Illustrated Sourcebook of Mechanical Components Part 6 pps

Generally speaking, Belleville springs are better suited than helical compression springs where longitudinal space is limited, in other words, where space for solid height is limited an

Robert 0 Parrnley

Add rubber befwsen

wosbers for gr@power

Let these ideas spur your own design creativity Sometimes commercial

Belleville washers will suit; other wise you can easily dish your own

0 will rotate if holding screw is shouldered

here for non-slipp;ng

non-s//p arrangement/

4, V-BELT PULLEY

Rubber sur face adhesive A

Alternate: Profecf corners and

11-12

Robert 0 Parmley

Pulley assembly

Mounting spring stabilizer

(Nut and washer

Force

Retain tapered coil spring

Knurled adjustment

~elleviIle springs are a versatile component that offer a wide range of applications

There are many places where these components can be used and their availability

as a stock item should be considered when confronted with a design problem that

requires a fast solution

11-14

New equations give stress-energy relationships and lead to the unusual arrangement

of nesting Bellevilles one inside the other

INCE 1867, when M i e n F Belfeville was issued a

S patent in Paris for the invention, Belleville springs

have never ceased to find wide application Now, more

and more, they are being used to absorb high impact

energies With the arrangements and new design

equations given here, you can easily design for maximum

energy-absorbing capacity

Also called conical disk springs, Belleville springs are

actually no more than conical washers They are very

compact, which leads to first of several advantages:

@ T h e y can absorb a large amount of energy at high

loads and with a comparatively short working stroke

They can return to the system practically all the energy

they absorb during the compression or impact stroke

With ring springs, in contrast, approximately 50% of

the input energy is dissipated as heat

Their load-deflection characteristics can be altered

simply by adding or removing individual washers, by

stacking washers in various parallel-series combinations,

1 Four arrangements of

Belleville springs The nested design

and by nesting them inside larger washers, Fig 1

Generally speaking, Belleville springs are better suited than helical compression springs where longitudinal space is limited, in other words, where space for solid

height is limited and where a helical spring would result

in a small index (the ratio of the mean coil diameter t o the wire diameter)

Design recommendations

You don’t need a maze of nomographs and tables to

calculate the impact load a Belleville spring must absorb With these new design equations, given below, you can predict the amount of induced stress directly But first, some important findings:

Nested springs-ne washer inside another-require

no more space than single springs and reduce the maxi-

mum stress by 14%

One-parallel-series arrangements of Belleville springs are more efficient energy absorbers than two- or three-parallel

has highest efficiency for absorbing

energy Contrary to popular opinion,

the one-parallel design is more efficient

than the two- or three-parallcl designs

series - contrary to some popular misconceptions

0 Final working stress is directly proportional to the

square root of the energy capacity and inversely propor-

tional to the outside diameter and square root of the

solid height

Oneparallel-series equations

To simplify analysis, it is assumed that the minimum

working height is equal to the solid height and that there

is no precompression for assembly Thus the total deflec-

tion is

Fs = H F - Hs

For symbols, see Box on page 93

The height-thickness ratio, B = h / t , determines the

shape of the load-deflection curve By varying the B

values, it is possible to obtain a wide variety of load-

deflection curves, Fig 2 The curves are plotted against the

deflection in terms of height as the spring washers are com-

NESTED ARRANGEMENT I

for most ratios of dish height (deflec-

tion to flat) to metal thickness

11-16

pressed from free height to solid height

ville springs are

Load

The conventional load and stress formulas for Belle-

where the constants C,, C, and Y are given by the equa-

The stress equation, Eq 2, givcs the value of the com-

pressive stress which occurs on the convex side at the inner diamzter The stress h2s a maximum value when

fi = h Hen-e the maximum (final) stress from Eq 2 is

The energy stored in one washer compresszd from

free height to the flat position is obtained by integration

of Eq 1:

-

For an assembly of N washers in series

Keeping in mind that H , = Nt and B = h / t , Eq 5

is rewritten to read

Z ( l - Q') YD,'

Combining Eq 3 and 6 gives the relationship between

final stress and energy capacity:

