# Belleville Spring

Belleville spring washers are used to provide an initial mechanical prestress.

## Machine Components

DAN B. MARGHITU, ... NICOLAE CRACIUNOIU, in Mechanical Engineer's Handbook, 2001

### 3.8Belleville Springs

Belleville springs are made from tapered washers (Fig. 3.15a) stacked in series, parallel, or a combination of parallel–series, as shown in Fig. 3.15b. The load–deflection and stress–deflection are

(3.42)$F=\frac{E\delta }{\left(1-{\mu }^{2}\right){\left({d}_{o}/2\right)}^{2}M}\left[\left(h-\delta /2\right)\left(h-\delta \right)t+{t}^{3}\right]$
(3.43)$\sigma =\frac{E\delta }{\left(1-{\mu }^{2}\right){\left({d}_{o}/2\right)}^{2}M}\left[{C}_{1}\left(h-\delta /2\right)={C}_{2}t\right],$

where F is the axial load (lb), δ is the deflection (in), t is the thickness of the washer (in), h, is the free height minus thickness (in), E is the modulus of elasticity (psi), σ is the stress at inside circumference (psi), do is the outside diameter of the washer (in), di is the inside diameter of the washer (in), and μ is the Poisson's ratio. The constant M, C1, and C2 are given by the equations

$\begin{array}{l}M=\frac{6}{\pi {log}_{e}\left({d}_{o}/{d}_{i}\right)}{\left(\frac{{d}_{o}/{d}_{i}-1}{{d}_{o}/{d}_{i}}\right)}^{2}\\ {C}_{1}=\frac{6}{\pi {log}_{e}\left({d}_{o}/{d}_{i}\right)}\left[\frac{{d}_{o}/{d}_{i}-1}{{log}_{e}\left({d}_{o}/{d}_{i}\right)}-1\right]\\ {C}_{2}=\frac{6}{\pi {log}_{e}\left({d}_{o}/{d}_{i}\right)}\left[\frac{{d}_{o}/{d}_{i}-1}{2}\right].\end{array}$
URL: https://www.sciencedirect.com/science/article/pii/B9780124713703500061

## Springs

P.R.N. Childs, in Mechanical Design (Third Edition), 2021

### 9.6Belleville Spring Washers

Belleville spring washers comprise a conical-shaped disk with a hole through the center as illustrated in Figs. 9.28 and 9.29. They have a nonlinear force-deflection characteristic, which makes them useful in certain applications. Belleville springs are compact and are capable of large compressive forces, but their deflections are limited. Examples of their use include gun recoil mechanisms, pipe flanges, small machine tools and applications where differential expansion could cause relaxation of a bolt (Fig. 9.30).

The force-deflection characteristics for a Belleville washer depend on material properties and the dimensions illustrated in Fig. 9.31. The force-deflection relationship is nonlinear, so it is not stated in the form of a spring rate and is given (Norton, 1996) by

(9.43)$F=\frac{4E\delta }{{K}_{1}{D}_{\mathrm{o}}^{2}\left(1-{\mu }^{2}\right)}\left[\left(h-\delta \right)\left(h-\frac{\delta }{2}\right)t+{t}^{3}\right]$

where

(9.44)${K}_{1}=\frac{6}{\pi \mathrm{ln}{R}_{\mathrm{d}}}\left[\frac{{\left({R}_{\mathrm{d}}-1\right)}^{2}}{{R}_{\mathrm{d}}^{2}}\right]$
(9.45)${R}_{\mathrm{d}}=\frac{{D}_{\mathrm{o}}}{{D}_{\mathrm{i}}}$

The load at the flat position is given by

(9.46)${F}_{\mathrm{flat}}=\frac{4Eh{t}^{3}}{{K}_{1}{D}_{\mathrm{o}}^{2}\left(1-{\mu }^{2}\right)}$

Eq. (9.43) has been used to illustrate the variation of force with deflection in Fig. 9.32 for a variety of h/t ratios. The curves have been normalized to the spring’s condition when flat for a steel Belleville spring. For an h/t ratio of 0.4, the ratio of force to deflection is close to linear. As the h/t ratio is increased, it becomes increasingly nonlinear. For an h/t ratio of 1.414, there is a nearly constant region for the force. For values of h/t greater than 1.414, the curves become bimodal where a given force could cause more than one possible deflection. Once installed and in the flat position, a trivial force would be required to trip it in one direction or the other. The deviation of force from the force to flat value with deflection is shown in Fig. 9.33 for a spring with Rd=2 and h/t=1.414. The force produced by an h/t=1.414 ratio Belleville spring is within ±10% of the force to flat value for deflections of 55%–145% of the deflection to flat. This is a useful characteristic if a nearly constant force is required over this range of deflection.

