STAR Hilman Rollers, an Estimate of the Load Stress
The following analysis first uses Mathcad's E-book, Roark's Formulas for Stress and Strain, and then a second look using the Mathcad Code of E. Deeg for Hertzian Stresses [D1].
Preliminary!!!!!
Hilman roller dimensions and numbers need to be checked
The preliminary result of both methods gives a maximum stress at the roller contact of ~0.9 GPa. This exceeds the yield stress of steel which is 0.25-0.6 GPa.
Weight of STAR based on guessed dimensions of the magnet steel
length of magnet iron bars
radial thickness of the magnet iron bars
iron bar fill factor around the circumference of the magnet cylinder
inner radius of the magnet cylinder
outer radius of the magnet cylinder
steel volume of the magnet cylinder
gravitational force of the magnet cylinder steel
OK now have better numbers for full STAR load
total mass less the mass of the end caps and end cap cal
number of load bearing rollers in each of the 4 corners
load on a roller
This should now include TPC, coils, barrel calorimeter, magnet steel and cradles. Does not incude end cap
Method 1, use Roark formula to calculate stress in the contact region. Roller dimensions guessed to be
3.5 inches diameter and 6 inches long when looking at
Table 33 Formulas for stress and strain due to pressure on or between elastic bodies
Case 2a Cylinder on a flat plate
Cylinder of large length compared to diameter
Cylinder on a flat plate
Provides an explanation of Table 33 and the notation used.
Enter dimensions, properties and loading
Diameter of cylinder:
Poisson's ratio for the cylinder:
Modulus of elasticity for the cylinder:
Poisson's ratio for the plate:
Modulus of elasticity for the plate:
Length of the cylinder:
Loading:
Constants
Load per unit length:
General formulas for width b and scmax
Width of rectangular contact area:
Stresses:
This occurs at a depth of
below the surface of the plane.
References
Ref. 5. Foppl, A.: "Technische Mechanik," 4th ed., vol. 5, p.350.
Ref. 6. Timoshenko, S. and J. N. Goodier: "Theory of Elasticity," 2nd ed., Engineering Societies Monograph, McGraw-Hill Book Company, 1951.
Ref. 44. Lundberg, Gustaf: Cylinder Compressed Between Two Plane Bodies, reprint courtesy of SKF Industries Inc., 1949.
Ref. 56. Sague, J. E.: The Special Way Big Bearings Can Fail, Mach. Des.,September 1978.
Method 2
Use Mathcad Code of E. Deeg for Hertzian Stresses [D1]. The code is shown along with some of Deeg's explanatory text plus my comments associated with this particular geometry.
redefine p to be consistent with the Deeg code.
Hertz's theory works with 2 ellipsoids. We represent the Hilman roller with ellipsoid 1 and the steel plate with ellipsoid 2.
Setting up the parameters for these two ellipsoids:
Young's modules for steel
Poisson's ratio for steel
Within Hertz's theory, the two elastic constants, Young's modulus and Poisson's ratio, are combined into one material constant q. For the present example, this means introducing the Hertz material coefficient
Assign
Since both surfaces are the same,
Angle between equivalent principal planes shall be:
To ~duplicate Hilman roller case make the 2nd ellipsoid the plane with both radii very large compared to other dimensions. Represent the Hilman roller with ellipsoid 1. Set r11 equal to the roller radius and set r12 to a large number such that the contact surface is on the order of the length of the roller which we assume to be ~6 inches. HW
Extreme radii of curvature:
The auxiliary angle W, a quantity introduced by Hertz for mathematical reasons, is then given by
The maximum stress and the corresponding deformation of the two bodies in the contact zone are given by
and
These equations require input of the force acting on the surfaces, but also contain three unknown quantities, the two semiaxes a and b of the contact zone and the function U(k), where k = b/a. Hertz derived the following equations for the two semiaxes:
and
where the factors f and g are
The functions U(k), V(k) and W(k) are the fundamental expressions in Hertz's theory. U(k) is a complete elliptic integral of the first kind. V(k) and W(k) can be reduced to algebraic functions of such an integral and its first derivative after the modulus. The latter statement is given by Hertz without proof. An attempt to reconstruct Hertz's derivation is made in [a].
To simplify application of his theory, Hertz calculated f(W) and g(W), and included them in his original papers. The functions U(k), V(k) and W(k) are
These expressions contain the ratio k as a variable. Because k = b/a = g/f, a transcendental equation for k can be found:
or its equivalent
This can be solved with Mathcad's root operator. Three seed values, each covering a specific range of the auxiliary angle W, were preselected. The appropriate one is chosen through Mathcad's conditional if function. The three seed values are
The program searches as follows:
and enters the value selected for k into
Because W has already been calculated, the values for f and g and, with them, the two semiaxes a and b of the contact ellipse are now determined.
Quantities depending on the semiaxes of the contact ellipse, a and b, are determined as well:
Area of the contact ellipse:
Hertzian stress:
Which is greater than the steel yield stress which is:
Deformation of the contacting surfaces:
Total compression:
stored compressional energy
If we wish to trace the perimeter of the contact zone, it is convenient to parametrize the contact ellipse over a dummy variable s:
Figure 1: Perimeter of the Hertzian contact zone
r12 was varied to get a contact length on the order of 8 inches which was assumed to be around the length of the Hilman roller.
[1D] Computing Hertzian Stresses in Fiberoptic Connector Modeling
by Emil W. Deeg, Dr. rer. nat.
Consultant, Lemoyne, PA
The original Deeg article/code in html is at:
http://www.mathcad.com/library/LibraryContent/MathML/fiberstress.htm
and the original Mathcad version is at:
http://www.mathcad.com/library/LibraryContent/fiberstress.mcd
References from the above article/code of Deeg
[a] Deeg, E. W. AMP J. Technol. 1992, 2, 14 - 24.
Deeg, E. W.; Bolhaar, T. AMP J. Technol. 1992, 2, 29 - 41.
Hertz, H. Journal f. d. reine und angewandte Mathematik 1881, 92, 156 - 171.
Hertz, H. Verhdlg. Ver. Bef. d. Gewerbefl. , Berlin, 1882, November.
Heerwagen, F. Zeitschr. Ver. Dtsch. Ingen. 1901, 45, 1701 - 1705.
Whittemore, H. L.; Petrenko, S. N. Friction and carrying capacity of ball and roller bearings; Technol. Paper Natl. Bureau of Standards No. 201. U. S. Government Printing Office: Washington, DC, 1921.
