


Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 

Enhancing Design Features of Asymmetric Spur Gears Operating on a 

Specified Center Distance Using Tooth Sum Altered Gear Geometry  

Avil Allwyn Dsa
1,*

, Joseph Gonsalvis
2
 

1
Department of Mechanical Engineering, Don Bosco College of Engineering, Goa, India 

2
Department of Mechanical Engineering, RNS Institute of Technology, Bangalore, India 

Received 28 January 2021; received in revised form 11 March 2021; accepted 12 March 2021 

DOI: https://doi.org/10.46604/peti.2021.7074 

Abstract 

Asymmetric gears have evolved from the rising demand for power transmission drives with high load-carrying 

capacity, surface durability, and service life. Direct design and S± profile shifted system are the most common 

approaches used for enhancing design features by geometry modification in asymmetric gears. This paper aims at 

establishing asymmetric gear geometry modification using tooth sum alteration for a family of gears running on a 

specified center distance as a feasible design approach. A complete mathematical treatment of the design approach is 

provided, and an in-house developed computer program is used for numerical simulation. The paper explores the 

influence of dynamic load factors, location factors for bending, specific sliding on load-bearing capacity, and surface 

durability on different tooth sum alterations. The study concludes that tooth sum altered asymmetric gear geometry 

can be employed as an effective design technique that offers designers flexibility in designing gears for surface wear, 

load-bearing, and tooth life.   

 

Keywords: asymmetric gear, altered tooth sum, specific sliding, dynamic load, wear  

 

1. Introduction  

In recent years nonstandard asymmetric gear design has been researched for devising ways to enhance the performance of 

gear drives. The published research indicates attempts to improvise design features by varying the drive or coast side pressure 

angle for varying the tooth thickness and addendum height for contact ratio. These changes are brought about either by altering 

the cutter geometry or by profile shifting. The major limitation of these methods is the need to have custom-made tooling for 

the former and the need to trade off variation in the center distance for the latter. A method to alter the tooth geometry for 

enhanced design features without custom-made tooling or varying the center distance would indeed be novel.  

Earlier, Kapelevich [1] proposed a method for the design of gears with asymmetric teeth. Several equations required for 

the design of asymmetric gears are developed for the synthesis of asymmetric gears. Litvin, Lian, and Kapelevich [2] proposed 

gear geometry modification of asymmetric gears for the reduction in transmission error by using a combination of involute 

gears and double crowned pinion with a larger drive side pressure angle. Karpat, Ekwaro-Osire, Cavdar, and Babalik [3] 

studied the dynamic characteristics of asymmetric spur gears by addendum modified gear geometry for the reduction in 

transmission error and dynamic factor. Karpat and Ekwaro-Osire [4] proposed tooth profile tip relief modification as an 

effective measure to reduce dynamic loads and wear. Sekar [5] made a comparative study of load sharing, fillet, and contact 

stresses in symmetric and asymmetric gear drives. Sekar and Muthuveerappan [6] estimated the tooth form factor and stress 

                                                           
*
 
Corresponding author. E-mail address: avil.dsa@dbcegoa.ac.in 

 
Tel.: +91832-2777344; Mobile: +91-9657043438

 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 
2 

correction factor for asymmetric gears using standard ISO-B methodology. Gonsalvis and Rayudu [7] reported the sensitivity 

of the contact ratio to tooth-sum alteration in symmetric gears, its influence on operating pressure angle, and profile shift 

coefficient. Avil and Gonsalvis [8] developed a methodology to evaluate dynamic load distribution in tooth-sum altered 

symmetric gears considering the altering root fillet geometry. Gonsalvis and Sachidananda [9] explored the possibility of 

altering the tooth-sum of symmetric gear-pair, to tailor the gear drive either for low, normal, or high contact ratio.  

While literature related to tooth sum altered symmetric gear geometry is scarce, however, the published literature reveals 

no reference on the benefits and problems of accommodating tooth sum altered profile shifted asymmetric gears running on a 

specified center distance. The present work attempts to explore the possibility of modifying the tooth geometry of an 

asymmetric spur gear-pair for design benefits by altering the tooth sum while holding the operating center distance constant, 

which also can be named as Tooth-Sum Altered Asymmetric Gear (TSAAG). The objective of this work is to establish 

mathematical expressions for the TSAAG geometry, check for kinematic compatibility and design superiority of TSAAG over 

normal contact ratio (NCR) asymmetric counterparts. A case study on altering the tooth sum of an asymmetric spur gear of a 

given module to obtain possible TSAAG combinations is modeled as a single degree of freedom spring-mass-damper system 

(SMD) [10-11] subjected to forced vibration, for evaluating the dynamic load factors, load-bearing ability and specific sliding 

characteristics with NCR asymmetric counterpart. 