For the usual spring materials where E = 30 X loR

psi and Q = 0.3, the final stress is

For all practical purposes, the stress at solid height,

which is the final stress, can be chosen at the permissible

stress-the maximum working stress for the spring

text continued, page 94

Di = Inside diameter, in

Do = Outside diameter, in

E = Modu-us of elasticity, psi

E X = Energy capacity, in.-lb

F = Deflection, in

F S = Total deflection, in ( F S = H F - - H X )

h = Dish height, in

H F = Free height of spring assembly, in

H = Solid height of spring assembly, in

ss = Final stress, psi

t = Thickness of washer, in

Y = Constant, see equation or chart for

For one-parallel series

11-18

This is particularly true for critical applicalions where

spring space is limited and loading is of an impact nature

In the design of Belleville springs, a main consideration

is to keep the final stress at a safe and reasonable level

Eq 8 shows that the final stress is inversely proportional

to the outside diameter and the square root of the solid

height; therefore, we make these two values as large as

space requirements allow Doubling the value of the

outside diameter (keeping all other variables constant)

will result in a 50% reduction in the final stress A

similar increase in the solid height will produce a 30%

reduction Eq 8 further shows that the final stress is

directly proportional to the square root of the energy

capacity

The values for the outside diameter, solid height, and

energy capacity are usually given within narrow limits for

a particular application The outside diameter is deter-

mined by the hole diameter into which the spring mu d

fit The value of the solid height is dictated by the

minimum working height and the energy capacity is pre-

scribed by functional considerations

The question now remains, what ratios of A = O D / ID

and B = h / t will result in the minimum final stress?

Visual examination of Eq 8 does not readily show the

stress effect of the two ratios However, by simplifying

Eq 8, a series of design curves, Fig 4, is obtained in

which the final stress factor is plotted wiah respect to

ratio A Ratio B acts as a parameter For this chart,

Eq 8 becomes

where S’, is called the variation index of the final stress

It can be seen from Fig 5 that the final stress is at a

minimum when the diameter ratio A = 1.7 for all values

of B Also, for a given B value, the final stress increases

at most by 3% in bhe range 1.5 4 A 6 2.0 Therefore,

we recommend that the diameter ratio should be kept

within the range of 1.5 to 2.0

Note also from Fig 4 that in the favorable diameter

ratio range the final stress increases with increasing values

of B This condition is true except for the height-thickness

ratio of B = 3, for in the range of A = 1.5 to A =

2.0, the final stress for B = 3 is less thm that for

smaller B values (of 1 < B L 2) Belleville washers,

however, have \been generally designed for energy ca-

pacity with B values less than unity because frequently

a short work stroke and a high degree of stability are

required

Example lane-parallel design

A set of Belleville springs, grouped into a oneparallel

series arrangement, are to absorb the recoil of a rifle

bolt The given requirements are:

Outside diameter, D o = 0.900 in

Solid height, H8 = 2.035 in

Stroke, F8 = 0.407 in

Energy-absorption requirement, EN = 100 in.-lb

Material = alloy steel, AIS1 6150

Step 1: Select the diameter ratio Based on the pre-

vious recommendations, a ratio of A = O D / l D = 1.7

is chosen Therefore from Fig 3 (or, for more accuracy,

from the equations of the constants) :

C, = 1.15, C, = 1.26, Y = 0.61

Step 2: Determine the height-thickness ratio I n this

case a stroke of 0.407 is required, which means that

Step 4: Calculate the thickness, t If the stress value

of 222,000 psi is acceptable, the thickness t can now

be calculated from either Eq 3 or 6

From Eq 3:

Ski, 5: Dctcrminc thc dish height, I?:

I& = B1 = 0.3(.055) = 0.011 itb

Step 6: Determine the number of washers:

The complete design data for this spring are listed

in the second column in Table 1 The data will be com- pared later to a nested arrangement

G :

I

0 2 0 30 4 0

Dicrneter ratio, A:OD/ID

4 STRESSES I N one-parallel Bellevilles

Consider the point A = 2.0 and S'# = 1.07 in Fig 4

The two curves B = 1 and B = 3 intersect at this point

which means that two spring designs can occupy the

same spring space (in other words, with the same Ha and

Do values), end have equal energy capacities and final

stresses-however, their total travel, FB, will be in the

proportion of 3 to 1 To illustrate this point, consider

the two springs listed in Table I1 with the same values for:

Outside diameter, Do = 2.300 in

Inside diameter, Di = 1.150 in

Total travel, k's 6 3.000 in

Energy requirement = 342 in lb

The two springs have been desigded to B = 1 and

B = 3 Final stress and energy capacity of both designs

are equal, but the springs differ in total travel and

numfber of washers The load-deflection diagram for

each design is shown in Fig 5 The energy capacities for

the springs are represented by the areas under the curves

The areas are equal to each other Note that there is an

energy content common to both designs The total travel

for design B (4.95 in.) is three times that of design A

(1.65 in.)

