by J. A. HARINGX 531.2:621.272.2:621.272.6

R 101 . 'Philips Res. Rep. 4, 49·80,' 194?

,ON HIGHLY COMPRESSIBLE HELICAL SPRINGS~, ~

.AND RUBBER RODS, ,.ANp-.:THEIR APPLICATIONFOR YIBRAT~ON·FREE MOUNTINGS, II*)

SummaryThis paper, the second of a series of six, comprises the calculationof the lateral rigidity of helical springs under compression and theirnatural frequencies for transverse vibration, starting oncemore fromthe simplification, mentioned in the first paper, \that the helicalspring may be treated as an elastic prismatic rod provided due allow-ance is made for the rigidity against shearing. Although, here too,this simplification leads to reliable results, the fact that. the springis actually a helically wound wire structure requires special attentionin respect of the spring ends, which are either rigorously fixed orelastically constrained by means of flat-wound end coils. Special

'precautions; without which the central line of the spring whencompressed takes a curved and oblique position, are indicated for I

the two cases separately. Further, 'particular attention is drawn tothe negative lateral rigidity of helical springs, occurring at certaincompressions, in connection/with its importance for compensatinganother positive rigidity. .'

RésuméCet article, le second d'une série de six, donne le calcul de Ia rigiditélatërale de ressorts hëlicoidaux soumis à une compression, et 'de leursfrêquences propres pour la vibration transversale., On part, iciencore, de la simplification proposée dans le premier article, d'aprèslaquelle Ie ressort hëlicoidal peut être considërê comme une barreêlnstique et prismatique, pourvu qu'on s'en tienne à l'interprétationcorrecte de la rigiditë au cisaillement, Bien' que cette simplificationconduise à des rësultats dignes de foi, Ie fait que Ie ressort soit con-'stituë par un fil enroulé en hélice nécessite une attention spêcialepour les extrêmitës du ressort, qui sant soit fixées rigoureusement, .soit encastrëes ëlastiquement au moyen des spires extrêmes enrouléesà plat. On a indiquë, sëparêment dans les deux cas. les prëcautiènsspéciales sans lesquelles la ligne centrale du ressort comprimé prendune position courbëe et oblique. En outre, on a accordéune attentionparticuliere aux valeurs négatives de la rigidité latérale des ressortshëlicoidaux <tui seprésentent.pour certaines compressions,par rapportà l'importance qu'elles prennent pour la compensation d'unc autrerigidité positive.

*) This is the second of a series of six papers to be published in these Reports and com-prising an unabridged translation of the author's thesis (Delft, December 1947), whichwas approved by Prof. Dl' Ir C. B. Biezeno, professor of applied mechanics, at :theTechnical University of Delft, Netherlands. For the first paper sée: Philips ResearchReports 3, 401-449, 1948.,

50 J. A. HARINGX

CHAPTER I.

"

ELASTIC STABILITY, LATERAL RIGIDITY ANDNATURAL FREQUENCIES OF HIGHLY COMPRES-SIBLE HELICAL SPRINGS '

PART B. LATERAL RIGIDITY OF HELICAL SPRINGS

10. The lateral rigidity of the helical spring under compreasion, the springbeing regarded as a, rod :

In the case where a helical spring is clamped at !lne end ana its otherend is free to movein a lateral direction (i.e., in the directionperpendicularto the central line of the spring in the initial state), a transverse forceexerted upon this free end will as a rule result in a cert~in lateral displace-ment. The lateral rigidity of the helical spring is now defined as the lateralf~rce required to produce a unit displacement of the spring end in thedirection of that force. .In section 6 we have seen that in respect of lateral buckling even a helical

spring having a pitch equal to the diameter may still be regarded as an .,elastic prismatic rod (at least when suitable allowance is made for therigidity. against shearing) without inadmissihle errors arising. Therefore,here again, we employ this simplification in calculating the later~l rigidityof the helical. spring under compression.We shall consider the general case where the spring clamped at one'

end is loaded at the 'other, end by an axial compressive force P, aIateralforce L and a. bending moment MI. and we shall try to calculate the lateral.displacement occurring. Using, the same notations as in section 3. we.now get (fig. 23) ,

'!

. I

NP"z .

5394'3

Fig. 23. The deflected centralline of an elastic prismatic rod replacing a helical spring,when the free end is simultaneously loaded with an axial force, it lateral force and abending moment.

a a(162) . .i'.

, 'HELICAL' SPRINGS, RUBBER RODS AND VIBRATION-FREE MOUNTINGS '51 .

, ,

M == Py + Mz + L(l-z); 'Q = P'lji+ L, N = P, (161)

and consequently [see eq. (1)]

d'ljidz

M· Py + Mz + L(l-z)

Further, we find for the slope of the tangent to the central line (fig. 5)" .

'dy (. .P) L- - = 'lji+ cp == 1+ - 'lji+-,

dz fJ fJ. (163)

so that from (162) and (163) we can derive the following differential equation -

, d2y+ P (1+ P)y _ -(1 + P) Mz + L(l~z),

- dz2 0 a fJ .' fJ a.' ,

, Taking again [see eq, (8)]

(164)

. P 'P'q2 = ~ (1+ op) ,

we have the solution

. o· .' Ml + L(l-z) ,y=Asmqz+Bcosqz- . , (165), ,P

iThe boundary conditions ('lp . 0 at z -:- 0 and y ='.0 at z = 1) are satis-fied when

o L ( P) 1. A = - P 1+ P -;j'

Ml lOL ( P) tan q1B=---+- 1+'..,,- --, (166).P cos q1 p. fJ q ,

. "

We can now easily calculate the lateral displacement t5 of the free endand its anglè of deflection 'ljiz'They are equal to~, .. .

o Ml (1 ) L ~(.' P) tanq1 .~ lt5 = - -- -,I +- 1 1+ -, -- -1 ,P cos q1, P . fJ q1, Ml q tan q1 L (1 ) ,

''ljiZ=P I+P/fJ + P cos q1-l,: '.

(167)

These equations mayalso be written in the form

(168)

where the coefficients Cl' c2 and Ca are determined by .'

., 52 J. A. HARINGX

I,

(169)

Again introducing the relative compression ~, and for the sake of brevitytaking .

You= 1--,flo.

we deduce from eqs (3), (8), (10) and (11) that, ,

(170)

p = Yo~,P 1-u~1+-=--,(J 1-~

. ..From' (169) and (171) it apJ>ears that the coefficients Cl' C2' C3 occurringin eq. (168) are related to one another and to the rigidity against axialcompressi~n 'ca= Yo/lo as *)

C~= i I (1+_~_Ca cot" iql) .c2 1-~Cl .

In connection with eqs (14), (15), (l'70) and (171), we obtain for a, circular-wire section' of diameter d .

(172)

;

V 2m + 1 10 /~ql =2(m + 1) Do H(l-u~), lu = m + 2 " ca= Yo= ~ E d

43' )

. 2(m+J) . 10 m+116noDoI '.

Thus, if there is no axialload (~ = 0 and I = 10) eqs (172) are reduced to

(173)

*() Of course, the relations (172) also hold for he~al springs under tension (~< 0).,. According to eq. (171) the quantity q is then îmagînary, However, putting q = iq',

tan t ql tanh t q'l 'we get iT := t q'l and cot2 ~ql = - cothê tq'l,

I .

,HELICAL SPRINGS, RUBBER-RODS AND VIBRATION-FimE MOUNTINGS 53.

