Noncircular Gears: Geometry and Visualization MODEL 

Development

by LTC David “Blake” Stringer, Ph.D.

ARL-TR-5865 January 2012

Approved for public release; distribution unlimited.
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Army Research Laboratory 

Cleveland, Ohio 44135 

ARL-TR-5865 January 2012 

Noncircular Gears: Geometry and Visualization MODEL 

Development

LTC David “Blake” Stringer, Ph.D.

Vehicle Technology Directorate, ARL 

Approved for public release; distribution unlimited.
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Noncircular Gears: Geometry And Visualization MODEL Development

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LTC David “Blake” Stringer, Ph.D.

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14. ABSTRACT

Circular gears are optimized to provide constant torque and speed ratio with low noise. In conventional automotive 

applications, a transmission provides the required power, torque, and speed settings using a finite arrangement of gear sets,

which limits the range of transmission gear ratios, resulting in a loss of efficiency. However, other transmission configurations 

operate over a continuous range of gear ratios, providing a continuous range of output speeds while maintaining constant input 

shaft speed. Circular gears are not always practical for these configurations. One solution incorporates noncircular gears. The

asymmetry of a noncircular gear pair results in instantaneously changing gear properties throughout gear rotation. This report 

presents a method for determining the required parameters to produce a noncircular gear pair of desired velocity and torque 

ratio distribution. Finally, to graphically depict the noncircular effects on gear motion, torque, and velocity distributions, a 

visualization model has been developed.

15. SUBJECT TERMS

Noncircular gear, gear geometry, gear analysis

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LTC David B. Stringer, PhD

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iii

Contents 

List of Figures iv

Acknowledgments v

Introduction 1

Methodology 1

Visulation Model 7

Conclusion 8

References 10

Nomenclature 11

Distribution List 12
iv

List of Figures 

Figure 1. Velocity profile during one revolution. ...........................................................................2

Figure 2. Dimensionless gear radius during one revolution. ..........................................................4

Figure 3. Gear rotation angle comparison.......................................................................................5

Figure 4. Noncircular gear dimensions. ..........................................................................................6

Figure 5. Two-lobe driven gear. .....................................................................................................6

Figure 6. Three-lobe driven gear. ...................................................................................................7

Figure 7. Visualization model capture 1. ........................................................................................8

Figure 8. Visualization model capture 2. .........................................................................................8

Figure 9. Visualization model capture 3. .........................................................................................8

Figure 10. Visualization model capture 4. ......................................................................................8
v

Acknowledgments 

This research was conducted in support of TPA TA-VT-2008-01, “Ground Vehicle Propulsion 

and Power Research,” between the U.S. Army Research Laboratory (ARL) and the Tank and 

Automotive Research, Development, and Engineering Center (TARDEC). 
vi

INTENTIONALLY LEFT BLANK. 
1

Introduction 

Circular gears have been used in conventional power train applications for over a century. They 

are optimized to provide constant torque and speed ratio with low noise. In conventional 

automotive applications, the transmission provides the required power, torque, and speed settings 

using a finite arrangement of gear sets. This finite number limits the range of transmission gear 

ratios, resulting in a loss of efficiency. However, other configurations have been used for power 

transfer, such as the continuously variable transmission (CVT) or, more precisely, the infinitely 

variable transmission (IVT). These configurations operate over a continuous range of gear 

ratios, allowing for a multitude of output speeds while maintaining constant input shaft speed. 

Circular gears are not always practical for these types of applications, necessitating another 

method of power transfer. One solution is to incorporate noncircular gears. 

Noncircular gears have proven effective for use in CVT/IVT applications since at least the 

1970s, due to their effects on gear ratio and torque transfer (1). The asymmetry of the gear pair 

results in instantaneously changing gear properties throughout the rotation of the gears. This 

requires more in-depth analysis to determine the gear parameters required to achieve a certain 

profile. Indeed, the geometric considerations for ensuring proper tooth contact and alignment are 

well documented (2, 3). Furthermore, the gear shapes, themselves, must be designed to achieve 

the desired velocity profile. This report presents a method for determining the required 

noncircular gear parameters to produce a desired velocity and torque ratio distribution. The steps 

of the methodology are introduced in detail. It is then demonstrated using an example from the 

literature. Finally, in order to better understand the noncircular effects on gear motion, torque, 

and velocity distributions, the results have been captured graphically through the development of 

a noncircular gear visualization model. 

