10.Mechanica..l Springs

8/10/2019 10.Mechanica..l Springs

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Chapter 9

MechanicalSprings



ME Introduction

- When a designer wants rigidity , negligible deflection is an acceptableapproximation as long as it does not compromise function. Flexibilit y issometimes needed and is often provided by metal bodies with cleverly controlledgeometry.

- These bodies can exhibit flexibility to the degree the designer seeks. uchflexibility can be linear or nonlinear in relating deflection to load. These devicesallow controlled application of force or tor!ue. "lexibility allows temporarydistortion for access and the immediate restoration of function.

- prings are mass-produced #and therefore low cost$, and ingeniousconfigurations have been found for a variety of desired applications.

- %n general, springs may be classified as wire springs, flat springs, or special-

shaped springs. Wire springs include helical springs of round or s!uare wire,made to resist and deflect under tensile, compressive, or torsional loads. Flat springs include cantilever and elliptical types, wound motor- or clock-type powersprings, and flat spring washers, usually called Belleville springs .



ME Introduction

Closed wound spring Compression spring



ME Introduction

Torsion spring Extension spring



ME Introduction

Flat spring



ME Introduction

Micro spring


http://slidepdf.com/reader/full/10mechanical-springs 7/149ME Overview

10-1 Stresses in Helical Springs10-2 The Curvature &ffect

10-3 'eflection of (elical prings

10-4 Compression prings10-5 tability

10-6 pring )aterials

10- (elical Compression pring 'esign for tatic ervice

10-! Critical "re!uency of (elical prings10-" "atigue *oading of (elical Compression prings

10-10 (elical Compression pring 'esign for "atigue *oading

10-11 &xtension prings

10-12 (elical Coil Torsion prings

10-13 +elleville prings

10-14 )iscellaneous prings

10-15 ummary


http://slidepdf.com/reader/full/10mechanical-springs 9/149ME Stresss in Helical Springs

t the inside fiber of the spring. ubstitution of

τ max / τ , T / FD 01, r / d 01, J / 2 d 3041, and A / 2 d 103 gives 4 1

5 3 FD F d d

τ π π

= +

6ow we define the spring index C which is a measure of coil curvature #for mostsprings, C ranges from about 4 to 1 $. D

C d

=(ence 4

5 s

FD !

d τ

π =

where % s is a shear"stress correction !actor and is defined by the e!uation

1 71 s

C !

C +=

- The use of s!uare or rectangular wire is not recommended for springs unlessspace limitations make it necessary.

- prings of special wire shapes are not made in large !uantities, and may not beas strong as springs made from round wire.



10-1 tresses in (elical prings

10-2 &he 'urvature (##ect10-3 'eflection of (elical prings










10-15 ummary


http://slidepdf.com/reader/full/10mechanical-springs 11/149ME &he 'urvature (##ect

- The e!uations in the previous topic are based on the wire being straight.(owever, the curvature of the wire increases the stress on the inside of the spring

but decreases it only slightly on the outside.

- This curvature stress is primarily important in fatigue because the loads arelower and there is no opportunity for locali8ed yielding. "or static loading, thesestresses can normally be neglected because of strain-strengthening with the firstapplication of load.

- %t is necessary to find the curvature factor in a roundabout way

- uppose ! s is replaced by another ! factor, which corrects for both curvatureand direct shear . Then, this factor is given by

3 13 4 B C !

C += −

&ergstr'sser !actor Wahl !actor


http://slidepdf.com/reader/full/10mechanical-springs 12/149ME &he 'urvature (##ect

(ence, the largest shear stress will be predicted by

45

B FD ! d

τ π

=

- ince the results of these two e!uations differ by less than 7 percent, the+ergstr sser is preferred. The curvature correction factor can now be obtained bycanceling out the effect of the direct shear.

6ow, ! s, ! B or ! " , and ! c are simply stress correction factors applied

multiplicatively to Tr 0 J at the critical location to estimate a particular stress. Thereis no stress concentration factor.




10-2 The Curvature &ffect

10-3 )e#lection o# Helical Springs










10-15 ummary


http://slidepdf.com/reader/full/10mechanical-springs 14/149ME )e#lection o# Helical Springs

- The deflection-force relations are !uite easily obtained by using Castigliano:stheorem.

ubstituting T / FD 01, l / 2 D# , J / 2 d 3041, and A / 2 d 103 results in

1 1

1 1T l F l

$ %J A%

= +

1 4 1

3 1

3 1 F D # F D# $

d % d %= +

- The total strain energy for a helical spring is composed of a torsional componentand a shear component.

where # / # a / number of active coils .


http://slidepdf.com/reader/full/10mechanical-springs 15/149ME )e#lection o# Helical Springs