Nested arrangements

Again, to simplify the analysis, it is assumed that

there is no diametral clearance between the nested springs

To have a meaningful comparison, both assemblies are

designed t o the same values for:

.Energy capacity, EN = ENo -t EN' (where super-

TABLE I SINGLE VS NESTED ARRANGEMENTS

Thickness, in

Dish height, in

Number of washers

Diameter ratio

Outside diameter, in

Inside diameter, in

Height-thickness ratio

Stroke, in

Solid height, in

Energy capacity, in.-lb

Material, AIS1

Final stress, psi

scripts o and i purtain to the oulcr and inncr springs,

For outer spring

For inner spring

For efficient design, the stress in the nested arrange- ment should be equally distributed, thus Sao = &' There- fore Eq 9 and 10 result in

E N = 1.346 E N o (12)

From the basic assumption that EN = ENo + EN', and

from Eq 11, it follows that

The relationship between the final stress of the single springs to that of the nested spring is obtained from Eq

0.055 0.01 1

37 1.7 0.900 0.530 0.20 0.407 2.035

100

6150 222,000

NESTEO ARRANGEMENT OUTER SPRING INNER SPRING

0.051 0.0102

40 1.7 0.900 0.530 0.20 0.408 2.040

74

6150 191,000

0.030 0.006

68 1.7 0.530 0.312 0.20 0.408 2.040

26

61 50 191,000

11 -20

that i s gained by the substitution of a nested arrange-

ment for a single spring is

The stress reduction is constant and applies generally

because it is independent of the solid height and outside

diameter of the single spring

I'xample >Nested design

Assume that the final stress of 222,000 psi in the first

cxample is excessive and must be reduced The use of a

riested arrangement will decrease the stress by 31,000 psi

to 191,000 psi A nested arrangement that is equivalent

to the first spring in energy and space conditions is easily

computed with the aid of Eq 3 and Eq 9 through 13 The

design data are listed in Table I

The solid height and total stroke of the nested design

are not exactly equal to those of the single spring because

the number of washers in each spring has to be a whole

number However, their differences are negligible, and

for comparison purposes the height and stroke are con-

sidered equal Note that the individual energy capacities

total 100 in.-lb in Table 1

Other parallel-series arrangements

A comparison is now made of a one-parallel series

with two-parallel and three-parallel series Again, for a

vlilid comparison, all spring assemblies have the same

The energy stored in the washers upon compression

,from free to solid height is

The energy stored in ,a two-parallel series, as shown

in Fig 1, with N , pairs of washers is

Combining Eq 3A and 6 A gives the following expres-

sion for the two-parallel series:

(I -Q2) 2 E H s E x (Rz2f4) Y )"[ c, 2 + C2] (7A)

The relationship of the final stress of the one-parallel

series to the final strew of two-parallel series, obtained

from Eq 7 and 7A, is

From the given condition that both assemblies should

have equal strokes, it follows that

Therefore, Eq 14 is rewritten as

L

A graph of the stress ratios for one and two-parallel

series arrangements are shown in Fig 7 Note that in

the practical range -of B , ( B 1), the one-parallel series

is more efficient &an the two-parallel series This is

particularly true for B values between 0.3 and 0.6 where

ail 8% stress savings can be realized

TABLE 11 .SAME PERFORMANCE - DIFFERENT TRAVEL

Diameter ratio

I

Final stress, psi 21 8,000

Energy capacity, in.-lb 1 342

SPRING

B

2.300 1.1 50 0.075 0.025 3.0 2.0

11 -22

Similarly, the h a 1 stress of the three-parallel series

(shown in Fig 1) is

In combination with Eq 7, and the fact that both

assemblies have equal strokes, it follows that the ratio

of stresses of one-parallel to three-parallel design is

L

-This equation is also plotted in Fig 6 Note that in

the practical B range, the one-parallel series offers a

better utilization of spring space than a three-parallel

series For example, two spring assemblies that corre-

spond to the points B = 0.4 and Ss/S, = 0.87 (a one-

parallel series, with B = 0.4, and a three-parallel series,

with B, = 1.2) will have equal strokes and energy ca-

pacities and occupy the same space package The final

stress of the one-parallel series, however, will be 13%

less This comparison is shown in Table III

G e d load-deflection formulas

load-deflection ( P / F ) calculations are

Formulas that can be used to good advantage for

For the usual spring materials where E = 30 X 10"