C3 C3 C2 m 2 ~ 4(2m+l) (10)2~Cl ='C2 "C1 = 4(2m+l) Do?1+ 3m Do)'

If, on the otherhand, an axialload is present, eqs (172) may, with the aidof the correction factors 81, 82, 83 and the transformation of 10 into I, be, ,written in a fo~m adapted to that of ,(174).With m = 10/3 we thus get

:: =,0'3845 ~1+ 0·766 (~JT81'c2' •

:-=iI82," Cl

:: __:0:10~7.m~1 + 3'066 (~J~83•Like the 'mutual ratios of Ca, Cl' c2 and C3 [eq. (172)], the factors 81'82, 83

are functions of the relative compression ~ and of the, slenderness lo/Do, of the spring. They can be determined by id~ntifying the right-hand. .memhers of the corresponding equations (172) and (175), and are graphi-cally plotted in figs 24, 25 and 26. " ,From eqs (167) and (168) we see that the' force L corresponding to a

displacen;'-ent c5 is determined either by the moment Mi or by the angle of

Ca . , ,m ç.' ,2m+1 (10 )2(è1= 2(m+l)?1 + 3m Do)' I,

,3

Cl .s~a~JI1 ' I -;057fi 4/ 1/LIj; Q3)Q2 L

~ -:/ !?V~ e- _.... •

~ --::.-- S =0

Zo. Do

I,

2 3 ' 4o53944

(174)

(175)

Fig. 24. The correction factor el as a function of the relative compression ~ and the slen-'demess la/Do. . '.. .' , ,

, I

54, J. A ,HARINGX

I

deflection:1f'z. Therefore, 'when we want to .calculateI the lateral rigidity' ,defined as L/(J, we must ,knowmore about Ml or 1f'Z' For instance, inpractice it often occurs' that the two, ends of the spring ~ust remainparallel, so that 1f'z is equal to zero. According to (168) the lateral rigidityis then equal to Cl and the ratio of the axial to the lateral rigidity is givenby the first eq. (175).When this ratio is wanted directly as a function of therelative compression~ for various values of the slenderness lo/Do we referto fig. 33 in section 13. Owingto eq. (168), for 1f'z = 0 the clampingmomentMz amounts to -c:;(J, where Cz can be calculated by means of the secondeq. (172) or (175). '

" .

'k00

laua2, 3. 0o

5391,.6

Fig. 25. The cori:ection factor 8z as afunction of the relative compression ~ andthe slenderness Zo/Do.

Fig. 26. The correction factor 83 as afunction of the relative compression ~ andthe slenderness lo/Do.

. On the other hand, it mayalso occur that the bending moment at the '.free end óf the spring is absent. Ixi.that case the lateral rigidity' LI (J is,calculated by putting Ml ÎÎ!- eq.·(168)equal to zeroand then eliminating 1f'Z'When we carry.~)Utthis calculation with the aid of (172) it appears thatthe ratio of the axial to the lateral rigidity can again be represented by thefirst eq. (175),provided'the length I is replaced by 21.This is obvious, since- for reasons of symmetry - the middle section of the spring (rod) with

\ ends constrained p~rallel cannot be subjected to a bending moment, andeach half of the spring is .under exactly the same conditions as the springwith a free end'not loaded by a bending moment. The lateral displacementis then half as great, which causes, it is true, that the lateral rigidity is of .twice the value. At the same time, however, the axial rigidity is also' 'doubled. .Here we have arrived at the final result with respect to the lateral

,1IF;LlCAL ~PRINGS. RUBBER RODS AND VIBRATION·FREE MOUNTINGS 55

, rigidity of helical springs *), but still we should like to ~ake two remarksin connection with the fact that the spring, instead of being an elasticprismatic rod, is actuaUy a h'elically wound wire structure. .

First of all, under certain circumstances it may he required to know, the compression of the spring corresponding to a given axial force with a,greater degree of accuracy than can be reached by the elementary calcu-lation (LIL = PIca). For the c~se that 'Only a compre,ssive,force Pis' ex:ertedwe can refer to eq. (93).,Usually, however, the helical' spring has no oppor-

I '

tunity to alter its number of coils when being compressed (for instance inthe case' of two ends constraui.ed parallel). Therefo~e, in addition to theaxial compressive force P there may,also he. a small torque W, which'has to 'provide' that n remains equal to no. Thus on account of (52) wehave, in addition to eqs (107) and (108), - '

R . cos a-=--.-'s, cos ao

(1?6)

and eq. (93), being no longer valid, has to b~ replaced by another one.In connectio'n with (71) and (72), we derive from (107),. (108) and (176)

for the actual radius R ' , .

R '- = 1 +! ~(2-~) tan2 ao+ ... ,~ - "

for the torque: W- ' ,

,_, WRo' ~~~1'-H-~(1-~)~tan2'ao~ ... ~ (178)

and for the compressive force P

p~~=~etan~o [l-t ~3-- ~,(1.-~) (2-~)~ tan2 ao + ..l (1.79)

The rigidity B against bending and the rigidity C against torsion in the'case of a circular wire section (diameter d; shear modulus G) are equal to~ ~.. ,. , "

B,.:.... In(~2: 1) Gd4, • C = ~ Gd4. (180)

*) The fact that the method of calculation for the lateral 'rigidity given here does indeedyield good results is proved not ouly by our own measurements to be descrilied iD. Isection13, but also by measurements of S. Gross and E.-Lchr given in: Die Federo,ihre Gestaltung und Berechnung, VDI-Verlag, 'Berlin (1938),p. 101. Unfortunately,our calculations cannot very well be checked against' the known measurements ofW. E. Burruèk, F. S. Chaplin and W. L. Sheppard, given in: Deflection of helicälsprings under transverse loadings, Trans. AIDer. -Soc, Mech. Engrs 61, 623-632;'1939, becau~e their method of fixing the spring ends has not sufficient~y been defined.

'/

(177)

1

I.

56 J. A. HARINGX

Thus, the spring dimensions 4t, the unloaded state being given, eq. (179)en'ables us to calculate the relation between the compressive force P and therelative compression e With the desired greater accuracy.A second point to be mentioned relates to the clamping of the lower

end of the spring. In the elementary method of calculation, chosen here, wehave tacitly taken the' point of view that the force P applied centrallycompresses the rod (replacing the spring) in the 'direction of its originalcentralline. Consequently the lateral displacement (J of the free end ~f thespring is only caused by a lateral force L and/or 'a bending moment Mz[eq. (167)]. On further investigation, however.vthis is found to be possibleonly when' the compressed helical spring (which is actually a helicallywound wire structure; see section 5) is in the state n,' and when at thesa~e time the central line of the auxiliary helix of radius R and slope acoincides with the centralline of the original helix of radius Ro and slope ao.For the clamped end of the spring wire, and likewise for the other end ifconstrained parallel, the transition from the slope ao to a is precluded.Furthermore it is hardly possible that the transition from the radius Roto R takes place unimpededly, because as a rule this transition is impossiblewithout a relative lateral displacement of both ends. Therefore, the helical..spring when being compressed cannot remain straight but is subjectto a lateral deflection without any lateral force occurring. For a relativecompression e the change (3= ao - a of the slope of the helix is to a' firstapproximation equal to aoe [eq. (131)]. According to eq. (~77) the change:

~ in the radius R - Ro, at least when the number of coils is not changedunder the compression, is given by *)

(181)

From this it follows that the smaller the slope ao, the less the fixing of thewire ends -affects the deflection of the spring.If the spring is only used with a 'definite relative' compression e, as

is the case with à resilient mounting, the ~ndesirable lateral deflection ofthe spring can be avoided by postponing the fixing of the wire ends until thespring has been compressed -to-the auxiliary helix of radius R and slope a or,what comes to the same thing; by making an allowance in advance forthe change (3 of the slope of the helix equal to aoe and for the change of theradius according to eq. (181). Only then are we sure that the centralliue ofthe compressed spring (rod) is straight and coincidentwith the direction of thecentrally applied compressive force. Fig. 27 shows how this can be realized.The helical spring is drawn in .the unloaded state and the end i~ clamped

*) For the case where there is no torque and the number of coils can vary freely, we. may refer to eq. (77).

53947Fig. 27. Actual construction of the clamped end of a right-handed helical-spring .. Forthe prescribed relative compression ~ the compressive force is transmitted axially.

