Analysis of Apex Seal Friction Power Loss 

in Rotary Engines

NASA/TM—2010-216353

September 2010

Robert F. Handschuh

Glenn Research Center, Cleveland, Ohio

A. Karl Owen

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

ARL–TR–5162

U.S. ARMY

RESEARCH LABORATORY
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Analysis of Apex Seal Friction Power Loss 

in Rotary Engines

NASA/TM—2010-216353

September 2010

Robert F. Handschuh

Glenn Research Center, Cleveland, Ohio

A. Karl Owen

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

ARL–TR–5162

U.S. ARMY

RESEARCH LABORATORY

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

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Alexandria, VA 22312

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2010-216353 1 

Analysis of Apex Seal Friction Power Loss in Rotary Engines 

Robert F. Handschuh 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

A. Karl Owen 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

An analysis of the frictional losses from the apex seals in a rotary engine was developed. The 

modeling was initiated with a kinematic analysis of the rotary engine. Next a modern internal combustion 

engine analysis code was altered for use in a rotary engine to allow the calculation of the internal 

combustion pressure as a function of rotor rotation. Finally the forces from the spring, inertial, and 

combustion pressure on the seal were combined to provide the frictional horsepower assessment. 

Introduction 

The United States Army currently uses a 28.3 kW Wankel engine as the propulsion system for a 

small, unmanned aerial vehicle. In 2009 the Army Vehicle Technology Directorate at the Glenn Research 

Center initiated a fundamental research program intended to provide information and recommendations 

for improving the durability and performance of this engine. To properly assess any potential 

improvements, a quantitative understanding of the frictional horsepower requirements is required. 

The rotary or Wankel engine is a fairly simple design of an internal combustion engine. There are no 

valves, camshaft, or connecting rods in this engine. The engine has two gears (one external that is fixed 

and one internal that rotates with the crankshaft journal), a rotor, a crankshaft, housings and support 

bearings. The number of parts to make an engine is very few in comparison to a typical internal 

combustion engine. 

Due to the crankshaft rotating the rotor (triangle-shaped component) that meshes with an external 

gear fixed to the housing and floating via a bearing on the crankshaft journal, the crankshaft rotates three 

times for every one rotation of the rotor. Apex seals are on the tips of the rotor ends (there are three) and 

provide the sealing from one cavity to the next. The seals are very important to the performance of the 

engine. 

The engine operates with the same four-stroke Otto cycle as a piston engine. The process is shown in 

Figure 1. In Figure 1 the four different strokes are numbered in the figure as intake, compression, power, 

and exhaust. Each cavity sweeps by the intake port, compresses the mixture, fires the mixture, then 

removes the exhaust every revolution of the rotor. 

The rotor of this engine holds the seal components. The seals are a critical aspect of the engine 

maintaining its performance. A break down of the seal will lead to leakage between the cavities and 

contamination of the incoming fuel-air charge and a loss of power. The seal is rectangular in cross section 

and wear resistant. An example of the seal and contact spring are shown in Figure 2. The seal is two-piece 

with a split diagonal feature in the rotational axis direction to allow for proper sealing of the engine 

cavities at all operating (temperature) conditions. 
NASA/TM—2010-216353 2 

 
NASA/TM—2010-216353 3 

Kinematic Analysis 

Dynamics of a Point on a Body With Translation and Rotation 

A kinematic analysis of the rotary engine mechanism will now be described by first developing the 

general equations as given in References 2 to 4. The shape of the housing—or the path of the apex seal 

during operation is called an epitrochoid. A fixed coordinate system will be used to describe the position 

(path), velocity, and acceleration of a point at the apex of one of the rotor locations is shown in Figure 3 

(there are three apex seals, 120 apart). 

