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AD478559 

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Distribution authorized to U.S. Gov't, agencies 
and their contractors; 

Administrative/Operational Use; MAY 1965. Other 
requests shall be referred to Frankford 
Arsenal, Philadelphia , PA. 


_ AUTHORITY 

USAFA ltr, 7 Jul 1971 


THIS PAGE IS UNCLASSIFIED 



















Summary Report: Analysis of Advanced Mechanical Time Fuze Components TR #65-9 May 1965 


SUMMARY REPORT 


ANALYSIS OF 

ADVANCED MECHANICAL TIME FUZE COMPONENTS 


By: 

Technik Incorporated 

TR #65-9 


50 Jericho Turnpike 

Jericho, New York 

May 1965 

For: 

Frankford Arsenal 

Philadelphia, Pa. 



Contract No. DA-30-069-AMC-17(A) 



CONTENTS 


Page No. 


1. INTRODUCTION l.l 

2. CONCLUSIONS 2.1 

2.1 Spiral Motor Spring 2.1 

2.2 Other Problems 2.3 

f 

3. SPIRAL MOTOR SPRINGS . 3.1 

3.1 Introduction 3.1 

3.2 Torque-Turn Characteristic* of Non-Spinning Motor Springs 3.3 

3.2.1 General Remarks 3.3 

3.2.2 Determination of Torque-Turn Characteristics In 

Linear Range 3.6 

3.2.3 Determination of Maximum Torque (T max ) 3.9 

3.2.3.1 Total Spring Torque 3.9 

3.2.3.2 Elastic Case 3.11 

3.2.3.3 Plastic Case 3.12 

3.3 Spin Effect on Motor Spring 3.17 

3.3.1 "Axially Symmetric" Solutions 3.18 

i 

3.3.1.1 Uncoupled Extensional Deformations 3.19 

3.3.1.2 Bending Deformations 3.24 

3.3.2 Eccentricity of Deformation 3.49 

3.3.2.1 Evidence of Occurrence of Eccentric De¬ 
formation 3.49 

3.3.2.2 Eccentricity of Spiral Center of Gravity 3.52 

3.3.2.3 Generalisation of "Axi-Symmetric" Bending 

Analysis 3.54 

3.3.2.4 Effect of Eccentric Deformation in Torque 3.56 




CONTENTS (omIUmoS) 




Pago No. 


3.4 Summary of RtiulU 

1.41 

4. 

PIVOT AND JOURNAL FRICTION 

4.1 


4,1 EUitlc Aulyili 

4.1 


4.2 Torques In Elastic Cass 

4.4 

5. 

BEAM HAIR SPRING PROBLEMS 

5.1 


5. 1 Effect of Setback on Hair Spring 

5.1 


5.2 Effect of Wedge Retainer an Hair Spring 

5.4 

6. 

TORSION HAIR SPRING 

4.1 


6.1 Torsional Frequency ( f ) 

6.2 


6.2 "Whirr* Frequency 

6.3 

APPENDIX A 

A.l 

APPENDIX B 

B.l 




NOMENCLATURE 


a r contact radius 
b * width of cross section 

C * constant of coil geometry coil spacing) 

^ - acceleration due to gravity 
h ' depth of cross section 

£z numerical index Indicating particular coil, >- * * m (subscript) 
j - numerical index indicating Iteration cycle (superscript) 

k ' spring constant 

r 

L - length 

' mass of spring per unit length 
number of coils of spring (or number of g's) 

a 

yp-pressure, ^pressure 
r ' spring radius 

coordinate of length measured along spring coll 
(A - radial deformation of spring 
variable numerical index 
A : cross sectional area of spring 
C* constant of Integration 
£.* modulus of elasticity 
/* force, - preload 
J > moment of inertia of spring section 
K‘ constants, defined as used, in text 
A7*spring bending moment 
M* value of M at attachment to barrel 
Ml maximum value of M (at attachment to arbor) 

P z force 


i 



£ , 

^« ratio of any coll radlua to radtua of last (largest) coll * 

c'i* t 7 

Q*Z'- Cpf> )J 
& 

]f'ie torque at spring arbor 

maximum linear value of T (occurs at 'fJl 
[J: strain energy of spring 
\*J- work done In coll deformation 
o(? angular measurement following spiral of colls 
S - linear deformation 
5"ex'- change in angle 
A change in radius 

eccentricity of spiral 
fT* 3.1416 

77: potential function * U - l/\f 


(T- stress 
<jj: yield stress 
p* radius of curvature 
yU* coefficient of friction 
^poisson's ratio 
shear stress 

(j?z angle of spring wind-up from position at rest in barrel 

point, <j> , at which linear action initiates (all coils are "free") 
<jP»point* C$> , at which linear action ceases (spring begins to extend) 
CO* spin rate 


ii 













1. INTRODUCTION 


This report will serve to summarise the analysis, and the conclusions 
drawn therefrom, in compliance with the requirements of contract #DA-30- 
069-AMC-17(A). A summary of these conclusions will be found in section 2 
of this report, and the full analysis is reported in sections 3 to 6. The sub¬ 
jects of this analysis are somewhat varied in view of the nature of the "Scope" 
of the contract, which calls for analysis in problem areas of more or less in¬ 
dependent content. The general areas covered are related to the effects of 
the dynamic projectile environment upon specific fuse components: 

(a) Spiral Motor Springs 

(b) Pivot and Journal Friction 

(c) Torsion Hair Springs 

(d) Beam Hair Springs 

Thus, the performance characteristics and various specific aspects of the 
behavior of these components in a spin or setback environment will be the 
subject of the following discussion. 

The initial analysis undertaken pursuant to contract #DA-30-069-AMC- 
17(A) was concerned with the analysis of spiral motor springs. This analysis 
had, as its objective, the determination of the effects of projectile spin upon 
the torque yield of the motor spring. 

It was determined that in order to obtain physically meaningful results, 

the non-spinning or "static" spring must be fully understood. At the time, 

the body of literature which dealt with spiral springs considered only those 

~ r*.4Ltit/y 

springs unconstrainedr by an outer barrel. A great deal of empirical evidence 


1.1 




was available which supported the view that the existing analytical approaches 
were inapplicable to "real" springs. 

Thus the "static" analysis was undertaken to determine the character of 
the relationship between arbor torque and the physical parameter s of the spring. 
This was determined, yielding the equation governing the "torque-turn" char¬ 
acteristic within the region of interest. This equation, in addition to being of 
great utility and interest, forms the basis for comparison in evaluating the 
spin effects. 

The next step in the analysis was facilitated by dividing the spin effect 
into its simplest constituents, i.e. , axisymmetric extensional deformation 
and axisymmetric bending deformation. The effect of these modes of defor¬ 
mation upon the torque output at the arbor is described in detail in section 3 
of this report, along with the "static" torque-turn characteristic. In this 
same section the possibility of the incidence of greatly eccentric deformations 

t' 

was introduced. 

The question of eccentric deformation received preliminary analytical 
and experimental study; as reported in section 3. This mode of deformation 
was not foreseen at the outset of the investigation, nor was the possible extent 
of its effects. These effects are Indicated, on an order of magnitude basis. 

Investigations of pivot friction under the influence of high centrifugal load¬ 
ing due to projectile spin are indicated in section 4, following the discussion of 
motor spring eccentricity. Calculations are performed for the fully elastic 
condition of the pivot and pivot seat. It is noteJ that under ordinary conditions, 
plastic action and accelerated wear may easily become important considerations. 


l.Z 




The beam hair-spring problems associated with this contract are in¬ 
cluded in section 5. 

Preliminary investigation of the torsion spring-critical speed problem 
is indicated in section 6. It is found that, for particular cases of spring 
geometry, the various modes of critical speed may be related to the tor¬ 
sional frequency parameters of the escapement. Similar relationships may 
be established for other configurations of interest. 



2. CONCLUSIONS 


2. 1 SPIRAL MOTOR SPRING 

The various stages of this investigation are listed below, for reference: 

(a) Study of torque-turn relationship for non-spinning spiral motor spring 

(1) Relationship in linear region 

(2) Maximum torque (utilizing "limit plastic" effects) 

(b) Axi-symmetric effects of spin environment 

(1) Effects of extensional deformations 

(2) Effects of bending deformations 

(c) Non-axi-symmetric effects of spin environment 

(1) Sources of spiral eccentricity 

(2) Eccentric ^effects. 