HELICAL SPRINGS, RUBBER J!.0DS AND VIBRATION·FREE JllOUNTINGS ' 57

hy turning the screw S. The other end of the spring, not drawn here, canhe clamped in a similar way. If the spring comprises a whole numher ofcoils no account needs he taken of the difference hetween Rand Ro' For'the rest, this difference is usually negligihle.

.•...

R

.' "

11. The helical spring with Hat-wound end .coils

, ~n practice the "clamping" of a he1ical spring is generally realized hy,providing the spring with flat-wound end coils pressed against a fixed ,surface (the supporting plane). It -will he ohvious that the spring wire .at the point of the "sharp" transition from the helical spring proper to the .flat-wound end coil (point A in fig. 28) 'is not rigorously clamped, hut thatits position is governed hy ,the elasticity of the end coil. Not only is theend coil free to "cur] up", i.e., to lose partly the contact with the support-ing plane with or without maintenance of cöntact point A, hut also thespring wire will he twisted., Just as is the case with the rigorously clamped spring end (section 10),

so here, too, with !he lower end constrained elastically the helical springmay undergo a lateral deflection without any lateral force L or any hendingmoment MI (fig'. 23) necessarily heing present. Mostly such a lateral

,58 J. A. HARINGX

.deflection is undesired. 'I'herëfore, before proceeding to determine the effectof the elasticity of the flat-wound end coils upon the lateral rigidity, weshall first ascertain what precautions can be taken to keep the central lineof the helical spring straight and coincident with the line of action of the.compressive force which is perpendicular to the'" supporting plane. Hereagain, as in the case of rigorously clamped ends, we shall start from -thesupposition that the helical spring - for instance as the supporting ele-ment of a resilient mounting - has to submit a given axialload, so that it isactive at a given compression ~.

z

. , ,

'Fig. 28. The "sharp" transition from thé Hat~wóundend coil to the helical spring properat A, in the loaded state. '

In section 5 we have seen that under certain: circumstances (state II)the helical spring .assumes the form of the auxiliary helix (radius R,slope a), thé centralline of which coincides with the line of action ~fthecompressive forcejs-axia). If, in view of our 'aim to keep the centrallinestraight, the conditions in question are actually to be realized, the mannerin which the helical spring proper is connected to the end coil supportedby the xy-plane (fig. 28) has to answer special requirements. The followingcalculation will show that the object in view is to be attained by initiallylaying the (flat-wound) end coil in an inclined plane, as indicated In fig. 31.The partienlar position of the plane of the end coil then en~,ures that thecentral line of the spring under the prescribed relative compression ~ is. I .

actually a straight line perpendicular to the supporting plane.The compressive force acting along the s-axia is .transmitted via the heli-

cally wound spring wire to the flat-wound end ~~il at the point A (fig. 28).

, ',HEr,IeAL SPRINGS, RVBBE~ RODS -AND \'IBRATIO:-;-FREE IIIOUN!INGS $9

, ,

As we shall prove later on,_this causes the end coil to he lifted, over a eer- 'tain length, away from the supporting plane. Ignoring the friction betweenthe end coil and the supporting xy-p,lane, we apparently have to do with anoriginally flat ring loaded perpendicularly to its plane only. Consequentlythe projection of the ring on the xy-plane retains the form of a circle. '

z

r"

Fig. 29. The flat-wound end c?il in the deformèd state.

Of course, the oalculation given in section 4 for a spring wire initiallybent according to a space curve is' also valid for the flat ring (fig. 29),provided we take .'

a=ao;:'-O. (182),

Moreover, as we .have seenç there are no displacements (] in the plane ofthe ring, so that we further put ...

Q= O. " (183)

With the displacement C.in the direction perpendieularto the plane ofthering we now find, using (182) and (183), for the curvature k and-the torsionw [eqs (32) and, (36)] -,

1, k=-,

R1

w = - (C'" + C'),R '2

(184)

and thus for the moments Ml and Ma [eqs (26) and (27)] *)

" (185)

,*) Thesè equations for the circular ring' can, of course, also he derived directly. See, e.g.,C. B. Biezeno and R. Grammel, Technische Dynamik, Springer, Berlin (1939), p, 339.

60 J. A. HARINGX

. 'The equilibrium of an .arc element (see fig. 30) gives us the followingrelations for the bending moment M1 and the transverse force Q in termsof the torque M3

1 11

Q::;: R (Ma + Ma)· (186)

zy ,,

53950

Fig. 30. An element of the Hat-woun:d end coil.

In the free part of the ring the transverse force has the constant value QO,which for the present is to be taken as unknown. Consequently the dif-.ferential equation is-

, (187)

When we provide the bending moment lW"l 'and the torque M3 at the end ofthe ring (0 = 0) with the superscript 0, the solution of (1~7) gives, in.c~nnection with the first eq. (186),

. .M3 = M~ sin 0 + (M~ --'-OOR) cos 0 + QOR,

""

M1 = M~ cos 0 - (M~ - QOR) s~:ó.0 . '

. (18~)

(189)and further

I •

, Owing to (185), (188) and (1~9) for C we obtain the differential equation. . .

R2 [ C' ]Cif' + C' =C QOR + (1 + A) ~M~sin.0 + (M~,- QOR) cos 0~', (190)

with the solution

C= Cl + C2 sin,e + C3 cos e+R2 [ (C . "']+C 0 QJR - t 1+ A) ~M~sin 0+ (M~"7QOR) cos,0~ .

. . (191), .

Provided the loading force is sufficient, the rmg is pressed against the,supporting plane at the loaded end (0 = 0), so that

r= 0 at 0 = O. (192)

HELICAL SPRINGS, RUBBER RODS AND VIBRATIOl'!:FREE lIIOUNTINGS 61

·Where - the deflected part of the' ring restores the contact' with the sup-, porting plane (say, e = -el)' not only C but also the slope Cf must beequal to zero. Th~refore ' ,

C= 0, Cf = 0 at e = -el' (193), "

,/ Further, at the same point e=- elwe meet two possibilities: either thefirst principal axis of the wire section [axis: 1) initially directed towards, thecentre of the flat ring points upwards, so that the remaining part of the,ring (-e2 < e < -el) "curls up" freely, or else it points downwards,with the remaining part of the ring pressed against the supporting plane.:Assuming that the former case arises, we state that the cross-section'e = -el of the spring wire is unloaded. Therefore the boundary condi-tions are

Ma = 0 at e='~el', (194)

All together we thus have five conditions [eqs (192), (193) and (194)]to calculate the three constants of integration Cl' C2, Ca and 'the unknownquantities el and QO.

The line of action of the force P (z-axis) intersects the supporting planeat a point of the z-axis, so that at the end of the spring wire (e = 0)M~ = -PR and M~ = O. From eqs (188), (189) and (194) it then followsthat

(195)- .

Further, owing to eqs (97), (191), (192) and. (193) the displacement C ofthe(circular) spring wire appears to be

, 'PR3[4m+3 ~n' , ~ ~'2m+l ~]C= -- - (1-:cos e) + sin e +! e )1- cos e ..C 4(m+ 1) 2. " , , ( 2(m+ 1) ,

.' ~ (196)

It- will be evident from' the preceding considerations that at the end, Ie = 0 the spring wire is pressed against the supporting plane by a force

P + QO = lP, which proves the condition (192) to he correct, Moreover,we still have to verify whether' the axis 1 of the cross-section e = -:-ndoes indeed point upwards. To a first approximation the angle between theaxis 1 and the xy-plane is equal to the direction cosine of that axis withrespect to the z..axis ('VI)' lri this cross-section we took MI equal to zero[cf. eq. (194)], and thus, according to eg. (185), è) is zero also, so that'VI= 'VI [eq. (37)]. Substitution of (182) in (35) then gives that at thecross-section e = - n

, C"'VI = -,

R(197)

..

62 ,------------------~--------------------------------~---- ,J. A. HARINGX

.or owing to (196) n PR21'1= --.

