Applications of Linear Equations with Constant Coefficients 295a sufficiently long period of time the solution depends only on the external
force driving the system and not upon the conditions under which the system
was started- that is , not upon the initial conditions.
We see from equation (27) that the steady state solution is oscillatory withfrequency w*- the same frequency as the forcing function j(t)- and with
amplitude A(w) = E/ JH(w). For 0 < w < oo as w --> 0, A(w*) --> E/d
and as w --> oo, A(w)--> 0. Also since b > 0, for all positive w, A(w) is
finite. Consequently, when there is damping present in a system (b > 0), the
amplitude of oscillation remains finite; whereas, when there is no dampingin the system (b = 0) and when resonance occurs (w* = w) the amplitude
of oscillation increases without bound until the system is destroyed. The
amplitude of oscillation A(w) will be a maximum when H(w) is a minimum.
Differentiating H(w) with respect tow, setting the result equal to zero, and
solving for w*, we find
* Jad - b^2 /2
w =.
aWhen w* has this value the forcing function is said to be in resonance with
the system. Resonance can occur only when ad - b^2 /2 > 0 which implies
b^2 < 2ad < 4ad which in turn implies b^2 -4ad < 0. Thus, resonance can occur
only if the corresponding free system (f(t) = 0) executes damped oscillatory
motion. Resonance will not occur if the free system executes critically damped
motion or overdamped motion. For damped oscillatory motion the resonance
frequency is
w* Jad-b^2 /2
FR= - =.
27r 27raThis frequency is less than the frequency of the corresponding free system
which is
F .:::... ,_J_ad-b^2 /_4
- 27r - 27ra ·
EXERCISES 6.1.2
l. A .6 kg mass is attached to one end of a spring. The other end is
attached to a movable support. The support is held fixed and the system
is permitted to come to rest. In the equilibrium position the spring
is stretched .3 meters. The mass is pulled down .2 meters below the
equilibrium position and released without imparting any velocity and
at the same instant a motor starts to drive the support with a forcef(t) = lOsinw*t kg-m/s^2.
a. What value of w* causes resonance?
b. If w* = 10 cycles/second, what is the general solution?