Driven and damped oscillations

Damped oscillations

Problem:

The differential equation describing the displacement from equilibrium for damped harmonic motion is
md2x/dt2 + kx + cdx/dt = 0.
(a)  State the conditions and find an expression for x(t) for underdamped, critically damped, and overdamped motion.
(b)  Show that for underdamped motion, the ration of two successive maxima in the displacement x is constant.

Solution:

Problem:

A damped oscillator satisfies the equation d2x/dt2 + 2γdx/dt + ω02x = 0,  where γ and ω0 are positive constants with γ < ω0 (under-damping).  Assume that the equation of motion of a particle of mass m is given by this equation.  At time t = 0, the particle is released from rest at the point x = a. 
(a)  Find x as a function of time t and sketch of x as a function of t. 
(b)  Find all the turning points of the motion and determine the ratio of successive maximum values of x(t).
(c)  Re-do part( a) for the case of critical damping, i.e. when γ = ω0.

Solution:

Problem:

Consider a damped harmonic oscillator.  Let us define T1 as the time between adjacent zero crossings, 2T1 as its “period”, and ω1 = 2π/(2T1) as its “angular frequency”.  If the amplitude of the damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 - (8π2n2)-1] times the frequency of the corresponding undamped oscillator.

Solution:

Problem:

A mass on a spring moves in one dimension and is subject to a velocity-dependent force.  The net force is F = -kx – bv.  Assuming is small, what fraction of energy is dissipated in each cycle?

Solution:

Problem:

The terminal speed of a freely falling object is vt (assume the drag force is proportional to the speed of the object).  When the object is suspended by a spring, while still influenced by the same drag force, the spring stretches by an amount x0.  Show that the frequency of oscillation of the object (when it is suspended by the spring) is ω, where ω2 = g/x0 – g2/(4vt2).

Solution:

Problem:

Consider a pendulum consisting of a rigid thin rod with length L and negligible mass supporting a ball of mass M.  The pendulum is immersed in a viscous medium which causes a frictional force F whose magnitude is proportional to the speed v of the ball, F = -μv.  It swings in a vertical plane under the influence of gravity.
(a)  Derive the equation of motion of the pendulum, allowing for arbitrary angles θ of deflection from the vertical axis.
(b)  Determine the fixed points for which d2θ/dt2 = 0 when dθ/dt = 0.  Determine for each fixed point the critical value of the drag coefficient μ above which there is no oscillation about the point for small displacements.

Solution:

Problem:

A grandfather clock has a pendulum of length L = 1 m and a bob of mass m = 0.5 kg.  A mass of 2 kg falls 0.7 m in seven days to keep the amplitude if the pendulum oscillations steady at 0.03 rad.
(a)  The quality factor Q of a damped oscillator is defined as
Q = 2π*(average energy)/(energy lost per cycle).
What is the Q of this system?
(b)  With no energy input from the falling mass, the pendulum obeys the small angle equation of motion
d2θ/dt2 + 2bdθ/dt  + ω02θ = 0.
Find b and ω0.

Solution:

Problem:

A mass m fixed to a spring of spring constant k and immersed in a fluid provides a model for a damped harmonic oscillator.  A circuit with inductance L, capacitance C and resistance R provides an electric analog to such an oscillator.
(a) Write out the circuit equation for the LCR circuit and Newton's second law of motion for the damped oscillator.  What assumption must be made about the dependence of the damping of the mass on velocity for the two equations to have the same mathematical form?
(b)  How are the constants m, k, and b (damping constant) related to the circuit constants L, C and R?  To what parameters of the electric circuit are the mechanical quantities x (displacement) and v (velocity) related?
(c)  Derive and expression for the displacement and velocity in the limit of weak damping.
(d)  What voltages measured in the circuit would give values proportional to the displacement and velocity of the mechanical oscillator?


Driven  and damped oscillations

Problem:

Consider a damped harmonic oscillator.  Let us define T1 as the time between adjacent zero crossings, 2T1 as its “period”, and ω1 = 2π/(2T1) as its “angular frequency”.  The damped harmonic oscillator is characterized by the quality factor Q = ω1/(2β), where 1/β is the relaxation time, i.e. the time in which the amplitude of the oscillation is reduced by a factor of 1/e.
(a)  After 8.6 seconds and 5 periods of oscillations, the amplitude of a damped oscillator decreased to 17% of its originally set value.  Determine the quality factor Q of the system.
(b)  A motor now supplies an external driving term M cosωt.  Determine the stationary solution for the now driven and damped oscillator.
(c)  For a system with an equation of motion

d2x/dt2 + 2β dx/dt + ω02x = A cos(ωt)

the quality factor Q is defined as Q = ωR/(2β), where ωR is the angular frequency for amplitude resonance.  Compare the shape of a plot of the amplitude of the oscillations versus the driving frequency Ω for the above driven and damped oscillator with a Lorentzian atomic line shape for Q = 4 and Q = 107.

Solution:

Problem:

A 0.500 kg mass is attached to a spring of constant 150 N/m.  A driving force F(t) = (12.0 N) cos(ωt) is applied to the mass, and the damping coefficient b is 6.00 Ns/m.  What is the amplitude (in cm) of the steady-state motion if ω is equal to half of the natural frequency ω0 of the system?

Solution:

Problem:

A particle of mass m = 1 oscillates without friction attached to a spring with k = 4.  The motion of the particle is driven by the external force F(t) = 3 t cos(t).  Find the equation of motion and solve it.  Discuss the physical meaning of the solution.

Solution:

Problem:

A mass m hangs in equilibrium by a spring which exerts a force F = -K(x - l), where x is the length of the spring and l is its length when relaxed.  At t = 0 the point of support P to which the upper end of the spring is attached begins to oscillate sinusoidally up and down with amplitude A, angular frequency ω as shown (x' = A sin(ωt)).  Set up and solve the equation of motion for x(t).

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Solution: