Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Time scale for the rapid transient

  1. Sep 16, 2011 #1
    The problem statement, all variables and given/known data

    Time scale for the rapid transient (overdamped bead on a rotating hoop).

    The governing equation m*r*φ'' = -b*φ' - m*g*sinφ + m*r*(ω^2)*sinφ*cosφ, can be reduced to ε(d^2φ/dτ^2) + dφ/dτ = f(φ).

    Using phase plane analysis it can be shown that the equation ε(d^2φ/dτ^2) + dφ/dτ = f(φ) has solutions that rapidly relax to the curve where dφ/dτ = f(φ). Here f(φ) = sinφ(γcosφ - 1), ε = (m^2*g*r)/b^2, τ = t/T, and T is chosen to be T = b/(m*g).

    a) Estimate the time scale T_fast for this rapid transient in terms of ε, and then express T_fast in terms of the original dimensional quantities m, g, r, ω, and b.
    b) Rescale the original differential equation, using T_fast as the characteristic time scale, instead of T_slow = b/mg. Which terms in the equation are negligible on this time scale?
    c) Show that T_fast ≪ T_slow if ε≪1. (In this sense, the time scales T_fast and T_slow are widely separated.)

    Relevant equations

    γ = (r*ω^2)/g
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?