Then using Castigliano:s theorem to find total deflection y gives

4

3 15 3$ FD # FD# &

F d % d %∂= = +∂

ince C / D0d , so

4 4

3 1 35 7 57

1 FD # FD # &d % C d %

= + ÷ B

The spring rate, also called the scale of the spring, is ' / F 0 &, and so

3

45d %

' D #

B

≅

≅






10-4 'o*pression Springs10-5 tability









10-15 ummary


http://slidepdf.com/reader/full/10mechanical-springs 17/149ME 'o*pression Springs

- There are four types of ends generally used for compression springs.- spring with plain ends (a) has a non-interrupted helicoid the ends are the sameas if a long spring had been cut into sections.- spring with plain ends that are s*uared or closed (b) is obtained by deformingthe ends to a 8ero-degree helix angle.- prings should always be both s*uared and ground (c) for a better load transfer.


http://slidepdf.com/reader/full/10mechanical-springs 19/149ME 'o*pression Springs



ME 'o*pression Springs

- Table 7;<7 shows how the type of end used affects the number of coils and thespring length.

Formulas !or the dimensional characteristics o! compression"springs$( a umber o! Acti*e Coils)



ME 'o*pression Springs

- "orys pointed out that s!uared and ground ends give a solid length + s of

+ s $ + t , a d

where a varies, with an average of ;.=>, so the entry d# t in Table 7;<7 may beoverstated.

- ,et removal or presetting is a process used in the manufacture of compressionsprings to induce useful residual stresses. %t is done by making the spring longerthan needed and then compressing it to its solid height. This operation sets thespring to the re!uired final free length and, since the torsional yield strength has

been exceeded, induces residual stresses opposite in direction to those induced inservice.

- prings to be preset should be designed so that 7; to 4; percent of the initialfree length is removed during the operation. %f the stress at the solid height isgreater than 7.4 times the torsional yield strength, distortion may occur. %f thisstress is much less than 7.7 times, it is difficult to control the resulting freelength.



ME Overview




10-4 Compression prings10-5 Sta.ilit/10-6 pring )aterials








10-15 ummary



ME Sta.ilit/

- Compression coil springs may buckle when the deflection becomes too large.The critical deflection is given by the e!uation

7 1

1; 7 1

eff

7 7cr C & + C λ

′′ = − − ÷

where ycr is the deflection corresponding to the onset of instabilit& .

- The !uantity - eff is the effective slenderness ratio and is given by the e!uation

;eff

+ D

α λ =

- C 1 and C 2 are elastic constants defined by the e!uations

( )7 1 .

C . %

′ = −( )1

1

1

1

. %C

% .

π −′ = +



ME Sta.ilit/

- , is the end/condition constant . This depends upon how the ends of the springare supported.

- Table 7;<1 gives values of 0 for usual end conditions. 6ote how closely theseresemble the end conditions for columns.

End"condition constants , !or helical compression springs-



ME Sta.ilit/

11 eff C λ ′

( )7 1

;

1

1

. % D +

% . π α

− < +

- bsolute stability occurs when, in &!. #7;<7;$, the term is greater than

unity. This means that the condition for absolute stability is that

- "or steels, this turns out to be

- "or s!uared and ground ends 0 / ;.> and

; 1.@4 D

+α

<

; >.1@ + D<



ME Overview





10-6 Spring Materials10- (elical Compression pring 'esign for tatic ervice

10-! Critical "re!uency of (elical prings

10-" "atigue *oading of (elical Compression prings






10-15 ummary



ME Spring Materials

- prings are manufactured either by hot- or cold-working processes, dependingupon the si8e of the material, the spring index, and the properties desired.

- %n general, pre-hardened wire should not be used if D0d A 3 or if d B @ mm.

- Winding of the spring induces residual stresses through bending, but these arenormal to the direction of the torsional working stresses in a coil spring .

- uite fre!uently in spring manufacture, they are relieved, after winding, by amild thermal treatment.

- great variety of spring materials are available to the designer, including plaincarbon steels, alloy steels, and corrosion-resisting steels, as well as nonferrousmaterials such as phosphor bron8e, spring brass, beryllium copper, and variousnickel alloys.



ME Spring Materials

- 'escriptions of the most commonly used steels will be found in Table 7;<4.

.igh"Carbon and Alloy /pring /teels



ME Spring Materials




ME Spring Materials

- The values of intercept A and the slope m of the line have been worked out fromrecent data and are given for strengths in units of kpsi and )Da in Table 7;<3.

Constants A and m o! / ut

A0d m !or Estimating MinimumTensile /trength o! Common /pring Wires



ME Spring Materials

Mechanical 1roperties o! /ome /pring Wires



ME Spring Materials

- Hoerres uses the maximum allowable torsional stress for static application shownin Table 7;<@. "or specific materials for which you have torsional yieldinformation use this table as a guide.

- Hoerres provides set-removal information in Table 7;<@, that , s&

G ;.@>, ut

increases strength through cold work.

Maximum Allowable Torsional /tresses (/ sy ) !or .elical Compression /prings in /tatic Applications



ME Spring Materials

- ome correlations with carbon steel springs show that the tensile yield strengthof spring wire in torsion can be estimated from ;.=> , ut .

