3.5 Whirling of Rotating Shafts
73
With the usual notations, Eq. (3.27) becomes
Z
r
=
η
2
1 − η 2
(3.28)
Equation (3.28) reveals that the speed of the shaft becomes critical when ω = p,
as z tends to infinity at that frequency.
In the complex representation, both real and imaginary parts are needed to describe
the motion.
3.6 Vibration Isolation and Transmissibility
Machines are often connected to the foundation by springs and dampers. The main
purpose behind it is to reduce the transmission of forces to the foundation.
A machine mounted on springs and dampers is shown in Fig. 3.10. Assuming the
support to be unyielding, the force transmitted to the support is
F T = c ˙
y + ky
(3.29)
Assuming steady-state motion, y = Y cos ( ω t − φ ) (see Eq. 3.14), one obtains
F T = −cω Y sin ( ω t − φ ) + kY cos ( ω t − φ )
(3.30)
The amplitude of F T , as given by Eq. (3.30), is
F T 0 =
(cω Y ) 2 + ( kY ) 2
(3.31)
or
F T 0 = kY
1 +
cω
k
2
Fig. 3.10 Spring-damper
supported machine
73
With the usual notations, Eq. (3.27) becomes
Z
r
=
η
2
1 − η 2
(3.28)
Equation (3.28) reveals that the speed of the shaft becomes critical when ω = p,
as z tends to infinity at that frequency.
In the complex representation, both real and imaginary parts are needed to describe
the motion.
3.6 Vibration Isolation and Transmissibility
Machines are often connected to the foundation by springs and dampers. The main
purpose behind it is to reduce the transmission of forces to the foundation.
A machine mounted on springs and dampers is shown in Fig. 3.10. Assuming the
support to be unyielding, the force transmitted to the support is
F T = c ˙
y + ky
(3.29)
Assuming steady-state motion, y = Y cos ( ω t − φ ) (see Eq. 3.14), one obtains
F T = −cω Y sin ( ω t − φ ) + kY cos ( ω t − φ )
(3.30)
The amplitude of F T , as given by Eq. (3.30), is
F T 0 =
(cω Y ) 2 + ( kY ) 2
(3.31)
or
F T 0 = kY
1 +
cω
k
2
Fig. 3.10 Spring-damper
supported machine