Applications of Linear Equations with Constant Coefficients 283angle is usually chosen with the restriction -n < ¢ :S n. With this restric-
tion when 0 <¢:Sn, ¢ is the angle by which (15) y =A sin (wt+¢) leadsthe function y = Asinwt and ¢/w is the time lead. See Figure 6.4. When
-n < ¢ < 0 , ¢ is the angle by which y = A sin (wt+¢) lags the func-
tion y = A sin wt and ¢/ w is the time lag. Since the period of the function
y = A sin (wt+¢) is 2n /w, the period of oscillation of a free undamped
system is P = 2n /w. The period is the time interval between successive max-
ima (minima) of the system. So for a simple pendulum the period is the time
interval from one time the bob is farthest to the right (left) until the next time
the bob is farthest to t h e right (left). And for a mass-spring system the pe-
riod is the time interval from one t ime the mass is at its lowest (highest) point
until the next time the mass is at its lowest (highest) point. The reciprocal ofthe period is call ed t he frequency. The frequency F = 1/ P = w/(2n) is the
number of oscillations per unit of time. A pendulum clock which is keeping
perfect time h as a frequency of one cycle per second (lc/s).
y
A
-----~~------------------,/ ',,,_ y = A sin rot
Lead Time
cp/ro-A
y =A sin (rot+<!>)
where <!> > 0Figure 6.4 Simple Harmonic Motion6.1.1.2 Free Damped Motion
tWhen there is a damping force acting on the system (b -j. 0) but no external
force driving the system (f(t) = 0 for all t), the form of the general solution
of equation (12) depends on the sign of the discriminant of the roots and,
consequently, the types of roots of the auxilia ry equation.
Case 1. When b^2 - 4ad < 0, the system executes damped oscilla-
tory motion. The roots of the a uxiliary equation are complex conjugates-
ri = -o+wi and r 2 = -o:-wi where a:= b/(2a) > 0 and w = J4ad - b^2 /(2a).
So the general solution of (12) is
(19) y(t) = e-at (Ai sin wt+ A2 cos wt)
where Ai and A 2 are arbitrary constants. As in the case of free undamped