Of the above listed topics, the non-spinning spring characteristics have 
V een fully explored, as indicated in section 3.2. Only in the event that it is 
found desirable to use the non-linear region of*, the torque-turn characteristic, 
would it be necessary to pursue the investigation of the non-spinning spring 
further than his been presently completed. 

Axi-symmetric effects of the spin environment are presented in section 3.3. 
These effects are discussed in great detail, and the analytical tools are devel¬ 
oped for their determination in any given case. Further work in this area would 
be concerned with the establishment of the relationship between torque, defor¬ 
mation and spring parameters; and, therefore, of torque-spin-sensitivity. 


2.1 


The incidence of non-axi-symmetric deformations and the torque per¬ 
turbations resulting therefrom, is pointed out in section 3.3.2. As is dis¬ 
cussed there, analysis of this phenomenon is, in many respects, an extension 
of the axi-symmetric analysis. It is true, however, that this extension is not 
minor in execution, although the basic concept of approach follows that already 
developed for the axi-symmetric case. 

Technik has found that in addition to axi-symmetric effects investigated 
within the scope of this contract the torque variations due to eccentric effects 
could be a dominant controlling factor in tne proper utilization of spiral springs 
in fuze design. The presence of these eccentric effects was first noticed from 
empirical evidence and then verified analytically. It was found that the existence 
of this eccentricity is intrinsically tied to the spiral spring geometry. The 
magnitude of the associated torque effects is dependent upon the parameters 
of the spring design and the spin environment. However, the specific relation¬ 
ship between eccentric behavior and physical parameters, has not yet been es¬ 
tablished. 

Extensive deformation of this sort would result in a major perturbation 
upon the torque-turn characteristic, above that induced by the axi-symmetric 
moder of deformation. At the present stage of Technlk's analysis, the exten¬ 
sion to eccentric deformation is a natural next step. As such, it involves an 
advance of the current state-of-the-art (as presented in section 3), which should 
be built upon the foundation of the current level of knowledge. Technik's pres¬ 
ent level of experience with this analysis will prove an invaluable asset in con¬ 
tinuing the investigation of all the ramifications of problems associated with 
spiral motor springs. 


2.2 



A summary of the analytical results of the spiral spring analysis will 
be found in section 3.4. 

2.2 OTHER PROBLEMS 

In addition to the major effort expended on the analysis of spiral motor 
springs, during the course of this contract several other independent prob¬ 
lems received Technik's attention as per the directions of the technical super¬ 
visor and the provisions of the contract "Scope". The results of these inves¬ 
tigations will be found in the respective sections devoted to the analysis (4,5 and 6). 

Briefly, these sections represent first-order analyses of torque losses 
in pivots and journal bearings under high centrifugal loading, of critical speeds 
in a torsional hair spring and of setback and fabrication technique effects on 
beam hair springs. 

In the section on pivot and journal friction, it is found that, in addition to 
conventional friction considerations, the high loads associated with high pro¬ 
jectile spin rates,may cause plastic deformation effects (brinelling). This type 
of deformation would influence the nominal elastic torque characteristics of 
the bearing and may lead to a situation designated aB accelerated wear, where 
the material is capable of shearing under the frictional loading developed. 

In section 5, it is shown that for the normal range of beam hair spring 
dimensions considered, resistance to setback loading is limited by the onset 
of plastic action rather than by beam buckling. Thus the indicated design pro¬ 
cedure is to increase the plastic carrying capacity of the spring cross-section 
with only secondary regard to the elastic stability of the beam. In addition, 
the effects of a wedge retainer upon beam hair spring deformation are shown 


2.3 



to be cxtenalve over the length of the epring* Thue* thle ehould be roc one Id* 
erod if it ie to be a method of fabrication. 

The final aection indicate! the unique reUtionehip that oxiete between all 
the tor!ion hair opring parametere. The oame parametere determine critical 
epced and natural frequency* thue knowledge of one may bo ueed to indicate the 
other* without the neceeeity of direct appeal to all implied parametore. 


3. SPIRAL MOTOR SPRINGS 


3.1. INTRODUCTION 

This report will summarise Technlk'a analysla. to date, with reapect 
to the behavior and deformation of aplral motor aprlnga in a apln environ¬ 
ment* The axis of apin Is taken perpendicular to the plane of the spring, 
as is conventional in mechanical time fuae practice; with the arbor axis co- 
linear with the projectile spin axis* This analysis is provided far under con< 
tract #DA-30-069-AMC-17(A). 




In order to fully understand motor spring characteristics in a spin 
environment, it is first necessary to understand these same character- 
istics in the non-spinning condition. It appears that even non-dynamic 
spiral spring theory has not previously been brought into agreement with 
the type of empirical data which is observed by spring makers and users* 

Thus the program has been divided into phases of investigation dealing 
separately with (1) non-spinning and (2) spinning environments. It was 
found that the "spin analysis" could advantageously be further sub-divided 
into so-called "membrane" and "bending" solutions representing extension- 
al spring deformation and bending deformation in the plane of the spring, 
respectively. 

The bending solution, which will be found to present the greatest phe- 
nomenalogical complexity, may be examined from the viewpoint of a super¬ 
position of eccentric effects upon an (almost) axially symmetric solution. 

The coupling of these two separate effects, which in combination with the 
other analytical results of this program represents the complete solution 
to the spinning motor spring problem. This report emphasises the various 
aspects of the symmetrical mode of deformation of the motor spring) limit¬ 
ing discussion of the eccentric effect to its preliminary aspects. The impor¬ 
tance of this latter mode of deformation was only recently determined during 
the course of experimentation. Later work will explore it more fully. 


3. 2 













3.2. TORQUE-TURN CHARACTERISTICS OF NON-SPINNING MOTOR SPRINGS 


3.2.1 GENERAL REMARKS 

Study of Frankford Arsenal drawings, photographs and experimental 
torque-turn characteristic^ which are presented in the following figure, 
indicates the following general conclusions regarding the non-spinning spring: 

(a) Experimental Torque-Turn characteristics indicate a discrepancy 
between wind-up and run-down curves. 

(b) The area between these curves is indicative of the energy lost to 
the system, i.e. , the excess of energy required to wind the spring over 
that released by the spring in unwinding. 

(c) Spring energy losses during the period of interest (the run-down 
cycle) are indicated by the area between the "ideal" run-down characteris¬ 
tic and curve; which ia a "mean" between the wind-up and the run-down. 

(d) The basic shape of the characteristics are often overlaid by cyclic 
perturbations corresponding to local disturbances in the geometry of the 
spring-barrel-arbor combination; indicative of additional energy losses due 
to friction between local points in contact. 

(e) The torque turn characteristic normally changes its character at 
some point in the wind-up or unwinding cycle, between a linear and a non¬ 
linear relationship. 

(f) There is another basic change in the shape of the characteristic at 


3.3 



the maximum number of turns (l.e., when the spring is wound "solid"), 


at which point the elope of the characteristic changes radically 

dT 


( 




very large). 


The element of area on the preceding curve, which is ~7'd c f » represents 
the elemental energy, • Thus the single cross-hatched areas shown be¬ 
tween the perturbed characteristic shape and the non-perturbed wind-up or 
run-down characteristics is a measure of energy loss due to friction imposed 
by local effects (interference between eccentric portions of spring). Likewise 
the energy loss between wind-up and run down is represented by the overall 
area between the two curves, whereas the available energy is that (double 
cross-hatched) enclosed by the run-down curve. 


In the following treatment, the effects of friction are not considered. 

Those effects, which are of great importance in their own right, are not 
first-order considerations in the present problem of the spinning motor spring. 
Further, this section will be limited to theoretical derivations of the necessary 
equations. 


