The Szorenyi Three Chamber Rotary Engine Concept

Abstract Currently automotive engines are reciprocating or Wankel rotary engine types. Reciprocating engines are bulky, heavy and complex, mainly due to the intake and exhaust valves and their associated cam-train. Wankel engines are rev limited due to the large eccentric load on the crankshaft, and have poor sealing of the apex seals leading to poor economy and undesirable emission gases. The Rotary Engine Development Agency (REDA) has designed a three-chamber rotary internal combustion engine concept using its patented Szorenyi Curve. It is an evolution of the four chamber engine design which was the subject of SAE Technical Paper 2017-01-2413 and SAE publication ‘So You Want to Design Engines: UAV Propulsion Systems’. This paper on the three-chamber engine concept includes an analysis of the major issues affecting the Wankel engine. The Wankel engine’s geometry results in excessive crankshaft deflection (due to the centrifugal force of the rotor which is eccentric to the crankshaft) and this limits its revs. However, the analysis reveals that the revs cannot be increased by increasing the diameter of the crankshaft. Also, analysis of the apex seal reveals that the shape of the stator accelerates the seal inwards during the power stroke and, at the same time, the seal experiences a large change in its contact angle with the stator surface. These effects combine to produce poor conditions for sealing the combustion chamber of the Wankel engine. The paper identifies that the Szorenyi concept engine design does not have these same issues because its symmetrical rotor enables high revving, and its continuously concave stator profile ensures an outward acceleration of the apex seal and much less change of contact angle with the stator surface. The paper concludes that the Szorenyi engine concept is a strong contender to replace Wankel and reciprocating engines in vehicles and light aircraft. Keywords: Rotary engine, internal combustion, concept, REDA, Szorenyi Curve

Introduction

The Rotary Engine Development Agency (REDA) has designed a new rotary engine concept. Known as the Szorenyi Three-Chamber (S3C) engine, it is a three-chamber internal combustion engine based on the patented Four-Chamber Szorenyi Rotary Engine (US Patent Number 6,718,938 B2). The four-chamber engine is described in a recent Society of Automotive Engineers (SAE) paper [1] and also features in a (SAE) book on Unmanned Aerial Vehicle (UAV) engines [2]. The S3C engine utilizes many of the patented features of the Four-Chamber Engine but avoids much of that engine’s complexity. The S3C engine is similar to a Wankel engine in operation. However, the Wankel engine is rev limited due to large eccentric loads on the crankshaft at high revs, and has inefficient sealing leading to poor economy and undesirable emission gases. The paper includes an analysis of the short-comings of the Wankel design and identifies the major advantages of the Szorenyi concept engine as its symmetrical rotor that enables high revving of the engine, and a continuous concave stator profile that improves sealing of the apex seals. The paper also identifies features of the Szorenyi engine concept that make it particularly suitable for light aircraft and UAVs.

The S3C Design Concept The S3C engine has a stator profile derived from the patented Szorenyi Curve. The curve is the geometric shape that contains a four-sided rhombus with corners that all follow the same trajectory when the rhombus is rotated about its geometric centre. Details of the curve and the development of the four-chamber engine can be found in the US Patent and the Ref A paper.

Figure 1: An internal view of the S3C engine

The internal lay-out of the S3C engine is shown in Figure 1. The stator profile is based on the Szorenyi Curve which is a family of curves that can contain a rhombus whose apexes remain on the curve in all orientations. The rotor has three segments whose width is the same as each side of the four-sided rhombus. The segments are arranged in a triangular pattern with the tips of each rotor segment (the apex seals) held in contact with the stator. The space between the face of the rotor segment and the stator form the combustion chamber. As the rotor rotates, the midpoint between the tips of the apex seals is at a constant radius from the crankshaft centreline. This radius becomes the crank-arm which connects the crankshaft to a pivot at the midpoint of the segment. The resulting engine assembly is shown Figure 2.

