Dynamics: Harmonic Excitation
|SDOF Systems under Harmonic Excitation|
When a SDOF system is forced by f(t), the solution for the displacement x(t) consists of two parts: the complimentary solution, and the particular solution. The complimentary solution for the problem is given by the free vibration discussion. The particular solution depends on the nature of the forcing function.
When the forcing function is harmonic (i.e. it consists of at most a sine and cosine at the same frequency, a quantity that can be expressed by the complex exponential eiwt), the method of Undetermined Coefficients can be used to find the particular solution. Non-harmonic forcing functions are handled by other techniques.
Consider the SDOF system forced by the harmonic function f(t),
The particular solution for this problem is found to be,
The general solution is given by the sum of the complimentary and particular solutions multiplied by two weighting constants c1 and c2,
The values of c1 and c2 are found by matching x(t = 0) to the initial conditions.
|Undamped SDOF Systems under Harmonic Excitation|
For an undamped system (cv = 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 near-zeros 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 divide-by-zero. 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.
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