3 . 4 












6 


3.2.2 DETERMINATION OF TORQUE-TURN CHARACTERISTICS IN 
LINEAR RANGE 

Man/ references may be found which deal with the spiral spring in the 
condition where adjacent colls are not in contact. (e,g., Chapter 27 of Wahl'o 
"Mechanical Springs") The relationships to be found in these references 
take the form: 

r- ¥ 

where T* torque delivered to arbor by spiral spring 
* bending stiffness of spring section 
l ' spring length 

s number of deflected turns undergone by spring 

/s * constant depending on prescribed boundary condition at 

outer boundary: fixed end ^ 1 4 fcC ~ 1.25 pinned end 

The problem which arises in the application of this relationship is gen¬ 
erated by the incompatability, in practice, of the definition of cf> with the 
requirement for non-contact, i.e., when O £ cf CQ some coils are in 
contact. Thus, in the normal course of events the Important position of 
zero torque occurs precisely in that range of values of where the clasi- 
cal solution does not apply. 





tmth w ^ 





Thus, in the barrel, the spring delivers zero torque at some non-zero 
number of turns; which is beyond the validity of the classical solutions. 



due to the barrel constraint and the resulting non-linearity of the solution 
for , the linear solution may be used only to find the slope ( ) 

for values of . Further, it is sufficient to arrive at the proper 


solution in this linear range, at this time, since the Bpring is always used 


in the range range for time fuze applications.) 


Given the slope, within the range of interest (i.e* 
one may integrate to find: 

r- f 7 * <? + c 


<TL - u £1 
def - * l 


which differs from the conventional expression only by the constant, C . 
Any boundary condition can determine the value of C , but since the zero 


boundary condition is submerged in the non-linear range of the solution, the 


3 . 7 







boundary condition at Cj^ will be utilized . 

Thus, using the notation at %*** • ex P re88 ion 

for the torque-turn characteristic is found to be: 

T’ - j- k ( 3U, - <?) 

In the above expression, which is valid for the linear regions of the char¬ 
acteristic, may be supplied from experimental measurements, or 

analytically as shall be discussed in the following section. It is to be noted 


that the limitations imposed by the theoretical assumptions do not seem to 
be important limitations in the correlation between experiments and these 
these theoretical results. 




9 


I 

I 

E 

i: 

i: 

[ 

e 

L 

r 

i: 

r 

c 

E 

I: 

0 

II 


3.2.3 DETERMINATION OF MAXIMUM TORQUE ( ) 


3.2.3.1. Total Spring Torque 

A free body diagram of the spring and of the arbor will illustrate 
the relation between the moment) , and the torque, 7**.* • Assuming 

a simply supported end at the barrel! 



3. 9 



Equilibrium equation* result In 


71. r ' Pf. 

7'*., Pf * 

or combining through the elimination of p , there results 

/' ( 


For the case where the end at the barrel 1s pinned r’, * & and 

/-v ^ 77(70 

Further it can be shown for the elastic case with a fixed end at the barrel 
that p- 6 and 


7lf V * /^y 

In general We can write 

7 ^-/ * My 


where for the elastic case 



* 


f-i/eP 

■=& 




Further, since the spring unloads and reloads elastically, even when it has 
rocei 'd a prior plastic deformation to achieve its form from the flat stock, 
this value of K, will hold in general. Cases other than pinned or fixed end 
can be considered separately but we can assume they will be within the range 
Indicated above. 


3. 10 





3.2.3.2 Elastic Casc 


The torque, 7 » which can be exerted by the spring, in its moat 

general position, is determined from that moment associated with the 
spring in its fully wrapped ( position as was shown in the pre¬ 

ceding section (2. 3. 1) . This , in turn, may be found on the assumption of 
an initially flat spring (i.c. , prior to assembly in the barrel), without 
radial pressure. Consider the fully elastic case: 


where 




£1 

'U * T 

r maximum spring moment at Cf> 





smallest radius of curvature of spring coil - arbor 
radius when spring is fully wound. 



However, it will be found from later numerical considerations that this 
case does not apply for the fuze springs under consideration. For these 
springs the plastic case must be considered. 


3 . 1 1 



3.2.3. 3 Plastic Case 


The spring when wound, from an initially flat state, and subjected 
to an enforced radius of curvature, p , however, may be in any one of 
the following states, with regard to its stress distribution: 




For the range of values encountered in time fuze motor springs, 
case III must be considered. Note that the maximum bending stress, 0^, , 
is equal to the yield stress, <T^ , in this case, as well as in cases II and IV 

In this case, however, there is an elastic core of thickness h ; thus II and 
IV are special cases of UI, with h' equal to zero and to h , respectively. 

Thus the moment at the arbor (for case in) can be shown to be given 

I 

b, the relationship 

M s b L(^*' 3 (i) J 


3.12 


where 


k s spring width 
h - spring thickness 
h - elastic core thickness 


The core thickness, h , may be found in terms of the radius of 
curvature of the spring considering the elastic deformation of the core, i.e,, 

*/? 

Substituting this relationship into the expression for the plastic moment, 
and noting that flpimric ~ T m ^ » the maximum spring torque is found 
to be: 

' K b [(*) ~ J 1 

It may be hoted that this expression can be written: 

T..,-K.n r ^ 

where ~ ^ ) j Bee case II, previous page. 

So that the condition ~ A^/ ■«// when f implies 


that: 


CW --/ 




_ £ A 

- Of t 


For initiation of 


plastic flow 


Thus: for 


/ ^ uj [(*)'- i f' 


for , 71 n. x = K & GJ T 

, T - < H 


3 . 13 


Note that subsequent to the initial deformation of the spring (pre¬ 
sumably during the manufacturing process) the spring upon unwinding and 
rewinding, will return to the value in a linear manner (even though 

may exceed K, M yield, for ) • This statement would be 

strictly true for an ideal elastic-plastic material, and for a "non-ideal" 
material is close enough for our present purposes. For an ideal clastic- 
plastic, the stress-strain curve has'the following shape: 



Thus, for multiple deformations the critical elements behave as; 



3. 14 



This elastic-plastic loading-unloading cycle on the outer fibers, com¬ 
bined with the elastic loading-unloading cycle on the inner fibers, results in 
the following stress distributions: 



stress di>tr)to* q / u / 7 wci 
/HI fit*./ iJs f e>r in *.'/'<c* J 

A*?- V 


/*/ 






■ 1 ‘ 


/<f 


s tress ctn trib* /<o* 
/* /ouJ&ct spri"i<-\ 

H 



■foie*./ S tress cJli friborttvi 
<*£ter a* •* <j 

h^e^r’ clijt~r’ib LA Z l O H 

H * Mf/.s-tu 


Note that the "apparent" loading and unloading cycle is: 




and that the residual stresses are unobserved. 


3 . 15 



Thus the value of is independent of the number of wind-up 

and unwind cycles imposed on the spring. The torque-turn characteristic 
( T' C-j> ) is linear for a range of values of torque up to and including 
This portion of the curve then is fully determined. The fully unwound por¬ 
tion of the curve falls in a non-linear domain and, being of less interest 
at the present time, is not treated in this report. 



3.3. SPIN EFFECT ON MOTOR SPRING 


Let us now consider the effect of spin on the static torque turn charact¬ 
eristic (as discussed in section 3. Z).The following assumptions will be util¬ 
ized in the ensuing study. 

(1) The spin environment may be represented by a centrifugal field 
only, i.e. , Coriolis effects are negligible. 

(2) The centrifugal field may be represented in a quasi-static manner, 
i.e. , transient effects and other inertial effects are not considered during 
the initial stages of th' investigation. 

(3) The bending and "membrane" effects due to the centrifugal field may 
be treated as uncoupled. 

(4) The axisymmetric effects may be treated as uncoupled from any 
eccentric bending effects. 

Assumptions (1), (3) and (4) are made at the outset in order to facilitate 
the investigations; they may actually be proven valid at a later stage in the 
investigation, thus becoming conclusions. Assuption (2) merely serves to 
indicate that the initial investigation seeks a steady-state solution to the prob¬ 
lem; transient effects will be considered later, and required modifications 
may then be applied to the steady-state solution. The steady-state solution 
yields insight into the mechanism under study and is necessary prior to the 
more complex transient solution in the course of new analytical investigations; 
the transient investigation may never be required. 


3.17 




3.3.1 "AXIALLY SYMMETRIC” SOLUTIONS 


A spiral is by its geometry, unsymmetrlcal. For axial symmetry, 

r ( c* + SoO x r ( oL ) 

must be everywhere true (i.e, , for all e><^ and for all Set ). For a spiral 
this is not true but it is almost true everywhere. That is: 


r (c<+ Zot J - rCot)+e 


wh 


C>C - any spiral angle 
^~ change in spiral angle 

s- _ r ds- 

<j r 

- r "pitch of spiral (constant in Archimedean spiral) 

<d cA 


For the special case of interest, where 
half of any coil is closely approximated by a 


£ -> o (large number of coils), 
semi-circle. 