2. The Geometry of TSAAG 

 

Fig. 1 Tooth thickness distribution factor 

An asymmetric rack cutter of module m, with standard proportions, is shown for reference in Fig. 1. The drive and the 

coast side pressure angles are denoted by 𝜙𝑝
𝑑  and 𝜙𝑝

𝑐  respectively. For convenience, the authors define drive side tooth 

thickness 𝑡𝑝
𝑑 , as a factor  of total tooth thickness 𝑡𝑝, along the pitch line. From the geometry of the rack we get 

d
p pt t

 
(1) 

1
c
p

d
p

tan

tan

 




  (2) 

The general form of expression for tooth thickness 𝑡𝑥 [1] at any arbitrary location for profile shifted asymmetric spur gear 

of module m with total profile shift coefficient 𝑋𝑠 [12] and shift factor   is given by 

p
x x x

p

t
t r

r

 
   

 
 

 (3) 

Pitch Line 

t
d

p
 t

c

p
 

t
p
 

m 

1m 

1.25m 


d

p
 

c

p
 

0.25m 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 

3 

 tan tan
2

d cc
p s p p

P
t X m      (4) 

   
d c

x p x p xinv inv inv inv   
 

    
    

(5) 

2.1.   Total profile shift in TSAAG system 

The need for profile shifting in TSAAG arises due to tooth sum alteration and the constraint of constant center distance. 

The geometrical consequences of profile shifting in a TSAAG system is either an increase or decrease in the tooth thickness, 

restoring physical contact between the teeth, and also an increase or decrease in the radius of the addendum circle maintaining 

the overall tooth height. For continual conjugate action, a calculated total profile shift coefficient 𝑋𝑠 is divided among the 

meshing gears. Mathematically total profile shift coefficient 𝑋𝑠, for a TSAAG system with design backlash B, operating on a 

working pressure angle 𝜙𝑤
𝑑  and radius 𝑟𝑤  with tooth thickness 𝑡𝑤 on the driver and driven is given as 

1
1 2

1

2 w
w w

r
t t B

z


    (6) 

Applying Eq. (3) for tooth thickness at working pitch circle, Eq. (6) can be written as 

1 2 1
1 2

1 2 1

2p p w
w w w w

p p

t t r
r r B

r r z

   
         

   
   

 (7) 

For any profile shifted system the ratio of tooth numbers is equal to the ratio of the working pitch circle radii 

2 2

1 1

a

w

w

r z

r z

 
  
 

 (8) 

Substituting Eq. (8) in Eq.(7), replacing (𝑧1
𝑎 +𝑧2

𝑎 ) = 𝑍𝑠
𝑎 and dividing throughout by (2𝑟𝑤1), Eq. (7) can we rearranged as 

1
1 2

1

1

2 2

a
p p w s

w

Bz
t t m Z

r

 

     
 

 (9) 

If the profile shift coefficient of the pinion is 𝑥1 = (𝜅𝑋𝑠), then the shift coefficient of the gear is 𝑥2 = (1 − 𝜅𝑋𝑠). Using Eq. (4), 

Eq. (9) is written as 

  1
1

1
tan tan

2 2

d c a
s p p w s

w

Bz
m X m Z

r
   

 
        

 
 (10) 

Rearranging Eq. (10), the general form of the expression for total profile shift coefficient can be expressed as  

 
2

2 tan tan

a
w s

s d c
p p

m Z B
X

m

 

  

  



 (11) 

The value of the total shift coefficient is constant for a TSAAG gear pair as long as the sum of the drive side and coast side 

pressure angles is unchanged. The coordinates of the drive and coast-side involute for profile shifted asymmetric gear are given 

by 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 
4 

cos , sinx x x xx r y r    (12) 

   2 tan tan
2

d c d d
x s p p p xX inv inv

z

 
            

 
 (13) 

2.2.   Pressure angles in TSAAG system 

An asymmetric gear pair with reference tooth-sum 𝑍𝑠
𝑟 and drive side pressure angle 𝜙𝑝

𝑑𝑟, operating on a center distance 𝑐𝑠
𝑟 

can be replaced by a tooth-sum altered 𝑍𝑠
𝑎 asymmetric gear-pair, operating at working pressure angle 𝜙𝑤

𝑑𝑎, keeping the center 

distance unchanged 𝑐𝑤
𝑑 = 𝑐𝑠

𝑟 (TSAAG system). If the tooth sum is altered by a factor α, then  

The relation between the number of teeth and radius of the working pitch circle is given by  

2 cos

cos

d
w w

d
p

r
z

m




   (15) 

Using the Eq. (15), Eq. (14) can be reframed as 

   1 2 1 2
cos

cos

a
d

a rw
w wd

p

r r r r





 
    
 
 

 (16) 

Using the relation (𝑟𝑤1 + 𝑟𝑤2) = 𝑐𝑤 , (𝑟1 + 𝑟2) = 𝑐𝑠  and defining 𝑐𝑤
𝑎 = 𝛽𝑐𝑠

𝑟 , Eq. (16) can be rearranged to express the 

operating pressure angles on the drive and the coast side.  