- The corresponding estimate of the yield strength in shear based on distortionenergy theory is

/ sy $ 0 5 +0 5 / ut $ 0 433 / ut 0 45 / ut

- amInov discusses the problem of allowable stress and shows that

for high-tensile spring steels, which is close to the value given by Hoerres forhardened alloy steels.

all ;.>@ s& ut , , τ = =



ME Spring Materials



ME Spring Materials

DC d =

45 s FD ! d τ π =

Bergstr2sser factor

music wire without set remo*ed

static load corresponding totorsional yield strength

10-5



ME Spring Materials

10 1 t i ( li l i



ME Overview




10-4 Compression prings

10-5 tability


10- Helical 'o*pression Spring )esign #or Static Service10-! Critical "re!uency of (elical prings







10-15 ummary



ME Helical 'o*pression Spring )esign #or Static Service

- The preferred range of spring index is 4 C 12 , with the lower indexes beingmore difficult to form #because of the danger of surface cracking$ and springswith higher indexes tending to tangle often enough to re!uire individual packing.

- This can be the first item of the design assessment.

- The recommended range of active turns is 3 a 15 .

- helical coil spring force-deflection characteristic is ideally linear. Dractically, it

is nearly so, but not at each end of the force-deflection curve.

- The spring force is not reproducible for very small deflections, and near closure,nonlinear behavior begins as the number of active turns diminishes as coils beginto touch.

- The designer confines the spring:s operating point to the central => percent ofthe curve between no load, F / ;, and closure, F / F s . Thus, the maximumoperating force should be limited to F *a %!F s .




- 'efining the fractional overrun to closure as 6 , where ( ) max7, F F ξ = +

it follows that ( ) ( )max

=7 7

5, , F F F ξ ξ = + = + ÷

- "rom the outer e!uality 3 / 70= / ;.734 J ;.7>. Thus, it is recommended that3 G ;.7>.

- %n addition to the relationships and material properties for springs, we now havesome recommended design conditions to follow, namely

3 71

4 7>

;.7>

7.1

a

s

C

#

n

ξ

≤ ≤≤ ≤≥≥

where n s is the factor of safety at closure #solid height$.




- When considering designing a spring for high volume production, the figure ofmerit can be the cost of the wire from which the spring is wound. The fom #figureof merit$ would be proportional to the relative material cost, weight density, and

volume

- "or comparisons between steels, the specific weight can be omitted.

- pring design is an open-ended process. There are many decisions to be made,and many possible solution paths as well as solutions.

- %n the past, charts, nomographs, and Kspring design slide rulesL were used bymany to simplify the spring design problem.

- Today, the computer enables the designer to create programs in many differentformats-direct programming, spreadsheet, ) T* +, etc.

( )1 1

fom relative material cost3

t d # Dγπ = −




- Commercial programs are also available. There are almost as many ways tocreate a spring-design program as there are programmers. (ere, we will suggestone possible design approach.

Design /trategy

- )ake the a priori decisions, with hard-drawn steel wire the first choice #relativematerial cost is 7.;$.

- Choose a wire si8e d . With all decisions made, generate a column of parametersd , D, C , M' or %', # a , + s , + ; , # + ; $cr , n s , and fom.

- +y incrementing wire si8es available, we can scan the table of parameters andapply the design recommendations by inspection.




Design /trategy

- fter wire si8es are eliminated, choose the spring design with the highest figureof merit.

- This will give the optimal design despite the presence of a discrete designvariable d and aggregation of e!uality and ine!uality constraints.

- The column vector of information can be generated by using the flowchartdisplayed in "ig. 7;<4.

- %t is general enough to accommodate to the situations of as-wound and set-removed springs, operating over a rod, or in a hole free of rod or hole.




Design /trategy

- %n as-wound springs the controlling e!uation must be solved for the spring indexas follows.

- "rom &!. #7;<4$ with $ / sy%n s C $ D%d ! B from &!. #7;<>$, and &!. #7;<7=$,

( ) max4 4

5 75 3 13 4

s& s B

s

, F D F D C !

n d C d

ξ π π

+ += = −

- *et s&

s

,

nα = ( ) max

1

5 7 F

d

ξ β

π +=

- ubstituting 0 and 5 and solving for C from !uadratic e!uation. The larger of thetwo solutions will yield the spring index. Thus

11 1 4

3 3 3C

α β α β α β β β

− −= + − ÷




Design /trategy




A-28




/ sy const(A)0d m

11 1 4

3 3 3C


− −= + − ÷

3 13 4 BC

! C

+= −




( ) max4 4

5 75 3 13 4

s& s B

s

, F D F D C !

n d C d

ξ π π

+ += = − τ 6

7D D2d

a 8d 9 ymax 0(:D ; F max )

t a 23

+ s d t

+ < + s 2 (=2 ξ )ymax

(+ < )cr 3$>;D0

( )1 1

fom relative material cost3

t d # Dγπ = −relative cost: Table 10-4




- There are vendors who stock literally thousands of music wire compressionsprings. +y browsing their catalogs, we will usually find several that are close.

