Number of Active Coils in Helical Springs

8/12/2019 Number of Active Coils in Helical Springs

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RP-S6-4

N um ber of A ctive Coils in Helical SpringsBy R. F. VOGT,1 MILW AUKE E, WIS.

D u e t o t h e t o r s i o n a l d i s p l a c e m e n t b e t w e e n c r o s s - s e c

t i o n s o f t h e h e l i c a l s p r i n g b a r o r w i r e i n t h e s o - c a l l e d d e a d

o r in a c t i v e c o i l s o n e a c h e n d o f t h e s p r i n g , t h e t o t a l d e

f l e c t i o n o f t h e s p r i n g i s g r e a t e r t h a n t h a t w h i c h c o r r e

s p o n d s t o t h e d e f l e c t i o n o f t h e f r e e c o i ls . T h e e x a c t

a m o u n t o f t h e s p r in g d e f le c t i o n c o r r e s p o n d i n g t o t h e

i n f l u e n c e o f t h e i n a c t i v e c o i l s c a n b e c a l c u l a t e d f o r

k n o w n l o a d i n g c o n d i t i o n s o f t h e s p r i n g a n d c a n b e c l o s e ly

e s t i m a t e d f o r a l l s p r i n g s s u b j e c t e d t o c o n v e n t i o n a l s p r i n g -

p r a c t i s e l o a d s . T h e d e f l e c t i o n s o f t h e e n d c o i l s a r e c a l c u

l a t e d i n t e r m s o f t h e d e f l e c t i o n o f a c t i v e c o i l s . A t e s t t o

d e t e r m i n e t h e a c t u a l n u m b e r o f a c t i v e c o i ls i s s u g g e s t e d

a n d e x a m p l e s a r e g i v e n .

THE predetermination of the correctnumber of active coils in helicalsprings is, in many applications,

very important. C en trifu ga l sprin g-loaded regulators, controlling the speedof prime movers, spring-loaded indicators, and many other apparatus requirehelical springs of which the correct num ber of active coils is essential.

If, in th e use of the conven tional helical-spring deflection equation

/ = deflection

r = mean radius of coild = diameter of wireP = load G = modulus of elasticity in torsionn = number of active coils

an error is made in determining the correct value of n, the factorG is usually adjusted to offset the original error. The factors/, r, d, and P are always fixed in value and can easily be measured and checked.

In such cases it is erroneously assumed that G is a variableamount for different helical springs, the variation ranging from

below 10,000,000 to 12,000,000.The fact, however, is that the modulus of elasticity in torsion

G is proportional to the modulus of elasticity in tension E and is

characteristic for each material and constant within the elasticrange:

1Assistant Chief Consulting Engineer, Allis-Chalmers Mfg. Co.Mem. A.S.M .E. Ro bert F. Vogt was born in Geneva, Switzerland,and had his primary and secondary schooling at Romanshorn, Canton School at St. Gall, and Swiss Polytechnicum at Zurich, Switzerland. His professional career began in the United State s in 1903.He has been connected with the Allis-Chalmers Mfg. Co. as mech anical engineer since 1907.

Contributed by the Special Research Committee on MechanicalSprings and presented at the Mechanical Springs Session of theAnnual Meeting, New York, N. Y., Dec. 5 to 9, 1932, of T h e A m e r i -c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s .

N o t e : Statements and opinions advanced in papers are to b eunderstood as individual expressions of their authors, and not thoseof the Society.

where ju is Poissons ratio . According to Httt te, 26th edition,for spring steel

E 30,000,000 lb per sq in.m = 0.275m 1/n 3.63G = 0.392 E = 11,700,000 lb per sq in.

Any discrepancy between these given values and experimentalvalues is due to an error in counting the number of active coilsin the helical spring under consideration.

It is the purpose of this paper to show how the correct numberof active coils can be determined both by analysis and experiment.

The number of active coils as used in the deflection formulafor helical springs does not always equal the total number ofcoils or the number of free coils. Particula rly, in most commercial helical compression springs we find tha t the number of activecoils must be more than the number of free coils, if we assumeG = 11,700,000 ~ 12,000,000 as correct and applicable to helicalsprings.

Tests of regulator tension springs, of which each end coil washeld at two diametrically opposite points, checked closely withG = 2/5 E when the number of active coils was counted fromthe middle of the two supporting points on one end of the springto the corresponding point on the other end, i.e., when the num

ber of active coils was taken as the number of free coils plusV2coil.

R o d U n d e r T w i s t

Let us consider, as shown in Fig. 1, a rod of the length L twisted by applying a force P per

pendicular to a rigid arm of lengthr, which is perpendicular to th e rod.The deflection of the point of ap

plication of the force P in the direction of P is expressed by theformula:

in which co is the angle of twist in rad ians.

H e l i c a l S p r in g W i t h R i g i d A r m s a t C o i l E n d s

If the rod referred to for Equation [1 ] is coiled into the shapeof a helical spring on which the rigid arms extend from the endsof the coil to the center line of the coil and are perpendicular tothe coil center line, the force P acting on the arms at the centerline of the coil in the direction thereof produces a deflection /of

a = pitch anglen numbe r of coils

467


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468 TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS

In this case all coils are active, i.e., subjected to the twist ofthe full torque moment P r and no coils or part of coils are inactive. The number of coils n is also the number of active coils.

