Two pendulums of length hang from a rigid horizontal support

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Action and Symmetry, Ch Normal modes as a generalised eigenvalue problem

Problem

Two pendulums of length

\ell=1.0\ \mathrm m

hang from a rigid horizontal support, with bobs of mass

m_1=1.0\ \mathrm{kg}

and

m_2=3.0\ \mathrm{kg}

. At a distance

h=0.50\ \mathrm m

below the support the two rods are joined by a light horizontal spring of stiffness

\kappa=50\ \mathrm{N\,m^{-1}}

, unstretched when both pendulums hang vertically. For small swing angles

\theta_1,\theta_2

from the vertical, take

g=9.8\ \mathrm{m\,s^{-2}}

. Small-angle expansion gives

\mathbf T=\begin{pmatrix}m_1\ell^2&0\\0&m_2\ell^2\end{pmatrix}

and

\mathbf V=\begin{pmatrix}m_1g\ell+\kappa h^2&-\kappa h^2\\-\kappa h^2&m_2g\ell+\kappa h^2\end{pmatrix}

. ((a)) Evaluate

\mathbf T

and

\mathbf V

numerically and find the two squared frequencies and mode shapes. ((b)) Starting from

(\mathbf V-\omega_r^2\mathbf T)\mathbf a^{(r)}=\mathbf 0

and

(\mathbf V-\omega_s^2\mathbf T)\mathbf a^{(s)}=\mathbf 0

with

\omega_r\ne\omega_s

, prove that

\mathbf a^{(r)\mathrm T}\mathbf T\mathbf a^{(s)}=0

for any symmetric

\mathbf V,\mathbf T

. ((c)) Verify the relation numerically for the two modes found in part (a), by computing

\mathbf a^{(1)\mathrm T}\mathbf T\mathbf a^{(2)}

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