Page 167 - Compression Machinery for Oil and Gas

Integrally Geared Compressors Chapter 4 155

analyses, lateral unbalance response, and torsional rotordynamics. However,
as previously discussed, IGC rotordynamic analysis must consider additional
influences based upon the loading from the gear reaction forces and the cou-
pling of the pinion and bull gear dynamics. API standards allow the gearing
system to be assessed as uncoupled components, but as the industry con-
tinues to evolve toward more aggressive performance (higher speeds and
higher efficiency), the need to consider coupled dynamics is becoming
evident.
The gear mesh adds significant stiffness near the center of the pinion rotor
and can affect the dynamic characteristics. Gear mesh stiffness is a complex
phenomenon, but API 684 [11] Section 2.8.3 provides equations to estimate
the stiffness coefficients in meshing gears as follows:

2
K ¼ CFWÞ cos βðÞ10 6 (4.6)
ð
2
K xx ¼ Kcos γðÞ (4.7)
2
K yy ¼ Ksin γðÞ (4.8)
K xy ¼ K yx ¼ Ksin γðÞcos γðÞ (4.9)
γ ¼ Aα + B (4.10)
2
In the above equations, C ¼12,057N/m is a constant, FW is the net face
width of the gear, and β is the helix angle. Also, in Eq. (4.10), α is the normal
pressure angle, A¼1 for downloaded rotors or A¼ 1 for uploaded rotors, and
B¼90 degrees for clockwise rotation or B¼270 degrees for counter-clockwise
rotation (looking into coupling end).
The following equations can also be considered. Eq. (4.11) represents the
localized deformation for gear tooth contact, which is based on the Hertzian
contact and accounts for gear tooth angle (ψ), loading conditions, and material
elasticity (E 1 and E 2 ). Dividing the normalized load (W/F) by the localized
deformation results in an estimate of gear mesh stiffness as provided by
Eq. (4.12):

W 1 1
9:000 +
F E 1 E 2
d ¼ 2 (4.11)
cos ψðÞ
2
W=F cos ψðÞ
∗
K ¼ ¼ (4.12)
d 1 1
9:000 +
E 1 E 2
AGMA 2001 [12] suggests mesh stiffness constant values anywhere from 10
to 20N/mm/μm. Dudley [13] states stiffness constant for mesh deflection are
not known with certainty, but stiffness values of 20–25N/mm/μm are reason-
able based on testing for 20 degrees pressure angle gears with low helix angles.