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Unformatted text preview: APPM 2360 Differential Equations Lab 2: Harmonic Oscillator with Modified Damping Michael Blanchard 810076309 Professor: David Bortz T/A: Jason Hammond 10AM 023 Ben Maples 810131335 Professor: Will Heuett T/A: Will Heuett 2PM 035 Aaron Willoughby 830039918 Professor: Will Heuett T/A: Sean Nixon 11AM 032 I. Introduction In order to gain a better understanding of ordinary differential equations we will take a deeper look at a model of a harmonic oscillator with modified damping. Because damping can take different effects on an oscillating system we will study systems that are over damped, critically damped and oscillatory damped. To start off we will solve an initial value problem analytically and examine the phase portrait. After we have an understanding of how these ODE’s are affected by the coefficients that represent the initial conditions we will compare two models and see how each condition impacts the long term behavior of the system. II. Body A. Harmonic Oscillator Model A simple harmonic oscillator can be described by the following ordinary differential equation (ODE): Equation 1: ) ( t F kx x a x m = + + This equation can be classified as a second order linear nonhomogeneous differential equation with forced and damped motion. In this equation, m represents the mass connected to the spring with units of mass (kilograms, slug, etc.); a represents the damping of the system with units of velocity force ( m s N ⋅ , ft s lb ⋅ , etc.); k represents the 2 restoring force of the spring used in the system with units of nt displaceme force ( m N , ft lb , etc.); F ( t ) is the external force acting on the system with units of force ( N , lb , etc.). The x terms of the ODE can be described as follows, x represents the acceleration of the mass with respect to time having units of 2 time nt displaceme ( 2 2 , s ft s m ,etc.), x represents the velocity of the mass with respect to time having units of time nt displaceme ( , , s ft s m etc.), and x represents the position of the mass at any given time with units of length ( m, ft, etc.) . The configuration of the mass spring system represented above can be described as a horizontal spring connected to a wall at one end with a mass ( m ) connected at the other resting on the ground. There is an external force F ( t ), such as a rocket, acting on the system pushing it either to the left or right. See figure 1 below. Figure 1. 3 Equation 1 can be written as a first order linear differential equation by using the substitutions x v = , and x u = . The resulting equation is displayed below as equation 2. Equation 2: + = = ) ( ' ' t F ku v m a v v u From this system of equations, a, m, and k are the same as in equation 1. The variable v is representing the velocity of the mass, v’ is representing acceleration, and u represents position....
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 Spring '07
 WILLIAMHEUETT
 ORDINARY DIFFERENTIAL EQUATIONS

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