Energy-Preserving Parameter Changes (Mass-Spring Oscillator)

If the change in or is deemed to be ``internal'', that is, involving no external interactions, the appropriate accompanying change in the internal state variables is that which conserves energy. For the mass and its velocity, for example, we must have

where denote the mass values before and after the change, respectively, and denote the corresponding velocities. The velocity must therefore be scaled according to

since this holds the kinetic energy of the mass constant. Note that the momentum of the mass is changed, however, since

If the spring constant is to change from to , the instantaneous spring displacement must satisfy

In a velocity-wave simulation, displacement is the integral of velocity. Therefore, the energy-conserving velocity correction is impulsive in this case.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.