of Microprocessors

538 MUSICAL ApPLICATIONS OF MICROPROCESSORS
~,
SUPPORT
SPRING
1 VIBRATION
MOTION
",,0(A)
(B)
1. Sum forces: RTF + I ~ 0
2. Substitute: I(P) + KV + MA ~ 0
. . dp d 2 P _
3. Get In terms of P. f(P) T K dt + M (jj2 - 0
4. Get in terms of A: HJf A) + KfA + MA = 0
A = ACCELERATION OF MASS
I ~ INERTIAL FORCE
M = MASS
F = FRICTION FORCE
R = SPRING RESTORING FORCE
P = POSITION OF MASS REL TO NEUT.
V = VELOCITY OF MASS
f ~ SPRING RESTORING
FORCE FUNCTION
(LINEAR OR NON-LINEAR)
K = FRICTION COEFFICIENT
(SECOND ORDER DIFF EQUATION)
(SECOND ORDER INTEGRAL EQUATION)
Fig. 15-5. (A) Spring-mass vibrator. (8) Forces on the mass. A, acceleration of
mass; I, inertial force; M, mass; F, friction force; R, spring-restoring
force; P, position of mass relative to neutral; V, velocity of mass; f,
spring-restoring force function (linear or nonlinear); and K, friction
coefficient.
change) it is experiencing. Its direction opposes the sum of the other two
forces.
After the various forces are identified and written in equation form, it is
customary to rewrite the equation in terms of the primary variable, P (position),
instead of velocity and acceleration. This is easily done, since velocity
is the time derivative of position and acceleration is the time derivative of
velocity. The result is a standard second order differential equation.
Our goal, however, is to simulate the physical process described by the
equation, not "solve" it in the mathematical sense. In order to get the
equation into an easily handled form for simulation, it is better to write it in
terms of acceleration, A, rather than position. If this is done, velocity is