You can study resonance in a driven oscillator in detail by modifying your computation for the spring–mass system with
friction. Let one end of the spring be moved back and forth sinusoidally by a motor, with a motion given by Dsin(ωDt) (see
Figure 7.59). Here D is the amplitude of the motor motion and ωD is the angular frequency of the motor, which can be
varied. (The free-oscillation angular frequency of the spring–mass system has a fixed value, determined by
the spring stiffness ks and the mass m.)
to the right (−D sin(ωDt)). The new length of the spring is L+ x − D sin(ωDt), and the net stretch of the spring is this
quantity minus the unstretched length L, yielding (s = x − Dsin(ωDt)). A check that this is the correct expression for the net
stretch of the spring is that if the motor moves to the right the same distance as the mass moves to the right, the spring will
have zero stretch. Replace x in your computer computation with the quantity [x − Dsin(ωDt)].
(a) Use viscous damping (friction proportional to v). The damping should be small. That is, without the motor driving the
system (D = 0), the mass should oscillate for many cycles. Set ωD to 0.9ωF (that is, 0.9 times the freeoscillation
angular frequency, ), and let x0 be 0. Graph position x vs. time for enough cycles to show that there is
a transient buildup to a “steady state.” In the steady state the energy dissipation per cycle has grown to exactly equal
the energy input per cycle.
Vary ωD in the range from 0 to 2.0ωF, with closely spaced values in the neighborhood of 1.0ωF. Have the computer
plot the position of the motor end as well as the position of the mass as a function of time. For each of these driving
frequencies, record for later use the steadystate amplitude and the phase shift (the difference in angle between the
sinusoids for the motor and for the mass). If a harmonic oscillator is lightly damped, it has a large response only for
driving frequencies near its own free-oscillation frequency.
(b) Is the steady-state angular frequency equal to ωF or ωD for these various values of ωD? (Note that during the
transient buildup to the steady state the frequency is not well-defined, because the motion of the mass isn't a simple
sinusoid.)
(c) Sketch graphs of the steady-state amplitude and phase shift vs. ωD. Note that when ωD = ωF the amplitude of the
mass can be much larger than D, just as you observed with your hand-driven spring–mass system. Also note the
interesting variation of the phase shift as you go from low to high driving frequencies—does this agree with the phase
shifts you observed with your hand-driven spring–mass system?
(d) Repeat part (c) with viscous friction twice as large. What happens to the resonance curve (the graph of steadystate
amplitude vs. angular frequency ωD)?