The following treatment will initially be 
problem related to the "quasi-symmetrical" 


limited to those facets of the 
spiral. 


3. 18 



3.3.1.1 Uncoupled Extensjonal Deformations i. 

f 

In view of assumption (3), "membrane" and bending effects may be 
treated separately. Membrane, or extensional deformation, effects will 
be treated first, where membrane effects are understood to imply effects 
traceable to net force resultants due to stress distributions across the 
cross-section of the spring. Bending effects, on the other hand, are under¬ 
stood to imply those effects traceable to stress distributions, having a net 
moment at the neutral axis of the spring cross-section. Thus any linear 
stress distribution is merely the superposition of the membrane and the 
bending stress distributions, as shown below. 


/'•# e* r stress chst- 



stress 



£<?•* 


stress 



In the following an energy approach will be introduced; which is only 
one of many techniques which will be utilized. 

Based on th* "quasi-symmetrical" assumption, one may develop, 
for th e extensional c ase alone, the following analytical model. Assuming 


3.19 



that each coil may be approxin ..tod b> a circular arc (l*e«» ^ d€r 

for Sek 'I'iT ), the following idealisation will bo utlllaodt 



Spri^ 




■ *i 


The element of spring length, for the i ’* coil, now appears as: 



Thus, for equilibrium: 


/V' /*t ^ V 


t + 


4 # 


and the radial growth ( ^ ) is given by 

A/r « 

A? 

where the membrane force, f\J , is constant for any one coil (l.c, , for 
each circular arc). 

The theorem on total potential energy of a body (which may be found 
on page 173 of Love's "Mathematical Theory of Elasticity") states that 


Ku 


3.Z0 



whore the total potential« Z Wg , is equal to the sum of the internal (etrain) 
energyi 1/ , of the deformed body, minus one-half the external work done 

on the body, j[ W (in this case, by the centrifugal forces) in deforming 
from its unloaded state to its loaded state of equilibrium. Since only "mem¬ 
brane" effects are now being considered the subscript y will be used. Thus: 

U, ■ i "Wn *0 


where 


«- i Jft 


Car r) 

N = stress resultant of membrane stresses 
A £ r spring stiffness in stretching 
l * length of spring 

S $ * arc length and angle respectively, measured 
around the spring 

f * radius to generic point on spring 
M * mass per unit length of spring 
CO t spin rate of spring 

LA - change in f under centrifugal force (** r * < 


Integrating 



for each coil, one finds for the 



Wfj, r 277^ C<J Z r* ts. 

'\ t * 

Thus: ' 

* 

W/y s 

n fit < 

where the coils are denoted by enumerators, / , / 4 i 4 ^ 

Likewise, one finds: 

Vn~ Je & 

ft/ 

3 . 21 



- o 


Substitution in s O • 

^ (N*r). - n Cr^u*). 

, *> * i •• 


/v) 4 * 

-S-- ->*/)£ 

Z Cr*")i 


CO 


:*> 


Substituting the coil displacement relation: 

Nr. 

~aT 

the general expression for all coils may be written: 


± (M'r). 




- *•* cO 


t (r'fi')l 


The above expression holds in general and, as can be shown, for the / 

M'n 


and j coils; 


f;'Vl 


M % r 

— L —-r ' * cO 

r. 3 N; 

) J 


or 


A/ 




CO 

















This Is the result that Is obtained by an equilibrium analysis of a 
hoop (or a thin cylindrical shell) of radius \T and body force equal to the 
value , The mechanism involved in the stretching of the spring is 

akin to that involved in the stretching of the hoop; an order*of-magnltude 
estimate of the value of /V s /l/ f (l.e., the uncoupled membrane force, alone) 
at the juncture of spring and arbor can be arrived at by 


where: 


/l/ : 

/ / 

/Y ~ spring membrane force at arbor 


f = arbor radius 




^ /4 ( m mfj per /< 


t<) 


The additional torque due to a spring membrane force of this magni¬ 
tude at the arbor is simply 


T m -- N, f, * * 3 

For constant CO , the value of ~]l/ is approximately constant throughout 
the unwinding cycle, except when the unwinding angle, , is close to the 
limits, zero and . At these points, and close to them, various param¬ 

eters are affected by the fact that coils are effectively taken out of operation 
by being pressed against either the arbor or the barrel. 

The membrane effect on the value of 75/ found for the particular 
case of the spring configuration being investigated 


b s # >f 
A - */" 


3140 rfrec 



0-seS-i* 


3.23 




is approximately ^ i*-oi , which if about 1.2% o£ the maximum static torque 
of 20.5 in-oz. A percentage deviation of this magnitude may not be consid- 
ered important. It is to be emphasised* 

however, that this represents only one portion of the total dynamic effect, 
so it will not be investigated in further detail until all the effects are ex¬ 
amined; and is subject to further experimental verification. 


3. 3.1.2 Bending Deformations 

3. 3.1.2.1 Axi-Sy in metric Deformations 

Axi-symmetrical" bending deformations represent a redistribu¬ 
tion of coils within the radial confines of the barrel and arbor, as the spring 
is spun up. In addition to these boundary constraints, spring bending de¬ 
formations must satisfy constant spring length and invarient total number 
of coils (since the spring is taken in a position of quasi-steady wind position) 
conditions. These reflect the conditions which must be imposed on the de¬ 
formation since membrane extension was found to be small. 

All deformations which represent geometrically possible coil 
distributions can be classified in the following manner: 

Class (1): Uniform distribution which will be assumed to correspond to the 
initial undeformed state of the spring. 

(Let ft, , , and represent arbitrary numbers such that 

-t M M ^ — total number of colls of spring, 


3. 24 




Then clast (2) and (3) deformations may be defined as: 







Class (2): 


Class (3): 


H, outer coils move outward (toward barrel) 

fl i inner coils move inward (toward arbor) 

fly coils remain stationary ( in class (1) distribution) 

outer coils move inward (toward median radius) 
>4 l inner coils move outward (toward median radius) 
fly coils remain stationary (in class (1) distribution) 



For further consideration each coil will be taken to have a con¬ 
stant radius ( r t ) with an intercoil spacing ( & r i ); which greatly simpli¬ 
fies the ensuing analysis while retaining the important basic physical param¬ 
eters. Thus the above figures are idealised as 



3. 25 


It is understood that in this schematic representation the coils are con¬ 
sidered "open" so that no membrane stress exists and that each coil occu¬ 
pies a full 360° (except possibly the first and last) transporting whatever 
material is required from its neighboring coil. 

The foregoing classification as to types of deformation, facilitates 
the utilization of the theorem of minimum potential energy. As stated in 
section 15 (Mechanics of Materials) of the McGraw-Hill Mechanical Design 
and Systems Handbook , the theorem is: "Among all states of strain which 
satisfy the strain-displacement relationships and displacement boundary 
conditions the associated stress state, derivable through the stress-strain 
relationships, which also satisfies the equilibrium equations, is determined 
by the "minimization" of TT where 

n L ujv ~ Lj p‘ u * a ^ 

where P* . , ft are the ^ y, £ components of any prescribed sur¬ 

face stresses." 

The function JT may be expressed, in the notation of this 

report: 

TT - U -ffl 

where 1/ s total strain energy of deformation from equilibrium position 
to any particular deformed position, and fy\f ^ work done by centrifugal 
forces during the same deformation. 

The following highly simplified example is illustrative of the man¬ 
ner in which the above theorem will be utilized in this report. 


3.26 



, 1 


Example > 

Let ue find the deformation of a eimple extension spring (spring con* 

stant • k )» under tho Influence of a force , with preload • 

//Z/y^Z/y S////S//M 



u*/**J*J 

pniif* 

pttlaoJtJ r*f» 
postfix 

Let us consider the equilibrated deformation which, In this case, 

satisfies the equation 

k 

The theorem of minimum potential can be applied to either the original or 
the preloaded system to provide the same result; except that in the former 
the preloaded equilibrium equation must be known and utilized. 

