A vehicle's shock absorber strut has mass

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch The spectral theorem, degeneracy and the matrices that will not diagonalise

Problem

A vehicle's shock absorber strut has mass m=2 kgm = 2\,\mathrm{kg}, damping coefficient c=8 N s m−1c = 8\,\mathrm{N\,s\,m^{-1}} and spring constant k=8 N m−1k = 8\,\mathrm{N\,m^{-1}}, so its compression x(t)x(t) obeys mx¨+cx˙+kx=0m\ddot x + c\dot x + kx = 0. ((a)) Show that the strut is critically damped, write the equation as x˙=Ax\dot{\mathbf{x}}=\mathsf{A}\mathbf{x} with x=(x,x˙)\mathbf{x}=(x,\dot x), and show A\mathsf{A} has a single repeated eigenvalue with only one independent eigenvector. ((b)) Construct the Jordan form of A\mathsf{A} and hence find x(t)x(t) given that the strut is released from a compression x(0)=0.05 mx(0) = 0.05\,\mathrm{m} at rest. Evaluate x(0.5 s)x(0.5\,\mathrm{s}). ((c)) At the strut's mounting bracket, the potential energy of small lateral displacements has Hessian H=(7−3−37) N m−1\mathsf{H} = \begin{pmatrix}7&-3\\-3&7\end{pmatrix}\,\mathrm{N\,m^{-1}} about its equilibrium. Classify the equilibrium.

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