This document proposes three new models for evaluating the mesh stiffness of spur gears. Model 1 considers gear bodies and teeth to be elastic and calculates mesh forces and tooth deflections at different positions. Model 2 considers teeth to be elastic and gear bodies to be rigid. It applies torque to the gear and measures angular displacement. Model 3 uses corner deflection values at arbitrary angles between teeth. Finite element analysis is used to validate the models.

1. Three new models for evaluation of standardinvolute spur gear | Mesh Stiffness
Machine Design & CAD-II
University of Engineering & Technology Lahore
Department of Mechanical Engineering

2. 2
Abstract
One of the main internal agitation reasons for the gear dynamics is time-variable mesh rigidity. For gear
dynamic analysis, an accurate measurement of gear mesh stiffness is essential. This is what we are talking
about. The research aims to create new models for the assessment of the rigidity of spur gears mesh. It
proposes three models. The model 1 proposed may give very precise mesh rigidity The result, however,
should be assumed that the gear bore surface is rigid. Read more about the proposal Model 1, the angular
pattern of deflection of the gear bore surface is found in our research The cosine curve is essentially followed
by a pair of meshing gears under a constant torque. Basic two more models are suggested on this observation.
Model 2 assesses the machine mesh Stiffness through the use of angular deflection at different
circumferential end angles Getting bore circle. For model 3, the corner deflection only needs to be used an
arbitrary circumferential angle of the end of the bore's surface region Can only be used between all teeth for
equipment of the same tooth profile. The simulations proposed Exactly tested and simple to use, in gear mesh
rigidity. The analysis of finite elements Used for validation of the precision of the models proposed.

3. 1. Introduction
Spur Gear:
Spur gears are a type of cylindrical gear, with shafts
that are parallel and coplanar, and teeth that are
straight and oriented parallel to the shafts. They’re
arguably the simplest and most common type of gear,
easy to manufacture and suitable for a wide range of
applications. The teeth of a spur gear have an involute
profile and mesh one tooth at a time. The involute
form means that spur gears only produce radial forces
(no axial forces), but the method of tooth meshing
causes high stress on the gear teeth and high noise
production. Because of this, spur gears are typically
used for lower speed applications, although they can
be used at almost any speed.

4. Material Of Spur Gear:
Spur gears can be made from metals such as steel or brass,
or from plastics such as nylon or polycarbonate. Gears made of
plastic produce less noise, but at the expense of strength and
loading capability. Unlike other gear types, spur gears don’t
experience high losses due to slippage, so they generally have
high transmission efficiency. Multiple spur gears can be used in
series The referred to as a gear train to achieve large reduction
ratios.
Profile:
An involute gear tooth has a profile that is the involute of a
circle, which means that as two gears mesh, they contact at a
single point where the involutes meet. This point moves along
the tooth surfaces as the gears rotate, and the line of force The
known as the line of action) is tangent to the two base circles.
Thus, the gears adhere to the fundamental law of gearing, which
states that the ratio of the gears’ angular velocities must remain
constant throughout the mesh.

5. Applications:
Spur gears are generally seen as best for applications that
require speed reduction and torque multiplication, such as ball
mills and crushing equipment. Examples of high-speed
applications that use spur gears – despite their high noise levels
include consumer appliances such as washing machines and
blenders. And while noise limits the use of spur gears in
passenger automobiles, they are often used in aircraft engines,
trains, and even bicycles.

6. 1.2 Types of Spur Gear
There are mainly two types of Spur Gears.
• External Spur Gear
• Internal Spur Gear
Internal Spur Gear:
Internal gears, in contrast, have teeth that are cut on the
inside surface of the cylinder. An external gear sits inside the
internal gear, and the gears rotate in the same direction.
Because the shafts are positioned closer together, internal
gear assemblies are more compact than external gear
assemblies. Internal gears are primarily used for planetary
gear drives.
External Spur Gear:
External gears have teeth that are cut on the outside
surface of the cylinder. Two external gears mesh together
and rotate in opposite directions.