The stresses in a Belleville spring are concentrated at the edges of the inside and outside diameters (Almen and Laszlo, 1936). The largest stress is compressive and occurs at the inside radius on the convex side. The edges on the concave side have tensile stresses given by Eqs. (9.48) and (9.49). These equations are quoted from Norton (1996).

(9.47)${\sigma }_{\mathrm{c}}=-\frac{4E\delta }{{K}_{1}{D}_{\mathrm{o}}^{2}\left(1-{\mu }^{2}\right)}\left[{K}_{2}\left(h-\frac{\delta }{2}\right)+{K}_{3}t\right]$
(9.48)${\sigma }_{\mathrm{t}\phantom{\rule{.10em}{0ex}}\mathrm{i}}=\frac{4E\delta }{{K}_{1}{D}_{\mathrm{o}}^{2}\left(1-{\mu }^{2}\right)}\left[-{K}_{2}\left(h-\frac{\delta }{2}\right)+{K}_{3}t\right]$
(9.49)${\sigma }_{\mathrm{t}\phantom{\rule{.10em}{0ex}}\mathrm{o}}=\frac{4E\delta }{{K}_{1}{D}_{\mathrm{o}}^{2}\left(1-{\mu }^{2}\right)}\left[{K}_{4}\left(h-\frac{\delta }{2}\right)+{K}_{5}t\right]$

where

(9.50)${K}_{2}=\frac{6}{\pi \mathrm{ln}{R}_{\mathrm{d}}}\left[\frac{{R}_{\mathrm{d}}-1}{\mathrm{ln}{R}_{\mathrm{d}}}-1\right]$
(9.51)${K}_{3}=\frac{6}{\pi \mathrm{ln}{R}_{\mathrm{d}}}\left[\frac{{R}_{\mathrm{d}}-1}{2}\right]$
(9.52)${K}_{4}=\left[\frac{{R}_{\mathrm{d}}\mathrm{ln}{R}_{\mathrm{d}}-\left({R}_{\mathrm{d}}-1\right)}{\mathrm{ln}{R}_{\mathrm{d}}}\right]\left[\frac{{R}_{\mathrm{d}}}{{\left({R}_{\mathrm{d}}-1\right)}^{2}}\right]$
(9.53)${K}_{5}=\frac{{R}_{\mathrm{d}}}{2\left({R}_{\mathrm{d}}-1\right)}$

The maximum deflection of a single Belleville spring tends to be small. They can, however, be stacked in combinations as illustrated in Fig. 9.34. If stacked in series, the total force will be the same as for a single Belleville spring, but the deflections will add. If they are stacked in parallel, the total deflection will be the same as for a single spring, but the forces will add. Series and parallel combinations are also possible. It should be noted that Belleville spring towers need some form of support, either by inserting them into a hole or over a rod. This, however, will reduce the available load due to friction.

The design or selection of a Belleville spring for a given application requires consideration of the diameter and h/t ratio as well as the type of material to be used in order to give the desired force-deflection characteristics (SAE, 1988). This process invariably involves some iteration. A material is selected, values for the diameter ratio are proposed, often based on given constraints, and values of the h/t ratio proposed either for a single spring or some combination.

In designing Belleville springs, it can be useful to estimate the thickness required to give a particular force in the flat position, which can be found, substituting E=207 GPa, μ=0.3, and K1=0.69 in Eq. (9.46), from

(9.54)$t=\frac{1}{1072}{\left(\frac{{D}_{\mathrm{o}}^{2}{F}_{\mathrm{flat}}}{h/t}\right)}^{0.25}$

where t is the thickness (m), h is the deflection to flat (m), Do is the outer diameter (m), and Fflat is the load at flat position (N).

Dimensions for a selection of Belleville springs manufactured according to DIN 6796 are given in Table 9.6.

Table 9.6. Dimensions for a selection of Belleville washer springs manufactured according 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
a
Force applies to the pressed flat condition and corresponds to twice the calculated value at a deflection h′−t.
b
Test force applies for loading tests to DIN 6796.

Example 9.4

A Belleville spring is required to give a constant force of 200±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.