, 8(m+l) C(198)

We therefore see that "i > 0, which means that the axis 1 does indeed. point upwards. If we now ensure that the remaining part of the ring

\. -(-02 < 0< -n) is free to lose contact with the supporting plane [partic-ularly paying attention to the necessary freedom of its tip, governed by'

_ the size of the angle' O2, the slope (1-~)aoand the thickness of the springwire], the boundary conditions (194) can indeed he applied.We are especially interested in the distortion of the ring at the end

o ' 0, which is given by the change in position of the system ofaxes1: 2, '3, as defined in section 4 and indicated in :fig.29. At the end 0 = 0the moment M1 is zero, so that this system ofaxes 1, 2, 3 coincides withthe system ofaxes I, 11, III [eqs (185) and (37)]. Substituting (182) and(183) in (31) and (35) we see that to a :first approximation the axes 1and 3 at the end 0' ,0 make angles wi!h the xy-plane equal to .

C" C''V1 = R' 'Va _- R', (199)

Owingto (196)we then have at the' el!d 0 = 0, .

n(4m + 3) PR2'V~ = - 8(m + 1) C'

PR2'Va = --- .

. , C_ (200)

As was to he expected under the given load, the two axes 1 and ,3 pointdownwards.According to eq~(93) the compressiveforceP is related to therelative compressionê, to a :first approximation, by PR~ = Cao~, andtherefore the angles of rotation at thc end 0 = 0 amount to [eq. (200);,R ~ Ro; m = 10/3]

(201)

Of the system ofaxes 1, 2, 3 belongingto the end of the helically wóund, part of the spring, wire the axis 1 lies along ,the x-axis and the axis 3'makes an angle a = (1....:.~)ao with the xy-plane, as indicated in fig. 28.This means that the (sharp) transition from the helical spring proper to theend coil is accompanied by an abrupt change in the position of the system,ofaxes 1, 2, 3 rigorously bound to the spring wire (compare thé positions

" ofthe axes 1, 2, 3 in :figs28 and 29). Accordingto eq. (201)the axis 1 turnsthrough an angle 1'V11 = 1'4.8aorand'the axis 3 through an angle a + l'Val =ao' Consequently, in the unloaded state the plane of the end coilwill be.perpendicular to the plane passing through the central line of the helicalspring and the point where the end coil joins the 'helix of slope ao,whilstits normal makes an angle {J = 1·48ao~with the centralline (see fig. 31).

53951

Fig. 31. The position of the flat-wound end coil with "sharp" transition to the helicalspring proper, in the unloadedstate. For the prescribed relative compression ~ the com-.pressive force is transmitted axially. ". .'" \ . '

..

HELICAL SPRINGS, RUBBE'R RODS AND VIBRATION-FREE MOUNTINGS .,' 63

Although thi's correction is particularly derived for the bottom end coil. (z = 0), it applies equ~lly well to the top end coil (z = I). If the .helioalspring has an odd number of half-coils the planes. of the two end coils arein the unloaded state parallel to each other, whilstm the case of a whole,.number of coils they enclose the angle 2fJ. The distance between the sup-porting planes under service condition amounts to I + d.

la

Whereas the difference between Rand Ro [see eq. (181)] is usuallynegligible, the 'correction in respect to the position of the flat-wound end

\ .coils becomes considerable when the slope ao of the helix and the relative.compression e at which the spring is active have a large ·value. The cor-rection, therefore, must especially be applied when we make use of highlycompressed springs' having a small number of coils and a large pitch.

"This' kind of spring will frequently occur in vibration-free mountings (seechapter IV, sectio~ 4). •

12. The effect of the Hat-wound end coils on the lateral rigidity of the helicalspring wider cempression 'I .

Ha-ring suéceeded in ensuring, by simple means, that th~ central Iine ofthe compressed spring remains straight and is perpendicular to the planesupporting the end coil, we can proceed to' correct the calculation givenin section 10 for' the case that the rod (replacing the spring) is not con-strained rigorously b~t elastically. The coefficient of constraint at thelower end of the rod, defined as the bending moment M per unit of angle 'IJl,is to be calculated. from the elasticity of the end coil, which is dependentupon the position of the plane in which the deflection takes place. For'

• instance, if the deflection occurs 'in the xz-plane the torque M~excerted at. . '\ \ ..

I.

64 J. A. HARINGX

, I

.point A of fig. 2&changes its value (M~~ PR) and we find th~ constraintto be equal to [see eq. (200) for m .. 10/3] -

M IPRI 8(m+l) C C-:;p= ~ = n(4m+3) R = 0'675 R',

(202)

In the case of a deflection in the yz-plane, however, the end coil is at theend e = 0 also loaded with a bending moment M~,and the constraintis then determined by the corresponding change of 'V3' Fortunately we may. put M~~ M~, so that el wil] differ only. slightly from st, Taking el =n + zl, sin el ~ -Lt and cos el ~ -1, according t~ (188), (189) and(194) we thus have, to a first approximation, .

M~Lt =-2-0'

M3(203)

Calculating the constauts of integration Cl' C2, C3 occurring in eq. (191)with the aid of the boundary co:O:ditions(192) and (193),. and employing(97) and (203) we find for 'V3at the end e = 0

C' M~~' n(4m + 3) M~R'V3=R =C - 4(m+ 1) ---C-'

The angle ofrotation ,caused by M~ is M~ R/C-1J3 so that inthis case theconstraint amounts to

• > "

M M~ = 4(m+l) C = 0'34 C.,'IjJ M~R/C-'V3. n(4m+3) R . R

(204)

Bearing in mind that the rigidity against flexure of the whole spring[ao/lo in eq. (15)] is proportional to the rigidity C = Glo of the spring wireagainst twist according to

1 Elo . m. m+l CIC 1.- --=0'18--,

no nDo 2m + 1 n(2m + 1) u; no s, no (205)

we can prove that the ratio e of tne 'flexural rigidity of the spring to theconstraint. of its end is equal to 0'27/no and 0'S3/no respectively [eqs (202)

Iand (204) with R ~ Ro]. These two values of e would make it necessaryto take into account the direction of the plane in which the deflection ofthe spring arises. Since, however, the end coil, is usually centred on theinside or outside by means of a rim, any phenomena of friction may easilyincrease the rigidity given in eq. (204) to the value occurring in eq. (202).This leads us, io take the constraint as being approximately equal to the

I .

HELICAL SPRINGS, RUBBER RODS MID VIBRATION-FREE MOUNTINGS 65. I

vaiue in eq. (202) for all directions, so that one single value for e suffices, viz.

0'27e= _'_'.

no(206),

When calculating the lateral rigidity of helical springs having a correctedposition of the flat-wound end coils we may again employ the differentlal

I equation (164) derived in section 10 and its general solution' (165). Asto the boundary condition for the "clamped" end z = 0 of the rod wecannot take "P = 0, as before, but have to put "P equal to the value1'48 Ro/e = 5'56 e Rono/C times the bending moment occurring [see eqs(202)and, (206)]. Moreover, we have to bear in mind that "Pl' the angle ofrotation of the supporting plane of the "free" end z = I, is not identicalwith the angle of deflection of the rod, but that it is greater by thevalue 1'48 Ro/r. = 5'56 eRono/C times the bending moment Mz (see fig. 23)..Thus, for the mutual relations of the coefficients Cl' C2' / Ca occurring in. eq, (168) we find

I Ca i'- ; ~1 - u; tan iql 1 (c~ = -;-? 1-; iqz l-eqltaniql-l~,I • ;

I c; = il(1 + __ C~), (207)SI-ES .c~ ~ ; Ca 1 - eqltan iql~. =r = il 1+ -- =r cot- iql . ,

. c2 . 1~~ cl 1 + eqlcot iql

the primes being introduced for the sake of distinction,For the meaning of u, iql and Ca we refer to eq. (173). The axial rigidity

