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Dimensions of Cardan Shafts

The dimensions of the shaft depend on many factors. The rules below will give an approximate selection. In borderline cases, please consult us. The questionnaires in chapter 12 will help you. We would be pleased to give you advice.

 

8.1 Selection of Joint Size for Stationary Drives

The part of the propeller shaft which determines its useful life is normally the joint bearing. So the joint size should preferably be determined from the transferable torque of the bearing. The calculation below is based on the standard roller bearing calculation, where the oscillating movement is regarded as replaced by a rotational one.

The dimension for the transferability of the bearing is the joint load rating T = C · R, where C is the dynamic transfer capacity of the bearing and R the distance of the bearing centre from the joint centre. The joint load rating is given in the data sheet for the shaft. Terf can be determined using the same equation. It applies to uniform operation, i.e. when the torque Md occurs throughout life Lh at rotation speed n and deflection angle ß.

Terf = Formel 14
Terf = necessary joint load rating in Nm
K = shock factor (see table)
ß = deflection angle of joint in ° (degrees). For angle < 3°, ß = 3° must always be used.
M = torque to be transferred in Nm
Lherf = necessary (required) life in h. This Lh at least is achieved by 90% of all shafts. The average Lh of all shafts is then 5 times as high.
n = rotation speed of shaft in rpm

Shock Factors

Drive Unit K with rubber coupling K without rubber coupling
Elec. motors
Motors with converter
Diesel engine 1-3 cylinders
4 or more cylinders
Petrol engine 1-3 cylinders
4 and more cylinders
Compressors 1-3 cylinders
4 and more cylinders
1
1
2
1,5
1,5
1,25
1,25
1,15
1
1
2,5
2,0
2,0
1,75
1,75
1,5

Example:

A working machine with a small mass moment of inertia, which assumes a torque of 1000 Nm at n = 1 450 rpm, should be driven by an electric motor via a shaft running under a deflection angle of 7°. The life should be 2000 h. What joint size is required?

Solution:

Electric motor and impact-free working machine gives an impact factor of 1.0. Then:

Formel 15

So Terf is found to be 1339 Nm. From the data sheet, we now select the shaft with the next highest value. If we are to use a shaft of design 008 for example, the type and joint size 008 195 are selected with a joint transfer capacity of 1460 Nm.

For the joint found, we now check that Formel 16

1000 Nm · 1,0 < 1460 Nm · cos 7° = 1449,1 Nm.

The condition is fulfilled, and the shaft can be used. It will achieve a life of:

Formel 17

In many applications, in particular in vehicles, the moment, the rotation speed and/or the deflection angle are not constant. We must then try to form classes to which moment, rotation speed and deflection angle can be allocated and determine their time proportions.

For an initial estimated joint size, the individual life can then be assessed for each class:

Formel 18

Where:

 

Lhn = individual life of class n, where n = 1,2,3...n
Mn = the moment allocated to class n
Tvorh = joint power factor of estimated joint size
nn = rotation speed allocated to class n
ßn = deflection angle allocated to class n

See above for other symbols.

From the individual life, the total life can be determined as follows:

Formel 19

where:
q = time proportion in %
Lh1...Lhn individual life in h.

 

8.2 Selection of Joint Sizes for Vehicle Drives

In this section, the following symbols are used:

MFG = function torque capacity (from data sheet)
MX = general dimensioning moment for a propeller shaft
MA,MB,MC = dimensioning moment for propeller shafts A, B, C
Mmot. = general proportional engine torque on propeller shaft
Mmot max = max. engine torque
MRad x = general proportional wheel adhesion torque at propeller shaft
s = joint bearing safety factor = 1,5 < s <2,0
k = shock factor (see table above)
µR = tyre coefficient of friction = 0,6 < µ < 1,0
= general gear efficiency
G = efficiency of engine gear
V = efficiency of transfer box
A = efficiency of final drive
iW = theoretical value for converter ratio
iWF = converter brake conversion
iG max = max. engine gear ratio (1st gear)
iG min = min. engine gear ratio (1st gear)
iV max = transfer box ratio (1st gear)
iV min = transfer box ratio (nth gear)
iA = final drive ratio
V = engine torque distribution ratio Tmot V / Tmot H
Rdyn = dynamic rolling radius of tyre
GV = front axle load; total front axle load
GV1 = front axle load, 1st axle
GV2 = front axle load, 2nd axle
GH = rear axle load; total rear axle load
GH1 = rear axle load, 1st axle
GH2 = rear axle load, 2nd axle

The function torque capacity MFG of the propeller shafts is given in the data sheets in this catalogue. This moment can be transferred by the propeller shaft for short periods at limited load frequency with 0° joint deflection angle.

