ANALYSIS OF RZEPPA AND CARDAN JOINTS IN MONORAIL DRIVE ... · PDF fileANALYSIS OF RZEPPA AND...

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Int. J. Mech. Eng. & Rob. Res. 2015 Sang June Oh and John T Woscek, 2015

ANALYSIS OF RZEPPA AND CARDAN JOINTS INMONORAIL DRIVE TRAIN SYSTEM

Sang June Oh1* and John T Woscek1

*Corresponding Author: Sang June Oh,[email protected]

The objective of this paper is to determine the best joint design in the driveshaft of monorail drivetrain systems. Two of the most common joints used in these types of drive train systems areRzeppa and Cardan joints. Constant velocity joint such as Rzeppa joint can operate at a muchlarger joint angle for desired torque transmission than non-constant velocity joints such as Cardanjoint. However, constant velocity joints are more expensive to manufacture than more versatileCardan joints. Therefore, there has been a constant debate in determining which joint typewould be optimal. The paper utilizes typical material characteristics and common operatingconditions of Rzeppa and Cardan joints. Then, the bending moment couples, which are generatedduring operation of these joints are analyzed with finite element analysis. This helps in determiningthe maximum allowable joint angle in using one joint type over the other. The paper furtherexamines the fatigue factor of safety and provides a general guideline in determining the betteroption between the two. It is determined from these analyses that Rzeppa joint is a better optionfor general monorail drive train systems. However, Cardan joint may be adequate if the jointangle is very small.

Keywords: Rzeppa joint, Cardan joint, Monorail train, CV joint, Non-CV joint

INTRODUCTIONNon-Constant Velocity (NCV) joint such asCardan joint (Universal joint, U-joint, or Hookejoint) and Constant Velocity (CV) joint such asRzeppa joint are couplings in a driveshaft usedto transmit power in various automotive andmachinery applications. Both allow parts of amachine not in line with each other limitedfreedom to move in any direction while

ISSN 2278 – 0149 www.ijmerr.comVol. 4, No. 1, January 2015

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Int. J. Mech. Eng. & Rob. Res. 2015

1 Department of Mechanical Engineering, California State University, Fullerton, Fullerton, CA 92831, US.

transmitting rotary motion (Seherr-Thoss et al.,1998). These two types have their ownadvantages and disadvantages, as well astheir own unique brand of characteristics,which makes them attractive for variousapplications.Therefore, the intricate workingsof both Non-Constant Velocity (NCV) andConstant Velocity (CV) joints will be exploredby narrowing the focus to two specific joint

Research Paper

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types: Rzeppa Joint (CV type) and CardanJoint (NCV type).

One method of transportation that utilizesboth of these joints in its drive system is astraddle-type monorail train (Sekitani et al.,2005). Various early designs of monorail trainsystems utilized the Rzeppa CV Joint totransmit power with constant velocity as thetrain moved through various turns andelevations. As time progressed, manufacturersstarted to develop Cardan NCV joint to replaceRzeppa joint in an effort to save cost. However,due to the nature of NCV joints, questions havearisen as to how reliable of a replacement theCardan Joint can be. Monorail manufacturershave stated that as long as the angle ofoperation is not too high, the Cardan Joint willbe adequate. For a long period of time,manufacturers generally indicate Cardan jointsshould be restricted to angles of 15 degreesor less (Carmichael, 1950). However, thisthreshold angle is difficult to pinpoint, since theoperating angles for different monorail trainsystems are not the same. Therefore, one ofthe objectives of this paper is to provide ageneral guideline and determine the thresholdangle based on the design of both joints andfurther justify whether or not Cardan joints canbe an adequate replacement for Rzeppajoints.

Rzeppa Joint is defined as a self-supportedconstant velocity joint that contains an outerand inner race, connected through ballspositioned in the constant velocity plane byaxially offset curved grooves. The balls aremaintained in this plane by a cage locatedbetween the two races (Wagner and Cooney,1991). There are multiple configurations thatfit this definition, each of which yield various

characteristics. This paper will be focusing onthe bell-type Rzeppa Joint, illustrated inFigures 1A and 1B.

