Forced Oscillations

Forced oscillation occurs when an external force causes an oscillator to oscillate. Accordingly, the applied force must also have a periodic course and can be described, for example, by $F(t) = F_0\cos(\omega t)$. Since the oscillator should also be damped, the following equation can be used to describe the movement: $$\ddot{x}(t) + 2\gamma\dot{x}(t) + \omega^2 x(t) = K\cos(\omega t)$$ This equation was again divided by $m$ and $F_0/m$ therefore replaced by $K$. We can now make the ansatz $$x(t) = A\cos(\omega t + \varphi)$$ where $A$ is the amplitude and $\varphi$ an arbitrary phase difference. Using the addition theorem for trigonometric functions results in $$[(\omega_0^2 - \omega^2)A\cos\varphi - 2\gamma A \omega \sin\varphi - K]\cos\omega t - [(\omega_0^2 - \omega^2)A\cos\varphi - 2\gamma A \omega \cos\varphi - K]\sin\omega t$$ Since this equation has to be true for all times $t$, the terms within the rectangular brackets have to be equal to 0. This leads to the following conditions: $$A(\omega_0^2 - \omega^2)\cos\varphi - 2A\gamma\omega\sin\varphi - K = 0$$ $$(\omega_0^2 - \omega^2)\sin\varphi + 2\gamma\omega\cos\varphi = 0$$ From the second equation, we can derive a condition for the phase difference $\varphi$: $$\tan\varphi = -\frac{2\gamma\omega}{\omega_0^2 - \omega^2}$$ Now we can solve the first equation for $A\sin\varphi$ and $A\cos\varphi$: $$A\sin\varphi = -\frac{2\gamma\omega K}{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}$$ $$A\cos\varphi = -\frac{(\omega_0^2 - \omega^2)K}{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}$$ In the final step, we can square both equations and add them up which results in the following very important formula for the amplitude $A$ (after replacing $K$ again with $F_0/m$. This equation is valid after the system of oscillator and exciter is not transient anymore. The two values ​​$\omega_0$ and $\omega$ denote the "natural" frequency of the oscillator and the excitation frequency. In this case, $\omega_0$ is also called the resonance frequency. of the oscillator. It is important to remember that a constant amplitude is set for all frequencies $\omega$. An interesting case occurs when resonance is present and $\omega = \omega_0$. Then the difference in the first term of the denominator vanishes, so the amplitude becomes maximum. If the oscillator were undamped, i.e. $\gamma = 0$, then the amplitude would become infinitely large, which is why one speaks of a resonance catastrophe. This can have devastating effects on bridges, for example. Strong winds and vibrations of the bridge can escalate each other so that the resulting amplitude causes the bridge to collapse. The road traffic regulations also prohibit walking in step over a bridge, since in the worst case a frequency close to resonance can occur, which can also lead to bridge damages.