We have seen that the general solution of our problem of dynamics can be expressed as a sum of all modal contributions. Certainly some of the terms of this sum are more important than others. Let us go back to the tensioned cable. For instance imagine that the loading is a point loading F(t) at a point says x_F, and that I want to compute the response of the cable y at a point x_R. Let us take zero initial conditions for the sake of clarity. The contribution of a mode N to this response will be the product q_N(t) phi_N(x_R). q_N(t) is given by the convolution product between the projected loading and the impulse response. The projection of the loading is just F(t) phi_N(x_F). To summarize, the contribution of mode N is phi_N(x_F) [F(t)*(impulse response)] phi_N(x_R). How large is that contribution when compared to the contribution of another mode? There are many terms in this contribution. First, it depends on phi_N(x_F), the value of the modal shape at the point of excitation. If the excitation is at a point where the modal shape is small then the contribution will be small. It may even be zero. So, this may be very different from one mode to another. Second, it depends on the convolution between the time evolution of the loading and that of the unit impulse response of the mode. As you know this may also vary a lot. For instance, imagine a harmonic loading. For some modes it might fall in the domain of quasistatic response, in others of inertial response and for one of them, in pure resonance. This term may of course vary hugely between modes. And finally, you have to combine it with the modal shape at the point of response phi_N(x_R). It all depends where the point is, but you can clearly imagine that even if the response of mode N is high it does little contribute to the motion at point x_R if phi_N is small there. I can do something similar for the part of the response that come from initial conditions. Here are the initial conditions in the modal space that we derived some time ago. The initial condition on q_N the displacement, and on (q_N)dot the velocity. Both involve the inner product between the shape of the initial condition and the modal shape. So, when we transfer into the modal space an initial condition it might give nothing on a particular mode and a lot on another one. Here is an example. When you play the harp you pull on a tensioned cable, and release it. The initial condition is like this. Here are the first two modes of the tensioned cable. You remember that the inner product is the sum over all the cable of the product of the two. Here the product of y_0 with phi_2 will be about zero. Only the first mode will move. It you pluck the string at a different location you will hear something different because there will be a bit of the mode 2 response, the second harmonic. Let us summarize. The general solution of our problem of dynamics involves modal superposition. This involves the sum of the response of all modes of the system. But we have just seen that the contributions of the modes to the response may vary a lot from one mode to another, for many reasons. The sum on all the modes might be actually dominated by a few ones. Let us introduce the idea of modal truncation. By this I mean retaining just the right number of modes that I need to have a good enough representation of the response. Of course, this will be an approximation of the reponse, but we always do approximations, for instance in using a tensioned cable model for a complex system. So, there is no use in refining so much the response. Instead, of summing over all modes, from N =1 to infinity, a total modal superposition, let us approximate the displacement as the sum over a limited series of modes, say N_1, N_2 up to N_p. These might be any numbers because there is no reason that the relevant modes are just the first ones. Here we have truncated the modal sum. This is a truncated modal superposition. Schematically, here are all my modes of my system, as bars of an endless xylophone. What is modal truncation? Well, for a given piece of music I only need to use a certain set of them. I can just disregard all the others. And play on a my subset, my truncated modal space. But of course, the subset might be different for another piece of music. How should we do the truncation? What are the modes that we should take into account in a particular case? I say in a particular case because, as you could see before, the respective contribution of modes varies a lot from case to case of loading or initial condition. Here are two simple rules. First, when you go higher and higher in mode numbers, you can expect that the contribution is going to decline at some point. Why? Because, once you reach high frequencies, much higher than the content of the loading, the response of each mode is going to be static. And the stiffness of the modes increase, to that their influence is going to be limited. That first rule would be to stop including modes far above the frequency content of the loading. This is frequency filtering. This means cutting the endless xylophone somewhere, but you may still have a lot left. The second rule is much more important and relies on your understanding of the physics. It says to remove all modes that for a reason of another, are expected to have a smaller contributions. This is removing bars in your cut xylophone. With that, you may be left with only a few relevant ones. Let us conclude. Describing the response as a sum over all modes did not seem to be very practical. First, because we cannot find all the modes of a system, and second because even if we could we cannot compute all the modal displacements. There is an infinity of equations. But in most cases only some of them really contribute to the dynamic response. There is no need to take them all into account. We can work in a truncated space. Next, you shall how efficient this is on some simple problems.
