Three carts A, B, C sit in a line on a frictionless track

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 carts A, B, C sit in a line on a frictionless track, coupled by four identical springs of stiffness

k = 10\,\mathrm{N\,m^{-1}}

: wall to A, A to B, B to C and C to wall. Cart A is a light connector of negligible mass; carts B and C have masses

m_B = 3\,\mathrm{kg}

and

m_C = 5\,\mathrm{kg}

. With displacements

x_1

(cart A),

x_2

(cart B),

x_3

(cart C),

\mathsf{M} = \mathrm{diag}(0, 3, 5)\,\mathrm{kg}

and

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

. ((a)) Explain why the equation of motion for

x_1

is algebraic rather than dynamical, and use the invertibility of the relevant block (the Fredholm alternative) to solve it uniquely for

x_1

in terms of

x_2

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

(x_2, x_3)

and solve

\mathsf{K}_{\mathrm{red}}\mathbf{v}=\omega^2\mathsf{M}_{\mathrm{red}}\mathbf{v}

for

\omega_1 < \omega_2

. ((c)) Find the mode shapes and verify, via the spectral theorem for

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

, that they are orthogonal in the reduced mass inner product.

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