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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 springmassdashpot 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 springmass system having (circular) frequency ....
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 Fall '09
 vogan

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