The bell-type Rzeppa Joint is primarily usedin applications where a drive torque isrequired, such as on the drive shaft of a car orin this case, the drive shaft of a monorail.Further observation of Figures 1A and 1Bshow how the outer race contains a set ofsplines, which will mate with a drive flange thatis mounted to a drive wheel. The inner race ofthe Rzeppa Joint is also splined, and this racemates with the external splines on a drive shaftthat is held in place by a snap (or retaining)ring, as seen in Figures 1A and 1B. The ballsthat ride within the six grooves are made ofpure steel. For the joint used in this study, theballs each have an overall diameter of 1.6870inches. Rzeppa joint design has manyappealing advantages (Qin et al., 2003). One

Figure 1A: Cross Sectional View of Bell-Type Rzeppa Joint

Figure 1B: Dissected View of Bell-TypeRzeppa Joint

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of these is its ability to transmit constantvelocity at a relatively high angle, close to 45degrees, and some as high as 51 degrees.Also, because of its rugged and compactdesign, it allows for a relatively high torquecapacity for a given swing diameter. Thisrobust design also allows for it to withstandhigh external axial forces of an intermittentnature (Wagner and Cooney, 1991). However,the robust design of the Rzeppa Joint leads tosome major disadvantages as well. Since it issturdy and heavy, the Rzeppa Joint is veryexpensive to manufacture, not only in materialcost but also in machining as well. Since thesejoints need to be either forged or casted, it isvery difficult to hold some of the demandingtolerances needed for various features, suchas the grooves for the balls and the sphericaldiameters of the cage and inner race. TheRzeppa Joint is typically made of a low tomedium carbon steel. For the analysis of thispaper, the chosen material is AISI 4130, takenfrom ASTM standard A322 for standard gradeHot-Wrought Steel Bar stock (ASTM, 2004).

The Cardan Joint is one of the mostcommon non-constant velocity joints used inthe automotive and power transmissionindustries. The SAE Driveshaft and UniversalJoint Design Guide defines Cardan Joint as anonconstant velocity joint consisting of twoyokes drivably connected by a cross throughfour bearings (Wagner and Cooney, 1991). Ingeneral, it is simple to manufacture a CardanJoint for use in any application. This allows fora greater flexibility in the design of the slip andtube yokes, ultimately yielding many differentstyles that will perform the same function. Oneof a very commonly used Cardan joint, “wing-bearing type” Cardan Joint will be chosen for

this paper. The “wing-bearing type” CardanJoint utilizes a slip yoke and tube yoke and anillustration of the entire assembly is shown inFigures 2A and 2B.

Primarily, these “wing bearing” types ofCardan Joints are used on automotive driveshafts and other applications. This particularstyle is attractive simply because of the ease ofattachment; rather than using a pin to connectthe yokes to the cross, fasteners are used tomate the yoke to the cross at the properattachment points. The drive yoke is allowed toarticulate independent of the shaft yoke due tothe roller bearings located within the cross.Typically, the driven yoke is splined in order tomate with the other end of the drive shaft.

Figure 2A: Cross Sectional View of “Wing-Bearing Type” Cardan Joint

Figure 2B: Dissected View of “Wing-Bearing Type” Cardan Joint

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One of the biggest advantages of CardanJoint is simplicity in manufacturabilityregardless of its size. This allows for use in awide variety of applications ranging from smallmachinery to large trucks. In addition to its easeof manufacturability, this design also yields alonger durability and minimum maintenancesince there are less rotating parts than RzeppaJoint, thereby minimizing many points offailure. One of the disadvantages of this jointtype is that it does not transmit the same outputvelocity at higher joint angles. This can certainlypose an issue for a monorail train drive system.This is why it is imperative that the thresholdangle be determined in order to validate theuse of the Cardan Joint in what would be aconstant velocity application.

