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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