In order 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. 9.33. 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 and 190 N, operating on both sides of its center.

From Eq. (9.54),

$t=\frac{1}{1072}{\left(\frac{{D}_{\mathrm{o}}^{2}{F}_{\mathrm{flat}}}{h/t}\right)}^{0.25}=\frac{1}{1072}{\left(\frac{{0.06}^{2}×200}{1.414}\right)}^{0.25}=7.88×{10}^{-4}\phantom{\rule{.25em}{0ex}}\mathrm{m}$
$h=1.414\phantom{\rule{.25em}{0ex}}t=1.414×0.788=1.114\phantom{\rule{.25em}{0ex}}\mathrm{mm}$

The minimum and maximum deflections are

${\delta }_{\mathrm{min}}=0.65\phantom{\rule{.25em}{0ex}}h=0.65×1.114=0.724\phantom{\rule{.25em}{0ex}}\mathrm{mm}$
${\delta }_{\mathrm{max}}=1.35\phantom{\rule{.25em}{0ex}}h=1.35×1.114=1.504\phantom{\rule{.25em}{0ex}}\mathrm{mm}$

δmaxδmin is greater than the required deflection range of 0.6 mm, so the force tolerance can be met.

From Eqs. (9.44) and (9.50–9.53),

K1=0.689

K2=1.220

K3=1.378

K4=1.115

K5=1

From Eq. (9.47), σc=−840 MPa

From Eq. (9.48), σti=355 MPa

From Eq. (9.49), σto=658 MPa

These stresses are well within the capability of a 1700 MPa uts material.

URL: https://www.sciencedirect.com/science/article/pii/B9780128211021000093

## Springs

Peter R.N. Childs, in Mechanical Design Engineering Handbook (Second Edition), 2019

### 15.6Belleville spring washers

Belleville spring washers comprise a conical shaped disc with a hole through the centre as illustrated in Figs. 15.28 and 15.29. They have a nonlinear force deflection characteristic, which makes them useful in certain applications. Belleville springs are compact and are capable of large compressive forces but their deflections are limited. Examples of their use include gun recoil mechanisms, pipe flanges, small machine tools and applications where differential expansion could cause relaxation of a bolt (Fig. 15.30).

The force–deflection characteristics for a Belleville washer depend on material properties and the dimensions illustrated in Fig. 15.31. The force–deflection relationship is nonlinear so it is not stated in the form of a spring rate and is given (Norton, 1996) by:

(15.43)$F=\frac{4\mathit{E\delta }}{{K}_{1}{D}_{o}^{2}\left(1-{\mu }^{2}\right)}\left[\left(h-\delta \right)\left(h-\frac{\delta }{2}\right)t+{t}^{3}\right]$

where

(15.44)${K}_{1}=\frac{6}{\pi ln{R}_{d}}\left[\frac{{\left({R}_{d}-1\right)}^{2}}{{R}_{d}^{2}}\right]$
(15.45)${R}_{d}=\frac{{D}_{0}}{{D}_{i}}$

The load at the flat position is given by

(15.46)${F}_{\text{flat}}=\frac{4{\mathit{Eht}}^{3}}{{K}_{1}{D}_{o}^{2}\left(1-{\mu }^{2}\right)}$

Eq. (15.43) has been used to illustrate the variation of force with deflection in Fig. 15.32 for a variety of h/t ratios. The curves have been normalised to the spring's condition when flat for a steel Belleville spring. For a ratio of h/t of 0.4 the ratio of force to deflection is close to linear. As the h/t ratio is increased it becomes increasingly nonlinear. For an h/t ratio of 1.414 there is a nearly constant region for the force. For values of h/t > 1.414 the curves become bimodal where a given force could cause more than one possible deflection. Once installed and in the flat position a trivial force would be required to trip it in one direction or the other. The deviation of force from the force to flat value with deflection is shown in Fig. 15.33 for a spring with Rd = 2 and h/t = 1.414. The force produced by a h/t = 1.414 ratio Belleville spring is within ± 10% of the force to flat value for deflections of 55% to 145% of the deflection to flat. This is a useful characteristic if a nearly constant force is required over this range of deflection.

The stresses in a Belleville spring are concentrated at the edges of the inside and outside diameters (Almen and Laszlo, 1936). The largest stress is compressive and occurs at the inside radius on the convex side. The edges on the concave side have tensile stresses given by Eqs (15.48) and (15.49). These equations are quoted from Norton (1996).

