# Force Analysis: Cam: Contact-Stress

 Before you can analyse Contact-Force and Contact-Stress, you need to reconfigure the model so that it transfers the forces correctly between the cam and the follower.

Cam-followers are manufactured from a limited number of steels [and some durable plastics], with rigid quality control and testing. Industrial Cams are manufactured by machine builders with different design and manufacturing quality standards and heat-treatment procedures. Thus, even though a cam is an easier machine element to analyse, it is left to the machine designer to choose the correct material, heat treatment and lubrication method to give the cam an operational-life as required by their customers. Heat-treatment is often sub-contracted, and its quality control is not specified.

## Nominal Contact Stress - Cylindrical Cam-Follower and Line Contact

As a cam rolls under a loaded cam-follower roller (or the cam-follower rolls over a cam) the cam and the cam-follower become stressed. The strains associated with the stress mean the cam and cam-follower become slightly flattened, The stresses at and below the contact-surface are quite complex. They are nothing like the stresses that you can calculate from Tension, Bending or Torsion.

Hertz was the first to calculate the stress 'field' and it is now frequently termed Hertzian Contact Stress [Reference Hertz (1881) has done the hard work to derive the equations we need to calculate the stress when there is contact between two bodies that are conformal or non-conformal.
[H. R. Hertz, "Über die Berührung fester elastischer Körper (On Contact Between Elastic Bodies)" Journal für die reine und angewandte Mathematik 92, 1881 pp. 156-171].
],
Hertzian Stress, or Contact-Stress.

MechDesigner calculates the Hertzian Contact Stress. and the Maximum Shear Stress which, as we will see, is quite important. It is convenient to consider to types of Hertzian Contact:

Line Contact : between two parallel cylinders, or between a cylinder and plate. This is equivalent to a cylindrical cam-follower and a 2D-Cam, for example.

Elliptical Contact : between bodies with more complex shapes. Their shapes are usually defined by their radii-of-curvature in orthogonal directions. When the principal radii are their direction are different, the nthe calculations are much more difficult. However, we can give very good approximations when the bodies are similar to a barrel cam-follower rolling over a cam-surface.

### Hertzian Stress at the Cam Surface [Line Contact]

 Distribution of Hertz Contact Stress - Cyclindrical Line Contact

With Line Contact, there are only two radii at the contact that we must know:

 • the Cam must be prismatic - an extruded shape -  and the radius-of-curvature may change continuously at the contact as the cam rotates..
 • the Cam-Follower must be prismatic, and cylindrical.

When the Cam and Cam-Follower lightly 'touch' each other, the contact is a 'line'. As we increase the contact force, the cam and follower deflect and the contact becomes a narrow rectangle across the cam and cam-follower.

 Contact Assumptions
 1 There is Plane Deformation [εx=0]
 2 The 2D-Cam and Cam-Follower are 'Isotropic' - the material properties are the same in all directions.
 3 The 2D-Cam and Cam-Follower are 'Linearly Elastic'.
 4 The dimensions of the contact area are very small in comparison with the radii of the cam and cam-follower.
 5 The shape of the deformed surface is a rectangle.
 6 At the contact interface, the bodies are approximately flat.
 7 The bodies are frictionless; only normal stresses arising during contact are considered. Relative displacements in the X and Y directions are neglected.
 8 The surfaces are free from surface debris and not lubricated.

Parameters to calculate Hertzian Contact-Stress for Line-Contact.

 L : Length of the contact. Typically, the minimum width of the Cam-Follower Roller and Cam [mm] E1 , E2 : Young's Moduli of the Cam and the Cam-Follower [N/mm2] ν1 ,ν2 : Poisson Ratios of the Cam and Cam-Follower [-] R1, R2 : Cam's Radius-of-Curvature, Cam-Follower Radius [mm] F : Contact Force between the Cam and Cam-Follower [N]

$\frac{1}{{R}_{e}}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}$

Equivalent Elastic Modulus:

$\frac{1}{{E}_{e}}=\left[\frac{{1-{\vartheta }_{1}}^{2}}{{E}_{1}}\right]+\left[\frac{{1-{\vartheta }_{2}}^{2}}{{E}_{2}}\right]$

Half-width in direction of rolling:

$b=\sqrt{\frac{8·F·{R}_{e}}{\pi ·L·{E}_{e}}}$

Mean Hertzian Stress:

${p}_{mean}=\frac{F}{area}=\frac{F}{2·b·L}$

Maximum Hertzian Stress:

${p}_{\mathrm{max}}=\frac{4·{p}_{mean}}{\pi }=\frac{2·F}{\pi ·b·L}$

Substituting for 'b' gives:

${p}_{\mathrm{max}}=\sqrt{\frac{F·{E}_{e}}{2·\pi ·L·{R}_{e}}}$

Pmax is at the middle of the contact patch, where the contact of a very light-load is initially made.