- )aximum deflection and maximum load are listed in the display ofcharacteristics.

- pring rates may only be close. t the very least this situation allows a smallnumber of springs to be ordered Koff the shelfL for testing.

- The decision often hinges on the economics of special order versus theacceptability of a close match.




Closed length o spring ! ree length o spring

3 1

3 4 B

C !

C

+=−

"q# $%-&%

'il quenching( temperedsteel

)).*




'+(T wire

ut m

A,

d =

Tempered carbon steel ire! s" # 0.50! $t

0.5

"q# $%-

8

( ) 4

5 B B

FD !

d τ

π =

( ) 1 11

1

3 7 1

3 3 B

C r ! C

C d −= =− "q# $%- )

Table 10-%

Table 10-6

/ 0*1 /&0*1




( ) 4 1

7@ 3 A A

D F !

d d σ

π π = +

( )( )

17 7 7

77 7

3 7 1

3 7 A

C C r ! C

C C d − −= =−

$%-

i F F '&= +

/ 0*1 - /&0*1




1art =? pull 4nob part 3? tapered retaining pin part ;?hardened bushing with press !it part 9? body o!

!ixture part @? indexing pin part >? wor4piece holder$ /pace o! the spring is @0: in 7D? =09 in D? and =(;0:)in long? with the pin down as shown$ The pull 4nobmust be raised ;09 in to permit indexing$




+ < space available in length




A-26



ME Overview




10-5 tability



10-! 'ritical 7re8uenc/ o# Helical Springs10-" "atigue *oading of (elical Compression prings




10-13 +elleville prings10-14 )iscellaneous prings

10-15 ummary



ME 'ritical 7ren8uenc/ o# Helical Springs

- The solution to this e!uation is harmonic and depends on the given physical properties as well as the end conditions of the spring.

- The harmonic, natural : fre!uencies for a spring placed between two flat and parallel plates, in radians per second, are

7,1,4,...'g

m m"

ω π = =

where the fundamental fre!uency is found for m / 7, the second harmonic for m /1, and so on. We are usually interested in the fre!uency in cycles per secondsince ; 6 < f , we have, for the fundamental fre!uency in hert8 #assuming thespring ends are always in contact with the plates $,

71

'g f

" =

%f the spring has one end against a flat plate and the other end free or when one



ME 'ritical 7ren8uenc/ o# Helical Springs

%f the spring has one end against a flat plate and the other end free or when oneend is against a flat plate and the other end is driven with a sine-wave motion, thefre!uency is

The weight of the active part of a helical spring is

73

'g f " =

( ) ( )

1 11

3 3a

a

d D# d " A+ D#

π γ π γ π γ = = =

where is the specific weight.

- The fundamental critical fre*uenc& should be greater than =@ to 3< times the fre*uenc& of the force or motion of the spring in order to avoid resonance with theharmonics .

- %f the fre!uency is not high enough, the spring should be redesigned toincrease 4 or decrease W




ME Overview




10-5 tability




10-" 7atigue 9oading o# Helical 'o*pression Springs10-10 (elical Compression pring 'esign for "atigue *oading




10-15 ummary



ME 7atigue 9oading o# Helical 'o*pression Springs

- prings are almost always sub?ect to fatigue loading.

- %n many instances the number of cycles of re!uired life may be small, say,

several thousand. +ut the valve spring of an automotive engine must sustainmillions of cycles of operation without failure so it must be designed for infinitelife.

- To improve the fatigue strength of dynamically loaded springs, shot peening can be used.

- %t can increase the torsional fatigue strength by 1; percent or more.

- Shot peening is a cold working process used to produce a compressive residualstress layer and modify mechanical properties of metals %t entails impacting a




stress layer and modify mechanical properties of metals. %t entails impacting asurface with shot #round metallic, glass, or ceramic particles$ with force sufficientto create plastic deformation.- %t is similar to sandblasting, except that it operates by the mechanism of

plasticity rather than abrasion each particle functions as a ball-peen hammer. %n practice, this means that less material is removed by the process, and less dustcreated.

- Nimmerli discovered the surprising fact that siBe? material? and tensile




strength have no effect on the endurance limits #infinite life only$ of springsteels in si8es under 7; mm.

- %t was observed that endurance limits tend to level out at high tensilestrengths. Nimmerli suggests that it may be because the original surfaces arealike or because plastic flow during testing makes them the same.

- Onpeened springs were tested from a minimum torsional stress of 745 )Da toa maximum of @1; )Da and peened springs in the range 745 )Da to 94; )Da.

