Machinery's Handbook, 31st Edition
354
Disc Spring Forces and Stresses For n p disc spring pairs arranged in series, the following equations are applied: F F S s n 2 total total p # # = =
L H n t h n 2 p p 0 # # = = + ^ h
(3)
Disc Spring Forces and Stresses Several methods of calculating forces and stresses for given disc spring configurations exist, some very complicated, others of limited accuracy. The theory which is widely used today for force and stress calculations was developed more than 65 years ago by Almen and Laszlo. The theory is based on the following assumptions: cross sections are rectangular without radii over the entire range of spring deflection; no stresses occur in the radial direction; disc springs are always under elastic deformation during deflection; and due to small cone angles of unloaded disc springs (between 3.5 ° and 8.6 ° ), mathematical simplifications are applied. The theory provides accurate results for disc springs with the following ratios: outside- to-inside diameter, D / d = 1.3 to 2.5; and cone height-to-thickness, h / t is up to 1.5. Force Generated by Disc Springs Without Contact Surfaces.— Disc springs in Group 1 and most of disc springs in Group 2 are manufactured without contact (load-bearing) surfaces, but have corner radii. A single disc spring force applied to points 1 and 3 in Fig. 6 can be found from Equa- tion (4) in which corner radii are not considered: (4) where F = disc spring force; E = modulus of elasticity of spring material; μ = Poisson’s ratio of spring material; K 1 = constant depending on outside-to-inside diameter ratio; D = disc spring nominal outside diameter; h = cone (dish) height; s = disc spring deflec tion; and, t = disc spring thickness. F K D 1 · E s h s h s t t − − + ^ j h · · 1 4 2 2 · · · h 2 3 µ = − ^ ` 8 B
D
F
1
H
2
3
h
F
d
Fig. 6. Schematic of Applied Forces It has been found that the theoretical forces calculated using Equation (4) are lower than the actual (measured) spring forces, as illustrated in Fig. 7. The difference between theo retical (trace 1) and measured force values (trace 3) was significantly reduced (trace 2) when the actual outside diameter of the spring in loaded condition was used in the calculations.
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