5.3 Connection with force (Transmission by friction)
5.3.3 Interference fit (press and shrink fits)
In a press or shrink fit the pressure between shaft and hub is caused by interference. The pressure increases the diameter of the hole in the hub and reduces the diameter of the shaft.
Chapter 1 describes the tolerances on shaft and hub that result in specific fits (intervals). Figure 5.9 left shows an axially loaded shrink fit. The right side picture shows a normal torsionally loaded shaft-hub connection. In the following we will describe the stresses and strains that appear due to the fit.
Side 105
[billedtekst start]Figure 5.9: A friction based connection. (W = friction force).[billedtekst slut]
Because of symmetry it is practical to use cylindrical coordinates (r, θ, z), where z is in the axial direction. Assuming axial symmetry, i.e., symmetric with respect to the z-axis the force equilibrium becomes independent of θ.
[billedtekst start]Figure 5.10: Free body diagram of infinitesimal piece of shaft or hub in cylindrical coordinates. Assuming rotational symmetry.[billedtekst slut]
The cut-out in Figure 5.10 shows the stresses under the axial symmetry assumption, which also allow for radial body force fr. Force equilibrium in the radial direction gives
removing higher order terms we have
Because of symmetric with respect to the z-axis the corresponding strains are (can be found in e.g. [6])
where vr is the displacement in the radial direction.
On the sides of the hub we know that there is plane stress and in the center of the hub we must have plane strain due to symmetry. In the following it is assumed that the thickness of the hub is such that we may assume plane stress throughout the hub, i.e.,
σZZ = 0 (5.14)
The constitutive equations (see [6]) under the assumption of plane stress are
where E is the modulus of elasticity and v is Poisson’s ratio. Using (5.11) and (5.12) in (5.19) and (5.20) yields the differential equations
Using (5.22) and (5.23) in (5.10) yields (after some manipulations)
The differential equation (5.24) is Euler’s differential equation. If we assume that the radial body force is given by
fr = ρrw2 (5.25)
where ρ is the density and ω is angular speed, then the solution to (5.24) is
where c1 and c2 are constants that are to be determined from the boundary conditions. Using (5.26) we may express the stress and strain as
Side 107
where the new constants c3 and c4 are related to c1 and c2 through
Expressing the radial displacement in c3 and c4 gives
We now have all the needed equations ready to express the relation between the interference and the corresponding pressure in an interference or press fit. First we make the assumption that fr = 0 and let the shaft be a tube, as shown in Figure 5.11. In the figure the inner radius ra is shown (in case of a solid shaft we use ra = 0), rb, is the outer radius of the hub and rf is the fit radius.
[billedtekst start]Figure 5.11: Definition of inner, outer and fit radii.[billedtekst slut]
Outer part (hub) First we look at the hub. The boundary condition is that the pressure is zero at the outer radius of the hub and at the fit radius we have the fit pressure pf. This gives
that yields
The radial deformation at the fit radius of the hub is
By inserting the values of c3 and c4
Inner part (shaft) The boundary condition is that the pressure is zero at the inner radius of the shaft and at the fit radius we have the fit pressure pf. This gives
that yields
The radial deformation at the fit radius of the shaft is
By inserting the values of c3 and c4
Side 109
[billedtekst start]Figure 5.12: Cut through shaft and hub before assembly.[billedtekst slut]
In Figure 5.12 we show a cut through the shaft and the hub before assembly.
The radius before assembly of the inner radius of the hub is rh and the outer radius of the shaft before assembly is rs. After assembly it is known that both of these radii must be displaced to the fit radius rf, this yields
rh + vrh = rf rs + vrs = rf⇒
rs – rh = vrh – vrs (5.48)
The radial and diametral fit is defined as
δr = rs–rh = vrh – vrs (5.49)
δd = 2δr (5.50)
the diametral fit corresponds to the fit specified in Chapter 1 and can be given as
The radial volume force is neglected in the calculations that result in (5.51). The reason for including fr is the possibility to include radial forces due to the rotational speed. In this case we can use the radial force as defined in (5.25). In many cases with low angular speed the influence from this term can be neglected. For completeness the diametral fit including the influence from the rotational speed is given. The calculations are identical to the ones that lead to (5.51).
it is noted that in the case of w = 0 then (5.52) becomes identical to (5.51). It should also be noted that the rotational speed will reduce the pressure in the fit, i.e., if a given fit pressure is needed, then we must for w ≠ 0 specify a higher fit δd to achieve this.
Assembly force and transmitted torque
For a press fit the assembly force, Fa, depends on the thickness of the outer member, the difference in diameters of the shaft and hub and of the coefficient of friction. The maximum shear stress is
where às is the static coefficient of friction. The maximal torque that can be transmitted is
where l is the length of the contact zone between the shaft and hub.
Shrink fits
To assemble shaft and hub with a shrink fit a temperature difference between shaft and hub must be established. The easiest way of doing this is by heating the hub. However, it is important to ensure that a temperature increase of the hub will not harm its heat treatment.
Gears are especially sensitive to this.
Assuming that there is a linear relationship between thermal strain and temperature, the following expression applies
Where α is the thermal expansion coefficient. The temperature difference ∆tm may of course be established also by cooling the shaft or by a combination of heating the hub and cooling the shaft.
Applying the temperature difference calculated from (5.55) will exactly allow for assembly. A small decrease in this temperature difference (e.g. by the two parts touching each other) will however make assembly impossible. It is therefore typical to add a handling deflection to the fit. The size of this handling deflection is typical
Smoothing out of surfaces
Especially when assembling shaft and hub by pressing it is important to take the associated smoothing out of surfaces into account. Due to the sliding of the surfaces on each other, the surfaces smoothen out and the interference is reduced. The reduction is based on experience and for press fits it amounts to
where Ra is the arithmetic mean roughness value (DIN 4768, DIN 4662, ISO 4287/1). For shrink fits the reduction is smaller and can be estimated to