Stator Cam plate Rotor segment Crank-arm pivot (hidden)

Apex seal

Figure 2: An exploded view of the S3C Engine

As the crankshaft rotates through one revolution, each of the three chambers completes the induction, compression, ignition and exhaust phases of the Otto cycle. Each chamber in turn draws the air in through a peripheral intake port (indicated by the lower arrow in Figure 2). As the rotor rotates (in a clockwise direction in Figure 1) the trailing apex seal of the chamber passes over the intake port thus trapping the air-fuel mixture and this is then compressed and ignited. The high-pressure gases of combustion generate a force that is offset from the crank-shaft centerline thus creating a torque and spinning the rotor to produce usable power. As the leading apex seal passes the peripheral exhaust port, indicated by the upper arrow in Figure 2, the gases are expelled. Then the leading apex seal passes the intake port (lower arrow in Figure 2), the intake stroke commences and the cycle is repeated. The rotor segment of the S3C engine is sealed by apex seals between the tips of the rotor segment and the stator profile, and side seals between the rotor segment and the side-plates. These seals are housed in channels and have a wave spring underneath them to hold each seal in contact with the stator and side-plates. The S3C engine uses cam followers attached to the sides of each rotor segment. The followers run around a cam-plate attached to the side-plate of the engine. This ensures that the apex seals at the tips of the rotor segment always remain in contact with the engine stator. The S3C engine architecture allows direct lubrication of the crank-pin, rotor segment cam follower bearings, as well as the apex and side seals. In the S3C engine, one rotation of the crank-shaft produces a complete engine cycle in each of the three chambers. Therefore a single rotor S3C engine is equivalent to a six cylinder reciprocating engine in terms of the number of powerstrokes per crank-shaft revolution. However, the Szorenyi engine does not need the cumbersome valve-train of the reciprocating engine but rather uses ports similar to those used in Wankel engines. The result is a simpler

engine with far less moving parts in a much smaller package than a reciprocating engine, and with smoother operation due to its perfectly balanced rotating parts.

Mathematical Modelling

The Royal Melbourne Institute of Technology (RMIT) University has conducted ideal mathematical modelling to determine the performance of a Four-Chamber Szorenyi, a reciprocating, and a rotary engine [3]. The model gave encouraging results but would need to be redone for the S3C engine because it uses a rounder shape for the stator profile and this changes the volume history. However, more than just ideal modelling is required and so REDA intends to conduct Computational Fluid Dynamics (CRD) modelling to determine the performance potential of the S3C engine.

Comparison of Engine Design Features

Analysis of the characteristics of the S3C engine concept conducted by REDA reveals that it has significant advantages over the reciprocating engine and the Wankel rotary engine in their wide range of applications. As discussed below, the Wankel engine design is compromised by its eccentric rotor and inefficient sealing of its apex seals. The reciprocating engine requires a cumbersome valve train, and an out of balance crankshaft which often requires a counter balancing shaft which adds weight, space and complexity to the engine. Wankel Engine Design Issues The Wankel engine is becoming more popular as a light aircraft and UAV engine. Wankel aviation engines are being produced by companies such as Austro, Rotron, UAV Engines, 3W International, Freedom Motors, and Advanced Innovative Engineering. However, the Wankel engine has severe design issues which directly result from the epitrochoid geometry on which the engine concept is based. This geometry results in a stator profile with a convoluted concave-convex surface which makes the apex seal prone to leaking gasses between chambers, and also leads to a small diameter crankshaft which is highly stressed due to centrifugal loads and so requires the imposition of a relatively low rev limit.

Figure 3: An Internal View of the Mazda Wankel Engine

Crankshaft centre

Rotor centre of gravity

Stator convex curve

Apex Seal Leakage in a Wankel Engine. Much effort has been expended to ensure that the apex seals of the Wankel engine provide efficient sealing. The apex seals have long been known to lift off the stator housing under certain conditions. The NASA paper of Reference 4 reported that “loss of contact between the apex seal and the trochoidal bore has been discussed by a number of researchers”. The resulting leakage of gases past the apex seal leads to unstable combustion, loss of power, and additional unwanted emissions. One of the causes of the leakage stems from the tortuous path that the apex seal has to follow. The epitroichoidal geometry of the Wankel engine produces an engine stator with both concave and convex curves. The sealing integrity of the apex seal is severly compromised when following these curves. This can be seen in Figure 3 where, with clockwise rotation of the rotor, the tip of the apex seal that is near top-centre is about to follow the convex curve of the engine stator. As it follows the convex curve the apex seal experiences a centrifugal acceleration forcing it away from the stator surface. This motion is only offset by the wave spring located under the apex seal.