For this paper, the components of Cardanjoints were chosen that can easily be obtainedthrough manufacturers. The drive yoke is aNeapco Wing Bearing End Yoke 7C Series,part number 10508J. The Shaft Yoke is aNeapco Wing Bearing Yoke Shaft 7C Series,part number 5505J. The Universal JointCross is a Neapco/Spicer Wing BearingUniversal Joint 7C Series, part number 5-7105X. These parts were all chosen for theanalysis because this particular driveshaft (7C)is a commonly used one in heavy trucks andmachinery. These parts are typically madefrom a low to medium carbon steel.

Since the Rzeppa Joint is being analyzedas AISI 4130, the Cardan Joint will also beassumed to be this same material forcomparison purpose in the analysis. Basically,the Cardan Joint boils down to a simple designwith complex characteristics (as opposed toRzeppa joint being complex design with simplecharacteristics). However, it is important to

understand the input and output torque ratio topredict the amount of stress that will be seenby the yokes, and ultimately determine the lifecycle of a Cardan Joint. This will help indetermining whether or not the Cardan Jointwould be an adequate replacement for theRzeppa Joint.

MATERIALS AND METHODSThe Rzeppa Joint and the Cardan Joint areboth designed to operate at a maximum angleof 45°. As previously mentioned, the RzeppaJoint will yield constant velocity across all ofthese angles, whereas the Cardan Joint willnot. Besides the obvious geometric designdifferences in both joints, the Rzeppa andCardan Joints yield a pair of torques on bothcouplings during angular navigation called“Secondary Bending Moment Couples”, whichmathematically explain the forces exerted onthese joints while transmitting torque at variousangles (Wagner and Cooney, 1991).

The Rzeppa Joint’s bending momentcouples at different angles are given by thefollowing equation:

2tan1

TCC ...(1)

where C is the secondary couple on the drivingmember, C1 is the secondary couple on thedriven member, T is the input torque and isthe angle between joints, also known as thejoint angle.

The Cardan Joint’s bending momentcouples are a bit more complex. Due to itsnature, the bending moments of the drivingyoke and driven yoke will depend on differentsituations during motion. The driving yoke’sbending moment couples are given by thefollowing equation:

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costanTC ...(2)

For the driven yoke, the bending momentcouple is given by the following equation:

sinsin1 TC ...(3)

where C is the secondary couple on the drivingmember, T is the input torque, is the jointangle, and is the angle of rotation of thedriving yoke from a position where it is normalto the joint angle plane. Please note that theorientation of the joint angle and the anglenormal to it is . The driving yoke can articulateboth horizontally and vertically, which yieldsthese two different angles. In order to justifythe validity, the comparison study will beperformed with = 45 degrees in the CardanJoint. This was chosen to see how effectiveCardan Joint can be, under worstcircumstances, in comparison to Rzeppa Joint.

Another factor in this design considerationis the fatigue life. In many industries, the easiestmethod of explaining fatigue life is through theFatigue Factor of Safety (FOS). This FOS isbased on the “stress-life” (Budynas andNisbett, 2011). One of the factors indetermining this FOS is the CorrectedEndurance Limit, Se, which takes into effectall surface finishing, temperature and shape,and is given by the following equation (Budynasand Nisbett, 2011):

efedcbae SkkkkkkS ...(4)

where eS is the Mean Endurance Limit of theparent material, ka is the reduction of thematerial strength due to Surface Finish, kb isthe size effect, kc is the Loading Factor, kd isthe temperature factor, ke is the reliability factor,and kf is the stress concentration factor. ThisCorrected Endurance Limit is then divided by

the Goodman Equivalent Stress to find the FOS(Budynas and Nisbett, 2011). The GoodmanEquivalent Stress is found by the followingequation:

uts

emeanampeq

S

* ...(5)

where amp is the amplitude stress, mean is theaverage, or mean stress, Se is the CorrectedEndurance Limit, and uts is the UltimateTensile Strength of the parent material. DividingEquation (4) by Equation (5) yields the FatigueFactor of Safety in Equation (6) below:

eq

eSFOS ...(6)

Ultimately, the goal is to determine whetheror not a simple joint design can replace acomplex one. From further observation ofEquations (2 and 3), an importantmathematical assumption can be made. Theeffort of the industry in attempting to replaceRzeppa CV Joints with more economicalCardan Joint is debatable. If the guideline forthe use of these joints can be found, it can beused as a benchmark for designconsiderations for all transportation driveassemblies.

