Special case : natural vibrations
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Recall that the finite element system of equations has the form
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We could also have written this equation as
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For natural vibrations, the forces and the displacements are assumed to be
periodic in time, i.e.,
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and
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Then, the accelerations take the form
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Plugging these into the FE system of equations, we get
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After simplification, we get
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If there is no forcing, the right hand side is zero, and we get
the finite element system of equations for free vibrations
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The above equation is similar to the eigenvalue problem of the form
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Since the right hand side is zero, the finite element system of equations
has a solution only if
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For a two noded element,
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Therefore,
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The determinant is
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This gives us a quadratic equation in which can be solved
to find the natural frequencies of the element.