(16) Three Spring-Coupled Masses

(16) Three Spring-Coupled Masses - Three Spring-Coupled...

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Three Spring-Coupled Masses Consider a generalized version of the mechanical system discussed in Section 4.1 that consists of three identical masses which slide over a frictionless horizontal surface, and are connected by identical light horizontal springs of spring constant . As before, the outermost masses are attached to immovable walls by springs of spring constant . The instantaneous configuration of the system is specified by the horizontal displacements of the three masses from their equilibrium positions: namely, , , and . This is manifestly a three degree of freedom system. We, therefore, expect it to possess three independent normal modes of oscillation. Equations ( 149 )-( 150 ) generalize to (202) (203) (204) These equations can be rewritten (205) (206) (207) where . Let us search for a normal mode solution of the form (208) (209) (210) Equations ( 205 )-( 210 ) can be combined to give the homogeneous matrix equation
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(211) where . The normal frequencies are determined by setting the determinant of the matrix to zero: that is, (212) or (213)
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