Page 66 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.1 Forced Harmonic Vibration 53

and
Cù)
~T
tan (f) = (3.1-6)
moj
1 -

These equations can be further expressed in terms of the following quantities:

= natural frequency of undamped oscillation
- 2moj^j = critical damping
c
^ = — = damping factor
^c
CCO _ C C^M) _ 0)
k c^. k ^
The nondimensional expressions for the amplitude and phase then become
Xk (3.1-7)
Fn ■ / (0 2
1 -

and

tan 4>= (3.1-8)

1 - 1 ^
These equations indicate that the nondimensional amplitude Xk/F^ and the phase
(/) are functions only of the frequency ratio (o/co^^ and the damping factor ^ and
can be plotted as shown in Fig. 3.1-3. These curves show that the damping factor
has a large influence on the amplitude and phase angle in the frequency region
near resonance. Further understanding of the behavior of the system can be
obtained by studying the force diagram corresponding to Fig. 3.1-2 in the regions
small, co/io^ 1, and co/co^ large.
For small values of ^ 1, both the inertia and damping forces are
small, which results in a small phase angle 0. The magnitude of the impressed
force is then nearly equal to the spring force, as shown in Fig. 3.1-4(a).
For (o/cx)^ = 1.0, the phase angle is 90° and the force diagram appears as in
Fig. 3.1-4(b). The inertia force, which is now larger, is balanced by the spring force,
whereas the impressed force overcomes the damping force. The amplitude at
resonance can be found, either from Eqs. (3.1-5) or (3.1-7) or from Fig. 3.1-4(b), to
be
(3.1-9)
cco„ 2^k