Modelling of the apex seal trajectory for a full engine cycle has been performed by REDA. The longitudinal acceleration of the apex seal (the centrifugal acceleration component in the line of the radial movement of the apex seal) and the lateral acceleration (causing side to side movement) has been computed at 2,400 rpm, the maximum power revs of the wankel engine. The longitudinal acceleration of the apex seal of the Wankel engine is shown in Figure 4. The peak negative acceleration can be seen to occur when the combustion chamber is at 600 after top dead centre. At this moment, the apex seal also experiences a large pressure differential from the chamber undergoing expansion of its combusted gases across to the following chamber which is part-way through its compression stroke. Modelling of the situation indicates that up to 7.5 atmospheres pressure differential occurs at this moment. The conditions experienced by the apex seal are very poor for maintaining efficient sealing.

Figure 4: Apex Seal Longitudinal Acceleration in the Mazda Wankel Engine The modelling performed by REDA also determined the acceleration of the apex seal of the S3C engine of the same chamber volume as the Wankel. The resulting longitudinal acceleration of the apex seal is shown in Figure 5. The seal experiences a fluctuation of acceleration similar to the Wankel engine, but because the stator of the S3C engine has a continuous concave profile, the acceleration is always positive and so ensures continuous contact of the apex seal with the stator. The situation is more favourable for the S3C engine

-­‐2,000  

3,000  

8,000  

13,000  

18,000  

TDC                                            BDC                                            TDC                                            BDC                                            TDC      Combustion  Chamber  Position  

Longitudinal  Acceleration  of  the  Trailing  Apex  Seal  of  the    Combustion  Chamber  in  the  Wankel  Engine  at  2,400  rotor  rpm      m/s^2  

 

because the orientation of the apex seal in the rotor segment of the S3C engine can be selected to avoid negative acceleration. In Figure 1, the apex seal is shown at 300, but the angle can be set much higher. In the modelling, the angle has been set at 700 to the rotor surface. In the Wankel engine the seal serves adjacent chambers and so must be set at 300 because it bisects the apex angle of the triangular rotor.

Figure 5: Apex Seal Longitudinal Acceleration in the Szorenyi Three-Chamber Engine

Although the longitudinal acceleration of the apex seal is more favourable in the S3C engine, the actual motion of the apex seal is also affected by the lateral acceleration imparted on the apex seal as it follows the stator contour and the friction force resulting from contact of the apex seal with the stator surface. The modelling does show that the lateral acceleration of the apex seal in both engines is roughly sinusoidal, but the acceleration of the apex seal in the Szorenyi engine is in one direction only whereas it changes from positive to negative in the Wankel engine. However, a full analysis of lateral, longitudinal and friction effects would be required to ascertain the motion of the apex seal and this is beyond the scope of this paper.

Modelling performed by REDA also reveals that, in the Wankel engine, the apex seal inclination (the angle between the apex seal orientation in the rotor and the stator surface) changes significantly during a rotor revolution. In the Wankel engine the apex seal inclination changes from +250 to -250 as it traverses the convex portion of the stator. In the S3C engine this change of inclination of the apex seal is a more moderate ±160. In the Wankel engine there is therefore a high possibility of the apex seal leaking due to the effects of the negative longitudinal acceleration of the apex seals, the large change of angle of incidence between the seal and the stator surface, and the high pressure drop across the apex seal at the time these conditions prevail. In contrast, the possibility of the apex seal leaking in the S3C engine is greatly reduced due to the positive longitudinal acceleration; the absence of side to side movement; and the smaller change in the apex seal’s inclination with the stator. Rev Limit of the Wankel Engine. A major disadvantage of the Wankel engine is the limited rotational speed of the rotor due to its highly stressed crankshaft and rotor bearings. In his book of Reference 5, Kenichi Yamamoto notes in regard to the load on the rotor bearing “the greater effect of gas pressure in the low speed range and that of centrifugal force in the high speed range.” This results in the Mazda Wankel engine being red-lined at 9,000 rpm of the