1

1

cos cos

cos cos

d d
w p

c c
w p


 




 







  
   

  


 
  

 

 (17) 

Reframing Eq. (16) to relate the working center distance of TSAAG with its tooth sum we get 

cos

2cos

a
d r
pa a

w sd
w

m
c Z





   
        

 (18) 

The standard center distance of the TSAAG is related to its tooth sum by the relation 

2

a
a a
s s

m
c Z

 
  
 

 (19) 

Subtracting Eq. (19) from Eq. (18), assuming that the TSAAG module remains unchanged 

  1
2

r
a a

w s s

m
c c Z





   
      

   
 (20) 

2.3.   Topping for TSAAG system 

 The TSAAG is a profile shifted system on fixed center distance. For a given profile shift, the ratio of addendum radius 

gain to that of tooth thickness gain along the pressure line is always constant and is approximately equal to 3 (refer to Fig. 2). 

This significant increase in addendum radius for attaining a nominal increase in tooth thickness along the pressure line may 

a r
s sZ Z  

(14) 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 

5 

r
1
 

r
w1

 

r
w2

 

r
2
 

a
w1

 

a
w2

 b
w1

 

b
w2

 

b
1
 

b
2
 

cause inappropriate root clearance. Similarly, a small amount of negative shift results in a considerable reduction in the 

magnitude of the addendum circle radius. This issue can be overcome by topping the addendum with a calculated positive or 

negative value. To ensure appropriate root clearance, addendum radii are topped with tooth topping coefficient Y. A TSAAG 

generated by a rack cutter with an addendum of 1.25 module and dedendum of 1 module in mesh is shown in Fig. 3.  

 

 

 

 

 

 

 

 

 

 

 

The radial root clearance 𝑟𝑐𝑙  is expressed as  

     1 2 1 2 1 22.25cl w wr b b m r r r r        (21) 

The radial distance between the root circle and the pitch circle is given by 

 

 

1

2

1.25

1.25 1

s

s

b X m

b X m





  


     

 (22) 

Using Eq. (22), Eq. (21) can be reframed as 

   0.25
a

cl w s
s

m r c c
X

m

  
  (23) 

Using Eq. (20) in Eq. (23), and defining (0.25𝑚 − 𝑟𝑐𝑙 ) = 𝑌𝑚, an expression for tooth topping coefficient is given by  

1
2

a
s

s

Z
Y X





 
   

 
 (24) 

The root clearance for any gear pair is established to be at least 0.25m. TSAAG system which is topped should satisfy the 

expression  

0.25clr Ym m   (25) 

TSAAG system works on the concept of altering the tooth sum of the reference gear pair. The parameters α and β enables 

designers to opt for tooth sum altered TSAAG system or center distance altered S± asymmetric profile shifted system. The Eqs. 

(1)-(25) are the most general set of expressions for TSAAG profile shifted system, S± profile shifted system, and standard spur 

 

 

 

Fig. 2 Profile shift 

 

Fig. 3 Topping for root clearance 

0.87 

2.62 
1.74 

0.6 
0.58 

0.89 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 
6 


x
 

T
o
o
th

 
c
e
n
te

r 

li
n
e
 (x

c
, y

c
)

d

 (x
c
, y

c
)

c

 

(x
d
, y

d
) 

r
x
 

f
x
 

gear system (STS). For a standard backlash-free gear-pair, α = β =1. For the TSAAG system, 0.96< α <1.04 and β =1. For the 

S± profile shifted system, 0.96< β <1.04 and α =1. TSAAG accommodates sums of various curvatures by keeping the center 

distance constant and changing operating pressure angles, whereas the S± asymmetric profile shifted system accommodates 

sums of various curvatures by keeping the sum of the base circle radii constant and changing operating pressure angles (refer to 

Fig. 4). Any two arbitrary gear pairs operate at the same working pressure angles as long as the ratio (α/β) is maintained 

constant. The magnitude of the total profile shift coefficient remains constant for any asymmetric gear pair as long as the sum 

of the drive side and coast side pressure angles are unchanged, referring to Eq. (11). 

