# K purely imaginary 4 either 2 or 3 holds but we cant

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k ^ . |<-----purely imaginary

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__________________ 4. Either 2 or 3 holds but we can't say which. [3] INHOMOGENEOUS EQUATIONS I drew the spring/mass/dashpot system and added a force to it: the little blue guy comes back into play. mx" + bx' + kx = F_ext (*) Also important will be the "associated homogeneous equation" mx" + bx' + kx = 0 (*)_h Input signals we will study: Constant Sinusoidal Exponential, Exp times sinusoidal Polynomial Exp times other (eg polynomial) Sums of these General periodic functions (via Fourier series) The general strategy in finding solutions is: Superposition II: If xp is any solution to (*) and xh is a solution to (*)_h, then xp + xh is again a solution to (*). Proof: Plug x into (*): k) x = xp + xh b) x' = xp' + xh' m) x" = xp" + xh" mx" + bx' + kx = (m xp" + b xp' + k xp) + (m xh" + b xh' + k xh) = F_ext + 0 as we wanted. In fact, if xh is the general solution to (*)_h then xp + xh is the general solution to (*). This is to be compared with Superposition I: If x1 and x2 are solutions of a homogeneous linear equation, then so is any linear combination c1 x1 + c2 x2 .
Superposition II splits the problem of finding the general solution to (*) into two parts: (1) find SOME solution to (*), a "particular solution," and then (2) find the general solution of (*)_h (which we have worked on for a while). [4] First case: harmonic sinusoidal response. Drive a harmonic oscillator by a sinusoidal signal: x" + omega_n^2 x = A cos(omega t) (**) There are two frequencies here: the natural frequency of the system and the frequency omega of the input signal. I showed what happens with a weight on a rubber band: for small omega the weight follows the motion of my hand; it passes "resonance," where the response amplitude is large; and when omega is larger the response is exactly anti-phase. Why? And what's this resonance?

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• Winter '08
• Staff
• Equations, general solution

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