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The University of Alabama at Birmingham (UAB)
Department of Physics
PH 461/561 – Classical Mechanics I – Fall 2005
Assignment # 8 Due: Thursday, October 20
- As studied in class, the damped harmonic oscillator allows three types of solutions:
Weak damping: γ< ω 0 : x ( t )= Ae −γ^ t cos(ω dt + φ)
(underdamping)
Critical damping: γ = ω 0 : x ( t )= C 1 e −γ^ t^ + C 2 te −^ γ t
Strong damping:
t t x t Ce Ce
−^ − − −^ + −
= +
2 02 2 02 0 : () 1 2
γ γ ω γ γ ω
(overdamping)
Exponential factors appear in all three solutions and determine the decay rate of the motion in each case. An inspection of the above equations reveals that the decay parameter that dominates the decrease in amplitude for each case is as follows:
Weak damping: γ < ω 0 : (decayparameter)= γ
(underdamping)
Critical damping: γ = ω 0 : (decayparameter)= γ
Strong damping: γ >ω 0 : (decayparameter)=γ− γ^2 − ω 02
(overdamping)
Note: In the case of strong damping, the decay parameter is chosen as the smallest of the two decay rates, because it dominates the decay for large t.
a) For fixed ω 0 , sketch the behavior of the decay parameter as a function of γ
for 0 <γ<∞.
Your sketch should: i. Verify that the decay parameter for an overdamped oscillator decreases with increasing γ. ii. Indicate the value of γ for which the decay parameter is maximum.
b) Explain the meaning of the maximum in the value of the decay parameter.
- Verify that the function x ( t )= te −^ γ t , is indeed a second solution of the equation of
motion for a critically damped oscillator ( γ = ω 0 )
- Find the rate of change of the energy E = 12 mx &^2 + 21 kx^2 for a damped oscillator and
show that the dE dt is (minus) the rate at which energy is dissipated by the damping force − bx &.
- A mass m subject to a linear restoring force − kx and damping − bx &is displaced a distance x 0 from equilibrium and released with zero initial velocity. Find the motion in the underdamped, critically damped, and overdamped cases.
- Solve Problem 4 for the case when the mass starts from its equilibrium position with an initial velocity v 0. Sketch the motion for the three cases.
- Solve Problem 4 for the case when the mass has an initial displacement x 0 and
initial velocity v 0 directed toward the equilibrium point. Show that for a large
enough value of v 0 (namely if v 0 > (γ +β) x 0 ;whereβ= γ^2 − ω 02 ), the mass
will overshoot the equilibrium in the critically damped and overdamped cases.
Sketch the motion in these cases.
- A mass of 1000 kg falls from a height of 10 m over a platform of negligible mass. One is interested in designing a spring/shock absorber system on which the platform will be mounted, such that the platform will reach a new equilibrium position 0.2 m below its original position as quickly as possible after the impact and without going beyond it (See Figure in the next page).
a. Find the spring constant k and the damping constant b of the shock absorber. ake sure the solution x ( t ) found satisfies the correct initial conditions and that the platform does not go beyond the new position of equilibrium. (i.e., ensure there is no overshooting).
b. Determine, up to two significant digits, the time it takes for the platform to position itself within 1 mm of its final position.