- The corresponding endurance strength components for infinite life werefound to be:npeened;

<eened;, sa / 137 )Da , sm / 4=9 )Da

, sa / 495 )Da , sm / >43 )Da

&immerli data




"or example, given an unpeened spring with , su / 735; )Da, the Perber ordinateintercept for shear, from &!. #@<31$, is

- "or the Poodman failure criterion, the intercept would be 447.7 )Da.

- &ach possible wire si8e would change these numbers, since , su would change.

=>a

, ,

, ,

su

sm

sa se 9.1>=

735;4=9

7

137

711 =

−

=

−

=




- n extended study of available literature regarding torsional fatigue found thatfor polished, notch-free, cylindrical specimens sub?ected to torsional shearstress, the maximum alternating stress that may be imposed without causing

failure is constant and independent of the mean stress in the cycle provided thatthe maximum stress range does not e!ual or exceed the torsional yield strengthof the metal.

- With notches and abrupt section changes this consistency is not found.

- prings are free of notches and surfaces are often very smooth. The failurecriterion, which is known as the /ines !ailure criterion in torsional fatigue: isused for notch free and polished specimens For this criterion: the stead& stresscomponent has no effect / sm < and / se / sa




;.@= su ut , , =

- %n constructing certain failure criteria on the designers: torsional fatiguediagram, the torsional modulus of rupture , su is needed. We shall continue toemploy &!. #@<>3$, which is

- %n the case of shafts and many other machine members, fatigue loading in theform of completely reversed stresses is !uite ordinary.

- ?elical springs: on the other hand: are never used as both compression ande9tension springs .

- %n fact, they are usually assembled with a preload so that the working load is

additional.




max min max min 1 1a m F F F F F F − += =

- The stress-time diagram of "ig. @<14 d: expressesthe usual condition for helical springs.

- The worst condition, then, would occur whenthere is no preload, that is, when τ min / ;.

- 6ow, we define

- Then the shear stress amplitude is 4

5 aa B

F D !

d τ

π =

- The midrange shear stress is given by the e!uation 4

5 mm B

F D !

d τ

π =




7D D2d




7

1

=

+ ut

m

e

a

, ,

, ,




7=+ut

m

e

a

, ,

, ,



We used three approaches to estimate the fatigue factor of safety in &x 7;<3




- We used three approaches to estimate the fatigue factor of safety in &x. 7;<3.The results, in order of smallest to largest, were 7.73 # ines$, 7.7= #Perber$, and7.79 #Poodman$.

- lthough the results were very close to one another, using the Nimmerli data aswe have, the ines criterion will always be the most conservative and thePoodman the least .

- %f we perform a fatigue analysis using strength properties as was done in Chap.@, different results would be obtained, but here the Poodman criterion would bemore conservative than the Perber criterion.

- Which criterion is correctQ Remember, we are performing estimates and onl&testing will reveal the truth/statisticall&


10 2 Th C t &ff t



ME Overview




10-5 tability





10-10 Helical 'o*pression Spring )esign #or 7atigue 9oading10-11 &xtension prings



10-15 ummary



ME Helical 'o*pression Springs )esign #or 7atigue 9oading




ut m

A, d =

;.@= su ut , , =

Table =<"> / sy <$9@/ ut




11 1 4

3 3 3C


− −= + − ÷

s&

s

,

nα =

( ) max1

5 7 F

d

ξ β

π +=




Sines-Zimmerli crietria



- .9tension springs differ from compression springs in that the& carr& tensileloading: the& re*uire some means of transferring the load from the support to the



ME ( tension Springs

loading: the& re uire some means of transferring the load from the support to thebod& of the spring: and the spring bod& is wound with an initial tension

- The load transfer can be done with a threaded plug or a swivel hook both ofthese add to the cost of the finished product, and so one of the methods shown in

this "igure is usually employed.

Types o! endsused onextensionsprings$




Ends !or extension springs$(a) sual design stress at A is due to

combined axial !orce and bendingmoment$

(b) /ide *iew o! part a stress is mostly

torsion at &$(c) mpro*ed design stress at A is due to

combined axial !orce and bendingmoment$

(d) /ide *iew o! part c stress at & ismostly torsion$

- %n this "igure the maximum tensile stress at A, due to bending and axial loading,is given by




is given by

( ) 4 1

7@ 3 A A

D F !

d d σ

π π = +

where # ! $ A is a bending stress correction factor for curvature, given by

( )( )

17 7 7

77 7

3 7 1

3 7 A

C C r ! C

C C d − −= =−

The maximum torsional stress at point B is given by

( ) 4

5 B B

FD !

d τ

π =

where the stress correction factor for curvature, # ! $ B, is

( ) 1 11

1

3 7 1

3 3 B

C r ! C

C d −= =−

- When extension springs are made with coils in contact with one another, theyare said to be close"wound .




- pring manufacturers prefer some initial tension in close-wound springs in orderto hold the free length more accurately.

- The corresponding load-deflection curve is shown in "ig. 7;-5 a , where &is theextension beyond the free length + ; and F i is the initial tension in the spring thatmust be exceeded before the spring deflects.