2.4   Contact ratio and interference avoidance criteria for TSAAG 

It is possible to accommodate gear pairs with different modules and tooth sums on a specified center distance. Operating 

higher module gears with lower tooth sums increases power transmission capability but reduces contact ratio and restricts 

speed range. Gear ratios achievable with any tooth-sum should satisfy the interference avoidance criterion [13] which also 

establishes the limiting value for the module. The necessary and sufficient mathematical conditions are given as 

maxra

f lr r

 


 

(26) 

and 

 1
2

ra rl

z
CR  


   (27) 

3. Safe Load Transmission Capability of TSAAG System 

  

Fig. 5 Load bearing in a TSAAG system 

Evaluation of power transmission capability based on material’s bending strength depends upon the tooth geometry. A 

tooth load intensity 𝑓𝑥 per unit face width along the pressure line at any arbitrary value of roll angle can be resolved into 

components along and perpendicular to the tooth centerline. The tangential component of the tooth force along the pressure 

  

Fig. 4 Difference between TSAAG and S approach 

 

a cos(
d

w
)
a
 b 

c
1

 

(
d

w
)
a
 

c abRb1+Rb2 

bc
1
+

2
  

a b 

c 

c
1

 

b
1

 cos(
d

w
)
a
 

(
d

w
)
a
 

(a) Center distance altered gears (b) Tooth sum altered gears 

(a) Tooth subjected to a load (b) A gear pair in mesh 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 

7 

line causes tensile stresses on one side of the root fillet and compressive stress on the other side. The effect of the radial 

component of the tooth force is to cause compressive stress on either side of the tooth. The net effective fillet stress at the root 

is compressive on one side and tensile on the other. These cyclic tensile stresses are responsible for the initiation and 

propagation of fatigue cracks. The expression for safe load-bearing F [13] is modified to find the maximum static load which 

could be safely transmitted by a gear of a given module and material to avoid bending failure in any kind of spur gear system 

(refer to Fig. 5). 

t dl

f
F m

k x


  (28) 

 md c
x

t f x

y y tan
k m k cos

Z A




 
  
 
 

 (29) 

x
sl

eq

x



  (30) 

where 𝑘𝑡 is called the geometric location factor for bending, A and Z are the cross-sectional area and the sectional modulus of 

the critical section. The Cartesian coordinates of the critical section on the drive side and the coast side are (𝑥𝑐
𝑑 , 𝑦𝑐

𝑑 ) and 

(𝑥𝑐
𝑐 , 𝑦𝑐

𝑐 ) respectively. Dynamic load factor 𝑥𝑑𝑙  is the ratio of location-based mesh stiffness 𝜆𝑥 to total mesh stiffness 𝜆𝑒𝑞 .   

4. Surface Wear in TSAAG System 

Gear surface wear is an on-going process that results in loss of material along the tooth profile with an increasing number 

of load cycles. The profile deviation increases as wear progresses, the effects of which are manifested in the form of an increase 

in noise, vibration levels, and backlash. These deviations generate overloads on the teeth for which the gearing may not have 

been sufficiently dimensioned when designed, which could cause the onset of the other modes of failure in the spur gear tooth. 

Surface wear is evaluated by the volume of material lost, and the degree of wear is described by wear rate, specific wear rate, or 

wear coefficient. Archards wear equation [14] for sliding contacts is a mechanical wear model popular for its simplicity and its 

ability to characterize wear under a wide variety of conditions. Flodin and Anderson [15] modified the generalized Archard’s 

equation to describe dry and boundary lubricated sliding surfaces of a gear pair in mesh.  

1 2
, , 1 0

1

2 r rx n x n w fc
r

v v
h h k a

v



   (31) 

Observing term within the parenthesis [𝑣𝑟1 − 𝑣𝑟2 ∕ 𝑣𝑟1] = 𝑉𝑠𝑠 called as specific sliding, and 𝜎𝑓𝑐 𝛼0 = 2𝑓𝑑𝑥 𝜋⁄  [12], Eq. (31) 

can be modified as    

, , 1

, , 1

4

4 *

dx
x n x n w ss

x n x n dl ss w

f
h h k V

F
h h x V k










 


  


 (32) 

where F is the static load transmitted per unit face width, ℎ𝑥𝑛 is the accumulated wear at any location after n cycles, 𝑓𝑑𝑥 is the 

dynamic load per unit face width, 𝑥𝑑𝑙  is the dynamic load factor at the contact location, 𝛼0 is the semi-contact width of the 

elliptical contact, and 𝑘𝑤  is the wear coefficient. The total profile error 𝜀𝑥 , dynamic load 𝑓𝑑𝑥 , and load factor 𝑥𝑑𝑙  at any 

location after n cycles is given by 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 
8 