C o m m e r c i a l S p r i n g s

Commercial springs are not equipped with rigid arms at theends. Many tension springs have loop ends, as shown in Fig. 2,which add a negligible amount of bending deflection to the tor

sional deflection givenin Equ ation [2]. Insuch springs all coils between th e base ofthe loops are activecoils. T he d eta ile ddiscussion of this type

;fig_ 2 of sprin g wil l beomitted in this paper.

In helical springs where the full length of the bar is within thecylindrical part of the spring, as is the case of commercial helicalcompression springs and also of many kinds of expansion springs,the activ e coils extend beyond the free coils. The free coilsinclude the coils of the spring which are not connected with brackets or yokes through which the spring load is app lied anddo not make contact with the end coils. The active coils includeall coils contributing to the deflection of the spring. The angle oftwist of the bar changes from maximum in the free coil to zeroin the end coils. The angle of twist within the end coils adds acertain amount to the total deflection of the spring, which can

be determined in many cases wi th complete accuracy and in allother cases with sufficient accuracy to satisfy fully the practicalapplications.

H e l i c a l S p r i n g s W i t h D i f f e r e n t T y p e s o f L o a d i n g

In order to illustrate the deflection of the end coils in helicalsprings, various ways of spring loading will be analyzed:

1 The Two-Point Loading.To an open-wound helical springthe load is applied by means ofyokes reaching on each end diametrically from one side of thecoil to the other (see Pig. 3).The load P is applied at themiddle of the yoke by means ofa pivot, so that each end of theyoke transmits the same pull P/2 on the spring.

As is shown in developingEquation [2], the effect of the

pi tch angle a is such that it may

be correctly assumed th at the endcoil is in a plane perpendicular tothe center line of the coil and thatthe forces are perpendicular to the

plane of t he coil.Referring to Fig. 3, the load

P/2 a t A has no part on the deformation of the end coil betweenthe points A and B. The loadP/2 a t B produces a moment ofP/2 2 r = P r at point Awhich is balanced by the same moment P r acting on theother side of the spring. All cross-sections of the spring bar

between A and B ' are under the influence of this moment P r.

If the end coil between A and B would be absolutely rigid, thecross-section at A wrould remain in the same position rela tive tothe end coil as it had before loading took place. Bu t the end

coil is as flexible as the free coils located between A and B ' andthe bar between A and B will twist in accordance with the respective moment acting on the bar at its cross-sections. Thismoment is no longer co nstant and equals P r for all cross-sectionsfrom A to B, but decreases gradually from the maximum of P/2 2 r = P r at A to P/2 -0 = 0 at B. Between A and B the

P r acting moment is M = P / 2 (r r cos


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RESEARCH RP-56-4 469

The general expression for the deflection of helical springswhich are not provided with rigid arms or loops at the ends andto which the load is applied in the common manner then takesa convenient form readily available to designers:

d I1 . 8 3 9

II1.122

D c= 7 .5 3 4 .1 4 0n'

19Vs6 . 0 7 5

195/sll 1/*

n - n ' + 1/2 = 6 . 5 7 5 12

P 8 V t

4580H/4

4600

f = 3 0 .7 6 0 1 .6 6 0Q

8 n-D* P f-d *

1 1,8 00 ,0 00 1 1,8 50 ,0 00

The springs were tested with a testing machine of 5000-lbcapacity. The dimensions were carefully taken with micrometers and averaged from many measurements taken of diameters

perpendicular to each othe r along the full length of the springs.The free coils were carefully determined and fixed by insertingspacers at the contact points with respective end coils.

For compression springs the forces are applied in the oppositedirection; the deflection is also in the reversed direction, bu t thecalculations and results are otherwise the same. The design andapplication of a commerical helical compression spring are suchthat the load condition ranges between the two cases given.The first condition is the more common.

The foregoing calculations are based on full-bar cross-section inthe end coil. This assumption applies to most commercialcompression springs, for that pa rt of the coil which must be considered in the calculation. The basic design of the spring isshown in Fig. 5.

The ends of such a spring are closed, 3/< of the end coil istapered from full cross-section to 1/ i thickness at the end, the

pitc h changes a t conta ct point B from p to d. Full cross-sectionof bar is maintained from B to A , suggesting a load division ofP/ 2 at A and P /2 at B. The variation is usually not far fromthis load division and comes within the range of the three- point loading of 3 X P/3, in which extreme case the differencewould only correspond to 1/ 3 1/ i = 1/ 12 coil for each end correction.

We must bear in mind that the resultant of these forces is inthe center line of the coil. If it were to fall outside the center line,the spring would bend out sidewise, which, in most compression-spring applications, does not occur to any appreciable extent.Within the range of free coils, i.e., from B to B ' (see Fig. 5), the

bar is subjected to a shear force equal to P /2 and a torque equal toP r. The effect of shear upon the deflection is small and can

be neglected. In a well-applied compression spring the torqueP r is uniformly the same all along the bar, P acting in line ofthe center line of the coil.

Due to the fact that the length of contact between the endcoils increases slightly during increase in deflection, therebyeffecting a slight decrease in active coils, the assumption of the2 X P/2 load division is more justified, as the error in allowingfor slightly less active coils than would correspond to an actual,

possibly different load dist ribution, is compensated by the tendency for a slight decrease in active coils during compression.It is therefore logical and practically correct to choose the 2 XP/ 2 load division.