7. 1.3 Spur Gear Design Fundamentals
•To design a spur gear, it is needed to create a tooth profile of the
spur gear.
• To design a tooth profile of the spur gear, the first step is to
know the components of the spur gear’s tooth profile. The
stereogram of the tooth profile.
• In all tooth profiles, the involute curve is the most commonly
used. The tooth profile of spur gears is composed of involute
and non involute tooth profiles. So, by means of parameter
modeling, the one tooth profile with the involute curve can be
built.
•The rotation matrix will be used to create a whole spur gear
until one tooth profile is completed. The spur gears are a very
common component. The
•fundamental functions about the spur gear are the key to
parametric modeling.
The stereogram of the tooth profile.

8. • The Tooth Thickness Function. In order to get the expression
of the involute profile, the thickness of circular teeth should
be calculated first. The base circle is an imaginary circle on
the involute cylindrical gear. When the generating line
forming the involute profile Thor the generating circle
forming the cycloid profile) rolls purely on the
circumference of the imaginary circle, the imaginary circle is
the base circle.
• The plain of the tooth profile with involute curve. (1)
Working profile; (2) lowest point of the working profile on
which contact may occur; (3) undercut; (4) fillet curve; (5)
base circle; (6) tangent at lowest point of working profile.

9.  Where Sty is tooth thickness in any circle, dy is the diameter
of any circle, Stref is tooth thickness of the reference circle,
dref is the diameter of the reference circle, αt is the pressure
angle, αy is the pressure angle in any circle, rb is the radius
of the base circle, and ry is the radius of any circle. The
variable can be thought as ry and the tooth thickness can be
easily deduced from the function. After that, it will represent
the involute profile part.
 The Involute Function. The definition of the involute curve is
the curve traced by a point on a straight line that rolls
without slipping on the circle. The circle is called the base
circle of the involutes. Two opposite hand involute curves
meeting at a cusp form a gear tooth curve. The length of base
circle arc ac equals the length of the straight line be:
Transfer of tooth thickness to the base circle.

10. • where αt is the pressure angle and αy is the pressure angle in
any circle. Function of α, or invα, is known as involute
function. Involute function is very important in the gear
design. The relation of ry and Sty can be deduced by the
involute function:
• Where m is the module of gear and z is the number of teeth
of gear.

11. 1.3.1 Mesh Stiffness
 Stiffness is the extent to which an object
resists deformation in response to an applied force
 The complementary concept is flexibility or
pliability: the more flexible an object is, the less
stiff it is
 The stiffness, k, of a body is a measure of the
resistance offered by an elastic body
to deformation
 For an elastic body with a single degree of
freedom(DOF) the stiffness is defined as
K =
𝑓
ϩ
 F is the force on the body is the displacement
produced by the force along the same degree of
freedom stiffness is typically measured
in Newtons per meter (display style N/m}

12. 1.3.2 Process of Parametric Modelling
Choose the Coordinate System:
First of all, what is the important thing is to find
the appropriate coordinate system of the tooth profile.
So, it is assumed that the origin is the center of the
pitch circle. According to this coordinate system, the
next step is to create the module of one tooth that is, y
axial symmetry in the tooth profile. The center of the
gear pitch circle is located at the origin (0, 0) of the
coordinate system. In order to show the axis of
symmetry of a tooth profile, we use the vertical dotted
line for identification. To complete the presentation of
a gear tooth, the dotted line in the middle of the image
is part of the tooth root circle.

13. Create the Involute Profile:
The method is determined to generate the profile of a gear as
a set of (x, y) coordinates, from basic data (module, number of
teeth, etc.). With the tooth thickness equation, it helps to
generate the profile of a gear as a set of (x, y). As shown in
Figure 6, the y-axis passes through the center of the circle and
bisects one tooth. Due to the equation, the arc can be computed:
Involute Curve

14. According to the abovementioned equations, ry is assumed as
the variable. Hence, the one tooth’s tooth profile is represented:
After computing this program, we can get the tooth profile
(above the base circle) expect the top land. The tooth profile.
But these four equations cannot represent the whole tooth
profile. These can only represent the tooth profile above the
base circle. These equations cannot represent the top land, and
the variable ry has a limiting condition.

