Dimensions for a selection of Belleville springs manufactured to DIN 6796 are given in Table 15.6.Example 15.7
A Belleville spring is required to give a constant force of 200 N ± 10 N over a deflection of ± 0.3 mm. The spring must fit within a 62 mm diameter hole. A carbon spring steel with σuts = 1700 MPa has been proposed.
Solution
Assume a 60 mm outer diameter to allow some clearance in the hole.
To provide a constant force, an h/t ratio of 1.414 is selected.
The variation of force of ± 5% can be met by choosing an appropriate deflection range to operate in from Fig. 15.33. If the deflection is limited to between 65% and 135% of the flat deflection, then the tolerance on force can be achieved. The nominal force of 200 N will occur in the flat position and the spring will provide a similar force, between 210 N and 190 N, operating on both sides of its centre.
From Eq. (15.54),
h = 1.414 t = 1.414 × 0.788 = 1.114 mm.
The minimum and maximum deflections are.
δmin = 0.65 h = 0.65 × 1.114 = 0.724 mm.
δmax = 1.35 h = 1.35 × 1.114 = 1.504 mm.
δmax − δmin is greater than the required deflection range of 0.6 mm so the force tolerance can be met.
From Eqs (15.44) and (15.50)–(15.53),
K1 = 0.689,
K2 = 1.220,
K3 = 1.378,
K4 = 1.115,
K5 = 1.
From Eq. (15.47), σc = − 840 MPa.
From Eq. (15.48), σti = 355 MPa.
From Eq. (15.49), σto = 658 MPa.
These stresses are well within the capability of a 1700 MPa uts material.
Example 15.8
A Belleville spring is required to give a constant force of 50 N ± 5 N over a deflection of ± 0.2 mm. The spring must fit within a 40 mm diameter hole. A carbon spring steel with σuts = 1700 MPa has been proposed.
Solution
Assume a 35 mm outer diameter to allow some clearance in the hole.
To provide a constant force, a h/t ratio of 1.414 is selected.
The variation of force of ± 10% can be readily met by choosing an appropriate deflection range to operate in from Fig. 15.33.
If the deflection is limited to between 65% and 135% of the flat deflection, then the tolerance on force can be achieved.
The nominal force of 50 N will occur in the flat position and the spring will provide a similar force, between 55 N and 45 N, operating on both sides of its centre.
From Eq. (15.52),
h = 1.414 t = 1.414 × 0.426 = 0.602 mm
The minimum and maximum deflections are
δmin = 0.65 h = 0.65 × 0.602 = 0.391 mm
δmax = 1.35 h = 1.35 × 0.602 = 0.812 mm
δmax − δmin is greater than the required deflection range of 0.4 mm so the force tolerance can be met.
From Eqs (15.44) and (15.50)–(15.53),
K1 = 0.689
K2 = 1.220
K3 = 1.378
K4 = 1.115
K5 = 1
From Eq. (15.47), σc = − 720 MPa
From Eq. (15.48), σti = 305 MPa
From Eq. (15.49), σto = 564 MPa.
These stresses are well within the capability of a 1700 MPa uts material.
Example 15.9
A Belleville spring is required to give a constant force of 10 N ± 1 N over a deflection of ± 0.15 mm. The spring must fit within a 16 mm diameter hole. A carbon spring steel with σuts = 1700 MPa has been proposed.
Solution
Assume a 14 mm outer diameter to allow some clearance in the hole.
To provide a constant force, a h/t ratio of 1.414 is selected.
The variation of force of ± 10% can be met by choosing an appropriate deflection range to operate in from Fig. 15.33.
If the deflection is limited to between 65% and 135% of the flat deflection, then the tolerance on force can be achieved.
The nominal force of 10 N will occur in the flat position and the spring will provide a similar force, between 11 N and 9 N, operating on both sides of its centre.
From Eq. (15.52),
h = 1.414 t = 1.414 × 0.18 = 0.255 mm
The minimum and maximum deflections are
δmin = 0.65 h = 0.65 × 0.255 = 0.165 mm
δmax = 1.35 h = 1.35 × 0.255 = 0.344 mm
From Eqs (15.44) and (15.50)–(15.53),
K1 = 0.689
K2 = 1.220
K3 = 1.378
K4 = 1.115
K5 = 1
From Eq. (15.47), σc = − 810 MPa
From Eq. (15.48), σti = 341 MPa
From Eq. (15.49), σto = 630 MPa
These stresses are well within the capability of a 1700 MPa uts material.
Table 15.6. Dimensions for a selection of Belleville washer springs manufactured to DIN 6796 from DIN 17222 spring steel
Notation |
Di (mm) |
Do (mm) |
h′ max (mm) |
h′ min (mm) |
t (mm) |
Force (N)a
|
Test force (N)b
|
Mass kg/1000 |
Core diameter (mm) |
2 |
2.2 |
5 |
0.6 |
0.5 |
0.4 |
628 |
700 |
0.05 |
2 |
2.5 |
2.7 |
6 |
0.72 |
0.61 |
0.5 |
946 |
1100 |
0.09 |
2.5 |
3 |
3.2 |
7 |
0.85 |
0.72 |
0.6 |
1320 |
1500 |
0.14 |
3 |
3.5 |
3.7 |
8 |
1.06 |
0.92 |
0.8 |
2410 |
2700 |
0.25 |
3.5 |
4 |
4.3 |
9 |
1.3 |
1.12 |
1 |
3770 |
4000 |
0.38 |
4 |
5 |
5.3 |
11 |
1.55 |
1.35 |
1.2 |
5480 |
6550 |
0.69 |
5 |
6 |
6.4 |
14 |
2 |
1.7 |
1.5 |
8590 |
9250 |
1.43 |
6 |
7 |
7.4 |
17 |
2.3 |
2 |
1.75 |
11,300 |
13,600 |
2.53 |
7 |
8 |
8.4 |
18 |
2.6 |
2.24 |
2 |
14,900 |
17,000 |
3.13 |
8 |
10 |
10.5 |
23 |
3.2 |
2.8 |
2.5 |
22,100 |
27,100 |
6.45 |
10 |
12 |
13 |
29 |
3.95 |
3.43 |
3 |
34,100 |
39,500 |
12.4 |
12 |
14 |
15 |
35 |
4.65 |
4.04 |
3.5 |
46,000 |
54,000 |
21.6 |
14 |
16 |
17 |
39 |
5.25 |
4.58 |
4 |
59,700 |
75,000 |
30.4 |
16 |
18 |
19 |
42 |
5.8 |
5.08 |
4.5 |
74,400 |
90,500 |
38.9 |
18 |
20 |
21 |
45 |
6.4 |
5.6 |
5 |
93,200 |
117,000 |
48.8 |
20 |
22 |
23 |
49 |
7.05 |
6.15 |
5.5 |
113,700 |
145,000 |
63.5 |
22 |
24 |
25 |
56 |
7.75 |
6.77 |
6 |
131,000 |
169,000 |
92.9 |
24 |
27 |
28 |
60 |
8.35 |
7.3 |
6.5 |
154,000 |
221,000 |
113 |
27 |
30 |
31 |
70 |
9.2 |
8 |
7 |
172,000 |
269,000 |
170 |
30 |