Ca - like the length I and the relative compression ; - relates. to thehelical part of the spring possessing no free coils. Owing to the elasticityof the 'end coils, however, the actual axial rigidity is less. As appears fromthe required oblique position f3 ofthe flat-wound end coils in the unloaded'state (see fig. 31)? the compression of the helicai spring is accompanied ateach end by the additional axial shortening f3R = 1'48 uo;R. In connectionwith eq. (67) and the relation L11= ;10 this additional axial shorteningis also eq~al to 0'23 L11/no, that is, equal to 0'23 times the compressionper free coil. Consequently the number of "effective" coils amounts tono + 0'46, so that the actual axial rigidity c~ is equal to

/

• I

I noca = ca no + 0'46· (208)

Besides the relative compression E and the slenderness lo/Do, the number. óf coils no will also occur as a variable in eq. (207), by the presence of e•• Thus the result 'becomes obscured, and we shall have to calculate the effect

• !I'

,66 J, A,' HARINGX

,\ '

of the elastieity of the end coilsfrom case to case *)"Let us take as an examplea helical spring With.lo/Do = l'S and no = 3 compressed to g = O'S; then'

. . • " • I

U = 0'615, iql = 0'829 and e = 0'09 ..Thus, according to (207) c~/ca =0'85, whereas according to (172) Cl/Ca = 1'22. This means that the lateralrigidity for flat-wound end coils (c~) is as much as 30% less than that forclamped ends (Cl)' Furthermore the axial rigidity c~ is 13% less than Ca.

, Particularly with heavy gauge wire, it is rather difficult to give a sharpbend to the spring wire where the helix joins the flat-wound end coil,(point A in fig. 28). A construction avoiding these difficulties is represented

" ,

in fig. 32. Instead of being flat along the entire periphery, -the end coil iswoundin süch a way as to return over its 'tip, passing gradually into thehelix. The end of the spring wire (diameter d) .then takes over the functionofthe point A ~fig. 28. Itis true, the wire'end (e = -ez = -2nin fig, 29)can no longer curl up freely, so that strictly speaking the condition (194)is' no longer complied with, but the effect upon the distortion of the flatring is so small as to need no further consideration. The top end coil beingmade in a similar manner, the distance between thë supporting planesunder service condition is 1+ 3d.The spring being compressed, it is possible that the circular spring wire

of the first coil will slide off the wire tip and take a position beside it.,In order to pr~vent this inconvenience we have to make sure not only

. '

53952

Fig. 32. Actual construction of the flat-wound end coil without "sharp" transition to thehelical spring proper. For the prescribed relative compression ~ the compressive force 'istransmitted axially. " •

*) Contrary to this statement it proves possible to simplify the determinJtion of the, effect of the flat-wound end coils. From eq. (17) or fig. 6 we first calculate the, factor À by which the initiallength of the spring proper should be enlarged to causebuckling at the. given compression ~ for ends constrained parallel, ql thus being equal

... to 1'&/.1. We further introduce the factorrz by which the length of the spring propershould be enlarged to bring about for ends constrained parallel a lateral rigidity equalto that of the' actual spring having flat-wound end coils. On account of the first eqs(172) and (207) we are able to compose a graph where the factor jz is plotted against ~I/À for various numbers of coilsn~,so that by replacing I by ,uI in the first, eq. (172)or la by plo in fig. 33 we easily find the lateral rigidity in question.

HELIêAL SPRINGS, RUBBER RODS A.."<DVIBRATION,FREE 1I1ÓUNTrnGS 6.7

that the line connecting the centres of gravity"of t~e two contiguous wire'sections runs parallel to the central line of the spring (direction of com-pression), but also that the compression per coil is less than.

Lil- <5'2 d.no

(209), ,

, ,

If this condition would not be satisfied, the contact pressure of the springwire (R:j tP = tCa LIL) would be so great that an imaginary beginning ofsliding off causesradially directed force components-which are capable ofexpanding the end coil in its "plane" *).,Wlien only half the end coil cantake part in the distortion (e.g. when a eentering groove is provided inwhich the end coil fits without being clamped] the factor 5'2 in eq. (209)

~ , 1

hás to be replaced by 10'4., We have shown that, when using flat-wound end coils, owing to theirelasticity the lateral rigidity must be less than. that calculated from theformula (172). On the other hand, however, it may occur that the lateralrigiditybecomes greater, for, if the end coils are not so carefully made asindicated in :6g. 31 or 32 it may happen that the fust free coil somewheremakes contact wit1J.the ènd coil. In that case the contact point will moreor less take over the function of the 'point A in fig. 28, such that we mayassume the constraint at the new end of the spring proper to be approxi-mately equal to that of the flat-wound end coil itself. Owing to the short-ening of the effective length of the spring the lateral rigidity will thenincrease to a .not inconsiderable degree.

To get an idea of what this means, let us assume that at each end of the.helical spring With lo/Do = 1'5, no = 3 'and ~ = 0'5 a quarter of a coilis eliminated. Then with a wire thickness d the free length l' equals. (1-~)lo__;2 X id, whilst 'this part of the spring (2t coils) in the unloaded, state has a length l~ equal to 0'833 looThe relative compression ~' is equalto l-l'jl(} and the corresponding axial rigidity is 1'20 ca'. Taking the wir~thickness d as equal to o·i Do, we deduce from the first equation (207)~ after replacing Ca by 1'20 Ca and providing the other coefflcients ofrigidity with a double prime - that C~/ca= 1'29. This means that owing tothe shortening of the spring the lateral rigidity is considerably- increased(C~/ca = 1'29 instead of c~/ca' = 0'85) and that it is even. greater than thelateralrigidity ofthe same spring with clamped ends (c1ica = 1'22), Further,in accordance with eq. (208) the ratios of the actual axial rigidities toCa are C~/ca = 1'01 and c~/ca . 0'87 respectively. ' '

The fact that with increasing relative compression this effect becomes ,

~) Since the lateral force L to ;be transmitted by the spring is apt to start this slidingmotion, it is advisable to limit the lateral displacement ó' of the spring to ó' < ~'6 dno·

. ,

68 .T. A. HARINGX •

more and more important is obvious from the test results to be describedin 's~ction 13 (fig. 35): '

13. Compeàsation 'of a positive rigidity by means 'öf the negative rigidityof a helical spring under compression

• I

In section 10 we have seen that the lateral displacement of the freeend of a spring having its other end clamped is generally due to a lateralforce L and/or a bending moment Ml [see fig. 23 and eq. ,(167)] andthat, to be able to speak of a certain lateral rigidity L/d, something hasto be definitely given in regard to Ml or "Pl'If the spring is .long enough and if it is compressed gradually, the lateral

force L required to cause a certain lateral displacement d for a given con-dition at the spring end (e.g., Ml = 0 or "Pl = 0) becomes smaller andsmaller, until at a certain stage the lateral rigidity has entirely disappearedand the spring 'is free to deflect without any lateral force being needed at all.The spring is then in an unstable equilibrium and buckles. From this' itfollows that the buckling of a helical spring under compression is also, tobe regarded as a particular case of the problem of the lateral rigidity.

The spring being compressed still further, an unrestricted 'deflection canonly be avoided by applying at the non-clamped end a reverse lateralforce. Thus, corresponding to a positive value. of d there is a negative valueof L; in other words, we have to do with a negative rigidity, Of course, initself, a spring with a negative lateral rigidity has no practical application,owing to the instability in absence of the reverse force. It can, however, beutilised to compensate wholly or partly some other normal positive rigidity.We then obtain a construction built up from' springs of ,l~rge axialrigidity, which as a wholeîs nevertheless very weak. Under certain cir-cumstances this may be most advantageous., Consequently it is worth while studying also the negative lateral rigidity,of helical springs. Confining this to the most important case of a compres-sion-loaded spring with ends constrained parallel ("Pl = 0), the lateralrigidity L/d is determined by Cl [see eq. (168)], which can be calculatedby means or'the first equation (172). In fig. 33 the. ratio of the lateralrigidity Cl to the axial rigidity Ca is plotted as a function of the compression

. ~ for various values of the slenderness lo/Do: The points of intersection onthe horizontal coordinate axis correspond to the critical relative compres-sions of fig. 6. Those on the dashed limiting curve indicate at what com-

, pressions the springs (having clamped ends) become useless by instability.They also correspond to the critical relative compressions of fig. 6, butnow for halved spring lengths. .

