NATURAL FREQUENCY ON ROTATING SHAFTS

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Theoretical background

The theoretical background contains all the facts for the respective area that has been found in books, on the internet and in documents at Saab. Here is theory for shafts and joints, calculations on natural frequency on tubes, power losses due to speed variation on a shaft, lubrication and performance, 4WD, Saab XWD.

Shafts

The most commonly used shaft to transfer torque between the gearbox and the rear axle on cars is the propeller shaft, also called cardan shaft. The name cardan comes from the famous Italian professor and mathematician Geronimo Cardano (1501-1576).
The first shafts that were used were solid shafts. Those aren’t very good because they are heavy, they don’t allow any axial movements between the gearbox and the rear axle and they transmit a lot of vibrations. Due to these problems the new shafts had a different design. They were hollow to save weight, and also had the ability to plunge (a sliding ability so that the length can vary). Even later the shaft was divided into two and three shorter shafts with joints mounted to the chassis between them. The reason for this was to prevent the shaft from working in the range of natural frequency which would make it vibrate.
The main task for an axle is to transfer torque and rotation. Two of the axles on the car that transfer most torque are the propeller shaft that transfer torque between the gearbox and the differential and the drive shaft that transfer torque between the differential and the wheel hubs.
A shaft in the driveline of a car should be designed to be able to transfer the given amount of torque during the whole life of the car without any maintenance. Other demands on the shaft are that it should neither produce nor transfer vibrations. It should also have low losses of energy and be as light as possible.
Shafts that have direct contact with the wheels are called unsprung mass. These shafts have even higher demands on their weight and vibration properties.
On some shafts there are mounted a vibration absorber to reduce the vibrations. A disadvantage with these is that they increase the total weight of the shaft.
The shafts in the driveline of the car are mostly manufactured from steel, but also aluminum and composites are used.
When calculating the dimensions of a shaft it’s important to take fatigue failure and stability into consideration. This means that calculations should be made with varying load and also different kinds of loads like bending, torsion and tension.
One method to use when you are dimensioning a shaft is to first do approximate calculations for both the shaft and the bearings. When that’s complete, continue with more accurate calculations. First when the shaft is setting the design load and then when the bearing is setting the design load. Calculations should be made for both static and dynamic loads.
When calculations are done on a shaft with statically stated bearings were the load is setting the dimensions you should do a force- and a torque-graph.
When calculating and the deformations are setting the dimensions, it should also include deformation relations in the calculations.
While calculating on a shaft with statically unstated bearings it should also include the geometrical relations. [2][3][5][11][12][13][14]

Joints

A joint is supposed to transfer torque and rotation between two shafts. There are many different kinds of joints for different applications. When choosing a joint for a specific use then following issues has to be considered: [3][6][11][13]

• Use
• Kind of joint
• Motion
• Forces and torque
• Driving conditions
• Running speed
• Design
• Dimensions and weight
• Environment
• Assembly and dismounting
• Replacement
• Length of life
• Price

Cardan joint (Hooke´s joint)

The cardan joint consist of a cross, where each shaft end is connected through bearings with the two opposite arms of the cross. There are different kinds of cardan joints on the market. In some designs the cross is replaced by a ball. One disadvantage with the cardan joint is that the rotational speed on the output axle is pulsating when you bend the joint. The size of the pulsations in rotational speed is depending on the bending angle and this causes vibrations. The nominal angle should be between 0,6 and 6 degrees according to GM best practice. The maximum angle should never exceed 20 degrees. During one revolution the second axle is going through two phases of accelerating and retardation. An equation for the angular speed on the second axle is shown below.
To counteract the pulsations on the second shaft mounted to the rear axle it is possible to assemble two cardan joints with a middle shaft. The result is three shafts and two cardan joints. The first and the third shaft then get constant speed. To get this working it is important that the U-shaped claws on the middle shaft-ends are mounted right depending on the angles in the joints, otherwise the pulsations will be even greater. The main reason that the cardan joint is used to a great extent in spite of the pulsations is because of its relatively high efficiency, and because it’s often used with small angels which doesn’t give very large pulsations. However it’s important that the joint isn’t working without any angel at all because the lubrication would then fail. [3][4][6][8][11][13][14][17]

CV-joints (Constant velocity joints)

CV-joints are able to transfer torque with constant angular speed even in quite large angles. These joints are more effective than cardan joints when it’s necessary with large angles. [11]

Ball joints (Rzeppa-joints)

The ball joint is the most commonly used joint on FWD cars. It consists of an inner and an outer half with tracks for balls that transfer the torque between the two halves. Except for FWD-cars this joint is also used on RWD and AWD cars. The appropriate angles for different kinds of ball joints varies. The nominal angle should normally lie between 1 and 6 degrees. The maximum angle varies for different joints and it can be up to about 50 degrees.
Some different kinds of ball joints are:

• AC (Angular Contact) Fixed Joint

This joint is useful in passenger cars and lighter vehicles. This joint can work up to 45 degrees. This joint is practical when high motions are needed for example on the front wheels.