Coordinate system S1 is the fixed frame of reference and coordinate system S2 rotates with the rotor 

and point P is the location of the apex seal. O and O′ are the centers for coordinate systems S1 and S2, 

respectively. 1 and 2 are the rotation angles of the crankshaft and rotor in coordinate systems S1 and 

S2, respectively. 1

 and 2 are the rotational velocities of the crankshaft and rotor, respectively. R



 is the 

position vector from coordinate system S1 to S2 . 

 is the position vector from O′ to the apex seal. R



 and 



 are fixed in magnitude but their direction is a function of the relationship between the coordinate 

systems. 

The position vector, r

 , for point P is given by 

      r R (1) 

Differentiating Equation (1) with respect to time results in the velocity with respect to the fixed 

coordinate system S1 as 

  







 

 

V r R (2) 
NASA/TM—2010-216353 4 

where 

       

  r ( (3) )

where  is the absolute rotation rate of coordinate system S2. Therefore, the apex velocity V



 is given by 

      





 

V R r ( ) (4) 

Point P′ is coincident with P in system S2 but fixed in this coordinate system. R



 is the absolute velocity 

of O′ and     is the velocity of P′ relative to O′ as viewed by a non-rotating observer. Therefore 

   

R represents the absolute velocity of P′. The term r ()



 is the velocity of P relative to O′ as seen 

by an observer rotating with the S2 coordinate system. 

Next the acceleration will be described. Differentiating Equation (4) with respect to time provides the 

following 

( ) ( ) (( ) ) () 



 









  

   





 

t t

R

t

V

t r (5) 

where the terms of equation become 

R R

t



   

 ( )

(( ) ) ( ) (( ) ) r r r t

      

 

   

()        (() )  () 

    

       

   r t

inserting these terms into Equation (5) results in the acceleration A



 being given as 

( ) (( ) ) 2 (( ) ) A  R          r    r



       



  (6) 

where 

R



 is the absolute acceleration of O′,  

 

 and  ( )    represent the acceleration of P′ relative to O′

as viewed from a non-rotating observer. In other words with  

 

 is the tangential acceleration and 

 ( )    is the centripetal acceleration of P towards the axis of rotation through O′. Vector ) (( )  r



 is 

the acceleration of point P relative to the S2 coordinate system as viewed by an observer moving with S2. 

Vector ) 2 (( )   r



  is the Coriolis acceleration. 
NASA/TM—2010-216353 5 

Application to a Rotary Engine 

For the rotary engine the following are given as known due to the timing of the mechanism between 

the rotor, crankshaft and the internal-external gearing. In this engine a 30-tooth external gear is fixed to 

the housing. A 45-tooth internal gear rotates about the fixed external gear with the crankshaft offset equal 

to the center distance difference of the meshing external-internal gears. This results in a 3:1 angular and 

rotational relationship between the crankshaft and the rotor. In the analysis, the crankshaft speed and 

rotation relationships are then given as: / 3 2  1 and   const 1 , and / 3 2  1 . 

The equation for the apex seal velocity becomes the following (since 0 (()r )  

 ; point P does not 

move relative to O′ for a rotating observer) 

      

 

V R (7) 

The acceleration is the following (since 0  



 , then 0  



 ) Equation (6) becomes 

   ( )    

 

A R (8) 

From Figure 3 the position vector for the apex seal is given as 





























   

   



0

sin cos

cos cos

1 2

1 2

R

R

r

 (9) 

and the velocity vector becomes 





























     

    



0

cos cos

sin sin

1 1 2 2

1 1 2 2

R

R

V

 (10) 

finally the acceleration vector is given by 





























     

      



0

sin sin

cos cos

2

2

1 2

2

1

2

2

1 2

2

1

R

R

a

 (11) 

Using Equation (9) developed for the apex seal path for a given crankshaft offset and the distance 

from the rotor center to the apex seal location, the path the seal takes for one revolution is shown in 

Figure 4. This is approximately the same geometry as the static housing that the seal is in contact with 

during operation and will be assumed in this paper to be the path of the apex seal. 

