Three carts sit on a frictionless track

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

Three carts sit on a frictionless track, connected in series by four identical springs of stiffness

k = 12\,\mathrm{N\,m^{-1}}

: wall to cart 1, cart 1 to cart 2, cart 2 to cart 3 and cart 3 to wall. Cart 2 is a light connector of negligible mass; carts 1 and 3 have masses

m_1 = 2\,\mathrm{kg}

and

m_3 = 3\,\mathrm{kg}

. Let

x_1, x_2, x_3

be the small displacements from equilibrium, so

\mathsf{M} = \mathrm{diag}(2, 0, 3)\,\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_2

is a static constraint rather than a dynamical equation, and use the fact that a square matrix is either invertible with a unique solution for every right-hand side or has a null space of dimension at least one (the Fredholm alternative) to justify solving it uniquely for

x_2

in terms of

x_1

and

x_3

. State the result for

x_2

. ((b)) Find the reduced

2\times2

mass and stiffness matrices

\mathsf{M}_{\mathrm{red}}

\mathsf{K}_{\mathrm{red}}

for

(x_1, x_3)

and solve

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

for the two frequencies

\omega_1 < \omega_2

. ((c)) Find the two mode shapes and verify, as the spectral theorem for the self-adjoint operator

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

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

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