### ODEs

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

(Force = Mass times Acceleration)

This is a second-order ODE relating the force on a mass at time to the second time-derivative of its position , i.e., . A physical diagram is shown in Fig.1.1. From this ODE we can see that a constant applied force results in a constant acceleration , a linearly increasing velocity , and quadratically increasing position . The initial position and velocity 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 for .

If the applied force is due to a spring with spring-constant , then we may write the ODE as

(Spring Force + Mass Inertial Force = 0)

This case is diagrammed in Fig.1.2.

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

(Spring + Friction + Inertial Forces = 0)

as depicted in Fig.1.3.

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

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