Machinery's Handbook, 31st Edition
Disc Spring Forces and Stresses 357 The difference between disc spring forces calculated by Equation (10) and the mea - sured forces varies from − 5.7 percent (maximum) to +0.5 percent (minimum). Disc spring forces calculated by Equation (4) and shown in manufacturers catalogs are less than mea- sured forces by − 11 percent (maximum) to − 6 percent (minimum). Force Generated by Disc Spring with Contact Surfaces.— Some of disc springs in Group 2 and all disc springs in Group 3 are manufactured with small contact (load-bearing) surfaces or flats in addition to the corner radii. These flats provide better contact between disc springs, but, at the same time, they reduce the springs outside diameter and generate higher spring force because in Equation (4) force F is inversely proportional to the square of outside diameter D 2 . To compensate for the undesired force increase, the disc spring thickness is reduced from t to t ′ . Thickness reduction factors t ′ / t are approximately 0.94 for disc spring series A and B, and approximately 0.96 for series C springs. With such re- duction factors, the disc spring force at 75 percent deflection is the same as for equivalent disc spring without contact surfaces. Equation (12), which is similar to Equation (10), has an additional constant K 4 that correlates the increase in spring force due to contact surfaces. If disc springs do not have contact surfaces, then K 4 2 = K 4 = 1. (12) where t ′ = reduced thickness of a disc spring h ′ = cone height adjusted to reduced thickness: h ′ = H − t ′ ( h ′ > h ) K 4 = constant applied to disc springs with contact surfaces. K 4 2 can be calculated as follows: (13) where a = t ′ ( H − 4 t ′ + 3 t ) (5 H − 8 t ′ + 3 t ); b = 32( t ′ ) 3 ; and, c = − t [5( H − t ) 2 + 32 t 2 ]. Disc Spring Functional Stresses.— Disc springs are designed for both static and dynamic load applications. In static load applications, disc springs may be under constant or fluc- tuating load conditions that change up to 5,000 or 10,000 cycles over long time intervals. Dynamic loads occur when disc springs are under continuously changing deflection be- tween pre-load (approximately 15 to 20 percent of the cone height) and the maximum deflection values over short time intervals. Both static and dynamic loads cause compres sive and tensile stresses. The position of critical stress points on a disc spring cross section are shown in Fig. 9. F K D E K s K h s h s t 2 a 2 2 4 2 l l µ − − ` ^ ^ h j h 8 · · · · · · · · t 1 4 1 4 2 3 l l = − + ^ h B K a b b ac 2 4 2 4 2 = − + −
F
F
1
1
0
0
H
2
2
F
h s
F
D d
Fig. 9. Critical Stress Points s is Deflection of Spring by Force F ; h − s is a Cone Height of Loaded Disc Spring.
Compressive stresses are acting at points 0 and 1, that are located on the top surface of the disc spring. Point 0 is located on the cross-sectional midpoint diameter, and point 1 is located on the top inside diameter. Tensile stresses are acting at points 2 and 3, which are located on the bottom surface of the disc spring. Point 2 is on the bottom inside diam- eter, and point 3 is on the bottom outside diameter. The following equations are used to
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