Notice that as you increase damping the

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Notice that as you increase damping, the pseudofrequency decreases, slowly at first, but faster as the damping approaches critical damping. At that instant, the pseudoperiod becomes infinite and you don't get solutions which cross the axis infinitely often. This is subtle but visible on "Damped Vibrations."
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_______________________________________ In the complex plane, the roots look like this: | -b/2 + i omega_d | * | | | | | -b/2 - i omega_d | * | | | I showed Poles and Vibrations, with B = 0. Question 1: If I move the roots the the left, 1. The amplitude of the solutions decreases 2. The pseudofrequency of the solutions decreases 3. The solutions decay to zero faster 4. None of the above, necessarily Question 2: If I move the roots towards the real axis, 1. The pseudoperiod increases 2. The amplitude of solutions decreases 3. The solutions decay to zero faster 4. None of the above, necessarily [2] The roots of s^s + bs + k , -b/2 +- sqrt( (b/2)^2 - k ), are :- -
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2/4 . complex roots . . | . . Re < 0 | Re > 0 .<----- repeated if k = b^ . | . . | . real < 0 . | . real > 0 .. | .. .. | .. ------------ ....... ------------> - b ^ | real, opposite sign |______ at least one zero root Here is a summary table of unforced system responses. One of three things must happen. I'll take m = 1 . k ^ . |<----- sinusoidal solutions . | . . | . . stable, | unstable, .<----- t e^{rt} too if k = b^2/4 . oscillating | oscillating . . | . stable . | . unstable, not oscillating not oscillating.. | .. .. | .. ------------ ....... ------------> - b ^ | most solutions grow |______ some nonzero constant solutions Question 3: You observe an unforced system oscillating, and notice that the time between maxima spreads out as time goes on. From this you can conclude: 1. The system can be modeled by the constant coefficient equation x" + bx' + omega_n^2 x = 0 where 0 < b < 2\omega_n. 2. This system is nonlinear. 3. This system is linear but the coefficients are not constant in time.
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  • Winter '08
  • Staff
  • Equations, general solution

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