Next the seal velocity as a function of crankshaft rotation angle found from Equation (10) is given in 

Figure 5. The velocity of the seal is presented as that in the fixed coordinate system. The velocity is given 

for the engine crankshaft speed equal to 7800 rpm. 
NASA/TM—2010-216353 6 

Next the acceleration of the apex seal is presented by evaluating Equation (11) as a function of 

crankshaft rotation angle at 7800 rpm. The acceleration is also shown in the fixed coordinate directions 

(see Fig. 6). 

The development so far has concentrated on the calculations made in the fixed coordinate system. At 

this point it would be useful to put the same calculation methodology into a coordinate system that is radial 

and transverse to that of the apex seal itself. This will be done via the coordinate system shown in Figure 7. 

From Figure 7 the radial and transverse directions are along vector 



 and perpendicular to this vector, 

respectively. Therefore, to express the acceleration in terms of a radial and transverse direction, the 

acceleration vector must be transformed into these directions. This can be written as the following: 













  

  

















y fixed

x fixed

transverse

radial

a

a

a

a

2 2

2 2

sin cos

cos sin

 (12) 

Using Equation (12) the acceleration shown in Figure 6 after transformation is shown in Figure 8. 
NASA/TM—2010-216353 7 

Combustion Modeling 

To effectively analyze the friction torque developed by the rotor seals, the effects of combustion 

chamber pressure on the rotor and its seals must be included in the analysis. Limited experimental data of 

pressures inside a Wankel engine are available and no data for the particular engine in question are 

publically available. Therefore, the required information was generated analytically. 

A widely-used commercial software package (Ref. 6) designed for performance calculations of an 

internal combustion engine was used to model this Wankel engine. These one-dimensional commercial 
NASA/TM—2010-216353 8 

packages do not contain specific Wankel engine configurations. Because they are not configured to 

directly model the rotary configuration, a number of assumptions are required that qualify the accuracy of 

the results. Nonetheless, the results can be used to provide a reasonable initial assessment of engine 

performance until high-fidelity three-dimensional modeling and experimental results can be obtained. 

The engine was modeled as a three cylinder 4-stroke reciprocating engine, as shown in Figure 9. 

This configuration required the engine to operate at one-third to actual speed with an added gearbox 

to provide the rated rpm. In addition, the simulation package does not directly allow modeling of a 

carburetor; however standard procedure is to use a fuel injection module that is properly configured to 

represent the operation of a carburetor. Other physical geometry required by the model, such as intake and 

exhaust port geometry was measured from the engine and input to the model. 

The following assumptions were made to assemble this model: 

1. The model fuel consumption was input as the listed brake specific fuel consumption (BSFC) at 

maximum rpm. 

2. Combustion was assumed stoichiometric. 

3. Stroke was assumed to be the same as the eccentric offset and the bore was calculated to provide 

the same volume. 

4. Top dead center clearance was used to adjust to the published compression ratio. 

5. Compression versus crank angle was input using formulas provided by Reference 7. 

6. Standard heat release (Weibe) and gas exchange models, as provided by the software, were used. 

With these assumptions, the simulation friction model was adjusted to result in the appropriate power output. 
NASA/TM—2010-216353 9 

These assumptions likely deviate significantly from the actual operation of the engine as this 

configuration most certainly has a different gas exchange characteristic than a typical 2-stroke and 

certainly the combustion characteristics in the very elongated combustion chamber with a “piston” 

moving horizontally as well as vertically will deviate from that of a standard piston. 

The resulting pressure trace versus crank angle is shown below in Figure 10. The rotor at 0

represents top dead center. In the coordinate system described earlier in this paper the orientation of the 

location of top dead center was transformed to the coordinate description shown in Figure 2 such that 

pressure from the cavity leading apex seal 1 (Cavity 3) was shifted 30 ahead. This made top dead center 

equal to when apex seal 1 had rotated 30 (or the crankshaft had rotated 90). 

