A taut cable of length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Function spaces and operator theory · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

L=0.90\ \mathrm{m}

, fixed at both ends, has uniform tension

T=45\ \mathrm{N}

and mass per unit length increasing linearly along its length,

\rho(x)=\rho_0(1+x/L)

with

\rho_0=1.0\ \mathrm{g\,m^{-1}}

. Its transverse modes satisfy

-Ty''=\lambda\rho(x)y

y(0)=y(L)=0

. ((a)) Using the two-term trial space

\{\phi_1,\phi_2\}=\{\sin(\pi x/L),\sin(2\pi x/L)\}

, form the stiffness and mass matrices

K_{ij}=T\int_0^L\phi_i'\phi_j'\,\mathrm{d}x

and

M_{ij}=\int_0^L\rho(x)\phi_i\phi_j\,\mathrm{d}x

. ((b)) Set up the generalised eigenvalue problem

\mathsf{K}\mathbf{c}=\lambda\mathsf{M}\mathbf{c}

and solve it for the two Ritz values of

\lambda

, which bound

\lambda_1

and

\lambda_2

of the true problem from above.

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