E x p e r i m e n t a l V e r i f i c a t i o n o f A n a l y s e s

A large number of tests substantiate the mathematical analysisand the general application of the results. To illustrate, thefollowing test records are presented. Errors in the dimensionsof bar diameter, coil diameter, and deflection, on account of theirlarge amount, are relatively small. The examples, therefore, areof especially high value as proofs of the analyses.

The following springs were made by the Railway Steel SpringCompany for the Allis-Chalmers Manufacturing Company:

Bar diameter (average) , in ___Coil diameter (average), in ___Free length .....................................Free coils .........................................Act ive co ils .................... . ..............

Total number of coils from t ip to tip of bar ( taper) . .

Lo ad, lb ............................................

Deflection........................................M od ul us ...........................................

F i g . 5

The theoretical number of active coils n and the modulus ofelasticity in torsion G may easily be determined by testing fordeflection two compression springs made of the same materialof equal bar and coil diameter, but with a different number offree coils, as follows:

Spring 1 Spring 2

Coi l d iame te r ...................................... D = DBa r di am et er ....................................... d = d Number of free coils ....................... n ' n " L o a d ........................................................ P ~ P Def lec t ion ............................................. / i = hNum ber of acti ve coils ................... m = n ' + x m = n" + xModulus of elasticity in torsion. G G

Since both springs are alike except for number of free coils, themodulus of elasticity in torsion is the same for both springs aswell as the effect of the end coils under the same load.

We find

and

If there should be any variation between the two springs indi, Di, or P, then/i or/2must be corrected to correspond to valuesfor d, D, and P adopted for the foregoing calculation. It will

be found th at x approximates the value of */ very closely and tha tG approximates 11,700,000 for any size and kind of steel spring

bar.

Appendix 1

T N o rder to find the tota l angle of twist of cross-section a t A under the influence of P r, the half-circle between A and B is

divided into differential lengths A (s) = r A


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acting on any cross-section of the arc A B is M = P /2 a =P/2 r (1 cos < p ) . (See Fig. 3.) The tw ist angle u betweentwo cross-sections separated by the distance A(s) amounts to32 M A(s) ml ,------- - = 4 m. The total twist angleth a t is, twist angle

tt a* Gof cross-section at A in reference to cross-section at B equalsthe sum of the angles of twist for all sections A (s) located between point A and point B. It is

If we assume the center 0 of the arc A B rigidly connected tothe cross-section at A , it will move with the cross-section A the

amount of r 16

-, which distance corresponds to thedi G

deflection of the center O of arc A B from its original positionunder the influence of load P/2 at point B.

For the three-point loading of the end coil, we find the deflection of the center of the arc of the end coil, proceeding thesame as in the case of the two-point loading, as follows:

Appendix 2

C a s e W h e n 4>< aT N calcu lat ing deflections of a po rtion of a circular ring ou t of

its plane by forces perpendicular to the plane of the ring, theknown solution of Saint-Venant can be used.2 If an incompletecircular ring is fixed at A and loaded by force P a t B (Fig. 6),

F i g . 7

then according to Saint-Venants solution the deflection at any po int C, defined by an angle , is given by the following equatio n:

2T he Saint-Venant solution can be found in Loves M athematicalTheory of Elasticity, pp. 456-457, or in "Strength of Materials,vol. 2, p. 469, by S. Tim oshenko. This m anner of solution was suggested by R. L. Peek.

in which C is the torsional rigidity and E l the flexural rigidity.This equation is satisfactory for any value of 0 between 0 and a.

T w o - P o i n t L o a d i n g

If there are two forces P/2 acting at A and B, the deflectionat B is readily obtained from the direct application of Equation

d * TT[a], substituting in it P/2 for P, a = 0 = i t , I = -----, and C =

j 64(l 7rG . Then the deflection of point B is:

oZ

Point 0 deflects

in which form the equation shows that the deflection of an endcoil is equal to the deflection of one-quarter of a free coil.

C a s e W h e n > a

If 0 is larger than a, the necessarydeflection can be obtained by usingSaint-Venants equation in conjunctionwith the reciprocity theorem.3 Fromthis theorem it follows that the loadP applied at C (Fig. 7) produces at B the same deflection as the deflectionat C produced by the load at B.Since Equation [a] gives the deflection at any point C in Fig. 6,we can get at once the deflection at any point B for the loadingshown in Fig. 7.

T h e e e - P o i n t L o a d i n g

Take now, as an example, the case of three loads P/3 put at points A , B, and C, 120 deg apar t (Fig. 8). Poin t A is consideredas fixed. The deflection of the point B consists of the two parts:(1) Deflection produced by the load P/3 at B and (2) deflection

produced at B by the load P/3 at C. The first part is obtained by substituting into Equation [a] P /3 for P and 2x/3 for theangles a and 0.

P R 3 5This gives: Fs du eto S = 7.50 - (assuming E = ~G).

( j t CL Z

The second part is obtained from the same equation by puttingPR3

4?r/3 for a and 2ir/3 for 0. This gives: Fsdue to C = 6.70 Crd

Hence the total deflection of the point B i s:

P R 3V b = U . 2 0 -

In calculating the deflection of the point C, we again have two parts : (1) Deflection at C produced by the load at C is obtained

Method proposed by Prof. S. Timoshenko.


