18.03 Lecture Notes - Lecture 18: Damping Ratio, Stopwatch, Sine Wave
Document Summary
Applications in engineering: a visit by professor kim vandiver. [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. Hm: we factored the characteristic polynomial to analyse this. Factor out the m and complete the square: p(s) = ms^2 + bs + k 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*. Kv: yep, and that"s the only situation in which you get vibrations. B/2m +- i omega_d omega_d = sqrt( k/m - (b/2m)^2 ) . So the general solution to this homogeneous equation is x_p = a e^{-bt/2m} cos(omega_d t - phi )