Methodology 

In circular gear applications, the speed and torque ratios are constant throughout the gear rotation 

according to the well-established relationships in equation 1. 

V

r



 2

1



T1

T2



r

1

r

2

, (1) 

where (Vr) is defined as the velocity profile between the driving gear and driven gear. The 

velocity profile is defined as the velocity or speed ratio of the gear pair over one revolution. It is 

also the inverse of the torque ratio. 
2

For noncircular gears, the same relationships hold, but the ratios of equation 1 are dependent 

upon the angles of rotation of the gears, (1) and (2). Thus, equation 1 can be written as

V

r



 2 2 

1 1 



T1 1 

T2 2 



r

1

 1 

r

2

 2 

, (2)

where the speed and torque ratios are functions of the angles of rotation. As will be developed, 

these relationships have a pronounced effect on the velocity profile.

In the case of circular gears, the angles of rotation are unimportant because they do not impact 

the ratios listed in equation 1. In noncircular gears, however, (1) and (2) drive the entire design 

of the gear geometry and must be calculated. Therefore, the desired velocity profile, (Vr), must 

be designed or known a priori. For purposes of this report, a velocity profile from the literature 

is assumed (4), and is depicted in figure 1. The angle of rotation on the x-axis belongs to the 

input gear (1). It is the independent variable in the methodology and drives all aspects of gear 

pair motion.

Figure 1. Velocity profile during one revolution.

The velocity profile has a desired minimum value of 0.5 and a desired maximum value of 1.5. 

These are indicated on figure 1 by the two dotted horizontal lines. The average velocity ratio, 

(Vr_avg), is thus 1.0. A value of 1.0 further indicates that both gears achieve a full rotation at the 

same time.

Figure 1 also shows two distinct areas of the profile: a linear portion and a nonlinear portion. 

The linear portion accounts for two-thirds of the rotation of Gear 1, as noted on the x-axis from 

0–240°. From 240–360°, the ratio is nonlinear and sinusoidal. Note that the absolute maximum 

and minimum values of the velocity profile at the transition points between linear and nonlinear, 
3

and vice versa, are slightly higher and lower, respectively, as a result of enforcing a constant 

slope of the velocity curve at the transition points. 

The linear portion of the curve takes the familiar slope-intercept form of equation 3. 

V  1 r _ linear m1

V

r _10

(3) 

The nonlinear relationship is slightly more complex. The desired function is a standard sinusoid 

according to the form of equation 4. The two transitions between linear and nonlinear portions 

must be smooth and continuous. This requires the linear and nonlinear functions to have the 

same speed ratios and the same slope at the points of transition. 

V  1 r _ nonlinear V

r _ avg

 asin 1   

(4) 

Equation 4 contains three unknowns: the function amplitude (a), frequency (), and phase shift 

(). An iterative technique, enforcing the required boundary conditions, easily determines the 

values of these coefficients. Once the expressions of equations 3 and 4 have been determined, 

the velocity profile can be used to calculate the radii of the two gears and, then, their angles of 

rotation. 

The radii of the two gears are continuously changing as the gears rotate. The radii can, therefore, 

be calculated as a function of the velocity profile. Assuming the length between the centers of 

the driving and driven gears as unity, r1+r2=1, and manipulating equations 1 and 2 yields 

equation 5: 

r

1



1

1

Vr

 1













(5) 

Equation 5 is a nondimensional representation of radius, presented as a percent of the distance 

between gear centers. These results are presented in figure 2, where the y-axis below the curve 

represents the radius of the driving gear as a percentage of the center distance. The y-axis above 

the curve depicts the radius of the driven gear. 
4

Figure 2. Dimensionless gear radius during one revolution.

