A two-cart system has M = pmatrix 1&0 0&2 pmatrix

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Oscillations and normal modes

Conditions of Validity, Ch Two quadratic forms at once: the generalised eigenvalue problem

Problem

A two-cart system has

\mathsf{M} = \begin{pmatrix}1&0\\0&2\end{pmatrix}\,\text{kg}, \qquad \mathsf{K} = \begin{pmatrix}16&-8\\-8&16\end{pmatrix}\,\text{N m}^{-1}.

((a)) Find the two squared frequencies

\omega_1^2, \omega_2^2

and the mode shapes

\mathbf{v}_1, \mathbf{v}_2

(first component

1

). ((b)) Find the generalised mass

\mu_n = \mathbf{v}_n^{\mathrm{T}}\mathsf{M}\mathbf{v}_n

of each mode and hence the unit-generalised-mass modes

\mathbf{u}_n = \mathbf{v}_n/\sqrt{\mu_n}

. ((c)) Write the kinetic and potential energies

T = \tfrac12\dot{\mathbf{x}}^{\mathrm{T}}\mathsf{M}\dot{\mathbf{x}}

and

V = \tfrac12\mathbf{x}^{\mathrm{T}}\mathsf{K}\mathbf{x}

as sums of squares of the normal coordinates

q_1, q_2

defined by

\mathbf{x} = q_1\mathbf{u}_1 + q_2\mathbf{u}_2