1 2
xn xn xnh h    

(33) 

 dx x r xf x    (34) 

dx
dl

f
x

F
  (35) 

5. Implementation Scheme and Methodology 

For the case study, a gear pair of module 2 with a drive side pressure angle 20˚ and tooth thickness distribution factor 

τ = 0.438 is adopted for a center distance of 108mm. Operating conditions created by four different tooth-sum alteration 

factors 0.96< α<1.04 and profile shift factors with β =1 are considered. The material properties, operating conditions, and gear 

geometry alterations described by Eqs. (11), (17), and (24), are tabulated in Table 1.  

Table 1 TSAAG geometry and material properties 

GR=1:1 α Material properties and operating conditions 

α 0.96296 0.98148 1 1.01852 1.03703* Density of steel 7700 kg/m
3
 

𝑍𝑠
𝑎 104 106 108 110 112 Damping factor (𝜉) 0.17 

𝜙𝑤
𝑎  0.4397 0.3968 0.349 0.294 0.2264 Young’s modulus (𝛦) 206GPa 

Β 0.144 0.144 0.144 0.144 0.144 Fillet strength (𝜎𝑓 ) 413MPa for 10
6 
 cycles 

𝑋𝑠 2.163 0.97 -0.1 -1.0273 -1.7885 Wear coefficient (𝑘𝑤 ) 5×10
-16

 m
2
N

-1
 

Y 0.1627 -0.0302 -0.1 -0.0272 0.2111 Speed 1000 rpm 

𝑡𝑝 4.715 3.847 3.069 2.392 1.839 *Case of interference 

The product of dynamic load factor and location-based geometric factor determines the safe load-bearing ability of a gear, 

referring to Eq. (28). The product of dynamic load factor and specific sliding is the deciding factor for accumulated wear 

depths over n cycles, referring to Eq. (32). Hence the determination of dynamic load factors is critical in the analysis of any 

gearing system. To evaluate dynamic load factors for TSAAG, a gear pair in mesh is modeled as two rigid disks connected by 

a spring-damper set along the pressure line (refer to Fig. 6). The single Degree of Freedom (DOF) dynamic model considers the 

influence of mesh stiffness, damping forces 𝐹𝑐 , friction forces, static transmission error at the mesh interface, and are expressed 

as a time-varying function. The gear pairs are assumed to be of unit face width and free of tooth profile errors. The differential 

equations of motion (EOM) for a single DOF spring mass damper system can be expressed as 

eq r eq r eq rm x c x k x F  
 

(36) 

1 2

1 2

e e
eq

e e

m m
m

m m



 (37) 

 

Fig. 6 Dynamic model for spur gear 

i
G
 

j
1
 

i
1
 j

G
 

j
2
 

i
2
 


1
 


2
 

R
b1

 

R
b2

 

 


1
, ω

1
 

{G}{1}{2}{3} 

i
4
 

j
4
 

{G} 

ω
2
 

T
2
 

j
3
 

i
3
 



Proceedings of Engineering and Technology Innovation, vol. 18, 2021, pp. 01-14 

 

9 

   2 1 1 2
1 1

1 2

1 1
n n

e i i e i i

i i
eq

e e

m m

k
m m

   
 

  




 
 

(38) 

2eq eq eqc m k  (39) 

where 𝑚𝑒𝑞, 𝑘𝑒𝑞, 𝑐𝑒𝑞 are mass, stiffness, and damping coefficient of the equivalent S-M-D system, 𝑚𝑒 is the effective mass, 

and 𝑥𝑟 is the relative displacement at the mesh interface.  

FEM analysis is adopted to create gear tooth stiffness 𝑘𝑠 , for TSAAG system as a function of its roll angle. The 

coefficients for the polynomials for various α values are tabulated (refer to Fig. 7 and Table 2). 

Table 2 Equation for stiffness of TSAAG 

𝑘𝑠(𝜃) = 𝑝1𝜃
3 + 𝑝2𝜃

2 + 𝑝3𝜃 + 𝑝4 

 p1 p2 p3 p4 

0.96296 -7304.8 9601.2 -4348.9 725.24 

0.98148 -3595.8 4013.4 -1631.5 284.15 

1 -1780.7 1580.5 -597.23 131.89 

1.01852 -937.43 536.73 -204.14 74.484 

1.03703 -1049.9 427.48 -133.25 54.627 

 

The equation of motion is numerically solved using the fourth-order Runge-Kutta linear iterative procedure. One mesh 

period is divided into a certain number of intervals and the solution procedure is started with a trial value of initial conditions. 

