MIT18_03S10_c13

MIT18_03S10_c13 - 18.03 Class 13, March 3, 2010 Forced...

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Forced linear equations [1] Superposition II [2] Harmonic sinusoidal response [3] Exponential response formula [4] Sinusoidal response using complex replacement I drew the spring/mass/dashpot system and added a force to it: the little sail comes back into play. mx" + bx' + kx = F_ext (*) Notice by the way that I can put the damper on the left in parallel with the spring: it still opposes velocity. If x' < 0 , F_dash > 0 and so on: so you get exactly the same equation. Also important will be the "associated homogeneous equation" mx" + bx' + kx = 0 (*)_h which we know all about after Lecture 12. Final comment on this: We can "reduce" this by dividing by m: x" + (b/m)x' + (k/m) = 0 If b = 0 we get solutions with circular frequency sqrt(k/m) . If b > 0 , you get exponentially damped sinusoids, with smaller circular frequency omega_d (or not oscillating at all, if b is big enough). In general, even if b > 0 , we call sqrt(k/m) the "natural circular frequency" of the system, and write omega_n for it. So in the underdamped case, when there is an omega_d , omega_d < omega_n . So the reduced homogeneous equation is x" + (b/m) x' + omega_n^2 = 0 . [1] The general strategy in finding solutions is based on "superposition." [Slide:] Superposition I: If x1 and x2 are solutions of a homogeneous linear equation, then so is any linear combination c1 x1 + c2 x2 . If the equation is of second order and neither of x1 , x2 is a multiple of the other, then c1 x1 + c2 x2 is the general solution. Now we have: 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 (*):
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MIT18_03S10_c13 - 18.03 Class 13, March 3, 2010 Forced...

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