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ODEs

Ordinary Differential Equations (ODEs) typically result directly from Newton's laws of motion, restated here as follows:

$\displaystyle f(t) \eqsp m\, \ddot x(t)$   (Force = Mass times Acceleration)

This is a second-order ODE relating the force $ f(t)$ on a mass $ m$ at time $ t$ to the second time-derivative of its position $ x(t)$ , i.e., $ a(t) \isdeftext \ddot x(t) \isdeftext d^2 x(t)/dt^2$ . A physical diagram is shown in Fig.1.1. From this ODE we can see that a constant applied force $ f(t)=f_0>0$ results in a constant acceleration $ a(t)$ , a linearly increasing velocity $ v(t)=\int a(t)\, dt$ , and quadratically increasing position $ x(t)=\int v(t)\, dt$ . The initial position $ x(0)$ and velocity $ v(0)$ of the mass comprise the initial state of mass, and serve as the boundary conditions for the ODE. The boundary conditions must be known in order to determine the two constants of integration needed when computing $ x(t)$ for $ t>0$ .

Figure 1.1: Physical diagram of an external force driving a mass sliding on a frictionless surface.
\includegraphics{eps/forcemassintro}

If the applied force $ f(t)$ is due to a spring with spring-constant $ k$ , then we may write the ODE as

$\displaystyle k\, x(t) + m\, \ddot x(t) \eqsp 0$   (Spring Force + Mass Inertial Force = 0)$\displaystyle .
$

This case is diagrammed in Fig.1.2.

Figure 1.2: Mass-spring-wall diagram.
\includegraphics{eps/massspringwall}

If the mass is sliding with friction, then a simple ODE model is given by

$\displaystyle k\, x(t) + \mu\, \dot x(t) + m\, \ddot x(t) \eqsp 0$   (Spring + Friction + Inertial Forces = 0)$\displaystyle .
$

as depicted in Fig.1.3.

Figure 1.3: Mass-spring-dashpot-wall diagram.
\includegraphics{eps/massspringdashpotwall}

We will use such ODEs to model mass, spring, and dashpot2.6 elements in Chapter 7.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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