(a) 8eometry o! the !orce F and extension y cur*e o! an extension spring(b) geometry o! the extension spring

- The load-deflection relation is then

F F '&+




i F F '&= +where ' is the spring rate.

- The free length + ; of a spring measured inside the end loops or hooks as shownin "ig. 7;<5 b can be expressed as

( ) ( ) ( ); 1 7 1 7b b + D d # d C # d = − + + = − +

where D is the mean coil diameter, # b is the number of body coils, and C is the

spring index.

- With ordinary twisted end loops as shown in "ig. 7;<5 b, to account for thedeflection of the loops in determining the spring rate ' , the e*uivalent number ofactive helical turns # a for use in &!. #7;<9$ is

a b% # # .

= +

where % and . are the shear and tensile moduli of elasticity, respectively

- The initial tension in an extensionspring is created in the winding process




spring is created in the winding process by twisting the wire as it is wound ontothe mandrel.

- When the spring is completed andremoved from the mandrel, the initialtension is locked in because the springcannot get any shorter.

- The amount of initial tension that aspringmaker can routinely incorporate isas shown in "ig. 7;<5 c.

(c) torsional stresses due to initial tension as a !unction o! springindex C in helical extension springs$

- The preferred range can be expressed in terms of the uncorrected torsionalstress i as




i

where C is the spring index

- Puidelines for the maximum allowable corrected stresses for static applications

of extension springs are given in Table 7;<=.

( ) =>a

C

C i

−−±=>.@

439.@

7;>.;exp

147τ

Maximum allowable stresses (% W or % & corrected) !or helical extensioni i i li i




springs in static applications




Table 1@/7




.* 1@/ 7




- The analyses in &xs. 7;<@ and 7;<= show how extension springs differ fromcompression springs




compression springs.

- The end hoo's are usuall& the wea'est part: with bending usuall& controlling .

- fatigue failure separates the extension spring under load.- "lying fragments, lost load, and machine shutdown are threats to personalsafety as well as machine function.

- "or these reasons higher design factors are used in e9tension/spring design

than in the design of compression springs .- %n &x. 7;-= we estimated the endurance limit for the hook in bending using theNimmerli data, which are based on torsion in compression springs and thedistortion theory.

- n alternative method is to use Table 7;-5, which is based on a stress-ratio of 6 τ min τ ma9 6 @. "or this case, τ a 6 τ m 6 τ ma9 .

Maximum allowable stresses!or A/TM A33: and type ;<3




- *abel the strength values of this Table as , r for bending or , sr for torsion.

- Then for torsion, for example, , sa / , sm / , sr 01 and the Perber ordinate intercept,given by &!. #@<31$ for shear, is

!or A/TM A33: and type ;<3stainless steel helical extensionsprings in cyclic applications

( )1 1

1

7 17

sa sr se

sm su sr

su

, , , , , ,

,

= =− − ÷




- o in &x. 7;-= an estimate for the bending endurance limit from Table 7;-5would be

- (ence

- Osing this in place of [email protected] )Da in &x. 7;-= results in # n f $ A / 7.;4, a reductionof > percent.

=>a, , ut r @.575$7579#3>.;3>.; ===

[ ] =>a

, ,

, ,

ut r

r e 347

7579

[email protected]

[email protected]

$10#7

1011 =

−

=−

=





ME Overview



10-5 tability10-6 pring )aterials



10-" "atigue *oading of (elical Compression prings10-10 (elical Compression pring 'esign for "atigue *oading


10-12 Helical 'oil &orsion Springs


10-15 ummary

- When a helical coil spring is sub?ected toend torsion, it is called a torsion spring



ME Helical 'oil &orsion Spri*gs

- %t is usually close-wound, as is a helicalcoil extension spring, but with negligible

initial tension . - There are single-bodied and double-

bodied types as depicted in "ig. 7;<9.

- Torsion springs have ends configured toapply torsion to the coil body in aconvenient manner, with short hook,hinged straight offset, straight torsion, andspecial ends. - The ends ultimatel& connect a force at adistance from the coil a9is to appl& ator*ue .

- The most fre!uently encountered #and least expensive$ end is the straight torsionend




end.

- %f intercoil friction is to be avoided completely, the spring can be wound with a pitch that ?ust separates the body coils.

- The wire in a torsion spring is in bending: in contrast to the torsion encounteredin helical coil compression and e9tension springs

- s the applied tor!ue increases, the inside diameter of the coil decreases.

- Care must be taken so that the coils do not interfere with the pin, rod, or arbor. - The bending mode in the coil might seem to invite s!uare- or rectangular-cross-

section wire, but cost, range of materials, and availability discourage its use.

- There are many stock springs that can be purchased off-the-shelf from a vendor.

- This selection can add economy of scale to small pro?ects avoiding the cost of




This selection can add economy of scale to small pro?ects, avoiding the cost ofcustom design and small-run manufacture.