(15.47)${\sigma }_{c}=-\frac{4\mathit{E\delta }}{{K}_{1}{D}_{o}^{2}\left(1-{\mu }^{2}\right)}\left[{K}_{2}\left(h-\frac{\delta }{2}\right)+{K}_{3}t\right]$
(15.48)${\sigma }_{t\phantom{\rule{0.24em}{0ex}}i}=\frac{4\mathit{E\delta }}{{K}_{1}{D}_{o}^{2}\left(1-{\mu }^{2}\right)}\left[-{K}_{2}\left(h-\frac{\delta }{2}\right)+{K}_{3}t\right]$
(15.49)${\sigma }_{t\phantom{\rule{0.24em}{0ex}}o}=\frac{4\mathit{E\delta }}{{K}_{1}{D}_{o}^{2}\left(1-{\mu }^{2}\right)}\left[{K}_{4}\left(h-\frac{\delta }{2}\right)+{K}_{5}t\right]$

where

(15.50)${K}_{2}=\frac{6}{\pi ln{R}_{d}}\left[\frac{{R}_{d}-1}{ln{R}_{d}}-1\right]$
(15.51)${K}_{3}=\frac{6}{\pi ln{R}_{d}}\left[\frac{{R}_{d}-1}{2}\right]$
(15.52)${K}_{4}=\left[\frac{{R}_{d}ln{R}_{d}-\left({R}_{d}-1\right)}{ln{R}_{d}}\right]\left[\frac{{R}_{d}}{{\left({R}_{d}-1\right)}^{2}}\right]$
(15.53)${K}_{5}=\frac{{R}_{d}}{2\left({R}_{d}-1\right)}$

The maximum deflection of a single Belleville spring tends to be small. They can, however, be stacked in combinations as illustrated in Fig. 15.34. If stacked in series the total force will be the same as for a single Belleville spring but the deflections will add. If they are stacked in parallel the total deflection will be the same as for a single spring but the forces will add. Series and parallel combinations are also possible. It should be noted that Belleville spring towers need some form of support, either by inserting them into a hole or over a rod. This however will reduce the available load due to friction.

The design or selection of a Belleville spring for a given application requires consideration of the diameter and h/t ratio as well as the type of material to be used to give the desired force–deflection characteristics (SAE, 1988). This process invariably involves some iteration. A material is selected, values for the diameter ratio are proposed, often based on given constraints, and values of the h/t ratio proposed either for a single spring or some combination.

In designing Belleville springs it can be useful to estimate the thickness required to give a particular force in the flat position which can be found, substituting E = 207 GPa, μ = 0.3 and K1 = 0.69 in Eq. (15.46), from:

(15.54)$t=\frac{1}{1072}{\left(\frac{{D}_{o}^{2}{F}_{\text{flat}}}{h/t}\right)}^{0.25}$

where t = thickness (m), h = deflection to flat (m), Do = outer diameter (m) and Fflat = load at flat position (N).

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),

$t=\frac{1}{1072}{\left(\frac{{D}_{o}^{2}{F}_{\text{flat}}}{h/t}\right)}^{0.25}=\frac{1}{1072}{\left(\frac{{0.06}^{2}×200}{1.414}\right)}^{0.25}=7.88×{10}^{-4}\phantom{\rule{0.5em}{0ex}}\mathrm{m}$

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),

$t=\frac{1}{1072}{\left(\frac{{D}_{o}^{2}{F}_{\text{flat}}}{h/t}\right)}^{0.25}=\frac{1}{1072}{\left(\frac{{0.035}^{2}×50}{1.414}\right)}^{0.25}=4.26×{10}^{-4}\phantom{\rule{0.5em}{0ex}}\mathrm{m}$

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),

$t=\frac{1}{1072}{\left(\frac{{D}_{o}^{2}{F}_{\text{flat}}}{h/t}\right)}^{0.25}=\frac{1}{1072}{\left(\frac{{0.014}^{2}×10}{1.414}\right)}^{0.25}=1.8×{10}^{-4}\phantom{\rule{0.12em}{0ex}}\mathrm{m}$

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
a
Force applies to the pressed flat condition and corresponds to twice the calculated value at a deflection h  t.
b
Test force applies for loading tests to DIN 6796.

δmax  δmin is not however greater than the required deflection range of 0.3 mm so the force tolerance is not met.

Altering the diameter of the spring however to say 15 mm or 13 mm does not give δmax  δmin > 0.3 mm. So a different h/t ratio would need to be considered.

