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Unformatted text preview: I. Impulse Response and Convolution 1. Impulse response. Imagine a mass m at rest on a frictionless track, then given a sharp kick at time t = 0. We model the kick as a constant force F applied to the mass over a very short time interval < t < . During the kick the velocity v ( t ) of the mass rises rapidly from to v ( ); after the kick, it moves with constant velocity v ( ), since no further force is acting on it. We want to express v ( ) in terms of F, , and m . By Newtons law, the force F produces a constant acceleration a , and we get F (1) F = ma v ( t ) = at, t v ( ) = a = . m If the mass is part of a spring-mass-dashpot system, modeled by the IVP (2) my + cy + ky = f ( t ) , y (0) = , y (0 ) = , to determine the motion y ( t ) of the mass, we should solve (2), taking the driving force f ( t ) to be a constant F over the time interval [0 , ] and afterwards. But this will take work and the answer will need interpretation. Instead, we can both save work and get quick insight by solving the problem approxi- mately, as follows. Assume the time interval is negligible compared to the other parameters. Then according to (1), the kick should impart the instantaneous velocity F /m to the mass, after which its motion y ( t ) will be the appropriate solution to the homogeneous ODE asso- ciated with (2). That is, if the time interval for the initial kick is very small, the motion is approximately given (for t 0) by the solution y ( t ) to the IVP F (3) my + cy + ky = , y (0) = , y (0) = . m Instead of worrying about the constants, assume for the moment that F /m = 1; then the IVP (3) becomes (4) my + cy + ky = , y (0) = , y (0) = 1 ; its solution for t > will be called w ( t ), and in view of the physical problem, we define w ( t ) = for t < 0. Comparing (3) and (4), we see that we can write the solution to (3) in terms of w ( t ) as F (5) y ( t ) = w ( t ) , m for since the ODE (4) is linear, multiplying the initial values y (0) and y (0) by the same factor F /m multiplies the solution by this factor. The solution w ( t ) to (4) is of fundamental importance for the system (2); it is often called in engineering the weight function for the ODE in (4). A longer but more expressive name for it is the unit impulse response of the system: the quantity F is called in physics the impulse of the force, as is F /m (more properly, the impulse/unit mass ), so that if F /m = 1, the function w ( t ) is the response of the system to a unit impulse at time t = 0. 1 2 18.03 NOTES Example 1. Find the unit impulse response to an undamped spring-mass system having (circular) frequency ....
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- Fall '09