Higher Equations ODEs Models

Higher Equations ODEs Models

1. Mass-Spring-Dashpot

   • Free Damped Motion
     • x"+2px'+[(w0)^2]x = 0
       • p = c/(2m)>0
         • w0= sqrt (k/m)
   • Forced Case
     • mx" + cx' + kx = F(t)
       • Mechanical Eletrical Analogy
         • Electric Circuits
           • dQ/dT = I(t)
           • LQ" + RQ' + (1/C)Q = E(t)
             • LI" + RI' + (1/C)I = E'(t)
           • Steady Periodic Current Isp
             • Isp = [E0*cos(wt-a)]/[sqrt(R^2 + (wL -1/(wC)^2)]
         • Electrical Resonance
           • I= E0/Z = E/sqrt (R^2 + (WL -1/WC)^2)
             • Maximum
               • wm= 1/sqrt(LC)
               • wL - 1/(WC) = 0
                 • C = 1/(LW^2)
   • If c = 0, then it's undamped. If c>o, then it's damped
     • Critical Damping
       • Ccr = sqrt (4km)
         • C > Ccr - overdamped
           • C = Ccr - critically damped
             • C < Ccr - underdamped
   • Free Undamped Motion
     • mx"+ cx' + kx = 0
       • w0= sqrt (k/m)
     • Undamped Forced Oscilations
       • mx"+ cx' + kx = Fcoswt
         • x(t) = Acoswt + Bsenwt
           • For small t, xc= transient solution; For large t, xc =0, xp = steady oscilations

2. Oscilation

   • Period T =2pi/w0
     • Frequency F= 1/T
       • Time Lag = a/w0

3. Simple Pendulum

   • o" + o' +ko= 0, where k = g/L

4. Wave Properties

   • Resonance
     • amplitude A = (F /k)/(k-mw^2)
     • Amplification Factor p= 1/|1-(w-w0)^2|
   • Beats
     • w~w0
       • x(t) = F/m(w0^2 -w^2)
         • x(t)= [2F/m(w0^2-w)]*[sin(w0-w)t/2]*sin(1/2)(w0+w)t