Product Design

Clamping Connection

• Frequently used to connect a shaft with a hub or to fix other parts together.

• The hole diameter is slightly bigger than the shaft diameter, creating a clearance fit (H7 / g6).

• Requires deformation of the hub and contact pressure to function.

• Not suitable for power transmission.

  • Uses load transmission by friction.

  • Needs an external force to generate clamping force.

  • Can be a one-piece (slotted hub) or a two-piece design (divided hub).

  • Designed for low to medium torsional moments.

  • Recommended for static or slightly swelling loads.

    • For higher loads, additional keys or wedges are used to prevent relative movement.

    • Calculation of combined connections is difficult due to unknown load distribution.

• Load transfer occurs solely through friction between components.

• Friction coefficients:

  • relevant coefficient of friction (CoF) for sliding is used.

  • Selection of kinematic CoF for sliding increases reliability.

Clamping Connection Types

• Clamping connection with divided hub.

• Clamping connection with slotted hub.

• Application defines the suitable type.

• Selection influenced by assembly conditions.

Clamping Connection with Divided Hub

• Preferred when axial positioning is impossible.

• Enables lateral fixing due to the two-piece design.

• Allows independent positioning from other elements on the shaft.

• Tensioning screws near the clamp center minimizes bending loads on the screws.

• Increased lever arms from screws positioned far from the center result in larger bending loads and hub deformations.

• The hub should be stiff enough to avoid significant deformation.

• Bolt and nut joints are preferred to generate clamping forces.

• Avoid fit bolts because they participate in the load transfer.

Calculation of Divided Clamping Connection

• An interference-fit between inner and outer parts is recommended.

• Small tolerances result in good pressure distribution (almost constant pressure).

• Large diameter differences result in small contact areas and high contact pressure.

• Pressure distribution is crucial for calculation.

• Factor K describes the influence by increasing the minimum pressure pc.

  • $FR_t$: tangential friction force in [N]

  • $A_c$: contact area in [mm]

  • $\mu$: friction coefficient

  • $K_A$: factor to consider the impact loads coming from the application

  • $M_T$: torsional moment in [Nmm]

  • $S<em>H$: safety factor (recommended $S</em>H = 1.5 2$

  • $D_c$: diameter of the contact area in [mm]

  • $l_c$: length of the contact area in [mm]

  • $K$: correction factor to consider a non-constant pressure distribution over the contact area

  • $p_{c,per}$: permitted contact pressure in [N/mm²]

• After determining $p_c$, define the screw size.

• Required clamping forces $F_{cl}$ generated by screws arranged on both sides of the clamping connection.

  • $A_{proj}$: projected area in [mm²]

  • $n$: number of screws

  • $F<em>S \geq F</em>{cl}$:

• Pre-tensioning force of the screw will be larger than the calculated value (for safety reasons).

Clamping connection with slotted hub

• $K = 1$ for constant pressure (ideal distribution)

• $K = \frac{\pi^2}{8} = 1.23$ for cosines-type pressure distribution

• $K = \frac{\pi}{2} = 1.57$ for line contact only

• Casted iron/steel: $p_{c, per} = 30…50 N/mm^2$

• Steel/Steel: $p_{c, per} = 50…90 N/mm^2$

Clamping Connection with Slotted Hub

• Requires axial assembly due to its one-piece design.

• Lateral assembly is not possible.

• A tight clearance fit is preferred to position the outer part properly.

• Screws generate contact forces and loads for hub deformation.

• The pivot point $D$ is difficult to define and dependent on the shape/stiffness of the component.

• A more detailed calculation such as an FEM-analysis is required to define $D$.

• The loaded area between inner and outer part is a line only

• The calculation of the contact pressure is not required here.

  • Line contact (poor contact pattern) results in smaller transferred torque loads.

Calculation of the connection with slotted hub

• Correlation between torque load $M<em>T$ and normal load $F</em>N$ for line contact

For the unfavorable case of a line contact, the correlation between torque load $M<em>T$ and normal load $F</em>N$ is

• The CoF can be taken from Table 10‑1: "Friction coefficients for dry contact" again.

  • $F_N$: total clamping force in [N]

  • $\mu$: friction coefficient

  • $K_A$: factor to consider the impact loads coming from the application

  • $M_T$: torsional moment in [Nmm]

  • $D_c$: diameter of the contact area [mm]

  • Required screw forces are dependent on the distances $l<em>1$ and $l</em>2$.