The values of the relative displacement and velocity after one mesh period are compared with the trail values. The solution is 

said to converge when the difference between the initial condition and the obtained values after one mesh cycle are within the 

preset tolerance. The value of relative displacement is used for finding dynamic load factors, referring to Eq. (35). The safe 

static load transmission capability is evaluated using the ratio of location-based mesh stiffness to total mesh stiffness. A flow 

chart of the in-house developed code is provided in Fig. 8. 

   
(a) α = 0.96296, κ = 0.5, β = 1 (b) α = 0.98148, κ = 0.5, β = 1 (c) α = 1, κ = 0.5, β = 1 

   
(d) α = 1.01852, κ = 0.5, β = 1 (e) α = 1.03703, κ = 0.5, β = 1 (f) Mesh stiffness of TSAAG 

Fig. 7 FEM analysis for the stiffness of TSAAG 



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10 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 8 Flow chart for evaluation of (𝑥𝑑𝑙 × 𝑣𝑠𝑠 ) & (𝑥𝑑𝑙 × 𝑘𝑡 ) 

6. Results and Discussion 

  

Fig. 9 Contact ratio vs. profile shift factor Fig. 10 Safe load vs. tooth sum alteration factor (α) 

Contact ratio is a geometric parameter of much importance which is an indicator of the influence of gear geometry on its 

performance. Since it is a pure geometric parameter, it is always a good starting point to begin an investigation that involves 

gear geometry modification. The investigation of the contact ratio provides information and direction in which the analysis 

needs to proceed. TSAAG is a tooth sum altered, profile shifted, addendum topped system, and all of these modifications affect 

the contact ratio. A TSAAG carrying positive profile shift exhibits a reduction in the contact ratio, owing to reduced tooth-sum, 

increased working pressure angle, and reduced radius of the addendum circle due to tooth topping. Limiting the study to gears 

with a contact ratio above 1.3 (refer to Fig. 9), it is observed that while the tooth-sum reduction by 4 is permissible with contact 

ratio falling to 1.36, a tooth-sum increase by 2 is permissible, without causing interference, raising the contact ratio above 2. It 

Compute  solving EOM 

Compute  

Re-compute cumulaitive  

 

Cumulative wear 

depth at each contact 

point 

n=n+1 

End 

Input  

Compute  

Cycles n=1, Wear depths  

Material 

Properties 

Start 



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11 

is important to note that +4 tooth alteration without any design backlash results in undercutting of gears (refer to Table 1) and 

would cause a certain part of the involute profile not to participate in load-bearing. This part of the involute profile would show 

an increase in the contact ratio mathematically but has no practical significance. Hence positive tooth alterations are advised 

only up to a profile shift value at which the first sign of undercutting appears. Similarly, the negative number of tooth 

alterations should be limited to a profile shift value at which the tooth becomes peaked. Attempting higher negative alterations 

would result in an unreasonable reduction in contact ratio. Introduction of design backlash reclaims profile shift to a certain 

degree, salvaging certain part of the addendum which would otherwise be topped. Hence one can expect negative number of 

tooth alterations without backlash to lower the contact ratio, but introduction of backlash shows an improvement of the same.  

The preliminary investigation of TSAAG based on contact ratio reveals the possibility of achieving high contact ratio 

(HCR) TSAAG with positive values of tooth alterations (α > 1) and low contact ratio (LCR) TSAAG, with negative values of 

tooth alterations (α < 1) and all gear pairs running on the same center distance. Experimental results by Kasuba [16] and 

Kahraman-Blankenship [17] have established that the dynamic loads decrease with increasing contact ratio in spur gearing. 

Even though TSAAG with HCR > 2 opens up the possibility of transmitting more power at low decibels without resorting to 

higher strength materials, it is important to interpret this result in the light of safe load transmission capability. 

