Page 69 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

56 Harmonically Excited Vibration Chap. 3

3.2 ROTATING UNBALANCE
Unbalance in rotating machines is a common source of vibration excitation. We
consider here a spring-mass system constrained to move in the vertical direction
and excited by a rotating machine that is unbalanced, as shown in Fig. 3.2-1. The
unbalance is represented by an eccentric mass m with eccentricity e that is
rotating with angular velocity co. By letting jc be the displaeement of the nonrotat­

ing mass (M —m) from the static equilibrium position, the displacement of m is
X + c sin o)t
The equation of motion is then

(M —m)x + m ( X + e sin ca 0 = —kx —cx
d r
which can be rearranged to
Mx cx + kx ^ (meoj^) sin cot (3.2-1)
It is evident, then, that this equation is identical to Eq. (3.1-1), where is
replaced by meco^, and hence the steady-state solution of the previous section can
be replaced by
meo)
X (3.2-2)
Moj^) +
and
CO)
tan (3.2-3)
k - Moj^
These can be further reduced to nondimensional form:
V2

M X
m e (3.2-4)

1 - (iF 2i:

Figure 3.2-1. Harmonic disturbing
force resulting from rotating unbal­
ance.