RESULTS AND DISCUSSIONEquations (1)-(3) were used for calculating theSecondary Bending Moment Couples for eachjoint. Table 1 below outlines the assumedoperating conditions of the theoretical Monorailtrain, with shaft torques at startup, accelerationand cruising speeds, along with the assumedoperating times for one year.

Table 2 below shows the results fromcalculating the bending moment couples for

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each joint with Equations (1)-(3) for the startupshaft torque. The joint angles in the first columnin this and the subsequent tables pertaining to

Bending Moment Couples were chosen toshow the small angle approximation mentionedin the previous section, and to outline a

1 370.28 370.22 261.81

2 740.78 740.33 523.65

3 1111.74 1110.21 785.58

4 1483.37 1479.76 1047.62

5 1855.91 1848.85 1309.83

6 2229.60 2217.38 1572.23

12 4509.01 4410.47 3153.13

15 5684.06 5490.38 3949.57

45 21213.20 15000.00 12426.41

Table 2: Bending Moment Couples at Startup Stage

Description Bending Moment Couples Cardan Joint, = 45° Bending Moment Couples Rzeppa Joint

C (Driving Yoke), in-lb C1 (Driven Yoke), in-lbJoint Angle (°) C = C1, in-lb

1 192.54 192.52 136.14

2 385.21 384.97 272.30

3 578.10 577.31 408.50

4 771.35 769.47 544.76

5 965.08 961.40 681.11

6 1159.39 1153.04 817.56

12 2344.68 2293.45 1639.63

15 2955.71 2855.00 2053.78

45 11030.87 7800.00 6461.73

Table 3: Bending Moment Couples at Acceleration Stage



Table 1: Operating Conditions of Theoretical Monorail

Startup 9% 2500 30000 150

Acceleration 4% 1300 15600 210

Cruise 87% 840 10080 250

Assumed hours per day 13.70

Assumed days per year 365

Assumed hours per year 5000

Description Frequency (% ofTotal Operation) Shaft Torque (ft-lb) Shaft Torque (in-lb) Shaft Velocity (rpm)

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perspective of how both joints react across thespectrum of various operating angles. 12° isalso shown in this table because this is thevalue at which tan and sin really begin toseparate in value and the small angleapproximation is no longer valid. However, thisdoes not mean that 12° is the “threshold angle”for the two joints. Tables 3 and 4 show theresults from calculating the bending momentcouples for each joint from Equations (1)-(3)using the acceleration phase and cruisestages.

The values from these tables were used astorques in a Finite Element Analysis (FEA) ofboth joints to ultimately determine the stressesin each joint and where certain failure pointswere expected. The FEA model of the RzeppaJoint is shown below in Figure 3, and the valuesfor the peak stress are given in Table 5 foroperating angles of +/-1°, +/–6° and +/–12°.These values were chosen because they arethe most common operating angles for thisapplication and they yield a broad spectrumof the various stresses seen by these two jointdesigns.

1 124.41 124.39 87.97

2 248.90 248.75 175.95

3 373.54 373.03 263.95

4 498.41 497.20 352.00

5 623.59 621.21 440.10

6 749.14 745.04 528.27

12 1515.03 1481.92 1059.45

15 1909.84 1844.77 1327.06

45 7127.64 5040.00 4175.27

Table 4: Bending Moment Couples at Cruise Stage



Figure 3: FEA Model of Rzeppa Joint

Startup Stage Torque Stress (psi)

Max Stress at 1° 961



Acceleration Stage Torque




Cruise Stage Torque




Table 5: Rzeppa Joint FEA

The outer race of the Rzeppa Joint was theonly part that was analyzed since the largeststress occur in this area during motion. The

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splines on the outer shaft were fixed, which isintended to represent the load wheel at a deadstop. The input torque was applied to the innersurfaces of the Rzeppa joint, and the value ofthis input torque is the bending moment couplecalculated from Equation (1). The bendingmoment couple was used since it is dependenton the operating angle and the initial drivetorque from the motor. The maximum stress isseen at the top of the splines (the faint greenarea in Figure 3).

