INTRODUCTION
Misalignment of meshing in pinion and gear is considered as a major contribution
of increasing the stress concentration on specific regions of the gear tooth.
This condition is related to the non uniform distribution of transmitted load
along the tooth face. The force that creates a torque with misalignment causes
high vibration with symptoms that sometimes cannot be explained (AlHussain
and Redmond, 2002). The meshing tooth in this condition will influence the
rigidity and the forces which is originated of complicated vibration and stress
distribution in the gears (Li and Yu, 2001). The importance
of stress analysis is focused on the determination of stress concentration regions
that failure or fracture may initiate at these regions (Ali
and Mohammad, 2008).
Several researchers had created a computerized FEM methodology of determining
the stress concentration on the spur gear tooth. Sfakiotakis
et al. (2001) had described a finite element procedure which simulates
the conjugation action of spur gear drives. Baud and Velex
(2002) then had created the experimental procedure to validate using a similar
model of dynamic tooth load from the AGMA. Kawalec et
al. (2006) then grouped these models in two main categories based on
their methodologies:
• 
Semianalytical: Where the empirical studies parametric
data is considered 
• 
Numerical models: A model of developed by using discretization
methods such as the boundary element method (BEM) and the finite element
method (FEM) 
The most recent studies made by Hassan (2010) that
study on transient stress of the spur gear using a super position mode calculation.
A 2D finite element model of three segment gear teeth was tested and the natural
frequency model proposed by Block Lanczos was used to determine the stress profile
at the root.
Regardless all the gear studies, it is mostly considered a stress state at
the root based on assuming the gear component meshing pinion to gear is in aligning
with each other. In actual condition, most of the component fitting, e.g., shaft
to gear, driven to slave gears etc., will affect to the misalignment error due
to tolerance, transmission, machining, assembly errors and others parameter
related.
In the gear contact sense, mesh misalignment implies the axial shifting of
the position of the meshing surfaces due to either deflections or errors in
the manufacture of the gears and their housing. The axial misalignment is essentially
results in a change in center distance of the shafts depends upon the plane
that it acts in. Axial misalignment parallel to the plane of action tends to
shift the load to the sub side of the tooth by increasing the separation at
one side of the tooth and reducing the separation at the other side of the tooth.
In this case, the shape and area of the theoretical active contact plane remains
the same as the ideal shape of the tooth (Houser et al.,
2006).
In this study one effect of axial misalignments that creates a parabolic function
at the surface of the gear flank in meshing condition and the stress distribution
at the contact region and tooth root of the spur gear is investigated. The comparison
of the load distribution factor to the face width of the gear mesh in misalignment
will also be considered.
ACTIVE GEAR CONTACT REGION
When gear meshes with perfect align condition of shaft, maximum torque is transferred
at the line of action in between pinion and the gear. The transmission force
F_{n}, which is normal to the tooth surface, as in Fig.
1, can be resolved into a tangential component, F_{t} and a radial
component, F_{r} as refer to equation:
There will be no axial force, F_{x}. and direction of the forces acting
on the gears are shown in The load is considered uniform as refer to Fig.
2a but when the axial misalign happens the load will be offset from the
contact area of the tooth flank as demonstrated in Fig. 2b.
The load of the active contact in case of axial misalignment can be specified
by coating the teeth by a thin layer of a soft material which explain the pressure
zone on the meshing tooth with refer to the experiment by Ameen
(2010). Result shows there are 4 major axial misalignment contact happen
due to axial misalignment in shaft contributes the non uniform load happen to
the gear teeth. The alteration of distribution load concentrated on one side
(B') as refer to Fig. 3 can be described as a parabolic function
based on the Eq. 3:

Fig. 1: 
Forces acting on a spur gear mesh 

Fig. 2(ab): 
Load distribution along tooth surface (a) Align and (b) Misalign 

Fig. 3: 
Theoretical active Load distribution along tooth surface 
Finally the equation:
Hence, Eq. 4 use as represents the active gear width at tooth
surface in misalignment condition with refer to the static condition. In actual,
gear always operates in dynamic motion. From Buckingham’s dynamic load
equation that consider as a small machining error and deflection of teeth under
load cause periods of acceleration, inertia forces and impact loads on the teeth
similar to variable load superimposed on a steady load (Buckingham,
2011). Those the equation:
and
where, F_{dy} is dynamic load, F_{t} is transmitted load, V
is the pitch line velocity, m/s, B is the face width actual length, mm, c is
the dynamic factor depending on machining error, e is the error in tooth shape,
21 is the value of constant k.
Hence:
where, a is the material contact constant depending on angle, E_{p}E_{g}
is the Modulus Young for pinion and gears.
To represent the theoretically active misalignment contact width (B'), it can
be divided into several segments of interval where each segment dynamic forces
is calculated as centre forces with the equation:
Thus, to determine the distribution forces on the gear contact during misalignment
would be based on the integration of Eq. 7 with Simpson's
rule method for numerical integration.
SPUR GEAR FEM MODEL
The physical gear model for FEM analysis in this study is a particular spur
gear pinion with tooth hobbing process manufacturing for transporting high load
of machinery transmission. The main characteristic parameter is defined as Table
1 below:
In accordance with investigations given in paper Atanasovska
and Nikolic (2000), expected maximum contact stress point on path of contact
is point B where point of passing from period with two tooth pairs in contact
to single meshed tooth pair period. The finite element types chosen for the
gear 3D pair is Autodesk Inventor CAD model isoperimetric structural solid element
defined by eight points gear modeling and surface contact element. The tooth
contact modeling is analyzed at the pinion gear only with assumption, the load
transferred in perfect form (the effect between pinion and gear approximately
the same). A 3D FEM gear pair models are derived like sweeped (copied) 2D model
in normal direction along the length equal to gears face width (Nikolic
and Atanasovska, 2004).
Table 1: 
Model characteristic parameter 

Singular element, spiderweb pattern and sweepmesh scheme is implemented
continuously along with several original strategies on free mesh size control
(Ariatedja and Mamat, 2011). The face width of the models
is then divided into 20 segments (each segments as 5 mm incrementally) which
give a possibility for accuracy determination of stress state and load distribution
along gear face width.
The boundary conditions on any 3D FEM model are defined by displacement constraints
at the direction normal to the surfaces which separate the modeled gear segment
from the rest of gear body (Atanasovska and Nikolic, 2000).
Model loading consist of the applied mechanical load which is modeled as the
load control and displacement control (Rahman et al.,
2008). To achieve statically stable models, the elementsteeth have displacement
constraints at the direction normal to teeth transverse plane at zaxis as Fig.
6. The external load is defined on the elementsteeth by few concentrated
forces at the path of contact direction at yaxis of Global Cartesian coordinate
system.
Two symmetrical models have been developed for the studied (align refer to
Fig. 4 and misalign conditionFig. 5) with
constant total forces F_{total} acting to the nominal surface segment
divided. Then, simulation in ANSYSFEM software in absolute align condition
with active gear contact (B) applied to the first model with nominal forces.
On second model, the theoretical active contact length B' (calculated as Eq.
4) with theoretical misalignment angle of 0.2°, 0.3° is applied
gradually to the discrete segment on the gear tooth face. The data for determining
deformation and principle stress state and load factor will be discussed as
comparisons between both results and compared to the previous studies from Atanasovska
and NikolicStanojevic (2007).

Fig. 4: 
Active gear mesh contact load at full length (B) 

Fig. 5: 
Active gear mesh contact at misalignment (B') 

Fig. 6: 
Displacement constraint of gear mesh 
RESULTS AND DISCUSSION
The numerical ANSYS FEA result in Fig. 7a and b
represent the equivalent stress state of Von misses criteria at active contact
region and the tooth root of gear surface is in absolute align condition. This
criteria is based on 3 dimension complex system of stress develop at any node
at the contact path within the surface body which the stress is acting in different
direction proportional to the change of magnitude through each node and point.

