Question

Consider a one-dimensional damped harmonic oscillator under the action of a prescribed force F(t), which the differential equation of motion is given by x'' + 2yx' + w0^2x = F(t)/m, where w0 is a ntural frequency of the oscillator, y is a positive constante and F(t) is given by F(t) = 0, when t <= 0 = F0, when 0 < t <= t1 = 0, when t > t1. Suppose further that the damping is subcritical, i. e., y < w0 and in t = 0 the oscillator is at rest at the origin of the OX axis. (a) Determine the function of motion of this oscillator during time 0 < t <= t1. Obtain an approximate expression for the result found assuming that t << 1/w0 and interpretate this result. (b) Suppose that t1 >> 1/y, show that x(t1) approx F0/(mw0^2) and vx(t1) approx 0. Using these results, as well as those found in the previous item, sketch the graph of x versus t between t = 0 and t = t1. (c) Determine the fuction of motion when t >= t1. As initial conditions, use the values of x and vx in the instante t = t1 calculated in the previous item, that is, assuming that t1 >> 1/y. (d) Sketch the graph of x versus t from t = 0 until t = 2t1. In this sketch, suppose that y = w0/(6pi) and t1 >> 1/y.

          Consider a one-dimensional damped harmonic oscillator under the action of a prescribed force F(t), which the
differential equation of motion is given by

x'' + 2yx' + w0^2x = F(t)/m,

where w0 is a ntural frequency of the oscillator, y is a positive constante and F(t) is given by

F(t) = 0, when t <= 0
= F0, when 0 < t <= t1
= 0, when t > t1.

Suppose further that the damping is subcritical, i. e., y < w0 and in t = 0 the oscillator is at rest at the origin of the
OX axis.

(a) Determine the function of motion of this oscillator during time 0 < t <= t1. Obtain an approximate expression
for the result found assuming that t << 1/w0 and interpretate this result.
(b) Suppose that t1 >> 1/y, show that x(t1) approx F0/(mw0^2) and vx(t1) approx 0. Using these results, as well as those
found in the previous item, sketch the graph of x versus t between t = 0 and t = t1.
(c) Determine the fuction of motion when t >= t1. As initial conditions, use the values of x and vx in the instante
t = t1 calculated in the previous item, that is, assuming that t1 >> 1/y.
(d) Sketch the graph of x versus t from t = 0 until t = 2t1. In this sketch, suppose that y = w0/(6pi) and t1 >> 1/y.

Consider a one-dimensional damped harmonic oscillator under the action of a prescribed force F(t), which the
differential equation of motion is given by

x” + 2yx' + w0^2x = F(t)/m,

where w0 is a ntural frequency of the oscillator, y is a positive constante and F(t) is given by

F(t) = 0, when t <= 0
= F0, when 0 < t <= t1
= 0, when t > t1.

Suppose further that the damping is subcritical, i. e., y < w0 and in t = 0 the oscillator is at rest at the origin of the
OX axis.

(a) Determine the function of motion of this oscillator during time 0 < t <= t1. Obtain an approximate expression
for the result found assuming that t << 1/w0 and interpretate this result.
(b) Suppose that t1 >> 1/y, show that x(t1) approx F0/(mw0^2) and vx(t1) approx 0. Using these results, as well as those
found in the previous item, sketch the graph of x versus t between t = 0 and t = t1.
(c) Determine the fuction of motion when t >= t1. As initial conditions, use the values of x and vx in the instante
t = t1 calculated in the previous item, that is, assuming that t1 >> 1/y.
(d) Sketch the graph of x versus t from t = 0 until t = 2t1. In this sketch, suppose that y = w0/(6pi) and t1 >> 1/y.

Added by Jose Antonio C.

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