UFC 3-220-01N
15 AUGUST 2005
force, e.g., Q = Qo sin (ωt), in which Qo is the amplitude of the exciting force, ω = 2πfo is
second), and t is time in seconds.
lf the model is oriented as shown in the insert in figure 12-2(a), motions
will occur in the vertical or z direction only, and the system has one degree of freedom
(one coordinate direction (z) is needed to describe the motion). The magnitude of
parameters are customarily combined to describe the "natural frequency" as follows:
fn = 1/(2xπ)x(k/m)1/2
(12-1)
and the "damping ratio" as
c/2x (k/m) -1/2
D=
(12-2)
Figure 12-2(a) shows the dynamic response of the system when the
amplitude of the exciting fore. Qo, is constant. The abscissa of the diagram is the
dimensionless ratio of exciting frequency, fo, divided by the natural frequency, fn, in
equation (12-1). The ordinate is the dynamic magnification factor, Mz which is the ratio
of A, to the static displacement, Az = (Qo/k). Different response curves correspond to
different values of D.
Figure 12-2(b) is the dynamic response of the system when the exciting
force is general by a rotating max, which develops:
Qo = mo(ē)4π2fo 2
(12-3)
where
me = the total rotating mass
ē = the eccentricity
fo = the frequency of oscillation, cycles per second
The ordinate Mż (fig 12-2(b)) relates the dynamic displacement, Az, to me ē/m. The
peak value of the response curve is a function of the damping ratio and is given by the
following expression:
Me(max) or Mż = 1/(2D)x(1 D2) -1/2 (12-4)
2D w 1 D2
For small values of D, this expression becomes 1/2D. These peak values occur at
frequency ratios of
__fo__ = w 1 D2
(12-5)
fn
12-2
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