It will be evident from the preceding considerations that, whereas theaxial rigidity of a helical spring remains constant for allioads, this is not

HELICAL SPRIN~S. RUBBER RODS AND VIBRATION·FREE MOUNTINGS 69

the case with the lateral rigidity. Thus, the axialload of the spring (whichneed not consist solely of a compressive ,force but may equally be a torqueor a combination of both) is a means of varying the rigidity in the lateraldirection. We shall therefore avail ourselves of this principle when an: ad-justable .spring rigidity is required, for instance, for a system with avariable natural frequency. Fig. 34 gives a diagrammatic representationof such a system. The natural frequency of the mass turning about a hori-zontal axis and supported resiliently can he varied by more or less com-pressing the two horizontal side springs.

For a given ,case fig. 35 shows th~ ratio of the natural frequency ('V) ofsuch a construction to that of the construction without side springs 0('1'0)as a f~nction of their relative compression e. The drawn curve represeilts.

2.S

C,

ë;;lofiJo=O

0.5 I--I-----:: I/. --,_ V-- Q75 V ./_...p;; VV /"

I-_-

~V V /

;..- .,./ ./ V /'1.25

I-" .....-:./ V V /'_~- .~" V /---- 1.75V V

r- V V-- ./ L~ V )1'

"- 2.f5 .........r--8:t----r--r- 2.5 ....Vr--~ ~ r-..::::t'-.. 2.75 S'~~ ~4""'" ~a6 ..... as 1.0..._ - ......

I~1\.' :"\ 3.5 -,r~r\_\ ~ I"

ZO/Oo=6 ~~ \ '\ "-\V ., ......

I~ \ \\' \ ,

1\ 1\ 1\

.,

2.4

2.0

1.6

. 1.2

O.S

0.4

o

-0.4

- -o.s

-1.2

. -1.6

539.53

Fig. 33. Ratio of the late~al rigidity Cl to th~ axial rigidity Ca of helical springs With endsconstrained parallel, as a function of the relative compression ê for various values of theslenderness 'D/Do.

/

70 J. A. HARINGX

the theoretically deduced relatio~. At the' moment of first buckling th~lateral rigidity of the side :springs is zero and t!_lenatural frequency returnsto its basic valve Vo (point A). Further, when the lateral rigidity is so

/ .

, I

Fig. 34. Diagrammatic representation of a spring system the natural frequency of whichis adjustable by compressing the horizontal side springs. ' , .

strongly negative that the rigidity of the vertical spring i~ fully compen-sated, the -construction as a whole is to be regarded as being in an unstablestate of equilibrium (point B). Since the natural frequency is proportionalto the square root of the total rigidity and this rigidity generallyapproachesthe zero point linearly with g, the curve at that point has a vertical tangent.Consequently round about zero the natural frequency is highly sensitiveto small changes in g. From this example we see that with the aid of theside springs the frequency can easily be redu~ed in the ratio of 1 to 4,corresponding to a reduction of tbc total rigidity in the ratio of 1 to 16.

1.6

1!. ...~

~. . ~I

~ r-,A" r-,- ., '\,.

1\118 -, s.

!.41.2

1.0

0.8

0.6

0.4

0.2

o 0.1 0.2 0.3, 0.4 0.5. . . 53 9SS

Fig. 35. The natUral frequency v at a compression ç of the side springs in relation to thebasic frequency '1'0 of the system without side springs:

--- theoretical curve for clamped spring ends,• • • •• test points for clamped spring ends,x x x x ·test points for flat-wound end coils., . ,

, ·71IJ;ELICAL SPRINGS, RUBBER RODS Mm VIBRATION·FREE MOUNTINGS

The results, of measurements taken with rigorously clamped spring endsare indicated by dots and show an excellent agreement with the theoreticalcalculatîons; In the case of springs having Hat~wound 'end coils, however,there are relatively large deviations, as may be seen from the points marked~th a cross. Their cause has been explained in section 12. Owing to theelasticity of the end coils the lateral rigidity is at fust less than that foundfor clamped spring ends and manifests itself in a lower natural frequency ')J.

When, however, the spring is compressed furthe~, the last coils partly,make contact with ,the Ilar-wound end coils. The resultant shortening ofthe free length of the spring is then accompanied hyan increase of thelateral rigidity. The measurements show that the two effects approximatelycancel out at a compression of g = 0'2. ,

It muk therefore be emphasized that the theoretical èalculations Will'agree with the practical results .only when the utmost care' is taken torealize th~ proper boundary conditions at the ends of the spring.

PART C. NATURAL FREQUENCIES OF HELICAL SPRINGS

14. Natural frequencies for longitudinal vibration of thé helical spring,under compression ' , .

An objection often raised against the use of steel springs for vibration-free- . . \ I

mountings is the small internal damping of this material, in consequenceof which the spring is easily liable to vibrate strongly at frequencies coin-ciding with its natural frequencies. Under certain circumstances this may.'indeed be a disadvantage, and it is therefore of the greatest interest toknow what natural frequencies are present,

In order to calculate these frequències we shall again suppose the helicalspring to be replaced by an elastic prismatic rod, which is equivalent to itin regard to both the rigidities and the distribution of mass. The axialforce P compresses the spring uniformly to the length 1= (1- g) lo andcauses the oross-section, initially lying at the level Zo above the bottomplane, to drop to the level z = (I-g) zo' Owing to the vibrations this cross_'section undergoes a certain' displacement , with respect tó its positionof equilibrium, so that its height above the bottom plane is then equal toz + , (fig. 36). When we again denote the axial rigidity per unit of instan-taneous length by y, the normal force N to be transmitted here is .

I , 0'N=P-y-.ozThe differential force dN between the top and bottom planes of an ele-

. J

ment of thickness dz imparts to that element an acceleration of -02'/ot2.'If e is the specific mass per unit of ~ctuallength, we thus get

. '

I

..

• r

72 J. A. HARINGX,

TI?s leads to the following differential equation for the displacement C02rC Y 02CO~2 ' e oz2' (210)

'We shall now consider the case where the rod (replacing the spring)acts as a connecting element between the mass m (fig. 36) and a foundationvibrating harmonically at the frequency ')I and the amplitude ao• The axiallo~d P is then equal to th~ weight mg,and the boundar~ conditions are

C = ao sin2nvtoC 02C

N-mg=-y-=m-oz ot2

_ at z = 0, ~ ,

at z = 1 • ~•

(2lJ )

, .

foundation53956

Fig. 36. The elastic prismatic rod replacing a helical spring, in the case of longitudinalvibrations. The rod acts as a supporting element of mass m.

r

.. ~Ignoring, as is usual, the transient phenomena," we take the synchronoussolution of (210)

C= (Cl sin qz.+ C2.cos qz) sin 2n')lt . (q = 2nv -Vel')') (212) ',I

Adapting this solution to the boundary conditions (211), the amplitudea ofthe m_áss'mis related to a~ID 'the-ratio

a 1(213)- -----,,---

ao cos ql - pql sin ql

Hence, resonance occurs when......cos ql- pql sin ql = 0 . (214),

Generally the mass m· will be very large 'compared to the mass elof the spring, and so p has a large value. 'Consequently, besides the very

small root ql =' Vl/p, the other roots ql can be taken as .being practically,equal to a multiple of st, The latter rO,otstherefore are practically independ-ent of the mass m, and we should have found them too, if the ends of thespring had been clamped (boundary conditions 1; = 0 at both z = 0 and

, z = l). The smallest root of (214) determines the res~nant frequency 'lIoof the system of mass m and spring. With the axial spring rigidity Ca = r/l'this frequency is equal to ,

Y- -1 r 1 Ca

\ 'lIo• . 2n epl'}.= 2n Ym . (215)

HELICAL SPRINGS, RUBBER RODS AND VIBRATION-FREE MOUNTINGS 73

The remarnmg roots correspond practically to the natural frequenciesof the spring clamped at both ends, which are

(k=1.2,3, ... ) (216) -

As can be calculat~d, a small internal damping of-the spring material[loss angle (J *)] reduces the infinitely large value ofthe amplitude ratio a/ao.occurring at resonance to th~, finite value

(a) 2'·-;; = n2k2p tan {J •Ores

(217)

This, however, always implies that a spring with a mass equal to 0'01 'T!'-(thus P = 100) and tand = 0'002 transmits the vibration to the mass at the

.' lowest natural frequency (k = 1) unattenuated. In such a case one can,if necessary, çoat the spring wile with a layer \of rubber or thermoplast,though this remedy is only partly effective. ' .