15. The next step is creating the module of the top land. Because the
generating method is the common method in manufacturing the
spur gear, so the graph of the top land can be thought as an arc
whose radius is ra:
where ra is the radius of the top circle:
In Sci lab, X is a variable. So, the programming idea is that the
last point in the involute curve is the start point in the top land
curve. Then, use formula to express the Y value with X and
plot.
Create the Non involute Profile:
The difference between Figures is the link from the tooth
profile above the base circle to the tooth profile below the base
circle. So, the key point is to make sure that the end point from
the tooth profile above the base circle has the same common
tangent with the start point from the tooth profile below the base
circle

16.  When manufacturing the spur gear by laser cutting, the
transition section from the involute curve to the non involute
curve will be destroyed if they do not have the same
common tangent. To create the part below the base circle, the
first step is to assume the shape of this part. Figure shows
that the next step is to create parts under the base circle by
finding the boundary of the same common tangent on the
profile curve, that is, the position where the base circle curve
intersects the profile curve. When the generating method is
used to manufacture the spur gear, the rack cutter’s pitch line
rolling on a pitch circle generates a spur gear. It is easy to
find the shape of the tooth profile’s part below the base circle
which looks like the circle arc. And the next step is to know
some basic parameters about this circle arc, such as the
origin of this circle arc, the radius of this circle arc, and the
radian from the start to the end of this circle arc. Firstly, get
the common tangency’s slope.
 With the value slope, the next step is to obtain the negative
inverse slope. Because the link point is the point where ry &
rb. So, it can be obtained the derivative at the link point:

18. The correct arc. The wrong arc.

19. After obtaining the value of the slope, it supposes this situation,
that is, the curve below the base circle. Figure shows the
detailed analyze of the arc. point A is the start point and point B
is the end point. The negative reciprocal slope can provide the
value θ:
The generating method.The analysis of the arc.

20. So, the next step is to obtain the radius r. To get the value r, we
need to obtain the value of vertical distance (H) between A and
B:
where rf is the radius of the tooth circle, H is the difference
between A and B in Y axis, and rc is the radius of this circle (A-
B). We can get the radius of this circle (A-B):
where C is the value of C point in X and Y axis.

21. After that, it is easy to compute the graphic curve in Sci lab.
And with the radius rf , the whole one tooth of the spur gear can
be computed.
Create the Whole Spur Gear:
Using the rotation matrix, as shown in equations, we can create
the whole spur gear.

22. 22
2. Proposed Model 1
 Model is the type suggested 1. For the assessment of rigidity
mains, the current model 1 is to be used. The gear bodies and
teeth are all considered elastic in this model. Increasing gear
mesh position measures gear mesh forces and deflection of each
tooth on the action line.
 This model is derived from the concept of rigidity: k 1/4 F = d,
whereby F is the force and d is the force movement in the
direction of the force.
 The model 1 can yield a precise mesh rigidity result, but for each
mesh position multiple variables must be calculated (one force
and two deflections in the single dental pair mesh period and two
forces and four difflections in the double dental combination
period.

23. • Form 2 is to be used for the measurement of rigidity of the
gear mesh.
• Gear teeth are supposed to be elastic while the transmission
structure is supposed to be rigid.
• The pinion body in this model is set, the torque (T) on the
gear body is applied, the corresponding angular shift (h) on
the gear body is determined at each gear angle.
• The effect of the gear has not been taken into account. Body
stiffness on a mesh that could produce inaccurate results.
• To analyze the efficacy of the three models listed above, a
recorded linear finite element analysis technique is used. The
linear analysis implies that a structure's displacement U is a
linear attribute of the applied structure.

24. • The load vector R; i.e. where the loads are aR rather than R, the
corresponding loads is aU. "The final element type is SOLID185
and the product shape is hexahedral.
• The Finite Element Models are used by ANSYS. Pinion mesh
nodes and equipment are pre-connected. In finite element
modeling, touch finite elements are thus not needed
• If one or two pairs are middle, six adjacent teeth are refined.
There's a web of bones.
• The modulus Young for the gear body in current Model 2 is 1000
times greater than that for the gear teeth, to model a rigid gear
body.
• Without adding computational costs, this environment will produce
good simulation results. The average angular displacement of all
nodes on the surface of the gear boor is determined to be the
angular displacement of the gear.
• In Model 1 proposed, all of the nodes on the surface of the gear are
linked to each other in order to simulate a rigid gear bore surface.

25. • The parameters of a pair of spur gears are given in Table 1.
Furthermore, the pinion and gear have an equal bore length of
17.5 mm. Several boron sizes will be used later in Section 5.
• The straight mesh rigidity is tested using the above three
versions. Close results can be achieved by the present model 1
and the proposed model 1.
• All models have been shown to be effective in determining the
rectilinear mesh rigidity. Model 2 of today is not so precise.
• The straight mesh steepness generated by the existing model 2
is almost double that of the two other models, showing that the
deflection of the gear body should be taken into account in the
mesh rigidity assessment.

