Machinery's Handbook, 31st Edition
360 Disc Spring Fatigue Life When the intersection points of the minimum and maximum stress lines fall inside the areas of each cycle line, only the approximate fatigue life can be estimated by extrapolat- ing the distance from the point of intersection to the nearest cycle line. The extrapolation cannot provide accurate values of fatigue life, because the distance between the cycle lines is expressed in logarithmic scale, and the distance between tensile strength values is expressed in linear scale (Figs. 10, 11, and 12), therefore linear-to-logarithmic scales ratio is not applicable. When intersection points of minimum and maximum stress lines fall outside the cycle lines area, especially outside the N = 10 5 cycles line, the fatigue life cannot be estimated. Thus, the use of the fatigue life diagrams should be limited to such cases when the mini mum and maximum tensile stress lines intersect exactly with each of the cycle lines. To calculate fatigue life of disc springs without the diagrams, the following equations developed by the author can be used.
10.29085532 – 0.00542096 ( σ max – 0.5 σ min )
(20) (21) (22)
Disc Springs in Group 1
N
=
10
10.10734911 – 0.00537616 ( σ max – 0.5 σ min )
Disc Springs in Group 2 Disc Springs in Group 3
N
=
10
13.23985664 – 0.01084192 ( σ max – 0.5 σ min )
N 10 =
As can be seen from Equations (20), (21), and (22), the maximum and minimum tensile stress range affects the fatigue life of disc springs. Since tensile stresses at Points 2 and 3 have different values, see Equations (16) and (17), it is necessary to determine at which critical point the minimum and maximum stresses should be used for calculating fatigue life. The general method is based on the diagram, Fig. 9 from which Point 2 or Point 3 can be found in relationship with disc spring outside-to-inside diameters ratio D / d and disc spring cone height-to-thickness ratio h / r. This method requires intermediate calculations of D / d and h / t ratios and is applicable only to disc springs without contact surfaces. The method is not valid for Group 3 disc springs or for disc springs in Group 2 that have contact surfaces and reduced thickness. A simple and accurate method, that is valid for all disc springs, is based on the following statements: if ( σ 2 max − 0.5 σ 2 min ) > ( σ 3 max − 0.5 σ 3 min ), then Point 2 is used, otherwise if ( σ 3 max − 0.5 σ 3 min ) > ( σ 2 max − 0.5 σ 2 min ), then Point 3 is used The maximum and minimum tensile stress range for disc springs in Groups 1 , 2 , and 3 is found from the following equations. For disc springs in Group 1 : (23) For disc springs in Group 2 : (24) For disc springs in Group 3 : (25) Thus, Equations (23), (24), and (25) can be used to design any spring stack that provides required fatigue life. The following example illustrates how a maximum-minimum stress range is calculated in relationship with fatigue life of a given disc spring stack. . 001084192 . 05 − σ = . 000542096 . 10 29085532 log N max min σ − . 05 min σ − σ = . 000537616 . 10 10734911 log N max − . 05 min σ − σ = . 13 23985664 log N max −
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