• These forces can be calculated based on the equilibrium of moment for the virtual pivot point D

  • For n screws the required pretension for the screw is

  • $F_{cl}$: pretension force for one screw in [N]

  • $S_H$: safety factor

  • $l_1$: distance between centre of the connection and virtual joint D in [mm]

  • $l_2$: distance between centre line of the screws and virtual joint D in [mm]

  • $n$: number of screws

• Hertzian pressure theory enables the calculation of the contact pressure.

• The following chapter gives an overview over different kinds of shaft-hub-connections. At first, the connections using a load transfer by friction are highlighted. Additionally, a few examples for form-fit connections are introduced.

Cone Clamping Elements

• Consist of conical ring-shaped elements compressed by an external load in the axial direction.

• The conical shape leads to a radial load component and a friction force in the contact area.
  During clamping a relative movement of components happens. The displacement of one of the connected elements (shaft or hub) is unwished. To overcome the displacement and to simplify the assembly of the clamping element, a self-aligning clamping element helps

• The element is used for the transfer of:

  • Static loads

  • Dynamic loads

  • Impact loads

• The function principle can be compared to conical interference fit assembly with additional cylindrical contact areas for shaft and hub. The calculation is nearly similar (see unit 11.4).

• The purpose of the element is to transfer a torque load.

• Nevertheless, other loads such as axial loads or bending loads will have an influence on the operation behaviour of the clamp:

  • In addition to the torque load the axial loads have to be transferred by friction in the cylindrical contact areas, too

  • Bending loads lead to a non-uniform pressure distribution over the cylindrical contact area to the shaft. Here, material properties have to be compared to the maximum contact pressure!
    When choosing a clamping element from a suppliers/manufacturers catalogue, the following criteria are to be taken into consideration:

  • Torque load, bending moment, axial load:

  • The maximum loads have to be calculated for the selection of the clamping element.

1. Outer/inner or intermediate element

2. Self-centering required

3. Self-locking (depending on the cone angle) required

4. Notch effect (depending on cone angle)

5. Runout performance

6. Assembly and disassembly

   • The costs of the joint, size and influence on joint parts

• The selection of the clamping element should be done with assistance of the supplier/manufacturer

• For additional outer loads (e.g., axial load), a resulting torsional moment has to be calculated

• Torsional moment to be transferred in [Nmm] for dynamic loading $T = K<em>A * T</em>{nom}$ for static loading $T = T_{max}$

• Axial load to be transferred in [N] for dynamic loading $F<em>a = K</em>A * F<em>{a, nom}$ for static loading $F</em>a = F_{a, max}$

• Shaft diameter in [mm]

• $T<em>{per}, F</em>{per}$: permitted load for the clamping element based on suppliers information on the contact pressure in [Nmm]/[N]

  Clamping Elements and Securing Elements

• In case of more than one pair of rings every additional pair takes less than 100% of the load for a single pair (see parallel key connections also)
  In general, two different types of clamping elements are known:

1. Clamping elements with external locking
   In order to generate a contact pressure shaft or hub are loaded by an external element such as a screw to provide a movement of the conical rings relative to each other.

2. Clamping elements with internal locking
   The external load applied does not stress shaft and hub. Only the contact pressure generated during tensioning of the clamping connection will be taken by shaft and hub.

Cone Clamping elements with external locking connection

Require at least one of the joined parts, shaft or hub, as a “constraint”

• For external locking, the clamping rings are available as standardized parts. All other parts (shaft, hub, tensioning element) are tailor made for the application.

The rings are available as full or slotted rings.

Cone clamping elements with internal locking connection

• Clamping elements with internal locking do not need one of the joined parts to generate the required clamping load. The forces circulate in the clamping element only.

• Shaft and hub are independent from the tensioning, only inner and outer diameter of these components have to be selected based on the clamping element geometry. Hence, the clamping connection itself is standardized completely for different applications.

• For internal locking, both self-alignment and self-centering is possible. These properties are important for assembly and operation.

• For high rotation speeds, an unbalanced connection would lead to large loads for all parts. To avoid it, a precise assembly of shaft to hub is required.

• The hub is positioned on the shaft. Taper locks is a combination of a screw and a clamping element with conical rings. Here, a conical helix substitutes the rings. The advantage of such a connection is the easy assembly and the flexibility in the positioning.

Star discs

• The shape of the basic disc can be compared to Belleville springs, it is not flat. Under external, axial load (blue arrows in Figure 11‑15) the discs are forced into an almost flat position. As a result, the inner diameter gets smaller while the outer diameter increases.

