Differential Equations for Engineers · Laplace transforms, discontinuous forcing, and impulses
Two masses m_1=m_2=1 lie on a frictionless line and are coupled by three springs of…
Problem
Two masses \(m_1=m_2=1\) lie on a frictionless line and are coupled by three springs of constants \(k_1=2\), \(k_2=3\), and \(k_3=2\), with the outer springs attached to fixed walls. Let \(x_1\) and \(x_2\) be displacements from equilibrium. Write the first-order system for \(\mathbf{u}=(x_1,x_2,x_1',x_2')^T\). Find the natural frequencies and the corresponding mode-shape vectors in the \((x_1,x_2)\)-plane.
Hint
Newton’s laws produce the second-order system \(\mathbf{x}''=-K\mathbf{x}\) with \(K=\begin{pmatrix}k_1+k_2&-k_2\\-k_2&k_2+k_3\end{pmatrix}\).
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