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Unformatted text preview: Example 5  free response of a rod with end spring Find the general form of the free vibration solution of a fixed/spring end homogeneous rod having an elastic modulus E, crosssectional area A and mass/volume ! . x = 0 x=L u(x,t) E, ! , A K The characteristic equation (CE) for the fixedspring rod problem was found in the lecture notes to be: tan ! L = "# ! L ( ) where ! = KL EA . The roots of the CE are those values of " L for which the left hand side tangent function intersects the linear function in " L on the right hand side of the CE. These roots are shown as the blue dots in the figure below. There are no known closedform solutions of the above CE. However, we can make some qualitative arguments about the roots that will help us in the numerical solution of the CE and in gaining some qualitative understanding of the response. These are listed below: a) The jth root of the CE, ( " L) j , is bounded by (see above plot): (2j ! 1) " 2 < # L ( ) j < j " This observation will be helpful for providing you with initial guesses for the...
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 Fall '09

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