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by substi tut ing P /3 for P and a = = 4ji-/3 into Equation [a],P R 3

which gives Fcdue to C 33.10 and (2) deflection at C pro-Gd

duced by the load at B. This is equal to deflection at B when theload is at C and is obtained from Equation [a] by substituting init P/3 for P and taking a = 4x/3 and = 2ir/3, which gives:

P R 3V Cdue to B = y Bdue to C = 6.70Gd 4

Then the total deflection at C i s:

COS


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472 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

B = spring loadSh = deflection I = length of spring barr = radius of coil a = pitch angle

In the deflection Equations [2], [3], [5], [a], [6], [7], and [8], it is assumed that the spring bar or rod is thin in comparison with the radius of the curvature, i.e., D /d c o . The error caused by this assumption, when the equations are applied to commercial helical springs where D /d = 3 or more is very small and for all practical purposes negligible.

Equation [8] is given to show the influence of the pitch angle. Other influences which affect the accuracy, and are not considered in this equation are caused by preventing the free ends of the spring from turning freely about the axis of the spring during compression or expansion and the pure shear deflection. The complications affected by properly considering all these facts are too great, and the change in end results too minute to warrant the application of these highly refined methods of calculation in practical engineering work.

As far as the deflection effect of the end coil is concerned, it is, for all practical purposes, sufficient to consider its torsional deflection only in the manner shown in Appendix 1. This is especially justified when we realize that the mathematically more complicated method employed in Saint-Venants solution

is also not absolutely accurate because the effects of such items as pure shear deflection, coil pitch angle, spring index D/d, weight, and end conditions of the helical springs are neglected. Another factor, which demonstrates the fallacy of striving for accuracy to the extreme in calculating the deflection of the end coils, is the unavoidable variation in cross-section shape and size, coil diameter, and pitch angle in commercial springs. Equa-

64 n r> P , , 64 (' + ) r3 Ptions / = ---- ------- and / = ---------- ---------- , respectively,

Or * a * ( j a 4

can be regarded as being accurate for all practical purposes.

Discussion

T. M c L e a n J a s p e r .6 The paper by Mr. Vogt on helical springs is exceedingly interesting. I am wondering if the values of E, 0, and l/ m are as constant for spring steel in general as is assumed in the paper. My reasons for asking this go back to some tests made in 1924 which were published in the Transactions of the American Society for Testing Materials of that year and some work presented in the Philosophical Magazine for October, 1923, which indicate that the state of the steel as well as the temperature at which the tests were made influences the values of the so-called elastic constants somewhat.

The only way that this should be determined for spring application is to make several tests on identically shaped springs made of different steels.

I am not familiar with the values of O to be assumed for steel when formed into helical springs and when using different steels,

and therefore the values presented in this paper may be an appropriate average for steel springs to be used at ordinary temperatures only.

W. M. A u s t i n . 6 The writer has had to apply helical compression springs, both large and small, to quite a variety of machinery and has often observed the influence of the end turns. Particu

larly, he has observed that the average spring designed to be made like the authors Fig. 6, except having a length relative to diameter several times longer than Fig. 6, will usually buckle badly when fully loaded.

Small springs often have their ends malformed. The spring maker winds enough wire on his mandrel to make two or more springs, and then cuts them apart. He then presses the end of the spring against the flat side of a rapidly turning dry grinding wheel. The heat generated makes the end turn red hot at some point about 3/s to V s turn from the end of the wire. The wire bends at this red-hot place and the end of the wire moves back against the next turn. He then dips the spring in water in an attempt to restore the temper to the heated part, and finishes the grinding.

The end turn, instead of tapering uniformly in thickness for 3/ 4of a turn to V i the diameter of the wire at the end, tapers for V i turn to a thickness about l / z diameter of the wire, then increases in thickness for another l/ t turn to 3/ i diameter of the wire, then tapers another x/ 4 turn to V 4 diameter of wire at the end. This last taper may lie against the next turn for most of its length.

The writer has often had to show the machine assembler (not a spring maker) how to cut off part of the end turn and regrind so that the spring will not buckle in service. Even if the spring were made according to the drawing as usually made, the center of gravity of the load would not be in the extended axis of the spring. If the spring is not more than three times as long as its diameter, the buckling is usually not very noticeable.

The writer prefers to make the end turn so that the end of the wire does not touch the next turn until the spring is compressed solid, and instead of making the ground end exactly perpendicular to the axis of the spring, to make it a helicord of small pitch relative to the pitch of the spring. If this is done, the end of the wire will take its proper share of the load without bending beyond the plane of the part, 3A of a turn away, where the tapering of the wire began.

Most springs are never completely unloaded in service, many of them never more than 1/ 2 unloaded. In cases like this the minimum load brings the end of the spring into a plane perpendicular to the axis. It is probable that the center of gravity of the load is not in the axis of the spring, at the time of minimum load, but as the load increases the center of gravity of the load

approaches nearer and nearer to the axis, and when maximum load is attained the ideal condition exists with the center of gravity of the load is in the axis of the spring.

It is then seen that the flat-ended compression spring and its modifications is at best only a compromise, more or less successful, so to load the spring that at no time during the compressing or releasing of the spring will any part of it be stressed beyond its safe load.