The FEA model of the Cardan Joint isshown below in Figure 4, and the peak stressvalues are given in Table 6.

The Cardan Joint was analyzed as a whole,with the splines on the yoke shaft suppressedfor easier meshing. From observation ofFigure 4, the “splines” on the yoke shaft, ordriven yoke, are fixed (simulating the loadwheel at dead stop) and the input torque(Cardan Joint Bending Moment Couple valuesfrom Tables 2-4) is applied to the wingbearings on the appropriate sides. The drivingtorque is applied from the driving yoke, movingcounterclockwise, and the driven torque isapplied to the driven yoke, moving clockwise.The maximum stress is seen in the joiningradius of the driven yoke (faint green area inFigure 4).

The stress values from Tables 5 and 6 wereplaced in a fatigue calculator to determine theFactor of Safety using a Reliability Factor (ke)of 94.95% (Assumed factor for General Useand Reliability Critical Parts). The FatigueFactor of Safety was calculated usingEquations (4)-(6) discussed in the Introduction.The Fatigue Factors of Safety for both jointsat each operating condition are shown in

Figure 4: FEA Model of the Cardan Joint

Startup Stage Torque Stress (psi)








Cruise Stage Torque




Table 6: Cardan Joint FEA, = 45°

Table 7: Fatigue Factors of Safetyfor Both Joints

CardanJoint FOS

RzeppaJoint FOS

Startup Stage Torque

Max Stress at 1° 25.04 13.24

Max Stress at 6° 3.66 2.19






Cruise Stage Torque

Max Stress at 1° 70.65 42.28



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been omitted here because the linear trend isthe same as the startup stage. Both the RzeppaJoint and Cardan Joint exhibit increasingstresses as operating angle increases. Thisis due to the direct proportionality of the jointangle and the bending moment couple (fromobservation of Equations (1)-(3). The smallangle approximation (tan sin ) mentionedin the introduction section is also quite validup to about 12°, however the bending momentcouples on the Cardan Joint begin to deviatewell before this stage.This in turn yields higherstresses on the both the driven and drivingyokes of the Cardan Joint.

From observation from Table 2, it isapparent that the bending moments on theRzeppa Joint are close to half the magnitudeof the Cardan Joint. This hints to a theory thatthe Rzeppa Joint’s stresses would be half asmuch as the Cardan Joint’s. This is a validprediction since Equation (1) for the RzeppaJoint’s bending moment contains a /2 termin the calculation, whereas Equations (2) and(3) for the Cardan Joint only contain a term.However, the analysis shows that the CardanJoint yields astress that is slightly lower thanthe Rzeppa at a 1° operating angle for startup,even though the bending moment couples arehigher (for the case of = 0, however).

At higher angles, the values of the bendingmoment couples on the Cardan Joint deviatefrom each other, as mentioned before. Thisyields higher stresses on the Cardan Joint,about 3,000 psi greater than the Rzeppa Joint.Ultimately this also constitutes a lower factorof safety as the operating angle increases, asseen in Table 7. While both will achieve infinitelife according to the calculator, it is safer to gowith the Rzeppa joint rather than the Cardan

Table 7. The goal of the fatigue calculator is tokeep the FOS above a value of 1 to maintaininfinite life.

The following two figures show the stressdistributions for both joints across all anglesfrom 1°-12° for the startup stage.