-­‐2,000  0  

2,000  4,000  6,000  8,000  10,000  12,000  14,000  16,000  18,000  

TDC                                            BDC                                            TDC                                            BDC                                            TDC      Combustion  Chamber  Position  

Longitudinal  Acceleration  of  the  Trailing  Apex  Seal  of  the    Combustion  Chamber  in  the  S3C  Engine  at  2,400  rotor  rpm      m/s^2  

crankshaft (3,000 rpm of the rotor). The revs are limited because the centroid of the rotor is at a constant radius from the centreline of the crankshaft and this produces a large centrifugal load on the crankshaft at high rotor revs. At high revs, the bending of the crankshaft can cause large deflections which shifts the rotor and cause the rotor seal to bottom out and make solid contact with the stator. It is an unfortunate feature of the Wankel engine that the crankshaft bending cannot be alievated by increasing its diameter as, counter intuitively, this has no effect. As shown in the analysis at Annex A, this occurs because increasing the crankshaft diameter also increases the three dimensions of the rotor as well as its radius of rotation and together this negates the effect of the increase in the crankshaft diameter. For a Wankel engine of particular dimensions the deflection of the crankshaft increases with the square of the rotor revs until a critical deflection is reached. The resulting rev limit in a Wankel engine is quite low – about 3,000 rotor rpm. This is equivalent to a four-stroke reciprocating engine rev limit of 6,000 rpm, which is typical of low powered automotive engines, but some high powered four-stroke engines are designed to rev to 15,000 rpm and beyond. So this range of revs is unavailable in the Wankel engine. The S3C engine is not rev limited in this way because its rotor is symmetrical and so there is no eccentric load on the crankshaft caused by the rotation of the rotor. With a perfectly balanced rotor, the S3C has the potential for high revs and so a high package density. Higher power for the same size engine has multiple advantages in aircraft applications of the engine. Additional Advantages of the S3C Engine In addition to the advantages of the S3C engine over the Wankel engine identified above there are other significant advantages of the S3C over Wankel and reciprocating engines discussed below: a. Ability to use Heavy Fuels. The rotary engine is well suited to improved combustion efficiency by using stratified-charge techniques [6]. This allows heavy fuels to be used in the S3C and Wankel engines and therefore makes these engines very suitable for use in light aircraft and UAVs 1. b. Ability to use Alternate Fuels. The Wankel engine, and by similarity the S3C engine, can be run on gasoline, kerosene, diesel, natural gas, liquid-petroleum gas, or hydrogen. Diesel can also be used in the S3C engine because the gentle curvature of the engine stator allows for high compression ratios. The Wankel engine is typically limited to a compression ratio of about 15:1 because, at that ratio, the face of the rotor comes into contact with the convex curvature of the stator. Consequently, the Wankel engine cannot use diesel fuel without supplementary compression of the intake gas. c. Ports, not Valves. The S3C (and Wankel) engine benefits from not requiring intake and exhaust valves and associated cams and cam-trains as in reciprocating engines. This leads to lighter, smaller and less complex engine - all desireable features of engines for aircraft applications. It also “would eliminate flow restrictions and dynamic limitations” [6]. However, the Wankle engine suffers from a large overlap of the intake and exhaust ports. As noted by Yamamoto in Reference 5, “During low speed or light-load operation, there arises a problem that the combustion will become unstable due to inclusion of a great amount of burned gas in intake gas caused by a longer overlap time of the opened intake and exhaust ports”. This occurs because the Wankel engine’s apex seals are set 1200 apart but, since there are four strokes every revolution of the rotor, the Wankel engine with peripheral ports has port