Describing the End +ocation

- %n specifying a torsion spring, the endsmust be located relative to each other.Commercial tolerances on these relative

positions are listed in this Table.

- The simplest scheme for expressingthe initial unloaded location of one endwith respect to the other is in terms of

an angle de!ining the partial turn present in the coil body as p 0;>< o,as shown in "ig. 7;-7;.

End position tolerances !or helical coiltorsion springs (!or D0d Gatios up to and

ncluding =>)

- "or analysis purposes the nomenclature of "ig. 7;-7; can be used.

- The number of body turns # b is the number of turns in the free spring body by




count. The body-turn count is related to the initial position angle 5 by

The !ree"end location angleis $ The rotationalcoordinate H is proportionalto the product Fl$ ts bac4angle is ,$ For all

positions o! the mo*ing endH 2 , constant$

integer integer 4@;b p # # β

= + = +o

where # p is the number of partial turns. The above e!uation means that # b takes onnoninteger, discrete values such as >.4, @.4, =.4, . . . , with successive differences of7 as possibilities in designing a specific spring.

&ending /tress

- torsion spring has bending induced in the coils, rather than torsion.




- Residual stresses built in during winding are in the same direction but ofopposite sign to the working stresses that occur during use. The strain-strengthening locks in residual stresses opposing working stresses provided theload is always applied in the winding sense.

- Torsion springs can operate at bending stresses e9ceeding the &ield strength ofthe wire from which it was wound (because of the residual stress that is opposite

sign of the wor'ing stress) - The bending stress can be obtained from curved-beam theory expressed in theform

=c !

σ =

where ! is a stress/correction factor . The value of ! depends on the shape of thewire cross section and whether the stress sought is at the inner or outer fiber .

- Wahl analytically determined the values of ! to be, for round wire,

1 13 7 3 7C C C C− − + −




( ) ( )3 7 3 7

3 7 3 7i o

C C C C ! !

C C C C − − + −= =− +

where C is the spring index and the subscripts i and o refer to the inner and outerfibers, respectively.

- %n view of the fact that ! o is always less than unity, we shall use ! i to estimatethe stresses.

- When the bending moment is = / Fr and the section modulus 0c / d 4041, weexpress the bending e!uation as

4

41i

Fr !

d

σ π

=

which gives the bending stress for a round-wire torsion spring.

De!lection and /pring Gate




- "or torsion springs, angular deflection can be expressed in radians or revolutions#turns$.

- %f a term contains revolution units the term will be expressed with a prime sign.

- The spring rate 4I is expressed in units of tor!ue0revolution #lbf E in0rev or 6 Emm0rev$ and moment is proportional to angle HI expressed in turns rather than

radians.

- The spring rate, if linear, can be expressed as

7 1 1 7

7 1 1 7

= = = = '

θ θ θ θ −′ = = =′ ′ ′ ′−

where the moment = can be e9pressed as Fl or Fr

- The angle subtended by the end deflection of a cantilever, when viewed from the built-in ends, is &0l rad. "rom Table -9-7,




( )1 1

33

@34 44 @3

e

& Fl Fl =l l . d . . d

θ π π

= = = =

- "or a straight torsion end spring, end corrections such as &!. #7;<3@$ must beadded to the body-coil deflection. The strain energy in bending is

1

1

= d9$

.

=∫ - "or a torsion spring, = / Fl / Fr , and integration must be accomplished over thelength of the body-coil wire.

- The force F will deflect through a distance rE c where H c is the angular deflection

of the coil bod&: in radians . pplying Castigliano:s theorem gives1 1 1

; ;1b b D# D#

c$ F r d9 Fr d9

r F F . .

π π θ

∂ ∂= = = ÷∂ ∂ ∫ ∫

- ubstituting / 2 d 30@3 for round wire and solving for E gives

@3 @3b b FrD# =D# θ = =




3 3b b

c d . d . θ = =

- The total angular deflection in radians is obtained by adding &!. #7;<3@$ foreach end of lengths l 7, l 1

7 1 7 13 3 3 3

@3 @3 @3 @34 4 4

bt c e b

=D# =l =l l l =D #

d . d . d . d . Dθ θ θ

π π π + = + + + = + ÷

- The e!uivalent number of active turns # a is expressed as

7 1

4a b

l l # #

Dπ += +

- The spring rate ' in tor!ue per radian is

3

@3t t a

Fr = d . '

D# θ θ = = =

- The spring rate may also be expressed as tor!ue per turn, ' S #units tor!ue0turn$ is

3 3d d




- Tests show that the effect of friction between the coils and arbor is such that theconstant 7;.1 should be increased to 7;.5. The e!uation above becomes

#units tor!ue per turn$. This e!uation gives better results.