With a joint deflection angle of ߺ, the function limit moment is reduced by the factor cos ߺ.

The function torque capacity MFG must be sufficiently larger than the dimensioning moment Mx.

MFG 1,5 · Mx

The dimensioning moments Mx for the propeller shafts between the engine and the final drive are calculated approximately from the moments of the torque Mmotx exerted by the engine and the adhesion moment Mradx exerted by the wheel, as follows:

Mx = ½ (Mmotx + Mradx)

For propeller shafts A between the engine and the gearbox, the influence of the high rotation speed part and the engine shock factor must be taken into account.

If a converter is fitted, some special features should be observed:

If the propeller shaft is installed between the engine with converter and the gearbox, the impact factor s = 1 must be used. If the propeller shaft is between the engine and gearbox with a converter in front, the effect of the wheel moment = 0.

If the brake conversion iWF < 1,4, its influence can be ignored, so iW = 1.

If the brake conversion iWF > 1,4, its influence must be allowed for by a factor of 0.76, so iW = 0,76 · iWF.

 

8.3 Selection System for Propeller Shafts in Vehicles for Normal Use

Road Vehicle 4 x 2

Bild: Straßenfahrzeuge 4 x 2

Selection torque for propeller shaft A between engine 1 and gearbox 2.

Formel 20

Selection torque for propeller shaft or multiple joint shaft B between gearbox 2 and differential 4.

Formel 21

Road Vehicle 6 x 2

Bild: Straßenfahrzeuge 6 x 2

Selection torque for cardan shaft A between engine 1 and gearbox 2.

Formel 22

Selection torque for propeller shaft or multiple joint shaft B between gearbox 2 and differential 4.

Formel 23

Road Vehicle 6 x 4

Bild: Straßenfahrzeuge 6 x 4

and Road Vehicle 8 x 4

Bild: Straßenfahrzeuge 8 x 4

Selection torque MA for propeller shaft A between engine 1 and gearbox 2

Formel 24

Selection torque MB for propeller shaft or multiple joint shaft B between gearbox 2 and differential 4

Formel 25

Selection torque for MB' for propeller shaft B' between differential gears 4

Formel 26

All-Wheel Drive 4 x 4

Bild: Straßenfahrzeuge 4 x 4

Selection torque MA for propeller shaft A between engine 1 and gearbox 2

Formel 27

Selection torque MA' for propeller shaft A' between gearbox 2 and transfer box 3

Formel 28

Selection torque MB for propeller shaft or multiple joint shaft B between transfer box 3 and differential gears 4

Formel 29

Selection torque MC for propeller shaft C between transfer box 3 and differential gears 4

Formel 30

All-Wheel Drive 6 x 6

Bild: Straßenfahrzeuge 6 x 6

Selection torque MA for propeller shaft A between engine 1 and gearbox 2

Formel 31

Selection torque MA' for propeller shaft A' between gearbox 2 and transfer box 3

Formel 32

Selection torque MB for propeller shaft or multiple joint shaft B between transfer box 3 and differential gears 4

Formel 33

Selection torque MB' for propeller shaft B' between differential gears 4

Formel 34

Selection torque MC for propeller shaft C between transfer box 3 and differential gears 4

Formel 35

These selections will avoid major dimensioning errors. However, they disregard important influences on the useful life such as deflection angle, rotation speed, loading, effect of dirt, temperature etc. For example, halving the deflection angle doubles the life, as 9.1 shows.

Please therefore use our questionnaire in chapter 12. We recommend the correct joint size using our computer program.

 

8.4 Critical Rotation Speed

The propeller shaft found from dimensioning specifications 8.1, 8.2 or 8.3 must now be checked for bending-critical rotation speed.

In general, propeller shafts run uncritically, i.e. their operating speed is below the critical speed. The critical speed for propeller shafts with steel tube is calculated from the equation:

where D = tube external diameter, d = internal diameter and l0 = free length between the joints or centre bearing assemblies all in mm.

If special propeller shafts are produced with steel rotating rod, calculate the critical rotation speed as

where D = rod diameter and l0 = free length, all in mm.