1 Heavy fuels are the the kerosene-type fuels such as AvGas, Jet-A, JP-5 and JP-8 used for aviation engines. JP-5 and JP-8 are used by the military. The ability of the S3C engine to use heavy fuels or diesel in light aircraft and UAVs is an advantage for military air operations as it minimises transportation of multiple fuel types and volatile gasoline.

timing overlap of 300. This results in poor volumetric efficiency and there is considerable exhaust gas recirculation because the intake and exhaust ports are simultaneously open for one-third of the stroke. The S3C engine geometry results in apex seals set 900 apart and therefore the ports can be set 900 apart. Consequently, port overlap can be avoided and gas recirculation can be controlled. d. Package Density Comparison with a Mazda Wankel Engine. A single-rotor S3C engine of the same total displacement of a Mazda Wankel engine, and operating at the same rotor revs (2,400 rpm for maximum power) has a core engine package volume 52% larger than that of the Mazda engine. But the S3C engine does not have the rev limiting issues of the Wankel engine. The rev limit of an S3C engine is yet to be determined, but if operated at 3,640 rpm the package size of the two engines is the same. Apart from a higher rev limit, the S3C engine could still replace a Wankel engine on its advantages of lower vibration, improved apex seal integrity, better combustion chamber shape, more flexible valve timing, and its ability to use diesel fuel. e. Superior Package Density to Reciprocating Engines. REDA has conducted modelling of the core engine dimensions and package volume2 of a single rotor S3C engine, the Rotax 912 engine, and the ULPower 350i engine. Both the Rotax and ULPower engines are popular four-cylinder horizontally-opposed reciprocating aviation engines. The Rotax is a small capacity high revving engine with a reduction gearbox to reduce the crankshaft revs to the lower propeller speed, and the ULPower is a large capacity low revving engine with direct drive from the crankshaft to the propeller. The modelling was done using their respective maximum power revs. The modelling of the S3C engine assumes a combustion chamber depth equal to its width, that is, a ‘square’ rotor face3. A summary of the modelling papameters and results is shown in Table 1. The modelling reveals that the core package size of an S3C engine of the equivalent displacement of the Rotax R912 engine is 33% of the package size of that engine, and 41% of the volume of the ULP350i engine’s core package size. In the modelling, the S3C engine is assumed to be operating at the same revs as the reciprocating engines. However, with a balanced rotor, there is potential for much higher revs and so even lower package volumes for the same power output when compared to the reciprocating engines. There is therefore the potential for large space and weight reductions by replacing existing reciprocating engines with the S3C engine. This is particularly advantageous in aircraft applications of the S3C engine.

Table 1: Core Engine Package Size Comparison

Engine Rotax R912 S3C equivalent to

R912

ULP 350i S3C equivalent to ULP 350i

Cylinders 4-cyl 3-chamber 4-cyl 3-chamber Engine displacement 1211 cc 1211 cc 3503 cc 3503 cc Chamber volume 303 cc 202 cc

(equivalent) 876 cc 584 cc (equivalent)

Compression Ratio 10.8 10.8 8.7 8.7 Cylinder configuration

opposed rotary opposed rotary

Carbureted or FI Carbureted fuel fuel fuel

2 The core engine here includes the crankshaft, crank-arm, con-rod, pistons, valves, cam-shaft and casing of the reciprocating engines. The equivalent in the S3C engine is the crankshaft, crank-arm, rotor and stator. The package volume is the cube that encases the core engine. The space required for the Propeller Speed Reduction Unit (PSRU), where needed, is not included in the package volume comparison. 3 The Wankel engine’s rotor depth is 44% of its width which is necessary to limit the deflection of the crankshaft and allow a sufficient rev range. The S3C engine does not have this limitation and the better shape combustion chamber is possible.

injected injected injected Cylinder cooling water air air air

Cylinder head cooling

air air air air

Max Power @ rpm

75 kW @ 5,800

≈75 kW @ 5,800

97 kW @ 3,300

≈97 kW @ 3,300

PSRU required Yes (2.43:1)