(i) Total System ( $> 0> ) 

w*r, n 
v-- ik a*!.) 1 


hence Tl ‘ t k ( S * S,)' ~ t/~S * £ * S.)] 

n* k T * n- f'7 -; fn -a «• tyn - tpi 

(frm tpiltir**" *f**tto*) 

It is now necessary to minimize fT with respect to all possible variations 
in % . This could be done by setting ^yj rsO . However, since this 


3. 27 



technique may not be applicable in thia report* we will conaidar the moat 
general illuatrative technique of actually preaenting the reeult graphically* 
For thia purpoae we plot JF i*. 5 "faf J* 



We see that the minimum occurs at 



which represents the desired solution. 

*• 

(2) Relative System ( $> 0 ) 

In this case we consider only the perturbation from the equilibrium 
system so that it is not necessary to utilize the preload equilibrium equa¬ 
tions (no matter what their complexity). Thus 

- F c> 

v*ikS l 


3. 28 




.. 1 


■ * >■ 




hence 


rr--^r-n 


fl * 


which plots similarly as 




& t£ 




and yields a minimum where J- ^ , which is the desired solution. 


For the purposes of the work of this report we will always con¬ 


sider the spin perturbations from the non-spin equilibrium position; which 


greatly improves and simplifies the minimisation procedure. 


3.3.1.2.2 Elimination of Class (3) Deformation 


In order for the assumed deformation pattern from class (1) 
to any other class, to be possible, the function rr must decrease. 


Thus: 


TT„, < o 


is required 


Since 


r7 0) " O 


i s the reference position. 


3.29 







■-•; >■'■%#& S ; 

. - ■' ^-r*' . /■ '*• *1 


However, 


7 ^ - - K, 


thus, the possibility requirement becomes 

[/c») ^ i*e ») 

but, for all deformations 


therefore it is necessary (but not sufficient) that 

V„_,> 


^ a-x) ' 

in order to satisfy the requirement. 


The work for the i u coil, in moving from the non-spin position 
to the perturbed position, is given by 

. i/7 »*i 

K r J ('"fid#) ^ fi LA; 


and the total work of all the coils due to the perturbation is, 


]/J - 2V f; 1 LA; 


where 


if * average / - coil radius 
&./ * change in ft coll radius. 


Now considering the class (3) deformation 


LA; *0 

> ", 

c o*h f 


► 

ii 

> 

to \ i* , 

s? 

Mi 

* 5 
V* 

(A; 7 0 


dtli } > 

f. 4 r " 

r ; - r <i) 


and using the absolute magnitudes of deformations, we can rewrite the work 
as ** 


y= 2ir*u> , [-5> l r' IU-J * Z r^/U;/] 
L 7*7 < u 


3.30 









Considering the upper bound 


v * z ffVt rji i /*/ r r t ','zw] 


However, considering the total coil length 

L.* zrZ *;• 

L f 2*7 jj 


(HJ* - Sft ) 




0 


) 


But from the requirement for constant length 


4 -L, 

2 t'i ’ Z Ci 


which requires 

i » { =° 

i-i 

or 

(*\) (+) 
Z M/ r 

< g ( i*t 


thus 



3,31 







since 


0 0 


r, *r,' 
W — o 


which is a direct violation of our requirement, so that class (3) deformations 
c annot exist. 


3.3.1.2.3 Axially-Symmetric Bending Deformation Pattern 


In accordance with the idealization set forth in preceding sections * 
the spring sys.em, with its governing equations is presented below for ref« 



TT^IS-W 


Potential function 


V« if 

U‘irei’2. # 

<./ * 

t 



Energy Expressions (See 
Appendix for discussion) 


^ "Constant Length Condition" 
insurss that length of deformed 
spring £ that of undeformed 
spring. 


' 3. 32 



Xt la required that TT be a minimum* within the bounds of the 
constraints of the problem. This requirement may be formulated as: 

M =0 sip. + m 

(mil i ) Ci t*) 

which together with the constant length condition, as written below, defines 
a linear set of equations: 

U'+Z*** 


From the above form of the constant length condition, one finds that for 

•> 

any 6 1** • 

hhi - 

and from the energy expressions: 

0>*) - zircj'-firp 




0 ’ 2*^~<rS 

f«; - t*st f 

iiL - wei ' 

Thus, the equations may be expressed: 

'«.■ - £/ u m = K 3 CR. 1 -/) ,of») 

* 

2 «,•* 0 

S" 

I* ^ Fi. 

K S £I 

I 

* -- fA C ***** Jeasily x <r rcsS'S£c.(<R*~/art*) 


where. 


Solving for J; I'f*) one find*: 

u , •- K /?/ (£*-/) ' 

Substitution of the above value of in the constant length condition equation 

2La ^i' : 0 y yields: 


Therefore 


*f-V <1 " 

r z r'k (r‘-o *1 c*". 

i• > S*< 

«• • 


so that all U■ can now be calculated. It will prove convenient to take r„ 

(at the arbor) since it is often sufficient to compute this value alone in the 
evaluation of spring torque. A sample calculation is presented in the Appendix^. 

In the event that coils bottom out against each other or the barrel, or 
if £/• then it is necessary to proceed with the "new configuration" step 
by step in an iterative procedure. To examine this iterative process let us 
proceed as follows: 


The following definitions apply to this analysis. 

total number of coils during j - iteration 
j = iteration index (j^~ iteration ^ present Iteration) 
( - coll index ( H i * m ) 


Then the "non-equillbrated" incremental force (acting on the / - so 11, during 

* f V 

the J iteration) is given by 

: (dw CJ> fT ) ^ ~ ( J W CJ> f-) 


3. 34 



and the additional coil displacement during the j ^ iteration (not the 
total /<* coil displacement) is . Thus the j** iteration trork done 

in coil displacement is 



-i (*o 



/*• 



<P 


Expressing the differential mass In terms of a differential coll length (of the 
form rdj), for the and preceding iteration: 

. <P cj> . . C V' ) , 

~ W (f.) d cj> • ) a cy 


Thus the expression for work becomes 


W 


‘i’ - !ir« 11~ <r<r] Ct*.)'* 

t*' 


or 




(■ c //V * 

Co,; ( r . 


where 





In order to minimise the potential , / / , for the y ^ iteration, 
analogous to the non-iterative procedure (which corresponds to ,ae J - I 

f / 9^ T T *0 

iteration), the strain energy, (/ , expression is likewise required. U 

represents the incremental strain energy associated with the change from 
the to the position of all coils, i.e., 


U e/> -ir£i2 (&) 

/> * 






.. ... ~i mm immjmmmm 


' '- liil 


***gB®S 




Likewise, the constant length condition must be employed for each 


Iteration. For the j“ iteration! it is required that 

Z c^f-o 

<*> 

*i* 

where it is to be noted that C^ t ) represents the additional displacement 


which occurs during the j* iteration. 


Thus the governing equations have been found to be expression* in 
the following form 

T ) ^ » t i") ^ ^ r i) 


r7 (J -t/"'--w 


// h / f**t f 


10 * 


W ( * Csf'cr;'^' 


**• 


u'^TffiZ c$) 

ic«y p - 


<i> 


Buereji k.xpres>0*s 


-o 




I- /V £Jt Vi o* 


Note that for the cas * J*t , the above equations reduce to those pre- 

ftj 


sented for the non-iterative case, since 5 i s 

It is required that // be a minimum, within the bounds of the con¬ 
straints of the problem. This requirement may be formulated as: 


'iTT 


<i> 


2 C*.) 


<l> 


- o - 


5/7 


<P 


)n <,> ; 


w 1 ’ * ; cu-) c p 


3^6 


i Arif'S 






which, together with the constant length condition, as written below, defines 
a linear set of equations: 




From the above form of the constant length condition, one finds that for any 

: 


3 Cu, ) < i ) _ _ . 

~ 7 > ~ 


and from the energy expressions: 

a**) $£)«’ * t-ir* *s CI> 4 e, \ el> 

* 2(r/ri 
& * ire I 

Thus the equations may be expressed: 


*[ A**?'-!! 