• Combination of star discs increases the transferred torsional moment. Again, the total torque is not equal to the torque of a single disc multiplied with the number of discs (see 11.1). The calculation follows the suppliers/manufacturers information.

Wedges and keyed joints

Wedges and tapered key are combined joint elements.

• An additional pretension resulting from an axial external load or an interference between shaft and hub is crucial for the load transfer overlaid.

Calculation of wedges and tapered keys

• The transferred torsional moments depend on the friction coefficient µ and the press in forces Fass.

• Equilibrium in horizontal direction:
  Friction Angle in degrees with $FR = \mu * F_V$
  Normal load on inner part in newtons
  Press-in force in newtons
  Angle of the wedge in degrees

• Based on this the transferred torsional moment is

• Transferred Torsional Moment in [Nmm], Friction Coefficient for Sliding, Diameter of the inner part in mm, Height of the recess for the key in mm The pressure in the joint is

• Loaded length of the joint in mm, Width of the joint in mm The permitted stresses for the parts result from their yield strength / breaking strength.

• For ductile materials like steel or casted steel yielding defines the limits. As soon as a brittle material is loaded, the breaking of the unit is limiting the permitted stresses.

• If the external load applied is larger than the friction force generated by the assembly force? Then, the wedge starts to slide until it is in contact to the groove sidewalls of shaft and hub. Now, an additional form fit exists. Certainly, this condition is not the one aimed when designing the connection. Sliding has to be avoided because it results in wear. Parallel key connection

• Contrary to wedges and tapers, the keys and keyways use form fit only. To avoid clearance in between the components, a fit for the key as well as for the groove of shaft and hub is necessary. A pretension between shaft and hub is not provided. The parts are standardized according to DIN 6885.

Dimensions for keys

• For a parallel key connection the only element that is standardized is the key. It does not make sense to define the dimensions of shaft and hub because the loads as well as the function and shape depend on the requirements. The dimensions for the key and the grooves in shaft and hub are listed in [DIN 6885]

Calculation of keys and keyways according DIN 6892

In the mentioned standard DIN 6892 a calculation routine is proposed that will be explained in detail here. The calculation has to be done for the weaker part. If both materials are the same, the shaft has to be checked first.

Design

A good design of the parallel key connection considers the following recommendations:●the stiffer the hub the better the contact pressure distribution reduces stress and more plane contact areacritical

Cylindrical Interference Fit Assembly

• All discussed connections require tensioning elements to transfer a load.

• If the contact pressure is generated by a deformation of the connected elements?
  Assembly For cylindrical interference fit assemblies the joining area is cylindrical. To achieve a friction force between the parts an interference fit is required. In other words: the outer diameter of the shaft is slightly bigger than the inner diameter of the hub. To enable a load transfer, the two elements have to be connected.

Forces for the assembly to overcome the friction forces are as follows:
Contact Pressure in [N/mm²], Contact Area in [mm²], Coefficient of Friction during assembly, see clamping connections
For lateral assembly of the interference fit cooling of the inner part and/or heating of the outer part changes the size of the components until the parts fit to each other. External forces are normally not required for this assembly method.
With Temperature in [K], Ambient Temperature in [K], Temeprature of inner part in [K], Required Temperature for Outer Part in [K], Maximum Interference between inner and outer part in [mm], Required Clearance for assembly
*For axial movement a chamfer is required to insert the part without damaging the surface

Design guidelines

*Consider notch effects and high wear and tear in edge contact zones

Calculation of cylindrical interference fit assemblies

*Valid for non terminal sections
*It is nothing more that ring shape with elastic stresses
Formula of transfered torsional Moment ( in Nmm) Dc ( joining diameter in mm)* lc(joining length in mm )* surface ( friction Coefficient or sliding) * Contact pressure in N/mm^2/
then for axial load

The smoothing

*The Smoothing depends on the diameter
*This strain is similar to the fraction of the deformation u and location
*Inner Parts
*Outer parts the equation describe what happens in general between two parts elastic moduli E as well as the Poission ratios ν are independent from each other.Under consideration of similar material properties for inner and outer partEI = EA = EvI = vA =vfor inner and outer part iswith

Conical interference fit assembly

*Conical interference fit assemblies
*Used to connect hubs with wheel and clutches with shaft extensions
*Assembly simple
donical geometry
*Cone ratio

Calculation conical assemblies

*Determine transferable torsional moment
*Cylindrical with mean contact area
Minimum and max interference