Yes (2.43:1)

No No

Core Engine: Width (cm) Height (cm) Depth (cm)

57.6 17.8 31.0

26.5 22.4 17.7

65.4 23.0 36.2

34.4 28.5 22.7

Core Engine Package Volume (litres)

31.8 10.5 (33% of R912)

54.5 22.2 (41% of ULP350i)

Design Refinement of the S3C Concept Engine The S3C concept engine requires refinement of its design. Lubrication of the concept engine is feasible but the optimum arrangement needs to be determined. Additional design work also needs to include component stressing, crankpin sizing, combustion chamber geometry optimisation, analysis of intake and exhaust port timing dynamics, and rev limit determination.

S3C Engine Applications

The Szorenyi Rotary S3C Engine is suitable to be used for all applications where a reciprocating or Wankel engine are currently used. Small or large versions of the engine could be made as well as multiple rotor versions. It is particularly suited for aircraft and UAV applications due to its multi-fuel capability, lower vibration levels than Wankel and reciprocating engines, and its compact dimensions. The lighter weight and smaller size of the S3C engine can translate to larger payloads or greater range or endurance of aircraft and UAVs than is achieved with current engines. The similarity of the stator shapes of the Wankel and S3C engines enables existing Wankel engine manufactures to readily produce an S3C stator assembly. The new S3C rotor assemble needs to be added. The water cooling system of the original Wankel engine can be retained, but a complete lubrication system for the rotor needs to be designed for the engine.

Conclusion The Rotary Engine Development Agency has created a new internal combustion rotary engine concept. Known as the Szorenyi Three-Chamber Rotary (S3C) engine, it offers significant advantages over the Wankel rotary and reciprocating engines. Compared to a Wankel engine, its major advantages are its balanced rotor, allowing higher revs, and its concave stator profile which leads to better combustion chamber sealing. The major advantages over the reciprocating engine are its simplicity due to the use of intake and exhaust ports, rather than valves and their cumbersome valve train; its more compact dimensions and lower weight; and its ability to use alternate fuels such as heavy fuels, natural gas and hydrogen. These advantages make the engine suitable for a wide range of applications, particularly light aircraft and UAVs. REDA is confident that the S3C engine has the potential to replace the reciprocating and the Wankel engine wherever they are currently used. The next steps in the

development of the S3C concept are to refine the design of the engine components, to design a lubrication system, and then to conduct computational fluid dynamics (CFD) modelling to demonstrate the potential of the engine.

References 1. P. King, The Development of the Four-Chamber Szorenyi Rotary Engine, Society of Automotive Engineers, SAE 2017-01-2413, 10 August 2017.

2. W. Kucinski, So You Want to Design Engines: UAV Propulsion Systems, SAE International, ISBN 978-0-7680-9175-5, 2019

3. L. Espinosa, P. Lappas, Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine, Nonlinear Engineering, NLENG.2017.0082.R2, 2018-07-06. 4. J. Knoll, C.R. Vilmann, H.J. Schock, R.P. Stumpf, A Dynamic Analysis of Rotary Combustion Engine Seals, National Aeronautics and Space Administration, Lewis Research Centre, Cleveland, Ohio, NASA-TM-83536, 1984

5. K. Yamamoto, Rotary Engine, published by Toyo Kogyo Company Ltd, 1971 6. Chol-Bum M. Kweon, A Review of Heavy-Fuelled Rotary Engine Combustion Technologies, Army Technology Directorate, Army Research Laboratory, Aberdeen Proving Ground, MD, ARL-TR-5546, 2011.

