In many mechanical problems a device is subjected to periodic external forces. For example, soldiers marching in cadence on a bridge cause periodic disturbances in the bridge, and the engines of a propeller driven aircraft cause periodic disturbances in its wings. In the absence of sufficient damping forces, such disturbances – even if small in magnitude – can cause structural breakdown if they are at certain critical frequencies. Consider the motion of an object in a spring–mass system without damping, subject to an external force $$F(t)=F_0\cos(\omega t)$$ where $F_0$ is a constant. In this case the equation of motion is $$my''+ky = F_0\cos(\omega t),$$ which can be rewritten as $$y''+\omega_0^2y=\frac{F_0}{m}\cos(\omega t)$$ with $\omega_0=\sqrt{k/m}$.
With the following simulation you can analyse the motion of an object in a spring–mass system without damping, subject to an external force $F(t)=F_0\cos(\omega t)$. The simulation also displays the solution of the initial value problem $$y''+\omega_0^2y=\frac{F_0}{m}\cos(\omega t),\quad y(0)=y_0,\quad y'(0)=v_0.$$
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The University of Queensland