26. • Since the current model 1 and the model 1 proposed offer very
narrow results, the specific benefits of the proposed model1 are
now highlighted.
• Model 1 uses contact points for forces and deflectors. For the
single tooth meshing period, it requires to measure one force
and two deflections and in the double tooth meshing period two
forces and four deflections.
• The proposed model 1 on the other hand does not at all use the
mesh frames. Instead, the gear bore surface is displaced
angularly by only one parameter.

27. 27
3. Proposed Model 2
 The proposed model 1 is shown to be efficacious with the finite
element analysis in the rectilinear system stiffness assessment. This
section will present two other versions, illuminated by the proposed
model 1, for the evaluation of steadiness of the gear mesh with the
finite element analysis.
 The proposed model 1 in Section 2 is shown to be efficacious with
the finite element analysis in the rectilinear system stiffness
assessment. This section will present two other versions, illuminated
by the proposed model 1, for the evaluation of steadiness of the gear
mesh with the finite element analysis.
 On the surface of the gear boron, eleven circles are seen. Each circle
is parallel to the surface of the gear bodies. The left image
corresponds to a two times the mesh of the diagraph (this moment is
the first point of the data. The right image is the same as a single
moment of the mesh of the diagram of the diagram of the diameter).

28.  Cosine curves, A cosða − B bisC, are thus used for angular
deflections in the final circular surface region, in which the
circumferential angle on the circle of the surface is a. In order
to reduce as possible the squares of the errors between the
data points and the fitted sinusoidal curve the variables A, B
and C are optimized by using the less square fitting form.

30.  When we try and error, we find that minimum angular deflection
achieved from the best fitting curve, will generate very good mesh
stiffness if it is called h, as we seek to make a max. angular
deflection, average angular deflection and some other angular
deflective values. The h value equal to the lowest value of the
fitted sine curve is selected for each gear mesh position. This h
value is replaced to achieve the straight mesh rigidity in this mesh
position.
 The model 1 suggested and the model 2 proposed. The T-driven
torque applied to the surface of the gear boron is 100 Nm in the
proposed model 2 and the angular deflections in 31 positions are
measured. The mesh rigidity derived from model 2 proposed is
very similar to that derived from model 1 proposed. The average
difference in mesh rigidity is about 1.5%. In the next equation the
difference is quantified:
The stiffness of the rectilinear mesh evaluated by using the
proposed model 1 is kt1 and kt2.

31. On the basis of the above-mentioned finite element analysis, we outline a
method which can be used to accurately calculate time difference between
mesh rigidity. A schematic and detailed steps of this experimental procedure
are provided as follows:
 Choose M measuring points, equally distributed around, on the end-circle
surface of the gear bore.
 Set the surface of the pinion boron.
 Make sure you touch each other's equipment and pinion teeth.
 To achieve torque T, apply a uniform force on the surface of the gear bore.
 Achieving M-points angular deflections.
 Adjust these M angular deflection values to the cosine curve and find the
minimum fitted angle deflection h.
 T and h replace to obtain the rigidity of the rectilinear mesh in that mesh
position.
 For other gear mesh positions, repeat steps.
This experimental procedure is simple to implement and is expected to provide
good measurements of the time-varying mesh stiffness. We will conduct such
experiments in our future work.

32. 32
4. Proposed Model 3
 We will define our proposed model 3 for the rigidity evaluation of
the gear mesh in this section. For increasing gear mesh location the
proposed M2 model includes angular deflections (at various
circumferential angles of the gear bores). This section includes
angular deflection at only one location of the gear boring in the built
model 3. The proposed model 3 however has a high precision
requirement with regard to the gear tooth profiles. If the tooth profile
error occurs on any gear tooth, it can not be guaranteed the
effectiveness of the model 3.

33.  We will demonstrate how the gear mesh rigidity can be
achieved when the gear tooth tip is meshed. The angular
change of the gear is called c, if the tooth tip of the first tooth
of the gear is in mesh. Relatedly, when there is a mesh at the
end of a second tooth, the angular shift is hm − c, etc. Hm
indicates the angle of rotation of the gear over the period of
one tooth mesh. A P-point on the end of a circle of the gear
bore arbitrarily chosen. The corner deviation from P is shown
by u at the tip of the first tooth. The device is hm βu, etc. The
angular deflection of point P is determined when the tip of the
first gear tooth is in the mesh. Similarly, angular deflections of
point P can also be measured if the tips of other gear teeth are
in a mesh.