• UF (Undercut Free) Fixed Joint

The UF joint can be used for similar applications as the AC joints. The advantage is that it can work with larger angles.

• VL (Verschiebegelenk Löbro) Disc Joint

The VL joint can work in great angles and does also admit some plunging motion. This is good when you have motions in the driveline or want to lower the assembly tolerances. This joint has very good performance on high rotational speeds. VL Disc joint is often mounted to the gearbox with a flange coupling

• VL (Verschiebegelenk Löbro) Monoblock Joint

The VL-Monoblock joint is mounted to the gearbox or to the wheel hub with a shaft. It has a very compact design which saves space and weight.

• DO (Double Offset) Joint

The DO joint is similar to the VL joint and allows high plunging motions up to 50mm and working angles up to 30 degrees. This joint has also similar properties as the tripod joints. One advantage to the tripods is the low rotational lash which improves the NVH (noise, vibration, harshness) properties. [7][11]

Tripod joint

One alternative to the ball joint is the tripod joint. It consists of an inner part with three short shafts with bearings and an outer part with tracks for the bearings. This joint demands very low forces for plunging. This prevents vibrations to be transmitted trough the joint and does also give lower power losses when working in large angels than the ball joint. One disadvantage with the tripod joint is that it has a play in rotational direction. This can cause NVH. The tripod joints are used for relatively small angles and for applications that requires great plunging motions. One example is in AWD vehicles where the tripod joint is used to prevent vibrations and movements from the engine and the drive train to be transmitted into the car.
Some different kinds of tripod joints are:

• GI (Glaenzer Interieur)

The GI joint is suitable as inner joint on the drive shafts on most vehicles. It has a working range up to 20 degrees and allows plunging up to 50mm. The low plunging forces in the GI joint gives good NVH properties.

• AAR (Angular Adjusted Roller)

The AAR joint has the same plunging length and can manage larger angles than the GI joint and has even lower plunging forces. This improves the NVH properties even more. [7][8][11]

Other joints

This paragraph is about new CV-joints that were found on the internet. These joints seem to be quite new. There have not been any facts found that these joints are being used in mass produced cars but if they are as good as the manufacturers say they should be suitable as a substitute to ordinary universal and CV-joints. All facts about these joints are collected from the respective manufacturer’s website.

Thompson Constant Velocity Coupling

The Thompson Coupling is essentially two Cardan joints assembled coaxially where the cruciform-equivalent members of each are connected to one another by trunnions and bearings which are constrained to continuously lie on the homokinetic plane of the joint.
At tests carried out by the manufacturer, The Thompson Coupling outlasted its rivals and it needed 9.3% less energy to do the same job, promising significant energy and fuel savings. [18]

Advantages according to the manufacturer

• The Thompson Coupling is the world’s first and only true CV-joint:
• Has all loads carried by roller bearings
• Has no sliding or skidding surfaces whatsoever
• Can tolerate axial and radial loads without degradation
• Constructed to any torque level
• Does not require special lubrication
• Does not require a dust boot
• No wearing components except replaceable bearings and trunnions
• Is suitable for automotive tail or propeller shaft applications
• Is less bulky than a double coupling or double Cardan joint. [18]

Disadvantages according to the authors

• The size? The minimum size is limited by the bearings.
• The price? No price comparisons between ordinary joints and the Thompson Coupling are made but the coupling is quite advanced and consists of many parts so the price should be quite high.
• The Thompson coupling does not allow any plunging movements.
• The accessibility. Because it’s a quite new invention there is no mass production yet and only one company that manufacture the coupling.

1 Introduction 
1.1 BACKGROUND
1.2 PURPOSE AND AIMS
1.3 DELIMITATIONS
1.4 OUTLINE
2 Theoretical background
2.1 SHAFTS
2.2 JOINTS
2.3 LUBRICATION AND PERFORMANCE
2.4 NATURAL FREQUENCY ON ROTATING SHAFTS
2.5 POWER LOSSES DUE TO SPEED VARIATION ON A SHAFT
2.6 VECTOR CALCULATIONS
2.7 HOW 4-WHEEL DRIVE WORKS
2.8 SAAB XWD
3 Implementation
4 Results
5 Conclusions and discussions
6 References
7 Search words.
8 Attachment
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Fuel efficiency in AWD-system