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Mass-Spring Oscillator Analysis
Consider now the mass-spring oscillator depicted physically in
Fig.D.3, and in equivalent-circuit form in
An ideal mass sliding on a
frictionless surface, attached via an ideal spring to a rigid
wall. The spring is at rest when the mass is centered at .
Equivalent circuit for the mass-spring oscillator.
By Newton's second law of motion, the force applied to a mass
equals its mass times its acceleration:
By Hooke's law
for ideal springs, the compression force
applied to a spring is equal to the spring constant
By Newton's third law of motion (``every action produces an equal and
opposite reaction''), we have
. That is, the compression
applied by the mass to the spring is equal and opposite to
the accelerating force
exerted in the negative-
the spring on the mass. In other words, the forces at the mass-spring
contact-point sum to zero:
We have thus derived a second-order differential equation governing
the motion of the mass and spring. (Note that in
Fig.D.3 is both the position of the mass and compression
of the spring at time .)
Taking the Laplace transform of both sides of this differential
To simplify notation, denote the initial position and velocity by
, respectively. Solving for
denoting the modulus and angle of the pole residue , respectively.
From §D.1, the inverse Laplace transform of is
, where is the Heaviside unit step function at time 0.
Then by linearity, the solution for
the motion of the mass is
If the initial velocity is zero (), the above formula
and the mass simply oscillates sinusoidally at frequency
, starting from its initial position .
If instead the initial position is , we obtain
Previous: Moving MassNext: Analog Filters
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/