Fig. 7(ab): 
Equivalent stress of von misses criteria at contact region
and the tooth root 

Fig. 8: 
Monitoring of tooth stiffness and load variation over face
width in align condition 

Fig. 9: 
Monitoring of tooth stiffness and load variation over face
width in align condition 
This result had validated the gear model with precise result because of the
fact that node with maximum direct stress value lie on the contact surface while
node with maximum equivalent stress lie under the contact of small deep grain
size. The formula given for the Von Misses criteria is:
where, σ_{1}, σ_{2} and σ_{3} is the
direct stresses for a pinion tooth in different cross sections along gear face
width.
From the contour, clearly we can see that the region in the curve section
tangential between the root radius to the tooth path contact showed the smooth
pattern of equivalent stress distribution. This result is expected as correspond
to the result of FEM analysis by Nikolic and Atanasovska
(2004).
As refer to the graph plotted at Fig. 8 and 9,
the contact stress at the gear mesh active contact region and the tooth root
showed a symmetrical pattern of curve with equivalent stress along contact length
of B.

Fig. 10: 
Equivalent Stress at Gear Tooth root in Misalign condition 
For the stress at contact region, the plotted clearly showed that the maximum
stress happens at the end of both side of the tooth flank. This result is opposite
to the value of the stress happen at the root of the gear.

Fig. 11: 
Monitoring of tooth stiffness and load variation over face
width in misalign condition 

Fig. 12: 
Monitoring of tooth stiffness and load variation over face
width in misalign condition 
Thus, according to the definitions based on common gear theory (Atanasovska
and NikolicStanojevic, 2007), the face load factor can be calculated as
ratio between maximum value when uninformed load distribution over gear face
width is taking into consideration and average value of the load distribution
over face width or calculation that neglects this influence and keep all other
calculation requirements unmodified. So, the face load factor KHβ for
contact stress and KFβ for tooth root stress at surface contact region
the face load factor can be calculated as:
Figure 10a and b showed the equivalent
stress at the contact region with angle 0.2° and 0.3° misalignment.
It is clearly seen that the stress distribution is acting to the subside of
one end at the gear face width and increase maximum at the edge of the gear
flank. This result is nearly similar to the contour analysis of FEA result from
Hassan (2010).
In Fig. 11, the effect of misalignment mesh to the stress
distribution at the root is presented. The numerical result from ANSYS showed
the deviation on subside end of the gear root in similar contour pattern as
the effect on the contact region of the face width. The equivalent stress increases
almost twice as the equivalent stress at the contact region. At both the entering
and leaving corners of tooth root, the stress levels get extremely exaggerated
due to what is called corner contact (Houser et al.,
2006). This corner contact may be harmful or may simply get polished out
so that the stress reduces closely to the surrounding values.
Graph in Fig. 12 is the contact pattern plotted with consideration
of influence of angular misalignment at the contact region of the gear tooth.
From the expression we can see that the face load factor at contact region KFβ
= 1.8304 and face load factor at tooth root KHβ = 1.248 calculated from
Eq. 9 the value of the factor is higher than the value when
the gear in align condition. Pattern of the contact stress is skewed to the
end of corner contact at the tooth flank of the gear. This value is increased
with the misalignment angle 0.3^{0} is measure.
CONCLUSION
Gear mesh misalignment comes in various modes from a number of sources that
can lead to a fatigue breakage to the gear tooth. In this paper the axial misalignment
effect of the tooth root and contact region at the flank of gear surface have
been identified. Some compromises parameter such as dynamic load and the deviation
angle of contact defined as a major contributor of increasing of stress on the
tooth root. The values of equivalent stresses and its distribution change with
the changing of misalignment angle, where the stress concentration is increased
at the contact region and on the tooth root proportional with increasing the
misalignment angle, this is occurring in the side of subside the load and decreasing
in the other side of the gear face. The face load factor calculation described
in this paper, also with the simultaneous incorporation of this influence in
gear load capacity calculations. This result also validates the previous result
from the other researcher where the increasing of the deviation misalignment
angle and load will cause the increasing of the face load factor and probably
could lead to a fatigue initiation at the maximum stress region and finally
leads to breakage of the gear.