowest and highest eigenfrequencies using trial vectors ${\left(\begin{array}{lll}3& 4& 3\end{array}\right)}^{\mathrm{T}}$ and ${\begin{array}{lll}3& -4& 3\end{array})}^{\mathrm{T}}$. Use also the exact vectors ${\left(\begin{array}{lll}1& \sqrt{2}& 1\end{array}\right)}^{\mathrm{T}}$ and ${\left(\begin{array}{lll}1& -\sqrt{2}& 1\end{array}\right)}^{\mathrm{T}}$, and compare the results.

Use the Rayleigh-Ritz method to estimate the lowest oscillation frequency of a heavy chain of $N$ links, each of length $a(=L/N)$, which hangs freely from one end. (Try simple calculable configurations such as all links but one vertical, or all links collinear, etc.)