The curve in figure 2 depicts the radial profile, which achieves the desired velocity profile of 

figure 1. There are two points where the non-dimensional radius equals 0.5, occurring at (1) 

values of 120° and 300°. These points indicate that the radii of the driving and driven gears are 

both one-half the length between gear-shaft centers. The result is a speed ratio of one at these 

two points, as is confirmed in figure 1.

Returning to the discussion of the angles of rotation, (1) and (2), the relationship between these 

two is critical to developing the shape of the gear and for ensuring conjugacy of the gear teeth. 

For each value of (1), there is a corresponding value of (2), which will generate the profiles in 

figures 1 and 2. The values of (2) as a function of (1) can be determined by calculating the area 

under curve of the velocity profile in figure 1 (4), which is accomplished by integrating 

equations 3 and 4. The resulting expressions are given in equations 6 and 7. Using these 

expressions, the relationship between the rotations of the driving and driven gears can be 

determined for a full revolution.

 21 m1V

r

1  0 d1 0

1_ linear _ max



(6)

 21 1_ linear _max asin 1   d1 1_ linear _ max

360



(7)

A convenient way to illustrate this relationship is to plot the values of (1) and (2) on the x-axis 

and y-axis, respectively. Plotting a reference line with a slope of unity provides a visual 

comparison of the angles of rotation. This is depicted in equation 3.

The correct interpretation of the curve in figure 3 is necessary to obtain any relevant information 

from it. Going back to figure 1, the velocity ratio is less than one between 0° and 120°. The 

driving gear is turning faster than the driven gear; thus, (1) is growing “faster” than (2). The 
5

curve in figure 3 should correspondingly move south of the center-line. At 120°, the velocity 

ratio is equal to one, which results in the maximum deflection south of the center-line. Beyond 

120°, the driven gear is rotating faster than the driving gear. This is indicated by the steepening 

slope of the curve and movement back toward the center-line. At 240°, the point of transition 

from linear to nonlinear velocity profile, the velocity ratio is still greater than one, such that (2) 

is growing faster than (1). Additionally, remembering that both gears reach a full rotation 

together, it holds true that both gears reach the transition point of 240° together, as well. This is 

indicated by the curve recovering the center-line at 240°. Soon after this point, the velocity ratio 

begins to decrease rapidly along the sinusoidal curve. However, the velocity ratio is still greater 

than one, causing the curve to continue moving north of the center-line. At 300°, the velocity 

ratio decreases to one, which again is the maximum deflection of the curve from the center-line. 

Beyond 300°, (1) is once again growing faster than (2). The curve moves southward until both 

angles meet at 360°, since both gears complete one revolution simultaneously.

Figure 3. Gear rotation angle comparison.

The shapes of the two gears themselves can be determined using the relationships in equation 8.

x

1i

 Lr

1i

cos  1i

y

1i

 Lr

1i

sin  1i

x2i Lr

2i

cos 180

o

  2i

y2i Lr

2i

sin 180

o

   2i

i  1,2,K , n

, (8)

where (n) is the total number of theta segments per revolution. Plotting the results of equation 8 

yields the geometry of the noncircular gear patterns, which are shown in figure 4.
6

Figure 4. Noncircular gear dimensions.

From figure 4, the gear geometry calculated using equations 1–8 is clearly noncircular. The 

driving gear (left) is somewhat kidney-shaped. The driven gear (right) has an egg shape. As the 

gears rotate, they will always mesh at the pitch point, as conjugacy has been enforced 

throughout. However, unlike circular gears, the pitch point, itself, is not stationary. From figure 

4, the pitch point slides along the line connecting the center-points of the two gear shafts.

The driven gear has one pointed “lobe,” giving it the shape of an egg. This is a consequence of 

the average velocity ratio having a value of 1.0. The number of “lobes” of the driven gear is the 

inverse of its average velocity ratio. A velocity ratio of 0.5 will result in two lobes. 

