MIT18_03S10_c18_transcript - 18.03 Class 18, March 14, 2010...

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18.03 Class 18, March 14, 2010 Applications in Engineering: A visit by Professor Kim Vandiver [1] Damping ratio [2] Measuring the damping ratio [3] Cell phones on vibrate [4] Extracting energy from a river [1] HM: In this unit, we've been studying the equation controlling a spring system m x" + b x' + k x = F_ext We began by thinking about the homogeneous case, the unforced or free system. If the system has any damping, then all these solutions die off; they are transients. This equation is surely too simple to be of any use in an engineering context, isn't it? KV: I use this every day. This is the oscillator equation. If it makes noise, it vibrates. How do you describe the solutions? HM: We factored the characteristic polynomial p(s) = ms^2 + bs + k to analyse this. Factor out the m and complete the square: p(s) = m(s^2 + (b/m) s + (k/m)) = m( (s + k/2m)^2 + (k/m - (b/2m)^2 ) If k/m > (b/2m)^2 the roots are imaginary, the system is *underdamped*. Is that an engineering term too? KV: Yep, and that's the only situation in which you get vibrations. So lets study that case today. HM: OK. So the root in that case are -b/2m +- i omega_d omega_d = sqrt( k/m - (b/2m)^2 ) . So the general solution to this homogeneous equation is
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x_p = A e^{-bt/2m} cos(omega_d t - phi ) Does that look familiar? KV: Well it's familiar, but now I begin to realize why students look at me and say hunh? I use different notation. When there's no damping you get the undamped, natural frequency omega_n = sqrt(k/m) Let's take omega_d and factor out the quantity omega_n : omega_d = omega_n ( 1 - (b/(2 omega_n m))^2 ) which I could write as = omega_n ( 1 - zeta^2 ) where zeta is called the *damping ratio*. HM: Let's try to re-express the original equation in terms of this zeta . Where's your expression for zeta? Ah, it's there; zeta satisfies b/(2 m) = omega_n zeta. So plugging back into the differential equation gives x" + 2 zeta omega_n x' + omega_n^2 x = 0 . Now you can see that critical damping occurs when zeta = 1 , underdamped when zeta < 1 . And we can write the solutions in terms of zeta : x_h = A e^{-omega_n zeta t} cos( omega_d t - phi ) By the way, what are the units of this zeta thing? I mean, omega_n has units 1/sec . What about zeta ? KV: Well, I'd look at the exponent, which must be dimensionless.
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This note was uploaded on 01/18/2012 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c18_transcript - 18.03 Class 18, March 14, 2010...

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