- lso &!. #7;-3=$ becomes

3 31@3 7;.1a a

d . d . '

D# D# π ′ = =

3 31

@3 7;.5a a

d . d . '

D# D#

π ′ = =

7 13

7;.54t b

l l =D #

d . Dθ

π + ′ = + ÷

- Torsion springs are fre!uently used over a round bar or pin. When the load isapplied to a torsion spring, the spring winds up, causing a decrease in the insidediameter of the coil body



MEHelical 'oil &orsion Spri*gs

diameter of the coil body.

- %t is necessary to ensure that the inside diameter of the coil never becomes e!ualto or less than the diameter of the pin, in which case loss of spring function wouldensue. The heli9 diameter of the coil DI becomes

where E Sc is the angular deflection of the body of the coil in number of turns,given by

b

b c

# D D

# θ ′ = ′+

3

7;.5 bc

=D# d .

θ ′ =

- The new inside diameter becomes

i D D d ′ ′= −

The diametral clearance J between the body coil and the pin of diameter D p will be




# b can be solved as

b p p

b c

# D D d D d D

# θ ′∆ = − − = − −′+

( )c pb

p

d D #

D d D

θ ′ ∆ + += − ∆ − −

which gives the number of body turns corresponding to a specified diametralclearance of the arbor.

- This angle may not be in agreement with the necessary partial-turn remainder.Thus the diametral clearance may be exceeded but not e!ualed.

/tatic /trength

"irst column entries in Table 7;<@ can be divided by ;.>== #from distortion-h $ i




energy theory$ to give

;.=5;.5=

;.@7

ut

& ut

ut

, , ,

,

= )usic wire and cold-drawn carbon steels

Fatigue /trength

- ince the spring wire is in bending, the ines e!uation is not applicable.

- The ines model is in the presence of pure torsion. ince Nimmerli:s resultswere for compression springs #wire in pure torsion$, we will use the repeated

bending stress # / ;$ values provided by ssociated pring in Table 7;-7;.

M T carbon and low-alloy steels

ustenitic stainless steel and nonferrous alloys

- s in &!. #7;-3;$ we will use the Perber fatigue-failure criterion incorporatingthe ssociated pring / ; fatigue strength




1

1

17

r e

r

ut

, , ,

,

= − ÷

the ssociated pring / ; fatigue strength , r

Maximumrecommendedbending stresses(% & Corrected)

!or helicaltorsion springs

in cyclicapplications as

percent o! / ut

- The value of , r (and , e ) has been corrected for si e: surface condition: and t&peof loading: but not for temperature or miscellaneous effects .




- The Perber fatigue criterion is now defined. The strength-amplitude componentis given by Table @<= as

11 1 17 7

1ut e

ae ut

r , , ,

, r,

= − + + ÷ where the slope of the load line is r / = a0 = m. The load line is radial through the

origin of the designer:s fatigue diagram.

- The factor of safety guarding against fatigue failure is a f

a

, n

σ =

- lternatively, we can find n f directly by using Table @<=

1 17

7 7 11

a ut m e f

e m ut a

, , n

, , σ σ

σ σ = − + + ÷ ÷




Angles ,? ? and H are measuredbetween the straight"end

centerline translated to the coilaxis$ Coil 7D is =@ mm$




E5$ =<"9@




E5$ =<"@@

E5$ =<"99

ot shot"peened




11 1 17 7

1ut e

ae ut

r , , ,

, r,

= − + + ÷

1

1

17

r e

r

ut

, ,

,

,

= − ÷






MEOverview

10 3 eflection of (elical prings








10-13 =elleville Springs 10-14 )iscellaneous prings

10-15 ummary



- "or example, using an h t ratio of G or larger gives an , curve that might beuseful for snap/acting mechanisms .



ME=elleville Spri*gs

- A reduction of the ratio to a value

between 1 41 and 1 causes thecentral portion of the curve to becomehori ontal: which means that the loadis constant over a considerabledeflection range .

- higher load for a given deflectionmay be obtained by nesting, that is, bystacking the springs in parallel.

- Mn the other hand, stac'ing in series provides a larger deflection for the same load: but in this case there isdanger of instabilit& .






MEOverview

10 3 eflection of (elical prings









10-14 Miscellaneous Springs10-15 ummary



MEMiscellaneous Spri*gs






MEOverview

( p g










10-15 Su**ar/

- %n this chapter we have considered helical coil springs in considerable detail inorder to show the importance of viewpoint in approaching engineering problems,



MESu**ar/

their analysis, and design.

- "or compression springs undergoing static and fatigue loads, the completedesign process was presented.

- This was not done for extension and torsion springs, as the process is the same,although the governing conditions are not.

- The governing conditions, however, were provided and extension to the design process from what was provided for the compression spring should bestraightforward.

- s spring problems become more computationally involved, programmablecalculators and computers must be used.



- preadsheet programming is very popular for repetitive calculations.

- s mentioned earlier, commercial programs are available. With these programs, backsolving can be performed that is, when the final ob?ective criteria areentered, the program determines the input values.

10.Mechanica..l Springs

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Transcript of 10.Mechanica..l Springs