Hello! Let us use modal superposition with truncation to solve some simple problems of the dynamics of beams in bending. You computed the first bending modes of a cantilevered beam. You know what they look like. The first mode and the second mode. And the third mode and so on. Imagine now, that we want to compute the response of the beam to a local impact, like this. Let us have an impact at the tip of the beam, and a rather soft impact first. By soft I mean that the duration is rather long. What will be the contribution of each mode? As you know this is the modal equation. The projected force fn is the product between the load and the modal shape. As the modal shape is 1 at the tip of the beam, we just have to solve m_N (q_N)ddot +k_N q_N = f(t) for each mode, where f(t) is the simple step evolution. How will the modal coordinate q_N evolve as a function of time? Solving a simple oscillator equation numerically is straighfoward. For instance, I can use a simple numerical scheme like here. q_N is discretized with a time step delta t. The modal acceleration is approximated with a centered difference formula. And we can compute qn step by step, as q_N^(p+1) is given as a function of the previous steps and of the loading. What do I get? Here is my loading. Here are q_1, q_2 and q_3. Look, q_1 oscillates but q_2 and q_3 are just about zero! The response is almost only on mode 1. Why is that? Because modes 2 and 3 are excited in their very quasistatic range. When the loading is ended the modal oscillators of mode 2 and 3, q_2 and q_3, just go back to zero. With that I can plot a very good approximation of the response of the beam as q_1(t) times phi_1(x). Here it is. If I added the other modes, nothing would change. A truncation with only one mode is enough. Let us now see what happens with the same impulse but on a much shorter time. A hard impact instead of a soft impact. The impulse is the same, which means that the sum of f(t) is not changed. The force is larger, but on a shorter time. Here are q_1, q_2, q_3. We have quite a different situation here. There is a lot of dynamics in modes 2 and even in mode 3. And here is the recombined motion with all the components. Quite different from the previous one. But even with just 3 modes, it is quite simple to get this rich response. This dynamics combining response on 3 modes is quite complex. What would happen if there was a bit of modal damping on each mode? How would I take that into account? In the modal equation all I need is to add a linear term proportional to the velocity, right here in the algorithm. That would be easy to approximate using centered differences. You can easily see how the time-stepping algorithm will be modified. How much damping should I take? Well, that depends on the cause of the damping, whether it comes from the mechanical interactions at the clamping, or the interaction with the environment. Let us take a high level on mode 1, and nothing on the other modes. Here are the modal responses. As you can see mode 1 is quite damped, but the other ones are unchanged. What would the superposition look like? Here is the beam response to the hard impact without damping. Here is the response with a damped first mode. At first the response is dominated by mode 1 but later only mode 2 remains. Finally, what would happen if I hit the beam at a different point? For instance, right here. Why right here? Because this is where mode 2 has no deformation. As a result the response does not contain any part of mode 2. And if the impact is close to the base of the beam, there is a lot of mode 2 and less of mode 1. Imagine that the impact is that of a ball, and that the beam is a bat. Baseball, or cricket, or any other game. A bat held in hands behaves like a free-free beam. You know the modes of a free-free beam. Here they are. If you hit the ball right in that zone here, the first two modes will have a same, small, contribution. Actually, if the beam is of variable thickness, like the baseball bat, we can achieve that mode 1, 2 and 3 have a zero modal shape at the same point, which is called the soft spot. This is how a base ball bat is designed. At the soft spot, if you hit the ball right here, the rebound is excellent because no energy is transferred into the vibration of the bat! But if you hit somewhere else, the bat shakes a lot. You have the same problem in tennis, in cricket, and many other sports. You should hit the ball at the right place. On this simple example of a cantilevered beam, you see that complex dynamics effect can be easily explained and predicted very simply with modal superposition and truncation. The system is a continuous system, but its dynamics is modeled by just a few oscillators.