In tension springs provided with hooks bent up out of the end turn and having tHe same diameter as the main body of the spring, the hooks have to stand the same bending moment as the torsional moment in the body of the spring. This means that the tension stress on the inside of the hook is about twice the shearing stress on the inside of the body of the spring because, for

1 D i r e c to r o f R e s e a r c h , A . O . S m i th C o r p o r a t i o n , M i lw a u k e e , W i s . 6 E n g i n e e r , W e s t i n g h o u s e E l e c . & M f g . C o . , E a s t P i t t s b u r g h ,M e m . A . S . M . E . P a . M e m . A . S . M . E .

If we transform this equation in terms used in Equation [2], we find


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Thus, with the additional fact that the wire is often damaged by making the hooks, accounts for the observed fact thattension springs, if they break, always break where the hook connects to the body of the spring. There is a way to reduce theexcessive stress in the hooks. It is to make the end turn a spiraland bend up the hook from the inner end of the spiral, makingthe mean diameter of the hook about one-half the mean diameterof the main body of the spring.

The au thor s tests on the two large springs would be much morevaluable if the springs had been loaded to near their maximumsafe loads instead of limiting the stress as ca lculated by the old formula to 34,400 for the small spring and 14,100 for the large one.In any event, I believe they should have both been loaded so as to

produce the same stress.It is quite generally known that Hookes law gives only the

first term of a rapidly converging series, so that Youngs modulus E and th e shearing modulus G both have higher values when determined by stress-strain measurements using low stresses than

they do when using stresses near but below the elastic limit.16 r3 P

The authors deflection r w = ----------- due to the torsion ind4 G

the end turn is the deflection due to torsion of the poin t B, Fig. 3, and no t the deflection of the pivot hole in the bar connecting points A and B. This would make the deflection of the

8 r3P pivot hold only In the au thor s analysis no account

d r (jt

is taken of the deflection at B due to the bending of the end turn by the load P/2 at B. This would produce a deflection aboutas large as that due to torsion.

In order to get experimental data on the deflection of the endturn, the writer had a piece of Vi-in. pretempered spring steelwire bent into 3/ t of a turn of 2n/ie mean diameter, as shown inFig. 11.

It was loaded at B with a 70-lb weight and the deflection at B was 0.29 in. If we let G = 11,400,000, the deflection per turnof the main body of the spring is 0.488, when loaded to 140 lb.The deflection at B is then seen to be 0.595 of the deflection ofone turn, and the deflection at the center of the bar connecting

A and B would be 29.7 per cent of the deflection of one turn,and for both ends the deflection due to the end turns is 59V2

per cent of one turn.

J. P. M a h a n e y . 7 At the beginning of his paper the authorshows that the value of G should be nearer 12,000,000 than10,000,000. This is true provided Poissons ratio is taken as 0.30to 0.335 rather than 0.365. In some instances attem pts have

7 Assistant Professor, Industrial Engineering, Virginia PolytechnicInstitute, Blacksburg, Va. Jun. A.S.M.E.

been made to prove G equal to the lower value by substitutingtest data in the conventional spring formulas, but, since itis generally admitted that these formulas are approximate forclosely coiled springs, such computations are not adequate

proof.The author states that the bar in the free coils is subjected to

a shear force of P/2. This is incorrect. The tota l load on the

spring is P; consequently, the single bar must trans mit thistotal load from one end of the coil to the other and the shear inthe bar will be P instead of P/2 .

Since present conventional spring formulas can be provedinaccurate, it does not follow that illogical corrections are acceptable. Adding one-half a coil to the number of free coilsadmits that a portion of the seated end coils deflect, which is

beyond comprehension. I t is true th a t torsional deformationextends beyond the free into a portion of the seated coils, forif this were not true, the first free coil at each end would notcontribute its full share of deflection. To infer tha t there isaxial deflection derived from the seated coils is a process of creating one error to compensate for another.

As the paper shows for balanced loading the load P on a com

pression spring ma y be resolved into two components of P /2each acting 180 deg apart. If the spring in Fig. 3 is loaded incompression, P/2 at B will produce torsional stress at A, andP/2 a t A increases this stress to the final value within the spring.The stress in the seated coil must build up to the proper value at

A in order that the active end coils may be completely effectivein contributing deflection. The stress within the seated coil

A-B produces deflection indirectly but its contribution to thetota l should not be counted twice. The authors mathem aticaldeduction clearly shows that the torque available in A-B issufficient to produce deflection equivalent to one-quarter of anactive coil provided it were free to move, which is of course im

possible.

R. L. P e e k , J r .8 H o w accurately the solutions given for two-and three-point loading apply to helical springs compressed between parallel plane surfaces requires further analysis. Following the treatment given in Loves Mathematical Theory ofElasticity, pp. 456-457, I have evaluated the force required

to keep the extreme end of the inactive turn in contact with the point A (Fig. 3), a condition that must be satisfied under com pression of this sor t. I find this force trivial in comparisonwith the reaction at A under two-point loading, and this consideration therefore does not affect the validity of applying theresult for two-point loading to compression between parallel plane surfaces. On the other hand, in such compression thechange in pitch angle of the active coils w'ill cause their axis to

be no longer normal to the paralle l plane surfaces applying load

and their deformation will not be that corresponding to a purelyaxial thrust. Whe ther this effect will appreciably change theresult, I have not ascertained.

