Design guidelines

1.1 Rotation angle of a single joint as a function of deflection angle ß

φ₁ = Input-rotation angle | φ₂ = Output-rotation angle

If a single joint is deflected by angle ß and rotated in this condition, rotation angle φ₂ of the output shaft differs from rotation angle φ₁ of the input shaft. The relationship between the two rotation angles is as follows:

As can be seen from the adjacent diagram, maximum lead occurs at about 45°, maximum lag at about 135°.

Fork position φ₁ = 0° is then obtained, when the input fork is located in the deflection plane of the joint.

1.2 Motion and torque characteris-tics of a sigle joint as a function of deflection angle ß

M_dI = Input torque
M_dII = Output torque
ω_I = Input-angular velocity
ω_Il = Output-angular velocity

When analyzing the motion and torque characteristics of a singular joint, it is found that with a constant angular velocity- and torque input, a fluctuating motion and torque curve is obtained at the output.

The reason for this fluctuation can easily be illustrated by following the torque characteristics at the fork position φ₁ = 0° and φ₁ = 90° as shown at left. Since the torque can only be transmitted in the spider plane, the spider however, depend-ing on the fork position, is always at a right angle to the input or output axis, output torque fluctuates twice per revolution between M_dl · cos ß and M_dl/cos ß.

The transmitted power, however, is constant, if you disregard friction losses in the bearings.

Therefore, the following applies:

For fork position φ₁ = 0° we obtain:

and for fork position φ₁ = 90°:

Section 2.2 illustrates that angular velocity and torque at the output of a single joint follow a sinusoidal pattern with a 180° cycle. Maximum angular velocity ω_{II max} coincides with minimum torque Md_{II min} min and vice versa. From this it can be deduced that a uniform output is possible, when a second joint, with a 90° phase shift is connected to the first joint by means of a shaft.

For ß₂ > ß₁ the following applies:

2.1 Single joint

As explained under 1.1, on a single joint the output velocity deviates from the input velocity. This means, the speed ratio is not uniform. This non-uniformity (fluctuation) can be calculated as a dimensionless value:

Fluctation rate

2.2 Universal driveline (2 joints connected in series)

If the preconditions listed in Chapter 1 for ob-taining a complete motion compensation cannot be met, it must be aimed for that::
U ≤ 0,0027

On drivelines with three-dimensional deflection angles, input and output shaft are not located in one plane. This results, if no special measures are taken, in a non-uniform output motion. The constantly repeating acceleration and deceleration unleashes inertia forces which can greatly reduce the life of the joints.

Example 1

Example 2

Example 1:
a) Rotate joint 1 counter clockwise by the offset angle
b) Rotate joint 2 clockwise by the offset angle.
The direction for viewing is, in both cases, from joint 1 to joint 2.

In Section 1.2 it was shown that the torque is transmitted only in the spider plane and that depending on the fork position, the spider can be perpendicular either to the input axis or the output axis.

The adjacent illustration shows the location and direction of the additional forces and moments on drivelines having a Z-arrangement, in particular for yoke angles φ₁ = 0° and φ₁ = 90°. This shows clearly, that the driveline center part is stressed by the torque which fluctuates between M_dI · cos ß and M_dI /cos ß in torsion and by the additional periodically alternating, moment M_ZII in bending. (See also Section 5.8).

According to the adjacent illustration, with the W-arrangement, an additional force, “S“ is introduced, caused by the additional moments M_ZII acting in the same direction.

To size universal drivelines properly, various conditions and factors must be considered. In view of the multitude of possible applications exact, generally valid rules cannot be provided. 

The max. permitted torques Md_max stated for the individual driveshaft sizes apply normally only for short-term peak loads.
Md_nom: Nominal torque for pre-selection on the basis of the operating moment.

Depending on the type of power input or installation, a driveline can be subjected to shock loads considerably above the rated torque. To take those into account, shock service factors must be implemented.
Following are some shock-service factors for the most common drives.

The decisive factor with regard to life expectancy of universal drivelines is usually the joint bearing. Therefore, in order to determine the individually required joint size, the life expectancy diagram shown later on should be used.

This diagram allows to:
a) determine the theoretical life expectancy of a selected driveline as a function of prevailing operating conditions, or
b) to determine the required joint size for a given life expectancy.

On machines or vehicles with changing operating conditions, at first, the individual life expectancy values (for each condition) must be determined from the diagram. Then the overall life expectancy L_hR can be calculated as follows:

In view of the multitude of applications, it is not possible to determine the suitability of a driveline by tests. Therefore, the selection and analysis of the required joint size is done by calculations. These are based on the computation of the dynamic load carrying capacity of full rotation needle – and roller bearings according to ISO recommendation R 281. The life expectancy diagrams shown in the catalogue are based on this recommendation and also on an equation formula especially suited for obtaining nominal life expectancy on universal joints. The thus obtained life expectancy lists the hours of operation that will be reached or exceeded by 90% of a larger number of equivalent universal joint bearings.