35. • At point P, Z-angular deflections are determined over a complete
rotation period of the gear. The number of teeth of the equipment is
here Z. If each pinion and gear tooth has the same gear shape, then
the mesh steadiness calculated when the tip of the gear's first tooth
is in mesh corresponds to the mesh steadiness calculated when the
Nth spit (N=2, 3, 4,.-or Z) is in mesh. Therefore, when the tip of the
first tooth of the gear is in mesh, the angular Z deflections at point P
can be viewed as the angular Z deflections.

36.  The angular differences between point P and the points that match. First
tooth tips, second dental tips,. . and the dent is u, hm · u, . (Z § 1) hm β u.
[...]. Attach these Z-squares to these Z-squares using a cosine method.
The minimal angular deflection h can be found in the curve of the cosine.
When the tooth tip of the gear is in the mesh, we can replace the T and h
with the rectilinear mesh steadiness.
 In this section, the pinion and the gear are ideal gears. The same tooth
profile applies to all pinion teeth. Then the teeth of the machine. On both
versions, the torque T used on the gear is 100 Nm. In model 2, corner
deflections of 31 positions are determined with a finite element approach,
which is distributed uniformly on the final surface circles of the gear boor
surface. For the proposed model 3, the angular deletions of 31 positions
are also indirect, because the number of teeth of the engine is 31. This is
the case for an even distribution on the end surface of the gear bore
surface.
 The mesh rigidity profile of a pair of spur gears with the finite element
method can be tested using this proposed Model 3, if all gear dents are
safe. Under this case, the proposed model 3 needs only one position of a
surface circle in the gear bore to measure deflection.

37.  The model 3 proposed is much easier to use than the model 2
that requires deflection measurement in several places of the end
surface circles of the gear bore. Model 3 is however sensitive to
errors in tooth profile. If there is a tooth profile error in any
gear's teeth, model 3 proposed is no longer useful.
 Based on the above analysis, the following experimental
procedure may be used to measure the time-varying mesh
stiffness of a pair of spur gears when all teeth are perfectly
healthy:

38.  Select an arbitrary position on the end surface circle of the gear bore.
 Fix the pinion bore surface.
 Make sure the gear and the pinion teeth touch each other. Record the
gear rotation angle at this moment. This rotation angle is used to
represent the gear mesh position.
 Apply a uniform force along the gear bore surface to generate a torque T.
 Obtain the angular deflection of the selected point in the Step (1).
 Rotate the pinion by a fixed angle of 360 degrees divided by Z. Repeat
Steps (2)–(5).
 Repeat step (6) until the gear rotates one cycle. By now, we obtain Z
measured angular deflection values.
 Fit these Z values using a cosine curve and find the minimal angular
deflection h from the fitted cosine curve.
 Substitute the T and h to get the rectilinear gear mesh stiffness
corresponding to the gear rotation angle recorded in step (2).
 Repeat steps (2)–(9) for other gear mesh positions.

39.  To order to calculate gear mesh rigidity this experimental technique only
requires 1 sensor. However, all teeth must have precisely the same tooth
profile and errors in tooth pacing do not occur. Throughout our future
research, we will establish experiments to further validate this model.
 We did not define a sensor type or equipment to be used to measure
angular deflections in the proposed experimental plans. For example, the
Video Gage technology from IMETRUM [42] can be used to calculate
angular deflection and can directly be accomplished by a 10-nanometer
resolution. In our future experimental research, there may also be other
methods for calculating angular deflections.
 However, the sensitivity of devices or sensors and deployment failures are
two essential issues worthy of consideration in experimental
measurements. The distortion angular is approximately 3 mm. To order to
reliably measure a 3 mm deflection, the measurement system requires a
resolution of 0.01 mm. Nevertheless, if the parameters of the gears vary
from those in this article, the resolution criterion for the measuring tool
the be different. In experimental measurement, the precision of installation
should also be carefully monitored.