2 


<i> _ r, 


V 


<i> 


A* 




c i > ** ** — f < i ) 

i = -fj- f. 




where 






one find*: 


. , tj) . _ (*) 

Solving for (u i ) (<**) and for (t4 n ") f 


Cu) 




,,>a <r> • / j * j? t/y ,,J 




uS°-- 


**■' «/>' 


£ *>,- v ' UV- ,1 

'*£*<*' 


Again it is found that for the case j »/ , the above expressions re¬ 
duce to those of the non-iterative case, i.e., 


for j •/ 

and 

thus 

Therefore 


s/"-/ 

S' W ‘/ 

a <n . £l.. • / 

" JM * / 


*e.‘[e, -‘1 •(>’«. 

■ k U 


which, as was expected, are the non-iterative expressions, tf . and 4 / , 
respectively. 


Note that, in general, the summation is over the h -1 coils which 
are not touching. The number of coils which are touching is variable over 
the iterations, i.e., f (j s * ) ^ CJ) . Thus a symbol of the form 
would be justified for use as the summation limit, but would add additional 
complexity to the notation without being necessary. 


3.38 






;jrai 




J 

’■v‘% 


The Iterative expressions for the displacements* t4 , are not simple 
functions of eO % or of CO* 1 * since 5 and Ji contain , Therefore* 

to solve*a true Iterative procedure must be utilised* so that tO^'* is always 
known from the preceding iteration. Then selecting the new CO the 

constants may be calculated and the displacements obtained. 


This is shown schematically below for two coils tho and jC ^ 
with radial position, r . plotted against spin rate cO . Note that the 
displacement, which is added to those preceding It, is associated with the j & 
iteration. The j ^ iteration, in turn, extends the range of the solution from 

-, ff~ty tii if j 1 |W ) ti-0 

the preceding limit CO to w by starting from the base £ ' e Yj ' + U. '\ 


iO 



w- r 


3.39 



In the same manner, the following iteratipn will extend the solution 


f ***+ t/.^and 

t i 


tj*t) tj) f/ 4,i 

to CO from CO by starting from the base Jy J 

< 1*0 

calculating U. . The only limitation upon this procedure is that each 
iterative step be sufficiently small so that the ratio 


(h> 

<!> v 




(j) fi-i) 

In certain special cases where « y 1 we find that 


C tJ> 

St ** 


p-c-gzn 


Therefore 


/ 


Thus he iterative'procedure may be pursued for any number of 

. / 

cycles, J , without limiting the break-off point to that at which the ef¬ 
fective outer coil is removed from the .system. The range of each cycle 


can be made as small as one pleases, thus yielding a mechanism for in- 

<j) <p 

suring the condition C4 f u f for each cycle of calculation. This special 


case was utilized in the special example computed in section 2.1.2.3. 1; 


but its validity would have to be verified in each application, so that it is 
better to use the more general relationship for ji. . 


















3.3.1.2. 3.1 Example (Spli>Induced bending deformation) 


Interpretation of the foregoing expression for the deformation* 
(J , will be facilitated by considering a numerical example. For this purpose 
the following set of physical characteristics will be assumed: 


A-O/ 



It will be assumed that the idealized spring for this example 
(i.e. , the spring with circular coils) may be represented by the coil radius 
distribution 

f 26 i , 

( 5? ' 

Thus the length of spring, / , is found to b. /*. 4/rr'i* ttrz.r { j 


and the inner coil ( r -. 125") is separated from the arbor by a clearance of 


3.41 



(. 125 - .11)= .015". Note that it is assumed here that the position of the 
spring coil is determined only by the position of its centerline, and that the 
physical thickness of the spring does not cause interference. 

With this inner coll clearance, it is implicit that the inner 
coil motion^r, from the fully wound position of the spring is likewise, 

/\f - < O/S ' 

Thus .for the non-spinning spring, the torque decrease from the fully wound 
position is given by (fixed, end case): 

AT:£I Tt- 

Using the numerical values 

/T * 3 0 */0 e psi' 

^ * 7~£ J> k * . O / 2 ■S’ * / 0~ ^ /Vf * 

t~ ~ . // it. 

one finds 

AJ-- 7-44 in-oe. 

Thus, this hypothetical spring has unwound to the point where its non-spinning 
torque is decreased by the amount 7.44 in-oz. This corresponds to an unwind¬ 
ing angle ( ^ ) as indicated in the linear elastic spring run down equation, 

( - cf) : er ( ~ / 

Cf) = C* 3.1* 

For this same spring, the maximum torque (at the fully wound 
position) is given by 

- i cy iay- i cj fyj 

3. 42 



Thus for a spring material for which the yield stress, a~y- 300 ,000 psi, 
the maximum (fully wound) torque is given by 

/*«+■/ ' / 7- 7/ /*» - oz . 

Thus the following non-spinning characteristic may be drawn 
for the spiral spring of this example. 



The remainder of the example will be concerned with the 
determination of the effect upon the spring, of spin induced bending defor¬ 
mation; where the initial condition of the spring is determined by the point 
on the non-spinning run down characteristic* 

/&. £7 j 7 . 


3. 43 








The profile of coil deformation under theee conditions is sh. *-v 
on the following page, as a function of K , Note that a form of iterative 
procedure must be used here since at K + 20(inches) , coil #10 contacts 
the outer constraint. For this reason, a second cycle of numerical calcu¬ 
lations is Initiated at this level of with a nine coil spring and initial 
radii (%) g which correspond to the final deformed radii of the first cycle 
calculation (r- *• U-,1 ). . 

This process is continued on the curve, with succeeding coils 
being effectively removed from the system by conforming to the shape of 


3.44 















■ ::SH SisH siin !s::E • :3:: »£:• s»»s £UH !3ESI 3sSS: K»i 

















the outer casing, with increasing /C . For a given spring configuration, 

^ £>>**•* r m * 

increasing a implies increasing spin velocity, CO , since A = y-y 
Thus although the deformation expressions utilized in this section are linear 
with /( , they are functions of the square of rotational velocity. 


Notice that the maximum values of LA- in the preceding example 

are: 


Cycle 

U, (max) 

Corresponding 

% of f; which • (Ji 

1 

.025 

.350 

7. 1% 

2 

.041 

.334 

12.2% 

3 

.063 

.311 

20.2% 


This indicates that it may be desirable to make the later cycles smaller in 
order to keep U about 10% of f- i or to utilise the full solution for^ t - . 


The motion of the inner coil is of the greatest interest, since the 
position of this coil indicates the bending torque being supplied to the arbor. 
The change in this torque is indicated in the following section. 


3.46 


3.3.1.2.3.2 Torque Sensitivity to Spin in Ex tropic 


The important torque values to consider in interpreting the 
spin sensitivity of the spring, are shown on the following curves. The torque 
associated with the spinning spring is found from the geometry of its inner coil 
(see preceding curve sheet, coil # 1). Since for th< present case ” 7 ” is initially 
equal to 10.27 in.-oz. , 

A 7 - /0-27 - El 

Thus (for this hypothetical example only ) the maximum torque increase due to 
spin induced, axisymmetric bending is 8% of the static value. 


3.47 



























3.3.2 ECCENTRICITY OF DEFORMATION 


3.3.2.1 Evidence of Occurrence ot Eccentric Deformation 

The possible importance of eccentric deformation of the motor spring 
was investigated by means of a preliminary test program. The results of 
these tests indicate the desirability of continuing an experimental program, 
as is presently under consideration. 

The test which was run, was specifically designed to be in the range 
of large values of K , without regard for other parameters such as torque 
output, inner to outer radius ratio, length to thickness ratio, etc. No at¬ 
tempt is now being made to predict the range of parameters under which 
eccentric deformation may be expected. A full analysis will be the subject 
of later work. 

The following photographs will indicate the nature of the eccentric be¬ 
havior. Photograph (a) shows the non-spinning spring and (b) shows the 
same spring (under stroboscopic light) spinning at 1500 rpm. The direction 
of the motion of the main mass of spring with respect to the center arbor 
and the outer end (reference) was found to be well defined and repeatable. 
This direction was found to be very close to from the outer end, meas¬ 

uring in the direction of the spring length. 


3,49 




43 


Attempts to bias the stationary spring with eccentricity in a direc- 

11 

tion other than ^ , were difficult. The dash dot arc on the following 
photo shows the relative direction in which the biasing agents were forced 
from their original (broken line) orientation. The final eccentric spring 
orientation, for this case, was only slightly affected. 


