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Mass-Spring Boundedness in Reality

To approach the limit of $ \omega_0 = \sqrt{k/m} = 0$ , we must either take the spring constant $ k$ to zero, or the mass $ m$ to infinity, or both.

In the case of $ k\to0$ , the constant force must approach zero, and we are left with at most a constant mass velocity in the limit (not a linearly growing one, since there can be no dc force at the limit). When the spring force reaches zero, $ x_1(n)=0$ , so that only zeros will feed into the loop on the right in Fig.F.37, thus avoiding a linearly growing velocity, as demanded by the physics. (A constant velocity is free to circulate in the loop on the right, but the loop on the left must be zeroed out in the limit.)

In the case of $ m\to\infty$ , the mass becomes unaffected by the spring force, so its final velocity must be zero. Otherwise, the attached spring would keep compressing or stretching forever, and this would take infinite energy. (Another way to arrive at this conclusion is to note that the final kinetic energy of the mass would be $ mv^2/2=\infty$ .) Since the total energy in an undriven mass-spring oscillator is always constant, the infinite-mass limit must be accompanied by a zero-velocity limit.F.5 This means the mass's state variable $ x_2(n)$ in Fig.F.37 must be forced to zero in the limit so that there will be no linearly growing solution at dc.

In summary, when two or more system poles approach each other to form a repeated pole, care must be taken to ensure that the limit is approached in a physically meaningful way. In the case of the mass-spring oscillator, for example, any change in the spring constant $ k$ or mass $ m$ must be accompanied by the physically appropriate change in the state variables $ x_1(n)$ and/or $ x_2(n)$ . It is obviously incorrect, for example, to suddenly set $ k=0$ in the simulation without simultaneously clearing the spring's state variable $ x_1(n)$ , since the force across an infinitely compliant spring can only be zero.

Similar remarks apply to repeated poles corresponding to $ \omega_0=\infty$ . In this case, the mass and spring basically change places.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University