A. M. W a h l .9 The exact solution of the additional deflection produced by the end tu rns of a helical compression spring isundoubtedly a very complicated problem, since it depends onthe exact shape of the end turns and on the distribution of loadthereon. The auth or has simplified the problem by assuming theend turns to have the full bar cross-section throughout theirlength. In add ition he assumes various distributions of load onthe end turns, finally choosing tha t which seems to agree best withtest results.

Since, in most practical cases, the deflection due to the end

8 Bell Telephone Laboratorie s, N ew York, N. Y.9Westinghouse Research Laboratories, East Pittsburgh, Pa.

Assoc-Mem. A.S.M.E.

round wire, the section modulus for torsion is ird3/16 and for bending is 23/32 , so if S t be the maximum tension stress in thehooks and S s be the maximum shearing stress in the coils, then


8/14

474 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

turns is but a relatively small part of the total deflection of the spring, a considerable error in estimating the effect of the end turns could be made without introducing much relative error in the tota l deflection of the spring. For this reason a rough approximation, such as the author has introduced, might be of value in practical work, provided it has been confirmed by a number of accurate tests.

The question of the effect of the end coils is closely bound up with that of the modulus of rigidity of the material. For years some spring manufacturers have used modulus values of 10.5 X 10 or 10 X 106 lb per sq in., as the author points out. It is well known that these values do not agree with modulus values obtained by means of torsion tests on ordinary spring steels. It has been the writers opinion that these modulus values have been used largely to compensate for inaccuracy in estimating the effect of the end turns and possibly for errors in the spring dimensions.

To illustrate this point, some tests made on different springs at the Westinghouse Research Laboratories will be mentioned. The method used was to measure deflections between prick- punch marks on diametrically opposite points of the coil in the body of a helical spring and is described in a previous publica

tion.10 The coil diameter and wire diameter were carefully measured at several points on each coil and the results averaged. By measuring deflections in the body of the spring, the effect of the end turns was eliminated. The values of effective modulus 0 could then be found from the known formula

Three springs from one manufacturer, having indexes of about ten, when tested in this manner, yielded the following values for the modulus:

Sp rin g N o ........................................... 1 2 3G X 10- lb per sq in .................. 11 . 4 5 11 . 4 6 11 . 5 0

Three springs having indexes of about 6.5 from another manufacturer gave the following values:

Spr ing N o ........................................... A B CQ X 10"Mb pe r sq in .................. 11 . 1 9 11 . 1 2 11 . 3 0

These values are all definitely higher than the value of 10 or10.5 X 106 as assumed by some spring manufacturers.

It should be noted that this method of determining the modulus assumes that the effect of the spring curvature is small, i.e., that the spring acts like a straight bar subjected to a torsion moment Pr. This of course becomes more nearly true for springs of large index. As far as spring deflections are concerned, this assumption is born out by previous tests by the writer,10wherein it was found that the ordinary deflection formula for helical round- wire springs was correct within 3 per cent for springs having

indexes varying from 2.7 to 9.5. In other words, a fourfold increase in curvature of a spring having a given wire diameter did not seem to have an appreciable effect on the modulus. The same thing is known to be true of curved bars in bending; i.e., in general, a curved bar in bending may be computed within a few per cent accuracy as far as deflections are concerned by using the fundamental methods applied to straight bars, although this is not true when stress calculations are made. The effect of curvature on deflection was also found to be small in the case of helical springs of circular wire by O. Gohner,11 who used more exact methods of calculation involving the theory of elasticity. The effect of curvature may be checked up experimentally by

* A. M. Wahl, F ur ther R esearch on Helical Springs of Roundand Square Wire, Trana. A.S.M.E., 1930, paper APM-52-18,

p. 217.11 O. Gohner, Die Berechnung zylindrischer Schraubenfedern,

Z.V.D .I., March 12, 1932.

it is found that n = 12, whence the added coils become 12 l lV i = Vs- But suppose G = 11.6 X 10 instead of 11.85 X 10* (a variation not at all unreasonable). Then we would find n = 11.65, from which the added coils would be found to be 11.65 11.5 = 0.15, a value which differs greatly from /a as found by the author. This example shows the necessity for an accurate knowledge of the effective value of G. This could be determined, as mentioned previously, by measurements between prick-punch marks in the body of the spring, after which the average dimensions of the spring would be accurately measured. In this connection, the writer has found it to be extremely difficult to obtain accurately the average wire diameter of a spring, without cutting it up after the test, since, due to coiling, the wire section becomes slightly oval.

The method of determining the number of active coils, as proposed by the author, consisting of using two springs similar in every respect except in number of turns, would no doubt give an approximation which would be useful in practical work. For purposes of checking the theory, however, it would be necessary to find the average dimensions of each spring accurately. This would involve more labor than would the testing of one spring, as suggested above. Furthermore, there is a possibility that the modulus would vary some between the two springs, and this again would involve an additional error. For these reasons it is the writers opinion that tests on one spring would be preferable in order to confirm the theory.

The value of Poissons ratio 1/m = 0.363 reported in the paper seems rather high for steel. Using G = 11.7 X 10a, E = 30 X 106, this would give 1/m = E/2G 1 = 0.283. Taking 1/m 0.3 (a value commonly used for steel) and E = 30 X 10, this would give G = 11.53 X 106, which is not far from the values obtained in the writers tests mentioned above.