URL: https://www.sciencedirect.com/science/article/pii/B9780081023679000159

## Anchor Bolts

Donald M. Harrison, in The Grouting Handbook, 2013

Load-generating washers (sometimes referred to as Belleville washers or Belleville springs) have been used on all types of machines since the 1700s. Certain industries use them widely, whereas other industries that could benefit from this technology reject it. A load-generating washer is a cone-shaped disk (Figure 2.22) that will flatten or deflect at a given rate. The deflection rate is usually very high, allowing the washer to produce and maintain very large loads in a very small space.

Load-generating washers are used in a variety of applications where high spring loads are required. They are particularly useful to solve problems of vibration, differential thermal expansion, and bolt relaxation.

Because of the conical design of these washers, a positive constant force is exerted against the anchor bolt nut. Combinations of these washers can allow the end user to increase or decrease the load generated by them on the anchor bolt (Figure 2.23).

Figures 2.22 and 2.23 show a pronounced dish. However, most Bellevilles actually have shallow dishes. Sometimes, it’s hard to tell which end is up. If that happens to you, lay the washer on a flat surface and look at it from the side.

URL: https://www.sciencedirect.com/science/article/pii/B978012416585400002X

## Springs

Peter R.N. Childs, in Mechanical Design Engineering Handbook, 2014

### 15.7Conclusions

This chapter has introduced the function and characteristics of a range of spring technologies. Design methods have been outlined for helical compression, extension, and torsion springs, leaf springs, and Belleville spring washers. There is no single design procedure that is suitable for all types of spring, and the procedures outlined here have not considered all of the important design parameters, which also include cost, appearance, and environmental considerations. Spring design invariably becomes a process of optimization in which various tradeoffs need to be balanced in order to provide the best possible total solution.

URL: https://www.sciencedirect.com/science/article/pii/B9780080977591000150

## Friction clutch

Heinz Heisler MSc., BSc., F.I.M.I., M.S.O.E., M.I.R.T.E., M.C.I.T., M.I.L.T., in Advanced Vehicle Technology (Second Edition), 2002

### 2.10Clutch (upshift) brake (Fig. 2.15)

The clutch brake is designed primarily for use with unsynchronized (crash or constant mesh) gearboxes to permit shifting into first and reverse gear without severe dog teeth clash. In addition, the brake will facilitate making unshafts by slowing down the input shaft so that the next higher gear may be engaged without crunching of teeth.

The brake disc assembly consists of a pair of Belleville spring washers which are driven by a hub having internal lugs that engage machined slots in the input shaft. These washers react against the clutch brake cover with facing material positioned between each spring washer and outer cover (Fig. 2.15).

When the clutch pedal is fully depressed, the disc will be squeezed between the clutch release bearing housing and the gearbox bearing housing, causing the input spigot shaft to slow down or stop. The hub and spring washer combination will slip with respect to the cover if the applied torque load exceeds 34 Nm, thus preventing the disc brake being overloaded.

In general, the clutch brake comes into engagement only during the last 25mm of clutch pedal travel. Therefore, the pedal must be fully depressed to squeeze the clutch brake. The clutch pedal should never be fully depressed before the gearbox is put into neutral. If the clutch brake is applied with the gearbox still in gear, a reverse load will be put on the gears making it difficult to get the gearbox out of gear. At the same time it will have the effect of trying to stop or decelerate the vehicle with the clutch brake and rapid wear of the friction disc will take place. Never apply the clutch brake when making down shifts, that is do not fully depress the clutch pedal when changing from a higher to a lower gear.

URL: https://www.sciencedirect.com/science/article/pii/B9780750651318500038

## Hot Embossing Technique

Matthias Worgull, in Hot Embossing, 2009

### 7.2.2.2Hydraulic Drive

If high molding forces are required, for example for large area molding, besides a large spindle drive a hydraulic drive unit can be used also as an alternative. Hydraulic drives are established drive mechanisms in macroscopic molding presses, for example to achieve the clamp force to close the molding tool during injection molding. The challenge in using this concept for micro hot embossing is to achieve the required range of molding velocities and the large bandwidth of molding forces. To fulfill these requirements the performance can be split off into two hydraulic cylinders: a first small cylinder responsible for the touch force and a second cylinder, here the main cylinder, responsible for the high molding force. The small cylinder is therefore integrated in the main cylinder (Fig. 7.4). This concept is implemented in the commercially available hot embossing machine Wickert WMP 1000.