The plot of safe load-bearing capability F vs α (refer to Fig. 10) shows an increase in load-bearing capability on either 

side of unity but the interpretation varies. The increase in load-bearing ability for LCR TSAAG with α < 1  is attributed to the 

increased tooth thickness and pressure angles. However, for HCR TSAAG, the increase in load-bearing ability comes due to 

the contact ratio rising above 2 as seen in the case of α = 1.01852. Further Increasing the value of α, reduces load-bearing 

ability due to the reduction in tooth thickness. The product of dynamic load factor and location factor for bending 𝑥𝑑𝑙 × 𝑘𝑡, 

referring to Figs. 11 (a)-(f), is the sole dimensionless parameter that decides the safe load-bearing ability of any gear. Normal 

contact ratio standard tooth sum (NCR STS) asymmetric gears and LCR TSAAG show one region of single tooth contact and 

two regions of double tooth contact. HCR TSAAG shows three regions of triple teeth contact and two regions of double tooth 

contact, indicating an improvement in the load-bearing capacity. The maximum value of 𝑥𝑑𝑙 × 𝑘𝑡 in either of the gears decides 

the safe load transmission capability of any gear drive. Dynamic load factor is speed-dependent and location factor for bending 

is geometry dependent.  

   
(a) α = 0.96296, κ = 0.5, β = 1  (b) α = 0.98148, κ = 0.5, β = 1 (c) α = 1, κ = 0.5, β = 1 

   
(d) α = 1.01852, κ = 0.5, β = 1 (e) α = 1.3703, κ = 0.5, β = 1 (f) Location factors 𝑘𝑡 for bending  

Fig. 11 𝑥𝑑𝑙 × 𝑘𝑡 for 0.96296 ≤ α ≤ 1.3703 



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The single tooth contact region is the largest for α = 0.96296 and starts reducing as alpha approaches unity. The dynamic 

load dominates the magnitude of 𝑥𝑑𝑙 × 𝑘𝑡 which is evident by observing the slope of the plots for α < 1. At any given speed, 

dynamic load in a gear system decreases with an increasing contact ratio. This is because of the narrow single or triple contact 

zone, which passes quickly, leaving no time for the system to respond. Hence for HCR TSAAG the value of 𝑘𝑡 dominates the 

magnitude of 𝑥𝑑𝑙 × 𝑘𝑡 . The plot of TSAAG with α = 1.01852 yields the lowest of the maximum magnitude of 𝑥𝑑𝑙 × 𝑘𝑡 

among TSAAG, which gives a clear impression of having the largest load-bearing ability. Also, HCR TSAAG is expected to 

have lower decibel levels due to reduced dynamic loads. 

While it looks very encouraging to know that it is possible to have improved load transmission capability with HCR 

TSAAG, it is also important to evaluate its surface durability. The product of dynamic load factor and specific sliding 𝑥𝑑𝑙 × 𝑉𝑠𝑠 

is the sole dimensionless parameter that decides the wear accumulation, refering to Eq. (32). The gear pairs are subjected to a 

load value to sustain 106 bending cycles. At κ=0.5, TSAAG as well as NCR asymmetric gears, irrespective of the value of α, 

attain equal approach-recess action. The plots of 𝑥𝑑𝑙 × 𝑉𝑠𝑠  for TSAAG operating under κ=0.5 with different α values, 

accumulated over 105 cycles are shown in Figs. 12(a)-(f). For TSAAG with α < 1 the slope of the curve is steeper in the 

region of single tooth contact owing to comparatively larger load and it reduces in the two teeth contact region. The region on 

either side of the pitch point covering a certain roll angle indicates a larger region of single tooth contact or lower contact ratio. 

The gears experience larger dynamic loads during the transition from single tooth to double tooth contact at lower speeds. The 

specific sliding in Fig. 12(f) shows significant improvement for α < 1 in the addendum and dedendum regions of both pinion 

and the gear. The improvement in specific sliding offsets the higher dynamic loads, keeping the accumulation of 𝑥𝑑𝑙 × 𝑉𝑠𝑠 

minimal up to 105 cycles. A comparison with asymmetric NCR gear indicates higher service life of TSAAG LCR with higher 

decibel levels at lower speeds due to reduced contact ratio.  

 

  

 (a) α = 0.96296, κ = 0.5, β = 1 (b) α = 0.98148, κ = 0.5, β = 1 (c) α = 1, κ = 0.5, β = 1 
  

 

(d) α = 1.01852, κ = 0.5, β = 1 (e) α = 1.03703, κ = 0.5, β = 1 (f) Specific sliding 𝑉𝑠𝑠  

Fig. 12 𝑥𝑑𝑙 × 𝑉𝑠𝑠 for 0.96296 ≤ α ≤ 1.3703 (continued) 

The accumulation of 𝑥𝑑𝑙 × 𝑉𝑠𝑠 for TSAAG HCR as the value of α approaches 1.03703 is significant over 10
5 cycles at the 

region near the root and addendum circles, where the slope of specific sliding changes drastically. Profile error introduced due 

to wear is accumulative over cycles of usage. The dynamic loads are lower as a result of a narrow region of three teeth contact 



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13 

which passes quickly. The dynamic load factor increases with the accumulation of profile error. Specific sliding considered 

constant for every cycle acts as an amplifier to the iterating dynamic load factors. Hence the accumulation of 𝑥𝑑𝑙 × 𝑉𝑠𝑠 

becomes large especially at locations having higher specific sliding. While the dynamic load in STS asymmetric gears and 

TSAAG LCR with α < 1 does not grow as much owing to lower wear rates, TSAAG HCR gears with α > 1, start with lower 

dynamic loads and build up to higher-value during the course of its life. These gears are expected to run with decibel levels 

lower than the STS asymmetric gears for an initial period, but the vibrations and noise levels increase over their service life.  