psi and Q = 0.3, the above equations are reduced to

The formulas are more convenient to use than Eq 1

and are acceptably 'accurate for B values less than dr

equal to unity, where the rate is essentially linear, as

can be seen from Fig 2

Other design recommendations

To simplify the analysis, it was assumed that there

was no initial spring compression and also that there was

no clearance between minimum operating height and

solid height However, in actual practice, it is recom-

mended that a small precampression be applied to pre-

vent looseness and that clearance be provided to avoid

loading to flat position The two recommendations are

easily satisfied by designing for a total energy capacity

slightly larger than actually required

Stress values given by Eq 2 and 8 are localized stresses

that occur at the inner diameter and not throughout the

entire cross section Therefore, caloulated stress values

may at times exceed the yield point of the spring mate-

rial and yet be permissible

TABLE 111 ONE-PARALLEL VS THREE-PARALLEL ARRANGEMENTS

Number of individual washers Outside diameter, in

Inside diameter, in

Diameter ratio, Height Thickness, in

Height-thickness ratio, Stroke, in

Solid height, in

Energy capacity, in.-lb Final stress, psi

ONE-PARALLEL SERIES

26 1.87 1.10 1.7 0.034 0.085 0.4 0.884 2.21

600 305,000

THREE-PARALLEL SERIES

48 1.87 1.10 1.7 0.055 0.046 1.2 0.884 2.21

600 266.000

SEM Applications

N Dale tong

w h e n a split lockwasher is called for in a screw fastening,

a flat washer is invariably necessary The ways of assembling

them illustrated below are strict requirements in military

specifications-especially for electronic equipment Corn- shown here can be depended on to pay off

mercial requirements usually vary-depending upon either the designer’s decision or product-cost restrictions For good quality and reliable service, however, the fastening methods

ASSEMBLY OF FLAT WASHERS AND SPLIT LOCKWASHERS Flut washers should be placed between

Nome fa/ maieriol / F / o i washer

11 -24

A standard off-the-shelf item with more uses than many ever considered

1 Coil spring stabilizer and compression brake

Cupped washers

4 Simple pulley and roller

I 1

Cupped washer)’

CURRed washers [sectioned

Plostic stem

9 Protection for step shoulders

Retaining Rings Aid Assembly, I

Retaining Rings Aid Assembly, II

Coupling Shafts with Retaining Rings

12-20 12-22 12-24

Energy Absorber Squeezes Rings to Cushion Shocks

Defection of Perpendicularly Loaded Split Circular Rings

Improve Design with Retaining Rings

RETAIN COMPONENTS on diecastings with

a simple-to-use grip ring Slipped over the

end of the shaft, the ring exerts a frictional

hold against axial displacement of the shaft

r -i

7 -

SHOULDER AND NUT are replaced by two

retaining rings A flat ring replaces the shoul- der, while a bowed ring holds the component

on shaft f o r resilient end-play take-up,

THREADED INTERNAL FASTENERS are costly because of expensive internal thread- ing operation Simplify by substituting a self-

locking retaining ring-see lower drawing

COVER-PLATE ASSEMBLY has been re-

designed (lower drawing) to avoid use of

screws and machined cover-plate Much thin-

ner wall can be used-no drilling or tapping

Free ring

Groove detail Roller Roller axle

Heavy-duty Two types of rings may be used on one assembly Here permanent-shoulder rings

provide a uniform axle step for each roller, without spotwelding or the like

Heavy-duty rings keep the rollers i n place

a casting with cored hole is secured with two rings: 1-spring-like ring has high thrust capacity, eliminates springs, bow washers, etc; 2-reinforced E-ring acts as a retaining shoulder or head Each ring can be dismantled with a screwdriver

~ ~ i l Triangular retaining nut eliminates the ~ ~ ~ ~ ~ ~ g

need for tapping mounting holes and using a large nut and washer Secure mounting of small motors and devices can be obtained i n this manner

These three examples show self-locking retaining rings used as adjustable

,stops on support members (pins made to commercial tolerances): A-external

riug provides positive grip, and arched rim adds strength; B-ring is

adjustable in both directions, but frictional resistance is considerable,

and C-triangular ring with dished body and three prongs will resist extreme

thrust Both A and C have one-direction adjustment only

when the vacuum is released, thus providing a support

during the "off" cycle Air or liquid is released when ball

is at rest and exits through the areas between the grip

points of the ring, which is adjustable at entry position

s+em -.m

Weight disc, I I

Ring

-Rubber stopper with internal threaded sleeve Drain hole

Triangular retainer nut positions and unifies components of

the tank drain assembly The triangular nut eliminates the

need for a large standard nut and lockwasher or spring-

type component and simplifies the design

Tamper-proof lock for a shaft in a housing provides location of the shaft and at the same time retains the key Heavy axial loading and permanent retention of the key are double values in this application Ring half\

Observation lid

"I*

u Ring half Observation lid on tubing makes it

possible t o inspect wiring at will

The two-part balanced retainer ring has identical semicircular halves, which are held together by the interlocking prongs at the free ends