The hot embossing machine consists of a massive frame with high stiffness. The movable crossbar is powered by a two-stage coaxial-arranged hydraulic cylinder with different diameters. Both cylinders are characterized by independent pressure generation and by an independent force and position controlling system. The measurement of force, normally measured in the flux pattern of the frame, is achieved here by the measurement of the hydraulic pressure. To eliminate external influences like the weight of the different tools or friction in the guiding elements, the force and position measurement systems are referenced by a reference step. This step has to be carried out only at the beginning of molding.

The small cylinder, called the touch force cylinder, responsible for the effecting of the touch force during heating of the polymer, provides a force in the range between 500 N and 30,000 N. The precision of the force control is typically in the range of a few 100 N; the length of stroke of this cylinder is in a range of a few millimeters with a precision of approximately 100 μm.

The main cylinder, called the molding cylinder, is responsible for the opening and closing of the tool and for the generation of the press force. The control of this molding cylinder is split into two different ranges, a coarse range for the closing of the tool and a fine range for the molding step. The coarse range is characterized by velocities in the range of 90 mm/sec, with a precision of a few decimillimeters. In the range of 50 mm before the molding tool is closed the coarse control mechanism is switched to a precise control mechanism supported by a high-resolution measurement of the crossbar position. To achieve a final high precision positioning, the velocity of the crossbar will be reduced. The achievable precision of the position is here in the range of approximately 100 μm. The maximum achievable press force of 1,000 kN can be controlled with a precision of 1,000 N.

The hot embossing process implemented by the hot embossing machine Wickert WMP1000 in combination with the basic molding tool (Section 8.4) can be illustrated in four steps in Fig. 7.5.

In detail a hot embossing cycle can be split into the following steps.

1.

Insert of the polymer film and closing of the molding tool by the main cylinder 1 up to the sealing point of the vacuum chamber (Fig. 7.5(a)).

2.

Evacuation of the vacuum chamber and providing of touch force by the small cylinder 2. During the providing of touch force the main cylinder 1 still remains in the position of step 1. If the basic tool with separated heating and cooling units is used, the heating plate has no contact with the cooling block because of the Belleville spring between heating and cooling plate (Fig. 7.5(b)). This air gap reduces the thermal mass of the heating block and with the implemented heating system heating rates up to 50 K/min can be achieved. Typical heating times for hot embossing processes are in the range of 120 seconds.

3.

After the molding temperature is attained, the molding process starts by the switching from touch force of the small cylinder 2 to the press force of the main cylinder 1. In a first step the cylinder 2 is displaced by cylinder 1. This means that cylinder 2 under retention of the touch force is opened under the same velocity that cylinder 1 will be closed. During this switching, the position of the crossbar remains constant.

4.

Velocity-controlled molding with the main cylinder 1 up to the desired press force. Under the act of the load the heating plates are pressed onto the cooling plates and generate a system with the required stiffness and plane-parallel configuration (Fig. 7.5(c)). The electrical heating is stopped immediately after the molding force is achieved and the heating plates are fixed by a magnetic clamping system to later provide a vertical demolding without any misalignment. In a following step the small cylinder 2 is locked with the main cylinder 1. In this configuration the load can be maintained over a selected period of time (holding time, or dwell pressure).

5.

By the contact of the heating plate with the cooling plate with a large heat capacity, the mold insert will be cooled in a short period of time down to demolding temperature. Typical cooling rates for molding PMMA are in the range of 180 seconds. If the demolding temperature is attained, the load is eliminated by a shift of cylinder 1. The control of this demolding step requires a precise motion of cylinder 1 over a distance of a few millimeters (Fig. 7.5(d)).

6.

In the last molding step the vacuum chamber is vented and the molding tool is opened completely. Cylinder 1 and cylinder 2 will be unlocked and the heating plates and the cooling plates will be separated by the springs.

Within the scope of an embossing cycle, the regulating distances of both cylinders can be programmed arbitrarily. For the closing of the molding tool, cylinder 1 can be positioned with a precision of 0.2 mm; the control of the force in this step is not possible. The final closing of the molding tool is achieved with a precision of positioning in the range of 100 μm. The small cylinder has a total stroke of about 50 mm and an identical precision of positioning.