Combination of Loads 

The rotary engine, when operating, has three different loads being applied to the apex seal that result 

in a frictional (parasitic) drag of the apex seals on the stationary housing. There is a spring force from a 

leaf-type spring located under the apex seal, the inertial force due to the mass of the seal being accelerated 

around the epitrochoid path and the combustion pressure forces. The loading is shown schematically in 

Figure 11. The spring force was experimentally measured for the system under study, and resulted in a 

force of approximately 44.5 N at approximately half of its compressed height. Also the mass of the seal 

itself was found to be 8.36 g. In the model developed the spring force was assumed to be constant and the 

inertial force due to the mass of the apex seal was found based on the analysis conducted above. 

Next the combustion loading on the seal had to be interpreted. As shown in Figure 10, the combustion 

loading pressure is very dependent on the rotor angle position. Each lobe of the rotor experiences this 

loading per revolution. 

The cavity pressure affects the apex seals before and after the seal location as shown in Figure 11. 

Therefore the pressure on each side of the apex seal was determined during the rotor rotation process. 

The cavity pressures are shown in Figure 12 as a function of rotor rotation angle. 

Using Figure 13 the radial and transverse forces on the apex seal can be found. The seal is assumed to 

be of length, L, in the third dimension and the pressure loading around the seal-housing contact location is 

assumed to be negligible. Also, the seal is assumed to remain in contact on the backside of the rotor-seal 

interface during the entire apex seal traverse of the rotor housing. 

The forces in the radial direction are given by the following for apex seal 1: 

 Fr  mseal Sr (13)
NASA/TM—2010-216353 10 

From Figure 13 this becomes: 

cont spring cav seal r  F  F  P WL  N  m S 3 (14) 

This can be rearranged to be: 

Fcont  mseal Sr  Fspring  Pcav3WL  N (15) 

The forces in the transverse direction are given by: 

 Ft  mseal St (16) 

Using Figure 13 this becomes: 

 Fcont  Pcav3h1L  Pcav1hL  N  mseal St (17) 
NASA/TM—2010-216353 11 

Solving Equation (17) for N results in the following: 

N  mseal St  L Pcav h  Pcav h  Fcont ( ) 3 1 1 (18) 

Inserting Equation (18) into Equation (15) and rearranging results in 

F   m a a P WL Lh F LP h cont  seal radial   transverse  cav    spring   cav  















   2 3 1 1 ( ) ( ) 1

1 (19) 

Using Figure 14 the friction torque can be found as apex seal 1: 

  cos  seal #1 Fcont T r (20) 

where  is given by: 























       

  

r

r R

2

cos

2 2 2

1 (21) 

and the power lost due to seal friction as a function of apex rotation location as: 

 



seal i

i

PTotal T #

3

1

 (22) 
NASA/TM—2010-216353 12 

Analytical Results 

The analysis developed in this paper will now be applied first to a single seal and then to the entire 

three-seal engine configuration. The main parameters that will be investigated will be the effect of friction 

and crankshaft rotational speeds. 

First a single seal will be investigated. In Figure 15 the seal contact force and cavity pressures 

(around apex seal 1) are shown as a function of rotor rotation angle. The crankshaft speed was maintained 

at 7800 rpm with the friction coefficient set to 0.04 and held constant. At this point the coefficient of 

friction chosen requires some discussion. Obviously this effect is related to many operational 

characteristics not contained in this analysis. These characteristics include sliding speed, contact load, 

temperature, pressure, etc. The assumption of using the friction coefficient equal to 0.04 is similar to the 

contact friction between smooth surface metallic components as described in Reference 8. Investigating 

this parameter is important enough to warrant a separate investigation. 

Using the information previously discussed permits the analysis of a single seal. Figure 15 shows that 

the large contact force between the apex seal and the stationary housing is directly linked to the cavity 

pressure ahead of the apex seal. 

