Edited by Jessica Shapiro
Regular readers may remember “Rugged bearings hold the weight of heavy equipment,” which appeared in Machine Design’s Basics of Design Engineering: Bearings section in our July 10th issue. That article has generated quite a bit of discussion among our readers. Several have asked for guidance, while a few offered insights on applying the equations. One reader’s comment prompted the following exchange.
Hertzian or not?
Reader Gary LaFayette of Sauer-Danfoss, Ames, Iowa, recognized that the equation on page 74 for B, the contact half-width between a shaft and its bushing, describes Hertzian contact.
“A shaft and a bushing are not Hertzian in nature because the diameters of the shaft and bushing are nearly the same and the contact area is large. The fundamental assumption used to derive the Hertzian stress formula is that the contact patch is small relative to the size of the bodies,” he said.
“An easy way to recognize that the equation is useless for conformal contact is to set D1 equal to D2. B goes to infinity, and Pmax goes to zero. This is obviously not a reflection of reality. When D1 equals D2, there must be a nonzero contact pressure developed when load is applied.
“There are no closed-form solutions for calculating contact area or contact pressure for conforming bodies. Many bushing manufacturers use a ‘projected’ area method to calculate contact pressure on a bushing. Of course, this doesn’t take into account how the clearance between the shaft and bushing affects contact pressure. To really understand contact pressure between conforming bodies, finite-element methods must be used,” LaFayette concluded.
Dr. David Krus and Mike Gedeon of Brush Wellman, the article’s authors, said the equation for contact area was correctly applied in this article, “provided there is some diametrical clearance between shaft and sleeve (D2> D1).”
“As Mr. LaFayette points out, there is an absurd result at D1 = D2, but that point is the onset of interference fitting, in which the bodies exert stresses on each other without an applied load. conformal contact is a regime not governed by hertz’s theory.
“To take the absurdity further, make D1 > D2, a true interference fit. Then the contact area half-width is the square root of a negative number and the equation is certainly misapplied.
“In any situation where D2 > D1 > 0, the article’s equations are often used to give a good approximation of contact stress and area. For example, in Fundamentals of Machine Component Design (Juvinall, Robert C., Marshek, Curt M., 2nd ed., John Wiley & Sons, 1991.), the authors apply the Hertz equations for B and Pmax to a bearing in Sample Problem 9.2 (p. 328). The example is a plain bearing ball and socket joint where the socket diameter is 10.1 mm and the ball diameter is 10.0 mm. The 0.1 mm clearance is representative of heavy-equipment design on small-diameter plain bearings. The authors go on to note that the Hertz equations do not apply when D1 = D2 but, after a sign change, can be applied to two balls in contact.
“Although we agree that finite-element analysis is a powerful tool for precise stress calculations, we suggest that FEA users first come to appreciate the underlying physical principles and empirical relationships.
“Each FEA solution is numerical and specific to the input variables including geometry, material properties, and boundary conditions. FEA can indicate whether or not a given set of variables will yield a desired result, but how does one choose the variables for the first FEA iteration? Without empirical relationships like Hertzian theory to suggest a starting point, engineers are left with a nearly infinite range of equally valid possibilities.
“This, of course, assumes the engineer or mechanic making the choices has access to three luxuries: enough information to define the boundary conditions accurately, enough time to do the modeling, and access to FEA software and experts. Even with complex operating conditions and tight budgets, the Hertz stress and contact area equations can be used to quickly select materials that have a good chance of succeeding.”
Too close for comfort
Mr. LaFayette disagreed:
“There are several authors who have considered the contact when diameter ratios are close to 1.00 because Hertz’s equations don’t apply (so called non-Hertzian contact). Chapter 5 of Contact Mechanics by Kenneth Langstreth Johnson, specifically Figure 5.4 (p. 118) shows how conformal contact varies from Hertzian contact. At ‘large’ clearances or light loads, Hertzian contact provides a close estimate, but as the contact patch grows due to tighter clearances or higher loads, the Hertzian formula predicts less stress than would actually be present. This is a result of Hertz neglecting higher order terms to make the closed-form solution tractable.
“In my opinion, whenever you present an equation, you must present the assumptions behind those equations and the valid range of those equations. Hertzian contact is not valid as the diameter ratio approaches 1.00. Perhaps the people at Brush Wellman have data to show the range under which their equation is valid. Based on Johnson and some of my own investigations, the limit where Hertz begins to diverge is probably somewhere in the vicinity of 1.01 to 1.003. Most bushing applications may be above this limit, but there are certainly those that are not.”
The authors clarified their approach to the article, saying “This has become a valuable discussion about the level of approximation needed for basic understanding versus what is required for highly accurate design. We agree that the equations presented are first-order approximations. We chose to sacrifice some accuracy to gain some simplicity. Since the approximations result in closed-form equations, they can be used to qualitatively explain relationships between variables available to designers.
“In addition to neglecting higher order terms, we also do not discuss 3D effects like shaft deflection or off-axis loading, critical concerns in heavy equipment design. This dimensional simplification again promotes basic understanding at the expense of accuracy.
“Our use of approximations is similar to 2D trajectory calculations based only on a projectile’s initial speed, launch angle, and gravity. Such approximations can not specify, for example, the exact path of a golf ball hit with backspin into a cross breeze. But the simplified approximation is helpful in explaining why, for example, a high chip shot and a low pitch might hit the green near the same spot and why one rolls farther than the other after landing.
“That is closer to the depth of discussion we intended. In future articles, space permitting, we will try to state our intentions more clearly.”