C‘> 


l 

[ 

r 

o 

r 



i 



3. 3.2.2 Eccentricity of Spiral Center of Gravity 


The natural spring bias in the -^r direction which is suggested, by 
the foregoing (preliminary) test, may indicate the importance of the geo¬ 
metric eccentricity of the center of gravity of a spiral. This can be illus¬ 
trated for a special case. 


Assume that the spring forms a spiral of Archimedes, i.e. , the 
radius is given by 

r = r. - ccf 

f 0 - outer (barrel radius 
2 UC - spacing between coils (constant) 


where 


Then a parametric representation of the spring is 



Let /7 number of coils (assumed an integral value for this discussion) 

i.e. , cP - n 2 
7 ***•'* 


Then the center of gravity of the spring, (% t ) is given by 


. 3.52 



where 



dh 


differential spring length = 


r 


j 


7 


The integrals aYe 

*l1T 


\ 


KJF ccf)dcf - nzrrfc- *cir) 

j 


Jl = ~ 


(r,-cy) s/«cf> ( r e -c<j>) d cf> = 


rtnff 

jl Jl J Cf '' C f) C ° J f C ^- C f )d ^ = 


w zir c (- zr 0 * mitre) 
2 mziy c l 


Thus 

and 


? = - 2C 

i Cr,-*cn) 


i. e. 


— Cot / s/>uc /h 0 ! 

* S ' ^ 7 


and the deviation of the c.g. ( ;f ^ ) from £ is given by 

-- "/-£/' J^Thl 

This is a small angle for most spring configurations, indicating that the 
eccentricity is located approximately at 



3,53 




T 


46 


[ 

I 

I 

I 

I 

I 


l 


3.S.2.3 G eneralisation of "Axi-Symmetrlc Binding Analysis 

In the energy analysis of the axl-sy mmetrlc bending deformation. 

It wae implicitly aasumed that the deformed shape of the colls remained 
circular, as in the Initial idealisation. Furthermore the initial idealisation 
wae aeeumed axi-symmetric, This analysis can be generalized as follows: 

(a) Allow r to be a continuous function of $ rather than discrete 
values f , thereby changing the sum on t to an integral on d & \ 

(b) Dccreaaethc range of each Kj , thereby changing the sum on j 
to an integral on d K (or on d ) j 

(c) Allow the initial idealization to be eccentric and allow the de¬ 
formed shape to be eccentric and non-circular . 

1- or the present, the implications of (c) will be considered in a pre¬ 
liminary manner, since this direction of analysis shows the greatest promise 
for the investigation of eccentric deformation. Initially the center of the un- 
deformed (circular) coils should be displaced from the center of the arbor 
(which is the center of spin). 


/ 'arbor j 

barrel Y 

J 



ce«/er <?{■ aadeforated 
cods 


J 







The deformed spring coils must be allowed to displace as well as deform 
(still within the confines of the arbor-barrel constraints). 


For this purpose the initial radius can be expressed 


r. 


(/• 


2 C 

c 


£e>i 




which assures that the initial eccentricity is equal to 2^ (see section 3.2.3). 
Then rather than a constant deformation ; around the circumference, a 
rigid body rotation plus a constant deformation plus an elongation of the cir- 



This deformation is expressable in harmonics of a f ourier expansion: 


U; -U * D it ' L jAI 


l ief bod 






C,lo HCyC* ftO* ^ 


racm/ rty 

*r0wfk displacement 

of- circle 

The potential function / f must be minimized with respect to the 
parameters t\ p and £ t now, rather than merely Ot . i.e., in 


3. 55 








functional notation: 


w- 

: W 

(?. 

, D:, £.) 


V* 

V 

(?1 

, a , £;) 


T7-■■ 

V- 

v 

T>(&, t 

!■ , Ei ) 

9 U - 

0 


!LJ to 

’in 

• 


) 


’ SEi 


The constant length condition, analogous to its previous form, is 
only concerned with radial growth, . The parameters D; and ti 4 
which indicate coil position and coil shape , do not influence the spring length. 
Thus the constant length condition appears as: 

± 

/ ■' 

which supplies the equation in R , where H -1 equations are sup- 

plied by - O (as in the case of £/• section 3. 1.2.3). The equa- 

tions - o and - O , hold for all values of i including i = W ; 

thus yielding a set of 3 >0 equations for R < } Di and 1 • 

3. 3.2.4 Effect of Eccentric Deformation on Torque 

The important thing about the inner coil with regard to spring torque 
position is its radius of curvature at its point of attachment with the arbor . In 
the case of eccentric deformation, the possibility of periodic fluctuations of 
the spring output torque becomes evident since the radius of curvature of the 
spring at its arbor interface, is determined by the direction of the spring ec¬ 
centricity. It appears at this time, that the direction of the eccentricity does 
not bear a constant relationship to the orientation of the arbor-spring connec¬ 
tion; but instead varies as the spring unwinds. 


3. 56 





I 

l 

l 

l 

I 


T 

t 

i 

i 

l 

l 


i 
* ' 





In preceding sections, it was found that the direction of the initial 
■piral eccentricity (and, baaed on this, of the final eccentricity) it approxi¬ 
mately 90° from the outer end connection; for the special cate of a spring in 
the shape of a spiral of Archimedes, having an integral number of coils. It 
would be of interest to inquire into the position of the Initial eccentricity for 
the case of an arbitrary (not necessarily integral) number of colls, as well 
as the non-Archimedian spiral configurations. This would be an imporv nt 
part of the investigation along with the extension and modification of the pres¬ 
ent axi-symmetric solution. 


As was pointed out in section 3.2.3. the spiral's initial eccentri¬ 
city can be accounted for by including additional terms in the expansion of the 
axi-symmetric spring deformation expression. These terms, in addition to 
allowing for initial (non-spinning) coil eccentricity, will provide for eccentric 
coil deformation, i.e. , in the axi-symmetric case, only radial coil growth 
was considered, whereas in this case coil elongation and rigid body coll mo¬ 
tion are possible. 


In order to obtain an estimate of the possible result of eccentric 
coil deformation, an illustrative example will be presented. 


Assume that an arbor and the first coil of a spire 1 spring may be 
idealized as shown below. 





3.57 





Then, letting the above schematic represent the axi-symmetric case for a 
particular configuration, the illustrative problem will consist of the deter¬ 
mination of the effect of non-circularity of the first coil, which would re¬ 
sult from non-axi-symmetric deformation. 

To this end a simple assumed deformed shape will be considered in 
order to avoid obscuring the meaning of the results of this illustrative ex¬ 
ample with premature complexity. (The results of this example will thus 
represent an order-of-magnitude indication of the possible extent to which 
eccentricities can affect spring torque output.) The assumed shape will be 
that of an ellipse, 

(a) the circumferential length is taken equal to that of the first circular 
coil; so that only deformations without extensions are being considered. 

(b) the minor axis is the same length as the diameter of the arbor. 

c/&/c>-firs/ coif 

\ m ~*— aptiis y f /r/c /T rj ■/ ca/ / 

\ / 

I 

H- 

/ 

/ 

} 



Thus, to be explicit, the symbols in the above figure represent: 

r- first coil radius when circular (axi-symmetric) 

& c arbor radius » semi-minor axis of elliptical first coil shap (with 
coil length unchanged) 

b * semi-major axis of elliptical first coil shape 


3.58 







The length of the semi-major axis, b , under these two conditions 
( (a) and (b) ), can be shown to be approximated by the following expres¬ 
sion in terms of axi-symmetric configuration, i.e,, arbor radius, a, and 
circular first coild radius, r : 


h z & (/-■%) + ? T 

The equation of the ellipse shown on the preceding sketch is 

£ + X* ./ 

a' b'- 

Thus the radius of curvature at any point may. be found to be given by 


If 


O) 


f - 


c* v 

0 +1, * 




C 


At the points of maximum and minimum curvature this becomes: 

/ W " 

f Co, °) * 


and 


t>y a 


13? 


Using the same numerical values as were used in the previously mentioned 
report (TR #64-11), take 

a - #11 m 

r - ,12" 

£1 s .375 # in 2 

which corresponds to the spring in the example examined in that report, 

routing at 34,300 rpm. 