The axial rigidity Ca ....:. r/l of a helical spring with diameter Do, wirethickness d and 'number of coils no amounts to [see eqs (14) and (15) withE = 2 (m + 1) G/m] , .

r ro G d4Ca =-= -= --3' (218)• / . l lo 8 noDo

whilst for the mass of the spring, denoti~g the specific weight-by s,~wecan write

n2

el = 4" noDod2s.g .

Using (218) and (219) we thus find for the natural frequencies according'to (216) V .kd Gg.

'lik = 2 - • (k ..:._1,2,3, ... ) (220)2n,noDo' . 2s

*) By the (mechanical)loss angle ö is understood 'the phase angle betw~en the stress' andthe strain of an elastic material with internal damping when it is subjected to a sinusoi-dal load.

(219)'

74 J. A. HARINGX

\

This, shows that the natural frequencies for longitudinal vibrations areindependent of the axialload.

In 'relation to the resonant frequency Vo the frequencies Vk [see eq~(215) and (216)] are

Vk -r: '1 / mass m-=nkyp=nkV "Vo ' mass of spring

(k= 1, 2, 3, ... ) (22~)

From this equation' it immediately follows that 'with a given natural fre-quency Vo and mass m of the hody supported resiliently a spring of theleast possible weight is favourable when high frequencies Vk are aimed at.Therefore we shall have to admit of the highest possible stresses in thespring material. The maximum shear stress r caused by theload P = mg-amounts to

S. PDo 8 mgDo..,;=---=---.

n d3 n d3(222)

Owing to (218) and (222) wemay therefore alsowrite for the frequencies (220)f I

k Ca"';I Vk = ----,-==, 2Y2Ggs m

Replacing ca/m in this equation by 4n2 v~ [eq. (215)] we obtain'" .

(223)(k = 1, 2, 3, ... )

2n2k, 2Vk = ,/- vo"'; •. y2Ggs.'

(k = 1, 2, 3, ... ) (224)

From (223) or (224) it is explicitly shown that high natural frequenciescan be obtained ,by stressing the material as highly as possible: But, ac-cording to (223), it is also shown that the greater the mass m and thesmaller the axial rigidity Ca, the more difficult it is. , '

In the case of a steel helicalspring G= 8300 kg/mm2 and s = 7'8 g/cm3,

whilst the acceleration of gravity g = 981'2 cm/sec2. Thus, expressingthe dimensions in mm and the-shear stress in kg/mm2, the natüral fre-quencies' for longitudinal vibrations are equal to [eq. (220)]

dVk = 3'64 . 10Sk -2' . (k = 1,2,3, ... )

, noDoor to [eq. (224)]

(225)

Vk = 0'55 kv~..,; •..I

(k'= 1,2,3, . ; .) (226)

15. Natural frequencies for transverse vibration of 'the helical springunder compression

As opposed to the longitudinal vibrations ,of' helical springs, whichaccording to eq. (210) ar~ governed by the s~me differential equations as

HELICAL SPRINGS, RUBBER RODS AND yIBRATION-FREE MOUNTINGS 75

applies to iongitudi~ally vibrating rods, we ~hall find that for the trans-verse vibrations we have to do with an' entirely distinct 'differential equa- .tion. This is due to the fact that the shear elasticity, which can always beignored with .normal rods, plays an important part in connection with the'transverse vibrations of helical spri~gs.. Fig. 37 shows an element of thé transversely vibrating helical springimagined as being replaced by an elastic prismatic rod. Again using, èto denote the specificmass pel; unit of instantaneous spring length and À. to

,I denote the radius of gyration of the cross-section with respect to the axisof ~end4ig, we find the equations of motion to be

'oN '-' +Q=O;o'ljl .

o2y. o'IfJ oQ'e-=N---,ot~ . OZ OZ

,(227)

z

53951

Fig. 37. An element of the rod replacing a helical spring, in the case of transverse vibrations.

Further, with the notations for the elastic deformations as introducedin section '2, we have) Q ,

cp=-,, {3

whilst the slope o(the tangent to the central line is equal to [èq. (6)]

o'ljl , M-=-,oz a

(228)

oy--=cp+'Ijl.

OZ(229)

./ '

76 J, A: HARJNGX

Bearing in mind tliat the normal force corresponds approximately to theaxialload P and using eq. (228), we can also write for the third equation

(227) (P) 02'IjJ' 02'IjJ{3 1+ - rp= eÀ2 --a-· (230){3 I. ot2 OZ2

Further owing to (229) we can write for the second equation (227) whendifferentiated with respect to z

02!p 02rp 02~ 02'IjJ:- e ot2 + {3 OZ2= e ot2 + P OZ2 • (231)

Eliminating rp from (230) and (231) we-find for 'IjJ the differential equation

04'IjJ_ (g+ eÀ2) 04'IjJ +.~eÀ2 04'IjJ+ P (1+ p) 02'IjJ+g (1+ P) 02'IjJ= O.oz4 {3 a oz2ot2 {3 a ot4 à {3 OZ2 a (3 ot2

(232)

In the calculation of the natural frequencies for longitudinal vibrationof the spring it appeared that, given a sufficiently large mass m, the rootsql of eq. (214) could be taken as being equal to a multiple of n, this showingthat these frequencies are practically independent of m and that we could •

, .just as well have imagined. the ends of the spring to be clamped. Since itmay be presupposed that the same applies in respect to the transversevibrations, we shall at once consider ihe case where. the spring has bothends clamped. Let us suppose that '

'IjJ = P(z) sin 2n'Vt , (233)

then substitution in eq. (232) yields for P(z) an ordinary differential• I '

equation of the fourth order. Putting, in the usual manner, P(z) = eer,we obtain the following critical equation for q

where

(235)

(234)

In the last two expressions 'V1 = t Y '1/el2 represents the lowest naturalfrequency for longitudinal vibrations [see eq. (216)]. Denoting the rootsof eq. (234) by ± ql and ± q2 respectively, wehave ..

(q1l)2 + (q2l)2 = el , (q1l)2. (q2l)2 = e2 , (~36)

whilst P(z) is equal to

P(z) = Cl sin q1z +Dl cos q1z + C2 sin q2z + D2 cos-qi!z. (237)

• 1 !

, .

y ('11)2e3 = {3 n2 'Ill '

and using the first equation (235) we thus getI

el2 ( P) ~. . dP d3P~' 0 • 0

- 4n2v2 - 1+- y = (el -e3) a:;+ l2 dz3 sin2 n'llt+ gl(Z)t +g2(Z),a 0 ~ • (239)

The solution fot y however has also to satisfy eq. (229). Eliminating fromthis equation the angle ep with the aid of (230) we see that the solution (239)satisfies the new eq. (229) only when hoth gl(Z) and g2'(Z) have a constantvalue. This would mean that - apart from the vibrations at frequency v - .-.'the spring as a whole would he displaced' at a uniform speed parallel toitself. Considering that the ends are clamped such a displacement isprecluded, so that in eq. (239) we have to take the constants of integrationgl(Z) and g2(Z) as heing zero.

