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. T_{erf} can be determined using the same equation. It applies to uniform operation, i.e. when the torque M_{d} occurs throughout life L_{h} at rotation speed n and deflection angle *ß*. T_{erf} | = | | T_{erf} | = | 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 | Lh_{erf} | = | necessary (required) life in h. This L_{h} at least is achieved by 90% of all shafts. The average L_{h} 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: So T_{erf} 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 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: 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: Where: L_{hn} | = | individual life of class n, where n = 1,2,3...n | M_{n} | = | the moment allocated to class n | T_{vorh} | = | joint power factor of estimated joint size | n_{n} | = | rotation speed allocated to class n | *ß*_{n} | = | deflection angle allocated to class n | See above for other symbols. From the individual life, the total life can be determined as follows: where: q = time proportion in % L_{h1}...L_{hn} individual life in h. **8.2 Selection of Joint Sizes for Vehicle Drives** In this section, the following symbols are used: M_{FG} | = | function torque capacity (from data sheet) | M_{X} | = | general dimensioning moment for a propeller shaft | M_{A},M_{B},M_{C} | = | dimensioning moment for propeller shafts A, B, C | M_{mot.} | = | general proportional engine torque on propeller shaft | M_{mot} _{max} | = | max. engine torque | M_{Rad 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) | *µ*_{R} | = | 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 | i_{W} | = | theoretical value for converter ratio | i_{WF} | = | converter brake conversion | i_{G max} | = | max. engine gear ratio (1st gear) | i_{G min} | = | min. engine gear ratio (1st gear) | i_{V max} | = | transfer box ratio (1st gear) | i_{V min} | = | transfer box ratio (nth gear) | i_{A} | = | final drive ratio | V | = | engine torque distribution ratio T_{mot V} / T_{mot H} | R_{dyn} | = | dynamic rolling radius of tyre | G_{V} | = | front axle load; total front axle load | G_{V1} | = | front axle load, 1st axle | G_{V2} | = | front axle load, 2nd axle | G_{H} | = | rear axle load; total rear axle load | G_{H1} | = | rear axle load, 1st axle | G_{H2} | = | rear axle load, 2nd axle | The function torque capacity M_{FG} 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 *ß*º, the function limit moment is reduced by the factor cos *ß*º. The function torque capacity M_{FG} must be sufficiently larger than the dimensioning moment M_{x}. M_{FG} 1,5 · M_{x} The dimensioning moments M_{x} for the propeller shafts between the engine and the final drive are calculated approximately from the moments of the torque M_{motx} exerted by the engine and the adhesion moment M_{radx} exerted by the wheel, as follows: M_{x} = ½ (M_{motx} + M_{radx)}_{} 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 i_{WF} < 1,4, its influence can be ignored, so i_{W} = 1. If the brake conversion i_{WF} > 1,4, its influence must be allowed for by a factor of 0.76, so i_{W} = 0,76 · i_{WF.} **8.3 Selection System for Propeller Shafts in Vehicles for Normal Use** **Road Vehicle 4 x 2** Selection torque for propeller shaft A between engine 1 and gearbox 2. Selection torque for propeller shaft or multiple joint shaft B between gearbox 2 and differential 4. **Road Vehicle 6 x 2** Selection torque for cardan shaft A between engine 1 and gearbox 2. Selection torque for propeller shaft or multiple joint shaft B between gearbox 2 and differential 4. **Road Vehicle 6 x 4** **and Road Vehicle 8 x 4** Selection torque M_{A} for propeller shaft A between engine 1 and gearbox 2 Selection torque M_{B} for propeller shaft or multiple joint shaft B between gearbox 2 and differential 4 Selection torque for M_{B'} for propeller shaft B' between differential gears 4 **All-Wheel Drive 4 x 4** Selection torque M_{A} for propeller shaft A between engine 1 and gearbox 2 Selection torque M_{A'} for propeller shaft A' between gearbox 2 and transfer box 3 Selection torque M_{B} for propeller shaft or multiple joint shaft B between transfer box 3 and differential gears 4 Selection torque M_{C} for propeller shaft C between transfer box 3 and differential gears 4 **All-Wheel Drive 6 x 6** Selection torque M_{A} for propeller shaft A between engine 1 and gearbox 2 Selection torque M_{A'} for propeller shaft A' between gearbox 2 and transfer box 3 Selection torque M_{B} for propeller shaft or multiple joint shaft B between transfer box 3 and differential gears 4 Selection torque M_{B'} for propeller shaft B' between differential gears 4 Selection torque M_{C} for propeller shaft C between transfer box 3 and differential gears 4 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 l_{0} = 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 l_{0} = 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: n_{operation} 0,6... 0,7 n_{krit} 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: Definition of imbalance: Imbalance U = u · r in gmm where u = unequalized individual mass on radius r Shifting of centre of gravity 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 n_{bal} = balancing rotation speed in rpm d = tube diameter in mm Example:^{ }Shaft of 44 kg, n_{bal} = 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 = · J_{m} and with *ß* = deflection angle of joint, = rotation angle position of propeller shaft (_{max} at 45º), n = rotation speed of shaft in rpm and J_{m} = mass moment of inertia of shaft centre part in Nms^{2}. 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 | n_{max} [ 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. |