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Unformatted text preview: Lecture 15 An Application of Eigenvectors: Vibrational Modes One application of ew s and ev s is in the analysis of vibration problems. A simple nontrivial vibration problem is the motion of two objects with equal masses m attached to each other and fixed outer walls by equal springs with spring constants k , as shown in Figure 15.1. Let x 1 denote the displacement of the first mass and x 2 the displacement of the second, and note the displacement of the walls is zero. Each mass experiences forces from the adjacent springs proportional to the stretch or compression of the spring. Ignoring any friction, Newtons law of motion ma = F , leads to: m x 1 =- k ( x 1- 0) + k ( x 2- x 1 ) =- 2 kx 1 + kx 2 m x 2 =- k ( x 2- x 1 ) + k (0- x 2 ) = kx 1- 2 kx 2 . (15.1) Dividing both sides by m we can write these equations in matrix form: x =- A x , (15.2) where A = k m B = k m parenleftbigg 2- 1- 1 2 parenrightbigg . (15.3) For this type of equation, the general solution is: x ( t...
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