# Forced oscillations

Suppose we are using a pendulum. From one extremity we just leave it, it reaches the other extremity and again reaches us at a certain time period. We will be able to apply the next force to it when it reaches us after a certain time period and hence we will be able to account for the damping force which it experiences. This simply means that we are applying a force at a frequency which is equal to the natural frequency of the vibrating body. But if we now apply a force at a frequency which is other than the natural frequency of the vibrating body, it is written in our book that its amplitude decreases. I searched for this doubt and I got that as the force is not in the same frequency as the natural frequency of the vibrating body the force is not applied at the time when the body is at the vicinity of the force. And so due to damping force its amplitude decreases, but we were said that a vibrating body acquires the frequency of the external periodic force, so the force should act at the correct point of time when the body is in its vicinity; as a result its amplitude should increase rather than decrease, but we do not find it so. Why? I request you give a lucid explanation to this because I am a tenth grade student. Please don't use any complicated terminologies.

• My professor explained it to me as: He compared the pendulum example to a child swinging on a swing and we apply a force to it. In order to reach the maximum amplitude( of the swing) the force frequency should match the child swinging frequency . Jun 20 '20 at 11:07

You seem to be considering what happens when you apply pulses of force (that is forces for very short periods of time). This is what you might do when pushing a child on a swing in order to increase the amplitude of swing. It then seems obvious that to get the maximum amplitude you need to apply the pulses at the natural frequency, $$f_0$$, of the swing (and at the 'right' point in its cycle) so that the natural oscillations are 're-inforced'.

The resonance curves that you see in textbooks are not drawn for pulsed forces (applied to a system with its own natural frequency). Indeed it is clear that you won't get the classic shape of curve if you use such forces. For suppose you apply a pulse at a frequency $$f_0/2$$, that is once every two cycles of natural oscillations. This will also lead to an increase in amplitude, so the resonance curve will have another peak at frequency $$f_0/2$$. There will be further peaks at each subharmonic ($$f_0/n$$).

Resonance curves, as usually understood, are drawn for the case of an applied force which varies smoothly throughout the cycle, a so-called 'sinusoidally' varying force. In this case there is only a single peak to the resonance curve. [It's more useful to consider a sinusoidal force that a series of regular pulses because it's extremely convenient to regard any shape of oscillation as a sum of sinusoids of different frequencies and amplitudes.] Unfortunately, though, with a sinusoidal driving force it's harder to get a simple intuitive feel for why the resonance curve has exactly the shape it does. Instead we use Newton's second law and proceed using mathematics.

So if I understand correctly we have a pendulum and we are pushing it for only a small time period every oscillation. Like pushing a child on a swing where the duration of the force is small compared to the period of the swing.

For example let's take a pendulum with a natural period of 1s. So if you leave it alone it completes a cycle every second. Now if we push it with a slightly larger period, say 1.1s, the following happens: when we push we will be late. So during some oscillations our push will be 'in phase' with the pendulum and during others our push will be 'out of phase'. With 'in phase' I mean that my push is in the same direction as the motion of the pendulum. In the example of the swing this would be pushing the child when it starts to move down. Out of phase means the opposite. It would be like pushing the kid when it comes towards me.

During the 'in phase' pushes I will be adding energy to the system so the amplitude will grow larger and during the 'out of phase' pushes I remove energy from the system and the amplitude will go down. Because my frequency of pushing doesn't match the natural frequency of the pendulum whether I'm in or out phase changes. Correspondingly the amplitude will grow bigger and smaller.

If you consider the largest amplitude that you achieve as a function of driving frequency you will get that the closer you are to the natural frequency, the larger the amplitude will be. In physics we often consider a simpler type of driving force $$F_{drive}(t)=A\sin(2\pi f_{drive}t)$$ where $$f_{drive}$$ is the driving frequency. It sounds more complicated but when you plug it into the math it actually becomes easier. You can then make the following plot

In the plot $$\zeta$$ means the amount of friction. On the horizontal axis you see the driving frequency where 1 corresponds to the natural frequency. On the vertical axis you see the amplitude after the system has reached steady state. Steady state means that the waited long enough so the oscillation doesn't change anymore. The nice thing is that, for a perfect pendulum, if you drive it using a sinusoidal force the motion will also be a sine wave (in steady state). In steady state the frequency of the motion is the same as the driving frequency.

Note: in the plot you can see that for very low frequencies the amplitude is still one. In this case the pendulum follows the driving force. It is like you are pushing a swing so slowly that you are basically moving the kid without any swinging happening.

Note 2: I'm not sure what the resulting frequency will be when you force for small periods of time like in your question. This case is actually considerably harder than the sinusoidal case.

• Applying impulses is not really "harder" than the sinusoidal case, just different. The impulse is equivalent to a change of momentum, which for a constant-mass pendulum means an instantaneous step change in velocity proportional to the impulse.. Jun 20 '20 at 14:59

A resonant driving force will cause the amplitude to increase until some damping force (such as friction) removes energy as fast as it is being added. There is an infamous example: On a very windy day a long suspension bridge near Tacoma started to oscillate (standing torsional waves). The changes in shape caused the forces from the wind to get into resonance. The damping forces were inadequate, and the bridge came apart.