40.  In this report, we propose three models for evaluating the involute norm
mesh rigidity of spur machines. Such three models are shown to be
efficient using an analysis of the finite elements. The model 1 as
proposed can not be used in experiments because the surface of the gear
bore requires rigidity. There is a potential for use in experiments in the
proposed models 2 and 3. We have presented an experimental strategy
based on our current model 2 in the last two paragraphs of Section 3. We
present an experimental model based on our idea in the last five
paragraphs of section 4: Proposed Model 3.

41. 41
5. Effect of Gear Bore Radius On The
Proposed Models 2 & 3
The effects of gear bore radius on the proposed models 2
and 3 are given as:
 The surface of the gear bore is presumed to be rigid, as
we defined in Section 2, in the proposed model 1.
 According to this statement, the proposed model 1
results in extremely precise rigidity of the gear mesh
and is not impaired by its precision.
 For evaluation of gear mesh stiffness angular
deflections in the proposed models 2 and 3 of the end
surface circle of the gear bore are required.

42.  The effect of gear boring sizes on the mesh rigidity result
derived from models 2 and 3 will be shown in this section.
 The size of the gear boring may be different provided a certain
number of teeth (the gear base circle is fixed). If the bore size of
the gear is small, the bodies of the gear are increasing. The gear
body is thin when the gear bore is very large.
 We will show that the suggested models 2 and 3 are successful
in assessing the rigidity of the system mesh that is insensitive to
the size of the gears.
 The pinion bore is fastened to 17,5 mm and chosen for three
sizes: 12,5 mm, 17,5 mm and 22,5 mm.
 The proposed 2 and 3 models for three gear pairs are discussed
in this section, as shown in Fig. Ten. The pinion is then fixed on
the surfacing of the gear bore and the torque (100 N to mm) is
applied evenly to the surface of the gear bore.

43.  The difference between the bore radius values of
12.5 mm, 17.5 mm and 22.5 mm is shown in the
figure.

44.  Graph between the Angular displacement of pinion
and Mesh stiffness is shown below, which explains
the effect of gear bore radius on the proposed
models 2 & 3.

45.  The standard result will be the mesh rigidity assessed using
Model 1 as suggested.
 The errors created by the proposed models 2 and 3 are
calculated by comparing the mesh rigidity difference between
the proposed models 2 and 3 and the proposed model 1.
 With the change in gear bore size, the errors caused by models
2 and 3 are increased. The error is quantified in the following
equation in each gear mesh position:
 where kt1 is the mesh stiffness obtained using the proposed
model 1 and kt2 represents the mesh stiffness evaluated
using the proposed model 2 or 3.

46.  The normal outcome is the mesh rigidity tested with the
proposed model 1. Models 2 and 3 errors are calculated by
comparing the difference between mesh rigidity between
model 2 and model 3 proposed with the model 1 proposed.
 The alteration of the borehole size raises the errors of model
2 and 3. In every gear mesh location the error is calculated in
the following equation:
 The table below shows the errors generated by the proposed
models 2 & 3.

47.  The typical result is the rigidity of the mesh measured with
model 1. Model 2 and 3 errors are measured in accordance with
the proposed Model 1 discrepancy between mesh rigidity and
Model 3.
 Changing the borehole size increases models 2 and 3 defects.
The error is calculated in the following equation in each position
of the gear mesh.
 We only concentrate in this study on assessing the mesh rigidity
of healthy gear pairs.
 In this study, tooth crack and pitting are not examined. For
future, we are developing our current model, representing tooth
crack and pitting effects on the assessment of rigidity for gear
mesh.

48. 48
6. Conclusion
 This study aims to develop new models to evaluate time-
varying gear mesh stiffness.
 The proposed model 1 can give a very accurate mesh
stiffness result comparing with other existing models. But in
the proposed model 1, the gear bore surface is assumed to be
rigid.
 In the proposed models 2 and 3, we do not require the
assumption of rigid gear bore surface.
 The proposed model 2 requires computing the angular
deflections of an end surface circle of the gear bore at
multiple circumferential angles.

49.  Finite element analysis and comparisons demonstrate that the
proposed models 2 and 3 can also give accurate result in gear
mesh stiffness evaluation and they are insensitive to gear bore
size.
 Comparing with the proposed model 1, the maximum error
caused by the proposed models 2 or 3 is 3.3%, which is small.
In the future, we will design experiments to implement our
proposed models 2 and 3.