Riley hobson bence · Problem 9.9

Normal modes

Mathematics — Methods · normal-modes-string-rayleigh-ritz

Problem

Three particles of mass

m

are attached to a light horizontal string having fixed ends, the string being thus divided into four equal portions each of length

a

and under a tension

T

. Show that for small transverse vibrations the amplitudes

x^i

of the normal modes satisfy

\mathsf{B}\mathbf{x} = (ma\omega^2/T)\mathbf{x}

, where

\mathsf{B}

is the matrix

\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}.

Estimate the lowest and highest eigenfrequencies using trial vectors

(3 \quad 4 \quad 3)^{\mathrm{T}}

and

(3 \quad -4 \quad 3)^{\mathrm{T}}

. Use also the exact vectors

\left(1 \quad \sqrt{2} \quad 1\right)^{\mathrm{T}}

and

\left(1 \quad -\sqrt{2} \quad 1\right)^{\mathrm{T}}

and compare the results.