7. Conclusions 

Altering the tooth sum of a reference NCR asymmetric gear pair offers a family of TSAAG gears for the same center 

distance with improved performance characteristics. Tooth sum is a macroscopic parameter that alters pressure angle and tooth 

thickness offering the flexibility of accommodating various tooth sums on the same center distance. Increased tooth-sum 

coupled with reduced pressure angle and tooth topping results in HCR TSAAG and decreasing tooth sum results in LCR 

TSAAG. Both LCR and HCR TSAAG have improved load-bearing ability and mesh stiffness. The effects of tooth-sum 

alteration on surface wear are influenced by meshing parameters such as dynamic load factors, specific sliding, and material 

properties. Wear rate increases for TSAAG, as α moves away from unity either towards lesser or greater values within the 

permissible limits. The specific sliding and dynamic load factors play a significant role in determining the overall dynamic 

load, surface wear build-up, and service life. The rate of change of slope of the specific sliding curve is a dominant factor 

influencing the wear rate and tooth life. HCR TSAAG offers up to 50% more load-bearing ability than NCR asymmetric gears, 

but higher wear rates increase dynamic loads which may reduce service life. LCR TSAAG has up to 13% more load-bearing 

ability with better surface durability but suffers loss in contact ratio which may cause noisy operation. In conclusion, tooth sum 

modified asymmetric gear geometry (TSAAG) is an effective design technique to introduce geometrical changes for design 

benefits while maintaining the center distance constant as they offer design benefits that are not achievable by NCR 

asymmetric gear design. 

Nomenclature 

α Tooth-sum alteration factor Β Design Backlash 

ß Center distance alteration factor 𝑐  Center distance 

�̈�𝑟 Relative acceleration at mesh interface 𝑐𝑒𝑞 Equivalent Damping coefficient 

�̇�𝑟 Relative velocity at mesh interface 𝐸 Modulus of elasticity 

κ Profile shift factor 𝐹  Total static load per unit face width 

λ Mesh stiffness per unit face width 𝐹𝜆 Spring force 

μ Coefficient of friction 𝐹𝑐  Damping force 

ϕ Pressure angle 𝑓𝑑𝑥 Location-based dynamic load per unit face width 

ρ Radius of curvature ℎ Accumulated wear 

𝜎𝑓 Allowable bending fatigue strength 𝑘𝑒𝑞   Equivalent Mesh stiffness 

𝜃𝑟𝑎 Roll angle of addendum radius 𝑘𝑡 Tooth geometric factor for bending 

𝜃𝑟𝑙  Roll angle of radius of limiting circle 𝑘𝑤 Wear Coefficient 

ξ Damping factor 𝑚 Module 

ε Tooth profile error 𝑟𝑏 Radius of Base circle 

𝜎𝑓𝑐 Contact pressure 𝑥 Profile shift coefficient 

𝑟𝑙  Radius of limit circle 𝑥𝑠𝑙  Static load factor 

𝑟𝑓 Radius of the fillet circle 𝑥𝑑𝑙  Dynamic load factor 

𝑟𝑥  Radius of any point on the profile 𝑥𝑟 Relative displacement at mesh interface 

𝑇 Total static torque per unit face width 𝑌 Tooth topping coefficient 

𝑣𝑟 Rolling velocity 𝑧 Number of teeth 

𝑉𝑠𝑠 Specific sliding 𝑍𝑠 Tooth-sum of gear pair 

𝑋𝑠 Total profile shift coefficient  



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Subscripts Superscripts 

1, 2 Driver and driven gears a  Altered 

𝑠 Standard r  Reference 

𝑤 Working d  Drive side 

𝑥 Any location along the pressure line c  Coast side 

𝑝 Pitch circle  

Abbreviations 

TSAAG Tooth sum altered asymmetric gear NCR Normal contact ratio 

HCR High contact ratio rpm Revolutions per minute 

LCR Low contact ratio STS Standard tooth-sum 
 

Conflicts of Interest  

The authors declare no conflict of interest.  

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