the resulting equation of motion is similar to the free vibration case but is a non-homogeneous differential equation its solution is the sum of two functions a complementary solution and a particular solution the complementary solution is the solution to the homogeneous form of the equation where the right hand side is equal to zero this is just the solution to the equation of motion for an underdamped system in free vibration that we saw earlier and the particular solution captures the effect of the external loads and is given by this expression r is the ratio of the frequency of the external force to the natural frequency of the model
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