class: Mathematical Modeling (need problem 13.1 ,overdamped case) 40 Mechanical

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class: Mathematical Modeling
(need problem 13.1 ,overdamped case)

40 Mechanical Vibrations 13. Overdamped and Critically Damped Oscillations On the other hand, if the friction is sufficiently large, then and we call the system overdamped. The motion of the mass is no longera decaying oscillation. The solution of equation 11.1 is where r, and r2 are real and both negative, 2mt If the friction is suffciently large, we should expect that the mass decays to its equilibrium position quite quickly. Exercise 13.1 shows that it does not oscillate. Instead, the mass either decays to its equilibrium position as seen in Fig. 13-1(a) or (b), or it shoots past the equilibrium position exactly once before returning monotonically towards the equilibrium position as seen Fig. 13-1(c): in or 0 0 0 Figure 13-1 Overdamped oscillations.

Explanation / Answer

given c^2 > 4mk

then r1 , r2 = (-c +- sqroot(c^2 - 4mk))/2m

where x = c1*e^(r1*t) + c2*e^(r2*t) is the solution

a. so, for x = 0

c1*e^(r1*t) = - c2*e^(r2*t)

also, at t = 0, amplitude = A

then A = c1 + c2

so

c1*e^(r1*t) = - (A - c1)*e^(r2*t) = -A*e^(r2*t) + c1*e^(r2*t)

c1(e^(r2*t) - e^(r1*t)) = A*e^(r2*t)

(1 - e^(r1 - r2)t) = A/c1

(1 - A/c1) = e^(r1 - r2)t

ln(1 - A/c1) = (r1 - r2)t

so, r1 - r2 = (-c + sqroot(c^2 - 4mk))/2m - (-c - sqroot(c^2 - 4mk))/2m

r1 - r2 = +-(sqroot(c^2 - 4mk))/m

when r1 - r2 > 0

then at t = ln(1 - A/c1)/(r1 - r2) the mass crosses x = 0 , once

if r1 - r2 < 0

then the mass will never cross x = 0 and always asymptotically reach x = 0

b. if initial position is xo

then the mass crosses intiial position at some time t

x(t) = c1*e^(r1*t) + c2*e^(r2*t

v(t) = c1*r1*e^(r1t) + ce*r2*e^(r2*t)

at t = 0

Vo = c1r1 + c2r2

at t = t, if mass crosses the x = 0 mark

t = mln(1 - A/c1)/sqroot(c^2 - 4m)

v(t) < 0

this can happen only if initially the mass had suffucent velocity Vo

hernec hte mass crosses the x = 0 mark only if the intitial velocity Vo is sufficently negative

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