Two carts of mass and sit on a frictionless track between three identical springs of…

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

Two carts of mass m1=2 kgm_1 = 2\,\text{kg} and m2=6 kgm_2 = 6\,\text{kg} sit on a frictionless track between three identical springs of stiffness k=6 N m−1k = 6\,\text{N m}^{-1}, one anchoring m1m_1 to a wall, one connecting m1m_1 to m2m_2, and one anchoring m2m_2 to a second wall. Small horizontal displacements x1,x2x_1, x_2 (from equilibrium) obey Mx¨=−Kx\mathsf{M}\ddot{\mathbf{x}} = -\mathsf{K}\mathbf{x} with M=(2006) kg,K=(12−6−612) N m−1.\mathsf{M} = \begin{pmatrix}2 & 0\\ 0 & 6\end{pmatrix}\,\text{kg}, \qquad \mathsf{K} = \begin{pmatrix}12 & -6\\ -6 & 12\end{pmatrix}\,\text{N m}^{-1}. ((a)) Solve det⁡(K−ω2M)=0\det(\mathsf{K} - \omega^2\mathsf{M}) = 0 for the two angular frequencies ω1<ω2\omega_1 < \omega_2, and find the corresponding mode shapes normalised so that the first component is 11. ((b)) Compute both the ordinary dot product v1⋅v2\mathbf{v}_1\cdot\mathbf{v}_2 and the mass inner product v1TMv2\mathbf{v}_1^{\mathrm{T}}\mathsf{M}\mathbf{v}_2, and state which one vanishes.

Browse the problem library