Three beads are threaded on a light string held taut between fixed end supports and…

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

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

Problem

Three beads are threaded on a light string held taut between fixed end supports and under uniform tension. For small transverse displacements

x_1, x_2, x_3

of the beads, each segment of string acts like a spring of effective stiffness

k = T/a = 8\,\mathrm{N\,m^{-1}}

(

T

the tension,

a

the bead spacing) coupling neighbouring beads (and the end beads to the fixed supports), giving

\mathsf{K} = k\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix}

. The middle bead has negligible mass, while the end beads have masses

m_1 = 1\,\mathrm{kg}

and

m_3 = 4\,\mathrm{kg}

, so

\mathsf{M} = \mathrm{diag}(1, 0, 4)\,\mathrm{kg}

. ((a)) Justify, using the invertibility criterion for a square linear system (the Fredholm alternative), that the middle bead's position is uniquely determined by

x_1

and

x_3

, and find it. ((b)) Find the reduced mass and stiffness matrices for

(x_1, x_3)

and solve for the two frequencies

\omega_1 < \omega_2

. ((c)) Find the mode shapes and verify their orthogonality in the reduced mass inner product, as guaranteed by the spectral theorem applied to the self-adjoint operator

\mathsf{M}_{\mathrm{red}}^{-1}\mathsf{K}_{\mathrm{red}}

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