So, we have solved the elementary brick of dynamics, the oscillator equation. We have the formula that tells us how an initial condition or a force is going to cause an evolution in time of the modal variable. But we need to understand how this works. Let us use our formula to see how an oscillator moves when we load it. In the formula that gives us the modal displacement q(t) we have three terms one that takes into account the force acting on the oscillator, one that takes into account the effect of the initial displacement of the oscillator and one that corresponds to the effect of the initial velocity of the oscillator. You remember that our oscillator here is not just a pendulum or a mass-spring system, it is any mode of any dynamical system! But to illustrate it you can imagine that q is the angle of a pendulum, that f is the force that I put on it, and q_0 and (q)dot_0 are respectively an initial angle and an initial angular velocity. Let us focus first on the effect of initial conditions, without any loading f. If we have just an initial condition in displacement q_0, the motion is just going to be q_0 cos(t). The oscillator is released and oscillates with a period 2 pi. If the initial condition is a velocity (q)dot_0 the oscillator oscillates with the same period, but the magnitude is now fixed by the initial velocity. More generally, if we take any initial condition combining q_0 and (q)dot_0 the solution is going to be q_0 cos(t) + (q)dot_0 sin(t). This motion maybe plotted in what is called a phase plane, where the two axis are the displacement q and the velocity (q)dot respectively. The initial conditions correspond to a given state point in that space. And then with time the state evolves at a constant velocity along a circle in the phase plane. This is really simple. Let us look at the other part of the formula, where the loading f(t) creates the motion. The solution is the convolution product between the force f and sin(t), which I can also write like this, with f(tau) sin(t-tau). There is a special case of interest, called the impulse loading. What is it? Imagine that the loading f actually stops after some time, say t_f. What happens then? Well then, the convolution integral stops at tau = t_f because there is no loading after that. Now, something special happens if the duration of that loading is very short. What do I mean by very short? I mean short versus the period of the oscillator, which is 2 pi. Why? Becauses in that case tau is very small versus the time of interest, t. I can then write that sin(t - tau) is close to sin(t). Let us go back to the response of the oscillator. Now because there is no more tau in it, I can take the sin out of the integral and we just have q(t) equals the sum of the loading over time, times sin(t). It is no more a convolution! It is just the sum of the loading (which is a quantity I know) times sin(t)! When you look at this response, you see that it is exactly the same as the response to an initial velocity (q)dot_0, where the pendulum is set into motion by a velocity, and then oscillates. But what is the equivalent of the initial velocity here, what sets the oscillator into motion? It is called the impulse of the loading, I, the sum of the loading over time. This is interesting because it means that we do not need to consider the fine evolution of the force, just the sum of it, its corresponding impulse. This type of loading, impulse loading, defined by a short duration is quite common in vibrations. For instance, when a plane lands, the first contact results in an impulse loading, that is going to set all the vibration modes of the plane into motion. In acoustics we have the same thing with cymbals here that are going to impulsively load the acoustic modes of your concert hall and of course, we have also impulse loading on free surfaces. When you dive, you impulsively load the sloshing modes of the pool! In these three cases, the time of the loading is short versus the typical time of the dynamics of the modal oscillator. Let us go back to our impulse load response. It is made of two parts. The magnitude is set by the impulse I the time evolution sin(t). I can call this the Unit impulse response, G(t), because it would be the response of my system to an impulse of unit magnitude I=1. Let us use this to understand better the general solution of the oscillator equation. Here is again the response of the oscillator, now to any loading f(t), as the convolution between f and sin(t), which is actually between f and G. f is any loading and G is the unit impulse response. What is that convolution? I can read it as follows: At a given time t, the response is the sum over all past tau times, of impulse reponses that were initiated by forces of magnitude f(tau). And these responses have been going on for a duration of (t - tau). This is exactly what convolution means - the sum of the effect of all the past history of loading. Let us summarize. We have been looking at the response of a free oscillator, without any loading, just initial conditions. The response was easy to understand - oscillations, at a magnitude that was set by the initial conditions, and at a period that was that of the oscillator, namely 2pi. Then we found that short loadings had exactly the same effect: a magnitude set by the level of the impulse, and then free oscillation. An finally, that led us to a better understanding of the response to any loading, by the convolution integral. But we still need to explore a very common case, that you find anywhere in vibrations, that of the harmonic loading, when the external force oscillates. The resonance issue! Let us do that next.
