Page 79 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

66 Harmonically Excited Vibration Chap. 3

3.5 SUPPORT MOTION
In many cases, the dynamical system is excited by the motion of the support point,
as shown in Fig. 3.5-1. We let y be the harmonic displacement of the support point
and measure the displacement x of the mass m from an inertial reference.
In the displaced position, the unbalanced forces are due to the damper and
the springs, and the differential equation of motion becomes
mx = - k { x - y) - c ( i - y ) (3.5-1)
By making the substitution
z = X — y (3.5-2)
Eq. (3.5-1) becomes
mz -h cz + kz = —my
= ma)^Y sin cot (3.5-3)
where y = Y sin cot has been assumed for the motion of the base. The form of this
equation is identical to that of Eq. (3.2-1), where z replaces x and mco^Y replaces
meco^. Thus, the solution can be immediately written as

z = Z sin {cot —(f))
^ _ mco^Y
(3.5-4)
]/(k - mco^y -h (cco)^
cco
tan (f) = (3.5-5)
k - mco^
and the curves of Fig. 3.2-2 are applicable with the appropriate change in the
ordinate.
If the absolute motion x of the mass is desired, we can solve for jc = z + y.
Using the exponential form of harmonic motion gives

y = Ve‘“'
2 = -■#» = (Z e “"^)e' (3.5-6)

c{k-y) k(x -y)

Figure 3.5-1. System excited by
-fkypKj motion of support point.