With the aid of (236), (237) and the' new quantity e to hOeintroducedaccording to

(238)

o J

HELICAL SPRINGS, RUBBER RODS AND 'VIBRATION-FREE JlIOUNTINGS 77..The lateral deflection y is calculated hy re:glacing Q and N in the second

=equation (227) hy pep and P respectively, then eliminating the angle ep, 0

with the. aid of (230) and finally integrating twice with respect to t.Putting

; i

/

q2 el .; e3 - (q2l)2 q2l (qll)2_- e3e=~· =-.-----,ql el - e3 - (qll)2' qi.l· (q2l)2- e3

eq. (239) can he written iD:the form

4n2v2 el2(1 + PIP) . . 0

. - ~ ( 2 l Y = (Cl cos qlz - Dl sm qlz +~J~-~ I+ eC2 cos q2Z - eD2 sin q2Z) sin,2nvt.

(240)

(241) .

The boundary conditions

'IjJ = 0, ~ at both Z = 0 and Z = l,°Y'= 0,

I

(242) 0

lead to the equ~tions

\ 0 Dl + D2 = 0, Cl + eC2 = 0 , I lCl sin qll + Dl c~s qll + 92 sin q2l +D2 co~q2l = 0 ,

Cl cos qll- Dl sin qll + eC2 cos q2l- eD2 sin q2l = 0 •

For a solution differing from zero the determinant of this system of equa-tions must he equal to zero, which leads to the condition .

(243)

78 J, A, HARJNGX

-which resolves itself jnto

From e,q. (243) it follows that the ratios Dl/Cl and D2/C2 are

Dl 1 D2 1 e sin qll- sin q2l=--=--

Cl e C2 e cos qll- cos q2lIf the first equation (244) is satisfied, then from (245) it follows that

Dl D2, - = - tan !qll, ,- = -taniq2l,Cl C2,

so that, owing to (241) and (243), y is equal to

~

cosq.dz-!l) cos q2(Z-tl)~ ...Y = const. - sUl2:n;'Vt.. cos MIL cos tq2l

(245)

(246)

(247)

In this case the deflection of the spring appears to be symmetrical withrespect to the middle cross-section z = tl. On the other hand the secondequation (244) lead~ in a si~ar manner to the asymmetrical deflection

. ~Sinq~ (z-tl) sinq2 (Z-tl)~ .'Y = const. sin 2:n;'Vt. (248), sin tqll sin tq2l

The radius of gyration A. -of a' ring of diameter Do with respect to' a

meridian is equal to i DoV2. Availing ourselves ofthe values for the springrigi~ties a, fJ'. y given ineqs (3) and (15), we obtain from (235), (238) and~= 'P/Yo [eq. (ll)] , '

, 4m + 1 ('V ')2 2(2m + 1) ( lo)2 ~ m+2 ~el = :n;2 - + - ~)1- ~ ,4(m+l') 'VI . m+l Do \, 2(m+'l)

• . I

e~ = 2m + 1 :n;2 (!.!\2[ m :n;2,(!_)2_ 2 (~)2 (1'-~) ~1- ' m +2 ~~],

m+l vIJ 8(m+l) ,1'1 po, ? 2(m+l))'m ('1.')2

I es , 2 (m + 1):n;2 'VI '

or with m, = 10/3

el = 8'16 CY~, 3'54 (;J 2~ (1- 0'615 .~},

e2= ~16'6 CY-35'0 (;)2 (1-~) (1- O'61~~)~CIf.

(249)

(250)

HELICAL SPRINGS, RUBBER RODS AND ~BRATION.FREE :MOUNTINGS 79

.Taking lo/Do and ~ as given, we state that according to (249) and (250)'the ,quantities el' e2 and e3 are functions of the variable V only. In con-nection with (236) ~nd (240) the same applies for the quantities q1l,. qzl -,and e, so that each of the two conditions' (244) represents in essence anequation in v. The roots of these equations thus give the natural frequenciesrequired and are most easily determined by graphical means. Accordingto whether we have to do with a root of the first or of the second eq. (244)the deflection of the spring is then symmetrical or asymmetrica1.

Given a sufficient 'degree of slenderness .la/Do it may happen that e2 is I

negative [see eq. (249)]. When we assume that always (q11)2 < (q21)2this means according to eq. (236) that (q11)2 is negative too. Puttingq1l ~ . iq~l and e= ie', we get .

(251)

and we write the conditions (244) in the form,

Of thè infinite number of natural frequencies, as a rule only the low~stwill be of interest to us, Now by allowing the frequency v,to increase fro~'zero on upwards, for given lo/.Dó and ~ we automatically find the lowestvalue of V at which one of the equations (244) is satisfied. With transversely-vibrating rods we know that the lowest natural frequency corresponds toa symmetrical deflection. It is therefore obvious to assume for the time beingthat this willlikewise be the case with the transversely vibrating spring. "The case of the symmetrical deflection corresponding to the first eq. (244)

• has bee~ further worked out and the result is given in fig. ,38. Here theratio of the lowest natural frequency for, transverse vibrations (VI) to thelowest natural frequency for longitudinal vibratio~s (VI) is plotted as afunction ofthe relative compression ~ and theslenderness ofthe spring lo/Do'. The dotted curve together with the coordinate axes gives the boundaryof the region within' which, in connection with a negative value of e2,we shall use the first condition (252) instead of the first condition (244).

. The dashed curve indicates where the curves plotted lose their significance .owing to the fact that then a lower natural frequency holds for the asym-metrical than for the symmetrical deflectïoit. This is the reason why thecurves have not been fully drawn beyond that dashed curve. Such a case,. however.. occurs only with ver} short springs and is therefore of minor'importance. More~ver the mutual differences are only small; for instance,with a spring having a slenderness lo/Do = 0 the lowest natural frequencyfor transverse vibrations in relation to that for Iongitudinal vibrations

../

80 J. A. HARINGX

lies at '-J!4(m + 1)/(2m + 1) = 1'50 instead of at Y2 '(m + 1)/m = 1'61 ~in~icat~d in fig.' 38.

0.8

0.6

0.4

0.2

PtV;I 9;;=0

.~...... ...... ······0' t ~ ·eI- ~ ~ ~ ........ ._~.

IJI--r-- -,L....- ./

_V V \t::- 4

t::::r::::t-- -~~

Vr-; ,.....t\'~ 7 r<

tO~ \ \

1.81.6

1.4

1.2

1.0

o aa 0.4 0.6 0.8 1.0S

53958 .Fig. 38., The ratio of the lowest natural frequency for transverse vibrations -(Vj") to thelowest natural frequency for longitudinal vibrations (Vl) as a function of the relativecompression ~ for various values of the slenderness 'o/Do. "

,While plotting fig. 38 we have partly made use of the possibility tocalculate, at least approxi~ately, the lowest natural frequency of a springby concentrating a part e of the spring mass (I}l) in the centre. When we dothis for .the longitudinal vibrations 'this mass is contained between two halfsprings each having twice the rigidity of the whole spring, and the natural.frequency' will then amount 'to ' '

VI = 2~ V:;;= '21nVe~2'

\ .In connection with formula (216) 'it thus appears that e must 'he equal to4/n2 ~ 0'4. Making the same calculation for the transverse vibrations andassuming in the first instance that an equal part of the spring mass has to be.concentrated in the centre, we find that the ratio V,l/VI will ha vt: to beequal to the square root of the ratio of the lateral to the axial rigidityfor each half spring. With sufficiently long springs this does indeed proveto be the case, so that there is some similarity between fig. 38 and the upperhalf of fig. 33. When the spring is loaded exactly to the buckling point thelateral rigidity of the half 'spring is reduced to zero and thus the natural.frequency for transverse vibrations will be zero too. That i~why the pointsof intersection on the horizontal axis in the two :figures agree, providedwe allow for a factor 2 with respect to the slendern~ss lo/Do'

(To be 'continued) Eindhoven, October 1948