There are also methods of obtaining the modified life expectancy. In this case varying survival probabilities, material quality and operating conditions are taken into account. The present technical know how does not allow statements to be made about variations in life expectancy performance resulting from differences in steel quality (grain, hardness, impurities). For this reason, no guidelines have been set in the International Standards.

As shown in 2.3 by taking certain precautions, a constant output can be obtained on a universal driveline. The center part, however, still retains a non-uniform motion; it is subjected twice per revolution to an acceleration and deceleration. The resulting acceleration torque caused this way is a function of the mass moment of inertia of the driveline‘s center part as well as of rpm and deflection angle. When regarding smoothness of operation and wear, the product of rpm and deflection angle should not be too high. For use in general mechanical engineering, appropriate guide values can be taken from the diagram below, which is designed for universal drivelines having a standard tubing of up to 1500 mm length.

For vehicle drive trains, these guide values must often be exceeded. Here, at most, up to 1.5 times the diagram value can be permitted.

As shown in 4.1, the center part of the an-gled driveline, when transmitting torque, is stressed periodically in bending by additional moment MZ_II. This incites the center part to vibrate. If the frequency of this bending vibration approaches the natural frequency of the driveline, maximum stress in all components, buckling of the shaft and development of noise will result.

To avoid this, long and fast running drivelines must be checked for critical bending vibration speeds. The critical, first order bending vibration speed of a driveline employing tubing can be roughly calculated as follows:

The critical bending vibration speed of a driveline is, as can be seen from the critical rpm formula, a function of tubing diameters and length of center part. By going to larger tubing diameters, the critical speed of a driveline can be increased. However, the diameter increase must remain within defined limits since a certain relationship between tubing dimensions and joint size must be adhered to.

For determining the required tubing diameter when maximum operating speed n_max and center part length L are given.

The double joint shafts of series 0.400.5 and 0.500.3 are intended for use in powered steering axles only.

As shown in the sketch below, when steering is activated, the axle system is rotated around pin center D. The double joint deflects at its two joint pivot points A and B. Since shaft II is fixed axially, shaft I must move in the direction S. This causes unequal joint deflection angles ß₁ and ß₂, and therefore, also a non-uniform (fluctuating) output motion.

The fluctuation can be kept very small provided joint center C is offset toward the fixed side by the compensation value X. This way, at a certain deflection angle (= synchronous motion angle ß_x) completely uniform motion is obtained, i.e., the two joint deflection angles ß₁ and ß₂ are equal.

ß_x = 30° to 35° would be an appropriate synchronous motion angle to select.

The center offset X required for smooth output can be derived
from distance a and synchronous motion angle ß:

Calculated center offset value X for individual joint sizes:

Sliding motion e at deflection angle ß, and also as a function of distance a and uniform motion angle ß_x, can be calculated as follows:

Max. slide motion e for the individual joint sizes:

Max. possible torque should be used for determining the required joint size. This could be the input torque, calculated from prime mover output, gear ratio and power distribution, or also the tire slippage torque, derived from allowable axle loading, static tire radius and coefficient of friction. 

Double joint shafts, when not centered, must have a bearing support at both shaft halves right next to the joint with one shaft half fixed axially and the other floating axially. 

Under torque, different force conditions exist at the joint spider pins and center piece with the double joint in an angled position than in a straight position. The reason for this is that the torque to be transmitted is not distributed evenly over the joint spider pins any longer. Also, as mentioned in Chapter 4, an additional moment occurs. This additional moment must be combined with the torque to be transmitted.

This resulting moment leads to higher compression loads and to a larger bending stress within the joint spider pins. The diagram below allows to take these factors into account. It shows the percentage the maximum allowable torque must be reduced in relation to the deflection angle.

Torque calculation for needle bearing equipped precision cardan shafts, pin and block cardan joints, ball and socket cardan joints, single.

The values M_max listed in the diagram represent limit values that may not be exceeded. They are admissible to the full extent only at small rotation speed and minor angle of deflection respectively during intermittent operation.

The transmissible torque varies depending on the size of the angle of deflection.

Recommendations for maintenance

An adequate lubrication shall be ensured for universal and ball-and-socket joints in permanent operation. Where drip oiling is not feasible, the joints have to be once daily lubricated (for lubricating points see arrow). Joints may also be enveloped in bellows; such bellows for these tow joint types may be ordered from us.

For utilization of cardan shafts under extreme climatical conditions (high and low temperatures) consult us first.