For an undamped system (c_{v} = 0) the total displacement solution is,
If the forcing frequency is close to the natural frequency, , the system will exhibit resonance (very large displacements) due to the nearzeros in the denominators of x(t).
When the forcing frequency is equal to the natural frequency, we cannot use the x(t) given above as it would give dividebyzero. Instead, we must use L'Hôspital's Rule to derive a solution free of zeros in the denominators,
To simplify x(t), let's assume that the driving force consists only of the cosine function, ,
The displacement solution reduces to,
This solution contains one term multiplied by t. This term will cause the displacement amplitude to increase linearly with time as the forcing function pumps energy into the system, as shown in the following displacement plot,
The maximum displacement of an undamped system forced at its resonant frequency will increase unbounded according to the solution for x(t) above. However, real systems will inject additional physics once displacements become large enough. These additional physics (nonlinear plastic deformation, heat transfer, buckling, etc.) will serve to limit the maximum displacement exhibited by the system, and allow one to escape the "sudden death" impression that such systems will immediately fail.