Figures 5A and 5B show that the stressesfor both joints increase linearly as the bendingmoment couples increase, and the onlydifference being the actual values of thestresses between the two joints. The stressfigures for acceleration and cruise stages have

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3x 10

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Angle (degrees)

Rzeppa Joint Stress vs. Angle at Startup

Str

ess

(psi

)

stress values at different angles least squares fit

Figure 5A: Rzeppa Joint Stresses versusAngle at Startup

Figure 5B: Cardan Joint Stresses versusAngle at Startup

0 2 4 6 8 10 12-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 104

Angle (degrees)

Str

ess

(psi)

Cardan Joint Stress vs. Angle at Startup

stress values at different angle least squares fit

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Joint especially if the operating angles aregoing to be higher than 12°.

It is also important to reiterate the fact thatall of the analysis of the Cardan Joint has beenset in one dimension; that is, , the anglenormal to the plane of the joint angle, has beenset to 45 degrees (the worst possible casescenario). This is another reason why it isimportant to understand the application andthe environment that these joints will be usedin and take into account all possible outcomes.

One possible alternative to this issue couldbe to use the Double Cardan Joint, whichutilizes two Cardan Joints coupled in thecenter, which eliminate the lopsided bendingmoment couples on the driven and drivingsides (Fischer and Paul, 1991; and Biancoliniet al., 2003). The only issue with using this jointin this particular application is the size; havingtwo Cardan Joints coupled at the center willprotrude much further than desired, thusrequiring two smaller Cardan Joints which maynot be strong enough to transmit the drivetorques. Another possible alternative is theThompson Coupling CV Joint, which tries toblend the best of both the Rzeppa and CardanJoints into one unit (Lin and Liao, 2007). Thisdefinitely could be a solution, but this joint willneed to be tested a lot more in order to validateits use in this application as it has not beenused in a Monorail train yet.

CONCLUSIONThe most important point to make from thisstudy is that if the joint angle low, sayapproximately well within –6° and +6°, and ifthe angle is minimal (near zero), the CardanJoint can be adequate. However, if at any pointin motion the operating angle exceeds 6° in

any direction, the Rzeppa Joint will be the betteroption in terms of fatigue and reliability.Therefore, the “threshold angle” can be saidto be approximately 6° in either direction ofangle. Exploring the differences betweenConstant and Non-Constant Velocity Jointsreveals that choosing between the two is notan easy decision. However, it can beconcluded unless the operating angle is verylow, Rzeppa Joint should be choice in moststraddle type monorail driving operation tohave more robust operation without anytroubling issues.

REFERENCES1. ASTM International & American Society

for Testing & Materials (2004), AnnualBook of ASTM Standards, AmericanSociety for Testing & Materials.

2. Biancolini M E, Brutti C, Pennestrì E andValentini P P (2003), “Dynamic,Mechanical Efficiency, and FatigueAnalysis of the Double CardanHomokinetic Joint”, International Journalof Vehicle Design, Vol. 32, No. 3,pp. 231-249.

3. Budynas R G and Nisbett J K (2011),Shigley’s Mechanical EngineeringDesign.

4. Carmichael C (1950), Kent’s MechanicalEngineers’ Handbook.

5. Fischer I S and Paul R N (1991),“Kinematic Displacement Analysis of aDouble-Cardan-Joint Driveline”, Journalof Mechanical Design, Vol. 113, No. 3,pp. 263-271.

6. Lin C C and Liao W K (2007), “TheDisplacement Analysis of a Concentric

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Double Universal Joint”, In 12th IF to MMWorld Congress on Mechanism andMachine Science, Comité Franáais pourla Promotion de la Science desMecanismes et des Machines Besanáon,France.

7. Qin G, Zhang Y Q and Wu C L (2003),“Dynamics Simulation Analysis Systemfor Ball Basket Constant VelocityUniversal Joint”.

8. Seherr-Thoss H C, Schmelz F andAucktor E (1998), Universal Joints and

Driveshafts, Beijing Institute ofTechnology Press, Beijing.

9. Sekitani T, Hiraishi M, Yamasaki S andTamotsu T (2005), “China’s First UrbanMonorail System in Chongqing”, HitachiReview, Vol. 54, No. 4, p. 193.

10. Wagner E R and Cooney C E (1991),“Universal Joint and Driveshaft DesignManual”, Society of AutomotiveEngineers, Warrendale, PA.

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