# Force Analysis: Cam: Nominal Contact-Stress

 IMPORTANT: 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.

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

As a cam rolls under a cam-follower roller (or a cam-follower rolls over a cam), there is a contact force, and the cam and the cam-follower become flattened, as does a car tire against a road surface. There is a so-called Hertzian Contact Stress, at, and beneath, the contact surface [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].
].

The Hertzian Contact Stress calculations give the contact-stress that is induced in flawless steels [no inclusions], and error-free contact such that the contact-load is constant across the cam and cam-follower  [See also Contact Assumptions, below]. The analysis also assumes that nothing is moving! MechDesigner is perfectly capable of calculating the Hertzian Contact Stress.

The Hertzian Contact Stress is sometimes called the 'Nominal Contact-Stress', especially with gearing calculations.

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

 Distribution of Hertz Contact Stress - Cyclindrical Line Contact

The Radius-of-Curvature of the Cam changes as the cam rotates, while the Radius of the Cam-Follower is constant.

When a Cam and cylindrical Cam-Follower Roller lightly 'touch' each other, their contact is a 'line'. When we increase the contact force, the cam and follower deflect [squash together, like a car tire against a road surface] and the contact line becomes a small rectangular across the width of 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.

The Hertzian Contact Stress beneath the Cam and Cam-Follower surface are a function of material properties and their radii-of-curvature.

The parameters that are fixed for each body are:

 L - Length of the contact, which is the minimum of the roller's width and cam's width[mm] - when the contact is a 'cylinder on a plate', or a 'cylinder on a cylinder'. E1 , E2 - Young's Modulus of the Cam and the Cam-Follower [N/mm2] ν1 ,ν2 - Poison's Ratio of the Cam and Cam-Follower [-] R2 - Cam-Follower's Radius [mm]

The parameters that are continually changing as the cam rotates.

 2b - Width of the contact [mm] F - Contact Force between the Cam and the Cam-Follower [N] R1 - Cam's Radius-of-Curvature [mm]

To calculate the Hertzian Contact Stress, at each point around the cam, it is useful to establish two 'equivalent' numbers for the material and geometry properties.

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

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:

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

Mean Hertzian Stress:

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

Maximum Hertzian Stress

pmax at the middle of the contact line

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

Substituting for 'b', and re-arranging, gives the Maximum Hertzian Stress, pmax, due to the Contact-Force, F, as:

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

Summary:

 • Hertz Contact Stress is proportional to the square root of the Contact Force. Thus,  2 × Contact Force : 1.41 × Contact Stress .

Note: Pmax is a compressive stress. It is often given as P0 in academic papers.

#### 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 compressive stresses directly below the centre-line of the contact.

Note: The horizontal axis does NOT imply a dimension. It is only stress.

Horizontal-Axis :  [Scale: 1.0 = Pmax ]

 o σz  : Stress in the vertical direction
 o σx  : Stress in the horizontal direction, in direction of rolling
 o σy  : Stress in the horizontal direction, in direction that is across the width of contact
 o τ45 :  Principal or Maximum Shear Stress that results from the maximum difference between the principal stresses.

Vertical-axis : [Scale: 1.0b = half contact half-width, b]

 o the distance below the surface is a linear scale of the contact half-width, b, with a range from z=0 to z=3b.

Maximum Value of Shear Stress

Shear Stress, τ45, is proportional to the Maximum Contact Stress, or Hertz Contact Stress. For line contact [cylinder on cylinder], the relationship is:

${\tau }_{\mathrm{max}}=0.3·{p}_{\mathrm{max}}$

Depth of Maximum Shear Stress [Line Contact]

$0.78·b$ below the centre-line of the rectangular contact 'patch'.

The image to the left shows that the stresses, σz, σx and σy are ALL compressive, which reduce below the surface. Because the volume of contact-stress is local to the contact, the surrounding material (the hinterland) does not want to move away to accommodate the expansion because the hinterland is not 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. At 0.78b below the surface, the Shear Stress is a maximum value [Line Contact]. The Principal Stresses, σx, σy and σz, at the depth of the Maximum Shear Stress, at a depth of 0.78b , relative to the maximum Contact Stress at the surface, are: σx(@o.78b) = -0.32Pmax σy(@o.78b) = -0.15Pmax σz(@0.78b) = -0.75Pmax These three values determine the Mohr's Circle diameters and their offset from zero stress. - see the image to the left. Maximum Shear Stress, τmax From the scale of the Mohr's circle, we can see that the Maximum Shear Stress[vertical axis] is half of the maximum difference to the Principal Stresses. [σz - σy  ]/2=  [-075-(-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 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'

Shear Stress and Traction with Rolling Friction - image above

Many elements in contact both slide and roll. 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.

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

The Hertzian Contact Stress exists beneath the two cylinders. The 'Maximum Shear Stress' is below the centre-line. There is also a 'Maximum Orthogonal Shear Stress' to the left and right of the centre-line.

As the cylinders roll, the contact point moves from the left to right, The Orthogonal Shear Stress at each point in the rollers changes its 'sign' from 'positive' to 'negative', as shown in the plot of Orthogonal Shear Stress.

The absolute value of Orthogonal Shear Stress is not as large as the Maximum Shear Stress. However, Orthogonal Shear Stress has a range [max – min] that is larger than the Maximum Shear Stress. 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 is a range of 0.512*Pmax.

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