Damped Harmonic Oscillators In the last session we modeled a spring-mass-dashpot system with the constant coefﬁcient linear DE .. . mx + bx + kx = Fext , where m is the mass, b is the damping constant, k is the spring constant and x (t) is the displacement of the mass from its equilibrium position. Fext dashpot spring mass x x=0 x We then assumed the external force Fext = 0 and used the characteristic equation technique to solve the homogeneous equation .. . mx + bx + kx = 0. (1) Restrictions on the coefﬁcients: The algebra does not require any restric tions on m, b and k (except m �= 0 so that the equation is genuinely second order). But, since this is a physical model, we will now require m > 0, b ≥ 0 and k > 0. The Damped Harmonic Oscillator: The undamped (b = 0) system has equation .. mx + kx = 0. At this point you should have memorized the solution and also be able to solve this equation using the characteristic roots. The solution is x (t) = c1 cos(ωt) + c2 sin(ωt) = A cos(ωt − φ). √ Here ω = k/m and the solution is given in both rectangular and amplitudephase form. The solution is always a sinusoid, which we consider a simple oscillation, and we call this system a simple harmonic oscillator. 18.03SC x t Figure 2: The output of a simple harmonic oscillator is a pure sinusoid. When we add damping we call the system in (1) a damped harmonic oscillator. This is a much fancier sounding name than the spring-mass dashpot. It emphasizes an important fact about using differential equa tions for modeling physical systems. It doesn’t matter whether x measures position or current or some other quantity. Any system modeled by equa tion (1) will respond just like the spring-mass-dashpot; that is, all damped harmonic oscillators exhibit similar behavior. We will see an important ex ample of this principle whe we study the case of an RLC electrical circuit.

Damped Harmonic Oscillators

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