A u t h o r ' s C l o s u r e

Answering Mr. McLean Jaspers discussion in regard to the constancy of the modulus of elasticity E and Poissons ratio m for spring steel at various temperatures, we may, according to Hiitte, for all practical purposes assume E and m and therefore G constant at temperatures between 0 F and 400 F.

Examples of springs applied at high temperatures are springs in steam indicators and on valves for internal-combustion engines and steam engines. As far as the author knows, the steam-indi- cator springs which are used for high-temperature steams and gases as well as for cold air have been accepted as accurate for

practical purposes without using any correction factors for the various temperatures to which they are exposed.The author, however, mainly considered springs used in at-

using the following method suggested by R. E. Peterson, of the Westinghouse Company. A heat-treated round bar of spring material is first tested in torsion, thus determining the technical value of the modulus G. This bar would then be wound into a spring, and heat treated, after which deflections would be measured in the body of the spring between prick-punch marks, so that the effective value of G could be found by use of the ordinary spring-deflection formula. The two values of G thus found

should be nearly the same if the effect of curvature is small.The writer would like to suggest that in determining the num

ber of coils to add to the free coils to find the active coils, it is necessary to know the effective value of G accurately; in other words, a small error in G would produce a big error in the number of added turns. For example, in the case of the authors spring II, if G is assumed 11.85 X 10, then from


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mospheric temperatures where accuracy in deflection values areessential.

Mr. Austin calls attention to irregularities in the shape ofcommercial compression springs especially in small sizes. How-

16 r3 Pever, he errs in his conclusion th at the deflection r o> = ------- -

dl Gis the deflection of point B (see Fig. 3). This deflection is derivedfrom the product r a which (as is clearly explained in the paperand in Appendix 1) is nothing else than the deflection of theoriginal end-coil center 0, which, as well as that of the pivothole between points A and B, is at a distance r from the centerof the bar cross-section subjected to torsion.

Mr. Austins claim that the bending effect of load P/2 onthe deflection of point 0 would be as large as th a t of torsion onlyis unfounded as may be seen from the Saint-Venant solution(see Appendixes 2 and 3) which includes the bending effect ofload P/2 at B and shows that the deflection of point O is evensomewhat less than that given by the author for torsion only.

In Mr. Austins experiment shown in Fig. 11 the deflectionof point B is claimed to have been 0.29 in. for a load of 70 lbat B; bu t according to Sain t-Venants solution this deflectionshould have been 0.231 in. for G = 11.4 X 106or 0.225 in. for G = 11.7 X 10.

Mr. Austin would have found more accurate and reliable results had he arranged his experiment according to Fig. 12. Thisarrangement consists of a helically and closely coiled spring-steel wire of one and a fraction of a turn . The coil diameter isabout 20 or more times the diameter of the wire which lattershould be about V< in. The wire and coil diam eter and th e deflection should be large enough to make unavoidable errorsnegligible in reference to the deflection. In Fig. 12 B A B is

exactly one full coil and BA = A B ' and each is one-half coil.In order to eliminate errors due to initial tension or deflection,the deflection ei e2for the load Qi Q2 is determined. The

1 --- 62deflection of a point B in reference to point A is / = ---------

2

for the load Qi Q2 at B. In order th at the stress in the wire iswithin the elastic limit of the spring steel Qi must be less than

15,000 lb where d and D are given in inches. The

sum of the differential deflections in the two half coils BA and

A B ' is alike and opposite in direction. For this reason the wirecross-section at A does not turn and therefore does not cause achange in the true deflection of B in reference to point A.

In Mr. Austins test, however, the cross-section at A, Fig. 11,

will turn and thereby increase the deflection of B, an amountcorresponding to the torsional twist in the wire within the copperclamp near point This clamped portion of the wire cannot

be held securely enough by the com paratively soft copper clampto prevent twisting of the wire and consequently the turning ofthe wire cross-section at A. This torsional displacement ofcross-section A of course increases the actual deflection due tothe twist in half coil BA which stamps a test made accordingto Mr. Austins arrangement, shown in Fig. 11, as unreliable.

The author has made a number of experiments according toFig. 12 in which he found the deflections to check very closelywith the Saint-Venant results.

J. P. Mahaney mentions that the pure shearing force in thefree coils due to the load P must be equal to P which is quite correct. However, according to the explanation given by the author in answer to A. M. Wahls discussion, the shearing force Pis divided into halves. One-half balances an excess of the sum ofthe torsional shearing-force components parallel to the axisof the spring and acting in a direction opposite to P, while theother half adds a pure shear deflection to the torsional deflection

8 n D3 Pas given by the conventional deflection equation / = ----- - -----a4 G

Mr. Mahaney, after admitting that torsional deformation extends beyond the free coils into a portion of the seated coils, elaborates considerably on his conception that since the end coils ina compression spring are not free to move they cannot contributeto the deflection of the spring. The deformation of the end coil,Mr. Mahaney claims, makes it possible for the first free coil tocontribute its full share of deflection. If this statement weretrue, the conventional spring-deflection Equat ion [2] as developed in the paper under the heading Helical Spring WithRigid Arms at Coil Ends would be faulty, as the first free coilsin this case do not have the benefit of torsional deformation inend coils and therefore wrould not contribute their full share ofdeflection. Obviously, such a contention is against sound reasoning as the development of the deflection Equation [2] includes thecontribution of the full share of deflection of all coils.