URL: https://www.sciencedirect.com/science/article/pii/B9780815515791500136

## General High-Pressure Experimental Technique

Scott Bair, in High Pressure Rheology for Quantitative Elastohydrodynamics (Second Edition), 2019

### 3.4.2Electrical feed-throughs

For the measurement of many parameters and effects such as pressure, temperature, force, and electrical properties a means to pass an electrical signal from the high-pressure region through the vessel to a recording device outside is useful if not essential. Generally more than one electrical connection is required. For example, a full strain-gauge bridge for strain measurement requires four connections and a manganin resistor for pressure measurement requires at least two. Sherman and Stadtmuller  have an excellent section dealing with electrical leads into a pressure vessel. Here, a number of electrical feed-throughs that have been useful to high-pressure rheology will be described.

The configuration shown in Fig. 3.5A is of the unsupported area (Bridgman) type. A stainless steel, 1.6 mm diameter, wire is passed through a hole in a short, 5–6 mm diameter, steel cylinder (the head) and is silver brazed in place. The head is supported by a nonconductive ceramic hollow cylinder (the washer) between the head and the flat surface of the closure. The ceramic material is usually alumina. The unsupported area is that of the hole in the closure so that the contact pressure between the washer and head and between the washer and closure exceeds the liquid pressure. These components are lapped together with a rotating motion and thread-locking compound is used as a sealant. These components are held in place by a stack of Belleville spring washers retained by a ceramic cap. As many as six assemblies may be arranged around a closure as shown in Fig. 3.5A and a plug using commercial pin receptacles can be constructed to fit the external ends of the rods. Internal lead wires may be soldered to a copper disc inserted into the spring stack. This feed-through is very rugged and reliable. On occasion, however, they show a peculiar trait. The ceramic insulating washer that is made of alumina may crack. Pressure tends to hold the crack closed so that no leak has been detected when this occurs but, for alumina, the crack becomes conductive making a closed circuit to the closure body.

The other types of feed-through shown in Fig. 3.5B and C utilize a commercial electrical cable that is generally used for thermocouple wire leads. It consists of one to six bare metal conducting wires packed into a stainless steel tube with magnesium oxide powder as an insulator. The wires may be thermocouple alloys or copper or nickel. This cable is provided with the tube having been tightly swaged around the powder and wires. If the tube is sufficiently long, perhaps 10 times the diameter, the magnesia powder will resist pressures to at least 1.3 GPa, the highest pressure for which these feed-throughs have been used in the author’s laboratory. Handling of the cable will cause powder to escape from the ends and this can be prevented by allowing the powder to absorb a mixture of acetone and two-part epoxy glue. Of course, the glue must be applied after the cable has been brazed into whatever component receives it as the temperature of the brazing process would decompose the glue.

In Fig. 3.5C, the cable is simply placed into a through hole in a steel closure of the shape of a commercial high-pressure plug. These two are then brazed together using high-temperature silver solder. The silver solder joint should always be placed at the internal end so that pressure will tend to tighten the joint. The nut that is used with a commercial plug can be used to retain the feed-through.

In Fig. 3.5B, the swaged cable is brazed in the same manner as above except that in this case it is received into a sleeve that functions as the mushroom in a Bridgman closure similar to Fig. 3.2C. A threaded hollow screw preloads the sleeve to initially energize the seal. The ends of the wires emanating from the external end of the swaged cable are connected to a commercial electrical plug for strain relief and convenience.

URL: https://www.sciencedirect.com/science/article/pii/B9780444641564000039

## MAGNETOSTRICTIVE MATERIALS

A. Flatau, in Encyclopedia of Vibration, 2001

### Transduction

Magnetostrictive materials are magnetoelastic in the sense that they do work in the process of converting between magnetic and elastic (or mechanical) energy states. However, magnetostrictive transducers are generally classified as electromechanical devices or electromagnetomechanical because their input and output are generally electrical and mechanical in nature. The conversion of magnetic energy to and from electrical and/or mechanical is transparent to the device user. A common two-port schematic appropriate for both magnetostrictive sensing and actuation devices is given in Figure 1. The magnetostrictive driver is represented by the center block, with the transduction coefficients Tme (mechanical due to electrical) and Tem (electrical due to mechanical), indicative of both the magnetoelastic attributes characterized by eqns (1) and (2) and the electromagnetic attributes associated with conversion between electrical and magnetic fields characterized by eqns (4) and (5).

