Hello. Here we are, with a lot of useful tools. First, we know that the dynamics of a system may be written in terms of modal coordinates. This modal decomposition and superposition allowed us to simplify the equation to solve. All we had to solve was a series of modal equations that were very similar. Then, we found out that all these equations were actually all the same. We only needed to solve the elementary oscillator equation. Schematically, as I said before, we went from concrete (that was the formulation in the physical space) to stones (that was the formulation in the modal space) and from stone to bricks when we showed that all modal displacements satisfied the same form of equation. Let us now use these bricks to build something. To keep things simple, I will be using the cable model, but we know the results are quite general. Here we are. In the physical space the unknown is y(x,t). We want to solve the equation of motion with initial conditions and external loading. Using modal decomposition this will be the sum of the modal coordinates times the modal shapes. Each modal coordinate is the solution of the modal equation, with initial conditions, with the projected loading, and this is related to the dimensionless oscillator equation our elementary brick. We know the solution of this elementary equation (q)ddot+q=f. It is the sum of the convolution product between the load and the unit impulse response G and of the response to initial conditions. q(t) = f*sin(t) + q_0 cos(t) + (q)dot_0 sin(t). I can use this solution and go back to each and every modal equation. For this I just need to use backward the change of variables I used before. Each mode had a different stiffness, a different mass. This resulted in a different time scale for each mode and a different scaling of the forces and of the initial conditions. I have then q_N(t) = f_N * G_N where G_N is the unit response on the mode N oscillator. Plus the response to the initial modal displacement and the response to the initial modal velocity. Here it is again. In these modal solutions I insert what I know about the modal loading and the modall initial conditions. Here is the loading, as a projection of the force. Here is the unit impulse response. Here is the initial modal displacement. And here is the initial modal velocity. And now by combining all the modal response I have the general solution of my problem of dynamics - the sum over of all modes of the q_N(t) phi_N(x). Quite a long formula, but quite general. Here is my formula again. This is a major result because it means that if I know the modes I know the solution of every problem of dynamics. Actually, linear dynamics, small motions, of course. Are we finished with dynamics? Alas not, because we have a problem here. We need to know all the modes of the system, and to compute the response on each and every mode. When I say all the modes I mean an infinite number of them. Remember that continuous systems have an infinite number of modes. For instance, the tensioned cable has an infinite number of modes and I would have to sum on all of them. There are some cases where I know all the modes. But even for this it might just impossible to compute the sum of all the modal responses. And in general this is just useless because to compute a response I first have to compute an infinite number of modes. What can we do ? Of course, there is a solution, and a very useful one. Used every day by engineers. Let us have a closer look at this sum.