Rotary Engine Development Agency Melbourne, Australia

January 2019

Annex A to The Szorenyi Three-Chamber Rotary Engine Concept

Page 1 of 2

The Influence of Crankshaft Diameter on Crankshaft Deflection in a Wankel Engine

The rotor revs in a Wankel engine are typically limited to an unusually low 3,000 rpm (9,000 rpm of the crankshaft). In the literature this is attributed to the high centrifugal forces at high revs causing excessive deflection of the crankshaft. This situation should be readily overcome by increasing the diameter of the crankshaft. The analysis below determines that this is not possible due to the relationship between crankshaft diameter, rotor size, engine displacement and combustion chamber proportions. The shape of the stator of the Wankel engine is epitrocoidal. It is deternined by two dimensions ‘e’ and ‘R’ as shown in the Figure 1. In the actual engine, ‘e’ is the radius of the crankshaft and ‘R’ is the distance from the centroid of the triangular rotor to the tip of the apex seal (the apex of the triangle).

Figure 1: The Construction of the Epitrocoidal Curve

The Cartesian co-ordinates of the stator are determined according to equations 1 and 2 below:

x = e.cos(3β) + R.cos(β) (1) y = e.sin(3β) + R.sin(β) (2)

The generating angle β is the angle of rotation of the triangular rotor of the Wankel engine. The angle 3β represents the crankshaft which rotates at three times the speed of the rotor. Selecting an R/e ratio of 7 (typical of the Mazda Wankel engine) produces the stator profile as shown in Figure 2. A triangle has been added to this to represent the rotor.

Figure 2: Wankel Engine Stator Profile

√3R    

R  

Annex A to The Szorenyi Three-Chamber Rotary Engine Concept

Page 2 of 2 The length of each side of the triangular rotor is √3R. The height of the triangle is R+R/2. The cross-sectional area of the rotor is half the base by the height = ½ (√3R).(R+R/2) = 1.3 R2. For this analysis the cross-sectional area can be expressed as k1R2 where k1 is a constant. The volume of the rotor is therefore k1R

2d where d is the rotor depth (and also the span of the supported crankshaft). The mass of the rotor is therefore k1ρR2d where ρ is the density of the rotor. This assumes a flat faced rotor, and although the shape of an actual rotor is curved, the rotor’s volume is still proportional to R3. So in this analysis the rotor mass is expressed as k2R

2d where k2 is a constant incorporating the rotor density, ρ. At engine rotor revs ω the centrifugal acceleration of the rotor is ω2e. The centrifugal force is F = mass x acceleration = k2R2dω2e. The force F is applied to the crankshaft throughout the rotation of the rotor about the crankshaft and causes the deflection of the crankshaft. The analysis of the Wankel engine geometry by Kenneth C. Weston [Energy Conversion, West Publishing Company, 1992] reveals that the displacement of the Wankel combustion chamber is 3√3Rde. In this analysis the chamber displacement must be kept constant, therefore the force F can be expressed as k2/(3√3)(3√3Rde)ω2R. Therefore F = k3ω

2R, where k3 = k2/(3√3).(chamber displacement). The force F causes a deflection δ of the crankshaft where δ is shown in the general literature as being proportional to Fd3/EI. The constant of proportionality depends on the manner in which the crankshaft is support, and E is Young’s Modulus (a constant). So for this analysis δ = k4Fd3/I, and I is the moment of inertia of the circular cross-sectioned crankshaft and so here I = ¼π(e)4 . For the Wankel engine the ratio R/e determines the proportions of the stator. (The ratio Mazda uses is R/e = 7.) So keeping the stator proportions the same, R/e = constant = k5,

so I = ¼π(R/ k5)4 = k6R4. Also, to keep the combustion chamber width and depth proportions

the same, the depth d can be written as k7R. Substituting for F, d and I in the equation δ = k4Fd3/I:

δ = k4(k3ω2R).( k7R)3/( k6R

4) therefore, δ = k8ω2 Hence in a Wankel engine maintaining the same displacement, combustion chamber proportions and stator proportions, the deflection of the crankshaft is independent of its diameter and only increases with the square of the rotor revs. The revs must be limited to ensure that the rotor does not bind on the stator wall or the side walls as this would cause failure of the engine. For the Mazda Wankel engine the rotor revs are typically limited to 3,000 rpm or 9,000 rpm of the crankshaft.

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