These equations apply for smooth tubes or rods. propeller shafts only achieve around 80-90% of this speed because of play in bearings and sliding pieces and additional dimensions. As the max. operating speed should lie 10-20% below this critical speed, the operating rotation speed selected is:

noperation 0,6... 0,7 nkrit

The maximum operating speed can be taken from the diagrams below.

Fig. 24:

Propeller shaft with steel tube

Propeller shaft with steel rotating road

If the maximum operating speed is not sufficient, a larger tube diameter or rod design with centre bearings should be used.

 

8.5 Balancing Propeller Shafts

Propeller shafts for drive shafts in the automotive industry are dynamically balanced. Balancing is the equalization of weight of eccentrically running masses (Fig. 25) in the propeller shaft to achieve quiet running and reduce load on the joints and bearings in the connected assemblies.

Fig. 25:

Bild 25

Definition of imbalance:

Imbalance U = u · r in gmm
where u = unequalized individual mass on radius r

Shifting of centre of gravity

Formel 38

where G = weight of part to be balanced.

Sensible Values for Permitted Imbalance

Practical experience has shown that as the rotation speeds increase, a smaller shift in the centre of gravity can be permitted. It is therefore sensible to take the product of rotation speed x shift in centre of gravity as a value for the permitted imbalance. DIN ISO 1940 "Requirements for balance qualitites of rigid rotors" is also based on this concept. A table there gives "quality classes" for different components, where it has been assumed that there is no point in balancing the different elements (wheels, rims, wheel sets, crankshaft components, shafts etc.) of a closed machine group, e.g. a vehicle, to widely differing quality classes.

According to DIN ISO 1940, propeller shafts should comply with class G40 ( · = 40 mm/s), and propeller shafts for special requirements, class G16 ( · = 16 mm/s).

Unless the customer specifies otherwise, the shafts are balanced at the maximum rotation speed to quality class G16. The permitted residual imbalance is determined from the equation below:

in g per side

where:
u = permitted unequalized individual mass per side in g
G = shaft weight in kg
nbal = balancing rotation speed in rpm
d = tube diameter in mm

Example:
Shaft of 44 kg, nbal = 3500 rpm, Tube diameter 90:
u = 99363 · 44 / ( 3500 · 90 ) = 13,8 g unequalized individual mass per side

As repeated clamping gives different values due to play, the values of the equation only correspond 65% to the value permitted under DIN ISO 1940. In test runs with repeated clamping therefore, 135% of the value given in DIN ISO 1940 is permitted, i.e. approximately double the equation value.

 

8.6 Mass Acceleration Moments - Influence of Rotation Speed and Deflection Angle

In order to achieve adequate smooth running of the propeller shaft, the mass acceleration moment of the centre part between the joints must not be too large. The mass acceleration moment depends on the mass moment of inertia of the centre part, the rotation speed n and the deflection angle of the joint. The permitted size of the mass acceleration moment increases with the moment transferability of the joint, i.e. as the joint power factor T increases, the permitted mass acceleration moment M also increases.

For propeller shafts in goods vehicles, depending on requirements, installation conditions and sprung mass system, the specific mass acceleration moment

M spec = 0,04 to 0,06 Nm/Nm

If sound radiation is taken into account (buses etc.), the specific mass moment of acceleration M spec. must be smaller; if humming noise is of secondary importance, M spec can be larger.

The specific mass acceleration moment M spec is the quotient of the mass acceleration moment of the centre part and the joint power factor T.

M spec = M / T

where M = · Jm

and

with ß = deflection angle of joint, = rotation angle position of propeller shaft (max at 45º),
n = rotation speed of shaft in rpm and Jm = mass moment of inertia of shaft centre part in Nms2.

The table below was produced from these equations and gives the max. n x ß value for propeller shafts of centre length 1.5 m as approximate values.

Joint Size nmax
[ rpm ]
n x ß
[ rpm · degree ]
196
200
253
375
376
411
490
491
590
600
610
620
680
700
710
5500
5500
5000
4800
4800
4600
4400
4500
4000
4200
4000
4000
3800
3700
3600
28000
34000
24000
21000
19000
19000
17500
17500
16000
18000
17000
16000
15000
16000
14000

How far these values can be exceeded depends on the requirements for smooth running and many peripheral conditions. With favourable sprung mass systems, the value can be exceeded up to 50 %.

 

8.7 Measures to Improve Smoothness of Running

To reduce the radiated noise (gears or axle noise), the propeller shaft can be fitted with a cardboard tube pressed into the shaft tube. This effectively damps the higher frequencies.

 
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