#### Stress below the centre of the Line Contact axis [x=0]

 Contact Stresses for Line Contact Poisson's Ratio Shear Stress The image to the left shows principal stresses [all compressive] relative to the Maximum Hertzian Contact Stress, Pmax , directly below the centre-line of the contact. Horizontal-Axis : Scale: 1.0 = Pmax :  Vertical-axis : Scale: b = half contact half-width σz : Stress in the vertical direction σx  : Stress in the horizontal direction, in direction of rolling σy  : Stress in the horizontal direction, in direction that is across the width of contact Ԏ45 : Maximum Shear Stress that results from the maximum difference between the principal stresses. Maximum Value of Shear-Stress, Ԏmax [Line Contact] - or the Maximum difference between any two Principal Stresses [see Mohr's Circle for more information]. For Line Contact, the relationship is: ${\tau }_{\mathrm{max}}=0.3·{p}_{\mathrm{max}}$ Depth of Maximum Shear-Stress [Line Contact] There is no symbol for this constant $0.78·b$ The image to the left shows that the stresses, σz,  σx and σy are compressive. Because the contact-stress drops to very low values near to the contact, the surrounding material (the hinterland) does not move away to accommodate the expansion because the hinterland is hardly stressed. Because of symmetry of loading, it can be shown that the x, y and z, stresses are 'Principal Stresses'.

#### Mohr's Circle for stress state at 0.78b below the Cam's Surface

 Mohr's Circles for the Hertzian Contact Stress at a depth of 0.78b, which is the depth of the Maximum Shear Stress. Principal Shear Stresses, σx , σy , σz The image to the left shows the Mohr's Circles for the Principal Stresses, , σx , σy , and σz , relative to the Maximum Contact Stress, Pmax at the urface at a depth 0.78b below the surface. σx @o.78b ÷ Pmax   = –0.32 σy @o.78b ÷ Pmax  = –0.15 σz @0.78b ÷ Pmax = –0.75 Maximum Shear Stress, Ԏmax From the scale of the Mohr's Circle, we can see that the Maximum Shear Stress[vertical axis] is one half of the maximum difference between Principal Stresses: [σz – σy  ] / 2 =  [ –0.75 – (–0.15) ] / 2 = –0.6 / 2 = –0.3 ${\tau }_{\mathrm{max}}=0.3·{p}_{\mathrm{max}}$ In many cases, surface roughness, friction, lubrication, thermal effects, and residual stresses will result in conditions that invalidate the exact results from Hertzian analysis.  Consequently, the stresses computed according to Hertz’s analysis are often regarded as guidelines that are correlated with experimental failure tests to find allowable stress limits.

#### Schematic of Orthogonal Shear Stress due to of 'Free Rolling' and 'Traction Rolling'

Orthogonal Shear Stress moving through the contact zone - image to left.

The Hertzian Contact Stress contact-stresses lead to Octahedral shear-stress and Orthogonal shear-stresses under the surface at the contact. The image to the left schematically illustrates how a cylinder rolling over a flat body in the absence of friction develops subsurface stresses. The stress created just below the centre of the contact is the maximum shear stress (octahedral). We have calculated this in some detail using the analysis above. It occurs at 45° to the contact surface.

Orthogonal shear stresses which are oriented parallel and perpendicular to the contact surface are created in-front of and behind the point of contact. The leading orthogonal stress has the opposite sign of the trailing stress. The magnitudes of the orthogonal stresses are always lower than the magnitude of the octahedral shear stress. However, the range of the orthogonal stresses is higher than the octahedral stress and is thought to be a more potent contributor to the development of contact fatigue damage. The Orthogonal Shear Stress is believed to be significant with respect to 'sub-surface Fatigue Cracks'.

The Orthogonal Shear Stress is greatest at approximately:

 • 0.5b below the surface.
 • 0.875b to the left and right of the central axis.
 • Its absolute maximum is 0.256*Pmax, which gives it full range of 0.512*Pmax.

Shear Stress and Traction with Rolling Friction - image above

Many elements that are in contact will slide and roll as they pass over each other. For example, gear-teeth, and flat-faced cam-followers sliding against a cam.

There is always a coefficient of friction and thus a friction force. This results in tangential, normal and shear stresses that are superimposed on the stress caused by the normal Hertzian Contact Stress. These stresses are illustrated schematically in the image to the above.

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