Page 86 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.10 harmoniC motion 83
Solution: The free-body diagrams of the four springs and four dampers are shown in Fig. 1.45(c).
Assuming that the center of mass, G, is located symmetrically with respect to the four springs and
dampers, we notice that all the springs will be subjected to the same displacement, x, and all the
#
#
dampers will be subject to the same relative velocity x, where x and x denote the displacement and
velocity, respectively, of the center of mass, G. Hence the forces acting on the springs 1F si 2 and the
dampers 1F di 2 can be expressed as
F si = k i x; i = 1, 2, 3, 4
#
F di = c i x; i = 1, 2, 3, 4 (E.1)
Let the total forces acting on all the springs and all the dampers be F s and F d , respectively (see Fig.
1.45(d)). The force equilibrium equations can thus be expressed as

F s = F s1 + F s2 + F s3 + F s4
F d = F d1 + F d2 + F d3 + F d4 (E.2)
where F s + F d = W, with W denoting the total vertical force (including the inertia force) acting on
the milling machine. From Fig. 1.45(d), we have
F s = k eq x
#
F d = c eq x (E.3)
Equation (E.2), along with Eqs. (E.1) and (E.3), yields
k eq = k 1 + k 2 + k 3 + k 4 = 4k
c eq = c 1 + c 2 + c 3 + c 4 = 4c (E.4)

when k i = k and c i = c for i = 1, 2, 3, 4.
Note: If the center of mass, G, is not located symmetrically with respect to the four springs and
dampers, the ith spring experiences a displacement of x i and the ith damper experiences a velocity
#
#
#
of x i , where x i and x i can be related to the displacement x and velocity x of the center of mass of the
milling machine, G. In such a case, Eqs. (E.1) and (E.4) need to be modified suitably.
■
1.10 harmonic motion

Oscillatory motion may repeat itself regularly, as in the case of a simple pendulum, or it
may display considerable irregularity, as in the case of ground motion during an earth-
quake. If the motion is repeated after equal intervals of time, it is called periodic motion.
The simplest type of periodic motion is harmonic motion. The motion imparted to the mass
m due to the Scotch yoke mechanism shown in Fig. 1.46 is an example of simple harmonic
motion [1.24, 1.34, 1.35]. In this system, a crank of radius A rotates about the point O. The
other end of the crank, P, slides in a slotted rod, which reciprocates in the vertical guide
R. When the crank rotates at an angular velocity v, the end point S of the slotted link and
hence the mass m of the spring-mass system are displaced from their middle positions by
an amount x (in time t) given by

x = A sin u = A sin vt (1.30)