A uniform rigid bar of mass and length rests horizontally on two identical vertical…

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 uniform rigid bar of mass

m=6\,\mathrm{kg}

and length

L

rests horizontally on two identical vertical springs of stiffness

k=30\,\mathrm{N\,m^{-1}}

, one under each end. Take as coordinates the small vertical displacements

x_1

and

x_2

of the two ends from equilibrium, and ignore horizontal motion. The bar's moment of inertia about its centre is

mL^2/12

. ((a)) Show that the kinetic energy is

T=\tfrac{m}{6}(\dot x_1^2+\dot x_1\dot x_2+\dot x_2^2)

, and write down

\mathsf M

and

\mathsf K

. Note that

\mathsf M

is not diagonal. ((b)) Find the two angular frequencies

\omega_1<\omega_2

and the mode shapes, and show that the modes are orthogonal in the mass product. ((c)) Normalise each mode to unit generalised mass. The left end is displaced by

10\,\mathrm{mm}

with the right end at equilibrium, and the bar is released from rest. Find the energy carried by each mode, and check that the two add up to

\tfrac12\mathbf x_0^{\mathrm T}\mathsf K\mathbf x_0

. ((d)) Someone computes the coefficients with the ordinary dot product,

\mathbf u_n\cdot\mathbf x_0

, reasoning that these two mode shapes are perpendicular in the dot product too. Find the total energy this gives, and explain why it is wrong.

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