If the other two seals are added to this diagram, the total friction power loss for a given speed and 

friction coefficient can be found. The results are shown in Figure 16. Here the frictional affects range 

from approximately 900 to 2200 W during a three crankshaft revolution period (one rotor rotation). 

Also, the information shown in Figure 16 can be displayed one component at a time. Shown in 

Figure 17 is the total power loss for all three-apex seals, for each component and the total. 

Next in Figure 18, the effect of friction coefficient on the frictional power loss is shown. A this point 

the actual friction coefficient is unknown, but believed to be between 0.01 and 0.04. In the analysis 

conducted this value was kept constant. In reality, this value would vary and be a function of the loading, 

speed, temperature and other conditions not accounted for in this analysis. 

The final comparison provided is the effect of crankshaft rotation speed on the frictional power loss. 

In Figure 19 the analysis was conducted for three speeds (1000, 4000, and 7800 rpm). The friction 

coefficient was held constant at 0.04. As shown in the figure, speed has a strong influence on the 

frictional power loss found. 
NASA/TM—2010-216353 13 

   
NASA/TM—2010-216353 14 

Conclusions 

Based on the results found in this study the following conclusions can be drawn: 

1. The dynamics of the rotor of a rotary engine have been described. The analytical treatment 

developed can be utilized for any rotary engine given the crankshaft, rotor design parameters, and 

combustion pressure. The apex seal path, velocity and acceleration were found. 

2. A combustion modeling computer code was utilized to predict the cavity pressure during the fourstroke simulation. The combustion pressure as a function of rotor position was used in the 

analysis. 

3. The effects of inertial, apex seal spring, and combustion pressure were analyzed separately and 

combined to determine the parasitic, friction losses in the rotary engine under study. 

4. Friction coefficient and crankshaft speed effects were investigated. Both factors had a drastic 

effect on the resultant power loss predicted. 
NASA/TM—2010-216353 15 

References 

1. C. Vilman, Deformation Analysis of Rotary Combustion Engine Housings, Michigan Technical 

University, NASA CR–188187, 1991. 

2. D. Greenwood, Principles of Dynamics, Prentice-Hall Inc., 1988. 

3. R. Hinkle, Kinematics of Machines, Second Edition, Prentice-Hall, Inc., 1960. 

4. H. Mabie and F. Ocvirk, Mechanisms and Dynamics of Machinery, Third Edition, John Wiley & 

Sons, 1975. 

5. R. Ansdale and D. Lockley, The Wankel RC Engine, Design and Performance, A.S. Barnes and 

Company, 1st Edition, 1969. 

6. GT-Suite, V7.0.0, November, 2009. 

7. K. Yamamoto, Rotary Engine, Sankaido Co., Ltd., 1981. 

8. P. Andersson, J. Tamminen, and C.-E. Sandstrom, Piston Ring Tribology, A Literature Review, VTT 

Research Notes, 2002. 
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4. TITLE AND SUBTITLE

Analysis of Apex Seal Friction Power Loss in Rotary Engines 

5a. CONTRACT NUMBER

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6. AUTHOR(S)

Handschuh, Robert, F.; Owen, A., Karl 

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WBS 877868.02.07.03.01.01.01 

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NASA/TM-2010-216353; ARL-TR-5162 

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Unclassified-Unlimited 

Subject Category: 37 

Available electronically at http://gltrs.grc.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

An analysis of the frictional losses from the apex seals in a rotary engine was developed. The modeling was initiated with a kinematic 

analysis of the rotary engine. Next a modern internal combustion engine analysis code was altered for use in a rotary engine to allow the 

calculation of the internal combustion pressure as a function of rotor rotation. Finally the forces from the spring, inertial, and combustion 

pressure on the seal were combined to provide the frictional horsepower assessment. 

15. SUBJECT TERMS

Internal combustion engines; Apex seals; Dynamics 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

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21 

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19b. TELEPHONE NUMBER (include area code)

443-757-5802 

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