Correspondingly, a value of 0.333 results in a driven gear with three lobes. These examples are 

easily calculated using this methodology and are presented in figures 5 and 6. Nevertheless, 

while it is theoretically possible to create a gear of several “lobes,” the manufacturing and tooth 

meshing requirements constrain their feasibility in actual practice.

Figure 5. Two-lobe driven gear.
7

Figure 6. Three-lobe driven gear.

Visualization Model

One of the most difficult aspects of noncircular gears to grasp is how the asymmetry affects the 

gear properties of torque and speed through a complete revolution. A visualization tool has been 

developed to address this. The tool is a Matlab-based script capturing the rotational motion of 

the gears as well as speed, torque, and velocity ratio. The driving shaft speed input and the 

driven torque output are both held constant at unity. The driven shaft speed output and the 

driving torque input are calculated using equation 9.

T1 T2

1



r

1

r

2

 2 1

r

1

r

2

, (9)

where () represents the efficiency of the gear mesh. This can have a wide range of values 

depending upon the quality of the gear material, surface finish, manufacturing tolerances, etc. 

For highly precisioned gears, the value assumed is 99%.

Equation 8 yields the x and y coordinates of the stationary gear geometries. Equation 10 

calculates the rotated x and y coordinates for the desired angles of rotation to create the rotational 

motion of the visualization model. 

x '

1i

y '

1i























cos1isin1i

 sin1icos1i

















x

1i

y

1i





















x '

2i

y '

2i























cos   2i sin   2i

 sin   2i cos   2i





















x

1i

y

1i





















(10)
8

Figures 7–10 provide four sequential stages of gear rotation produced using the visualization 

model, along with the corresponding instantaneous values of torque, speed, and speed ratio.

Figure 7. Visualization model capture 1.

Figure 9. Visualization model capture 3.

Figure 8. Visualization model capture 2.

Figure 10. Visualization model capture 4.

Conclusion

This report has presented a methodology for determining the geometric and dynamic parameters 

of noncircular gears in rotation. The velocity profile was introduced as the parameter that 

ultimately drives the resulting design of the gear. The linear and nonlinear velocity profile 

functions were determined using an iterative solution method. The radii of the gears were 

determined. The relationship between the angles of the driving and driven gears was calculated. 

This resulted in the calculation of the gear geometries. Finally, the methodology was automated 

into a visualization model to graphically show how the gear parameters fluctuate using 
9

noncircular gears. This type of analysis forms the basis for analyzing more complex driven 

systems using noncircular gears, such as the IVT. 
10

References 

1. Ferguson, R.; Daws, L.; Kerr, J. The Design of a Stepless Transmisssion Using Noncircular 

Gears. Machine and Mechanism Theory 1975, 10 (6), 467–478. 

2. Litvin, F.; Fuentes-Aznar, A. Noncircular Gears: Design and Generation; New York: 

Cambridge University Press; 2009.

3. Litvin, F.; Fuentes, A. Gear Geometry and Applied Theory; 2nd Ed. New York: Cambridge 

University Press; 2004.

4. Kerr, M. All Gear Infinitely Variable Transmission, U.S. Patent 6,849,023, 2005.
11

Nomenclature 

m slope of linear velocity profile 

ri gear radius 

Ti gear torque 

Vr velocity ratio 

i gear angle of rotation 

i gear rotational speed
12

NO. OF 

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8725 JOHN J KINGMAN RD STE 0944 

FT BELVOIR VA 22060-6218 

1 CD OFC OF THE SECY OF DEFNS 

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THE PENTAGON 

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1 US ARMY RSRCH DEV AND ENGRG CMND 

ARMAMENT RSRCH DEV & ENGRG CTR 

ARMAMENT ENGRG & TECHNLGY CTR 

ATTN AMSRD AAR AEF T J MATTS 

BLDG 305 

ABERDEEN PROVING GROUND MD 21005-5001 

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FT HUACHUCA AZ 85613-5300

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TOTAL: 15 (13 HCS, 1 ELEC, 1 CD)