The transduction process relating the electrical and mechanical states can be described with two coupled linear equations. These canonical equations are expressed in terms of mechanical parameters (force F, velocity v, mechanical impedance Zm), electrical parameters (applied voltage V, current I, electrical impedance Ze), and the two transduction coefficients:

(7)$V={Z}_{e}I+{T}_{em}v$

(8)$F={T}_{me}I+{z}_{m}v$

Common magnetostrictive transducer components include a magnetic circuit, a solenoid for transduction of magnetic-to-electrical energy and vice versa, mechanisms for DC magnetic and mechanical (prestress) biasing. A generic magnetostrictive device configuration is shown in Figure 2. The permanent magnet provides a DC magnetic field. The magnetic circuit passes though the magnetostrictive driver, end caps made of magnetic materials, and the permanent magnet. Belleville spring washers are used to provide an initial mechanical prestress. The solenoid can be used to provide both AC and DC magnetic fields for actuation purposes. Alternatively, for sensing applications, the change in voltage induced in the solenoid is to detect a change in strain and/or in the force applied to the device.

Figure 3 depicts a major hysteresis loop from an actuator similar in design to that shown in Figure 2. Superimposed on the major hysteresis loop are five minor hysteresis loops collected by driving at 0.7 Hz with an applied of ±5 kA m−1as the DC magnetic bias was increased from 5 to ±45 kA m−1 in increments of 10 kA m−1. AC operation of magnetostrictive actuators is typically achieved through the use of a DC magnetic field to provide bias operation so that it is centered about the steepest portion of the major hysteresis loop. This region is called the burst region, and the DC magnetic field amplitude required for operation about the middle of the burst region is called the critical field. All subsequent data are from magnetically biased transducers.

Figure 4 shows frequency response functions of acceleration per input current, where the change in the transducer's axial resonant frequency varies from 1350 Hz to over 2000 Hz, reflecting the effect of DC bias on the elastic modulus. The reduction in elastic modulus below that at magnetic saturation is known as the delta E effect. Note that in Figure 4 only the axial resonant frequency of the magnetostrictive driver shifts and that the structural resonance of the device housing at 3300 Hz is not affected by the changing DC field.

Figure 5 illustrates the sensitivity of major strain-applied field hysteresis loops to variations in mechanical load or prestress. While this attribute allows tailoring of devices preloaded for optimized performance under a constant load (such as acoustic source applications), this introduces a parameter that must be optimized for operation under variable load conditions which are typical of vibration control applications. Actuators using tailored magnetostrictive composite drivers can minimize device sensitivity to variations in the external load.

The upper traces in Figures 6–8 are Bode plots of strain per applied field, strain per magnetization and magnetization per applied field, respectively. The Bode plots were obtained using a swept sinusoidal signal at a relatively low signal excitation of the device (e.g., driving the device to produce maximum strains at resonance that are less than one-third of the device full strain potential, thereby minimizing the presence of undesired harmonics). The lower traces are minor hysteresis loops of these quantities recorded at (from left to right) frequencies of 10, 100, 500, 800, 1000, 1250, 1500, and 2000 Hz.

Information on performance is used to implement calibration, input and/or output linearization, and control schemes to facilitate the optimized use of magnetostrictive devices. For example, information from Figures 3 and 5 might be coupled to optimize the DC magnetic bias in real time to produce specified strains under varying mechanical loads. Figure 4 demonstrates the ability to tune a system's resonant frequency in real time, which is the basis for a patent pending, tunable magnetostrictive vibration absorber design. This can be coupled with information on frequencies that minimize losses by using hysteresis loop data from Figures 6–8 to tailor the frequency operation for the most efficient electromechanical performance. Additionally, this suggests the ability to target operating conditions for minimization of internal heating under continuous operation, which is of particular concern for ultrasonic operation.

Another significant loss factor that is associated with the transduction of electric to magnetic energy under dynamic operation is eddy currents. Eddy current power losses increase with approximately the square of frequency and thus have a significant impact on the operational bandwidth of devices. Laminations in the magnetostrictive core help to mitigate the effects of eddy currents, however, materials such as Terfenol-D are brittle and costly to laminate. Materials such as insulated magnetic particles or the silicon steels in common use in motors and power systems are suitable for the magnetic circuit components that make up the transducer housing, as they simultaneously support flux conduction and offer high resistivities. Magnetostrictive composites that use nonelectrically conducting matrix material yield reductions in eddy current losses. They have been proposed for extending device output bandwidth by an order of magnitude, from roughly 10 kHz to close to 100 kHz. Such composites offer great promise as high-frequency magnetostrictive drivers.