The fact that the end coils are held so that they can moveonly axially and parallel to their plane does not prevent the barof the end coils from twisting due to the torque applied. Thusthe axial deflection of the spring is increased proportionally tothis twist and corresponds to one-fourth of an additional freecoil per spring end beyond the deflection of a spring with rigidarms at free coil ends.

The author fully agrees with R. L. Peek, Jr., that in cases ofcompressing helical springs between parallel plane surfaces, thedeformation of the spring as a whole and in particular of the endcoil, will be different from the deformation as calculated in accordance with assumptions made in the analyses in the paper.This difference will vary with the different shapes of the springends as furnished in commercial helical compression springs.

However, w^hen we consider the error range due to (a) usingthe conventional spring-deflection equation instead of theSaint-Venant equation given in Appendix 3, (6) unavoidablevariations in spring-bar and coil diameters of commercial springswhich appear in the equation in the fourth and third power, respectively, (c) change in pitch angle and coil diameter duringcompression, (d) uncertainty as to spring end loading conditions,(e) neglecting the influence of the spring index D/d , and (/) uncertainty as to the actual value of the modulus of elasticity E orG, respectively, the variation of the actual deflection of the endcoil from the one calculated, and given as being equal to the de

flection of Vi coil due to maximum torque, is so small in comparison to other discrepancies that its disregard is fully justified.This is very apparent when we realize that a 5 per cent error indetermining the end-coil deflection results in an error of less than


10/14


0.5 per cent in reference to the total deflection of a spring withfive active coils.

An objection-free determination of the actual deflection ofthe end coil of a commercial helical compression spring with anaccuracy within such a small error range would be very difficult.

Referring to A. M. Wahls discussion, the values of E, G, and m were taken from the latest edition of Hiitte, 1931, first

volume, p. 689, where the following data for spring steels isgiven:

E = 2,100,000 kg per sq cm or E 30,000,000 lb per sq in.G = 822 ,000 kg pe r sq cm G = 11,700,000 lb per sq in.

G / E = 0.392 v m = 3 .63 , f rom G = E / 2 (1 + 1 /m)

These data have always corresponded with spring tests madeunder the consideration of the proper number of active coils (regardless of small or large number of active coils), as given in theauthors paper, and were therefore accepted by the author as

being dependable. Hiit te is considered one of the outstandingsources of reliable engineering information.

Mr. Wahl questions the accuracy of determining the value ofG by testing two springs as suggested by the author, and in his

example assumes G = 11.5 X 10 instead of 11.85 X 106, inwhich case Mr. Wahl calculates the effect of the end coils to bethat of 0.15 free coils instead of 0.5 as demonstrated in this

paper. Mr. Wahls analysis is, on this point, incorrect and deceiving.

In the authors example, G is determined from actua l values ofn ' = 11.5, d 1.122, D = 4.14,/ = 1.66, and P 4600 and (inconformity with the th eory developed in the paper) n = n ' + ' / 2.

If, in the example, the value of G had been different, say11.5 X 10, then the defle ct ion/ would have been 1.715 in. instead of 1.66 in. as it actually showed in the test, and n = 12and no t 11.65. The num ber of effective coils is fixed by thespring design and does not depend on the value of G.

The value of G cannot vary much for commercial spring steel.

The skeptical engineer, however, can determine its value andconcurrently the actual effect of the end coils, with satisfactoryaccuracy, by the method of testing two springs of equal dimensions but with greatly differing numbers of coils, as suggested bythe author in the last part of his paper.

Mr. Wahl, in referring to the influence of the spring index onspring deflection, mentions that the conventional spring-de-flection equation for helical round-wire springs is correct within3 per cent for springs having indexes varying from 2.7 to 9.5.

The author determines the effect of the spring index on thespring deflection definitely by adding the direct shear deflectionto the torsional deflection of the helical spring. The deflection

t P Lof direct or pure shear for the spring is /" = yL = - L = ,

G Fi Gd 2 * 7Twhere L = 2Rirn, Fa = ------- (for circular cross-sections from4-1 .2

Hiitte), P = spring load, or / " .= ----- ----- - . About half G o2

of this deflection is already included in the conventional springequation, as in helical springs under load P about one-half theshear load P is balanced by the total sum of torsional shearing-

stress components parallel to the spring axis, as explained byDr.-Ing. A. Rover in Z.V. D.I ., Nov. 20, 1913, p. 1907.

By using the conventional spring-deflection equation, it isassumed that a curved bar has the same torsional deflection asa straigh t bar of the same length. The stress distribution in thecross-sections of the straight and curved bars is, however, slightlydifferent and causes a small difference in deflection, amounting

to one-half the deflection due to pure shear. The deflection ofthe curved bar is less than that of the straight bar, when the pure shear deflection is considered for both.

The total deflection of the helical spring under load P is:

or

or

7 = shear angle in radianst = shearing stress

R = mean radius of coil D = mean diameter of coil

d = diameter of spring wire or bar n = number of active coils

P = spring load G = torsional modulus of elasticity/ = tota l spring deflection

L = effective length of wire or bar /" = deflection due to pure shear.

Applying the extended deflection equation in conjunction withaccurate spring tests will result in finding more uniform valuesfor G.

Plotting the shear deflection, in per cent of torsional springdeflection, against the spring index D /d we find the curve givenin Pig. 13.


11/14

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14/14

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Documents

Number of Active Coils in Helical Springs