Thus, from (l) and (3), one finds: 


3.39 










S' .1257 

ffab)* *0962 

j> (ct t o) c * 1435 

Then the change in moment at these same points, £rorn that of the coil in 
its circular (i.e,, non-eccentric) position, is given by : 


/ - J— - i- - 7. OC />'' 

SI i r 

Co,*) J 


£1 


1 


C<*,o) 


J- 

r 


i 


f(«,o) 


1.2 7 


This means that the first coil, if it remained circular , would impose 
a constant torque producing moment,/^/ , upon the arbor. However, that 
same coil, if it had an elliptical shape imposed upon it by the eccentricity 
of its deformation, could impose moments which vary from the value M by 
the preceding AM values. Thus with £1- , 375, as stated previously, the 
imposed torque could vary from 

M m „ - M * 12.36 

•f)*” el'll 

* M - f.zz />. -*?• 

Jj+'+etr'n 

depending on the relative arbor-first coil orientation (where Si , at this 
spin rate, for the example is approximately 11.4 in-oz). 

Then assuming that the position of the arbor relative to the first coil 
varies continuously around the circumference during a single turn of the 
arbor in the unwinding cycle, the resultant torque characteristic appears 
as shown on the following curve. 


3.60 



0 



& ir« / 'Spn'ij */-+£. -fertsfic !« Jfnv'ir'o**«?*^ 

£<rsftfr/c ■/roi 


3.61 


✓ 







3.4. SUMMARY OF RESULTS 


(a) For the non-spinning spring in a barrel: 


where 


~ ~ l ^ ^ ~ I 


) 


S- spring torque at any value 
(J - angle through which spring has been turned 
Ci£ - angle turned to reach "solid spring" state 
SI s spring section stiffness 
/ - spring length 

^ < 7 fixed end at barrel 
[/■*£ ) pinned end at barrel 

[b<K 


k •- 


7 " ' 


1 L< 


l S. F>P 

_• f it > J ' / *•*•*» 

b k - width and thickness of spring section 
(T S' yield stress and modulus of elasticity of spring 
P - arbor radius 

>,£ L 

(b) Additional torque in spinning spring, due to extensional deforma* 


tion only (membrane): 




, <f i <f •'f—. 


where 


M - mass per unit length of spring 
- spin rate 
r r j>: arbor radius 


3.62 





(c) Maximum bending torque in spinning spring is given by radius 


of curvature of spring at arbor, as in static case 


T- (° f ( a ) ab ° ve ) 


(d) Change in bending moment in spinning spring is determined 

/ 

by means of the relationship: 


AM • £f fi 


where 


and 


O, - radial deformation of first coil 
f - coil radius <sr constant 

£ £/(/?. -o 


AT i, AM 

In tin* event that the colls "bottom-up" the iterative procedure to obtain des- 
i ribed in this report is utilized. 


3.o J 








4. PIVOT AMD JOURNAL FRICTION 


Technik has found that torque losses associated with pivots and journals 
are attributable to a complex set of phenomena* of which the "conventional" 
definition of "friction" provides only one portion. Thus the analysis which 
was performed* and is described herein, serves to point up the additional 
work necessary for the full understanding and description of the phenomena 
involved. 

The analysis which is presented here is the conventional "Hertslan" elas¬ 
tic contact deformation analysis. For the purpose of generating quantitative 
information, this analysis is combined with a torque evaluation based on an 
assumed coefficient of friction* although the value to be assigned to such a 
coefficient is greatly dependent upon the nature of the physical interference 
phenomenon involved. 

4. 1 ELASTIC ANALYSIS 

For the present, although various geometries differ (i.e. , journals and 
pivots) and may require somewhat different treatment* the pivot will be ex¬ 
amined in order to expose certain basic problem areas. Consider a spheri¬ 
cal pivot point in a spherical seat, as shown below: 






Then from Hertzian theory, the radius of interface pivot-seat contact 
is 

a * /to? l/?- / *•- — ) -- ?.W * /o'* 

r £ ( je,< / 

and the maximum interface pressure, v , is given by 



The stresses at the points (1,2,3) shown above, may be shown to give 
rise to the maximum shear stresses (for Treeca yield criterion); 

lif.ooe ,>c 

Note that the maximum shear stress at the center of the circle of con* 
tact (1) is . 1 , whereas the normal stress is . For the ductile ma* 

terials being considered, the maximum shear stress is a good indicator of 


4.2 




the incidence of plaatlc action. This value mutt be compared to shear 
yield ( 7^ ) which can be taken to be ( (J^ * tensile yield of the ma¬ 

terial. 

Thus, for many hardened steels used for pivots, the stress at point (1) 
does not indicate plastic action while the stress at (3) is probably of a 
marginal nature in this respect. The stress, , at (2), however, in- 
dicates a plastic sone below the surface of the spherical seat. It is ques¬ 
tionable at this time, without further investigation, whether this none ex¬ 
tends to the surface (at (3) ). If it does not, the plastic sone will be elastically 
contained and should not be accompanied by large deformations. 

If, on the other hand, the sone reaches the surface to become unconflnfd, 
large deformations may accompany the action, with resulting large frictional 
areas and associated torques. 



4.3 








4.2 TORQUES IN ELASTIC CASE 


The pressure distribution between the spherical pivot and seat for the 
elastic (Hertzlan)analysis, is; 




thus the torque is given by 


T< 


? 




Taking y4 * constant 


T- ft- « > 


Thai for a = 2.86xl0* 3 (16.4# loid) 

27 y* >/# 


- 1 


<•* •/£ 


or 


T- ns 

It may thus bo noted that the torque y , lost at the pivot is directly 
dependent upon the value of y , the coefficient of friction. A frequently 
encountered value of this coefficient is (say) y r .i. In this case 


7'».*$/ 


/«. - o e 




and values of y may be found, as great as, or even greater than, unity 
(seizing). The actual limit on attainable values of y is imposed by the 
shearing strength of the material. 

If, in general, the frictional force, y f . *t any point exceeds the 
shearing stress of the material, the material will, in fact, shear; resulting 








in a greatly accelerated wear process. The process will continue, always 
causing the pivot and seat to conform to each other and Increasing the contact 
radius, until the friction force, ^4p , no longer exceeds the shear strength 
at any point. Thus, the contact radius having been increased, the frictional 
torque losses will likewise Increase. 


To visualize the accelerated wear process, assume the following highly 
idealized rotating pivot-scat configuration, i.e. , conical and flat, respec¬ 
tively : 


P 

The initial pressure, y , at the point of contact approaches infinity, thus 
for any value of yi* , ^ yr ? T, , where £ is a constant, ie., that value 

of shear stress which will cause the material to shear. As the material 



shears the cone is truncated and the scat is simultaneously "worn" so that 
a finite contact radius, a , is attained: 



4.5 









Thus the interface pressure reduces as a Increase?, until thsvalue of 
at no point exceeds S, ; since, if this value is exceeded, shearing will 
occur changing the pressure distribution, ^ , increasing the contact radius 
, and reducing y* f below the £ value. At this time "accelerated 
wear" is considered to cease and the more conventional wear ensues. 

Thus the question of torque losses rather than encompassing only consid¬ 
erations of geometry, material and "friction" linked by elastic analysis 
should be broadened. Included in the investigation should be the additional 
analyses concerned with 

(a) plastic deformation of contacting parts - Increase in contact area 
and friction losses over elastic case; 

(b) accelerated wear process - shearing of material and associated 
friction torque increase. 


4.6 








5. DRAM HAIR SPRING PROBLEMS 


An outline of preliminary results of two hair spring problems investigated 
by Icchnik will be presented in the following pages. 


5. I UK U T Ol SET MAI K ON HAIR SPRING 

Statement: Gt\en the beam shown below, find the maximum "g" 

loading that it tan support. Establish general criteria as well as specific 
results tor the general range of dimensions presented. 



t = . 00330 - . 00008 (in) 
h = .013- . 001 ( in) 

£ = 25 x 10 b ( # / in^) 

^ = .3 


/ = .3 {HI in 3 ) 

. 3 ^ ^ ■ 45 (in) 

200 x 10 3 f ^ 250 x 10 3 (#/in^) 

8,000 * ft 4. 25, 000 (desired) 








In the solution of this problem we will at first assume that there is no initial 

warping of the spring; further work under the scope of the proposed task will 
remove this restriction. 

Remarks : The following analysis indicates that the limitation on 
the capacity of the hair spring to resist set back forces may be provided by it b 
plastic deformation rather than its buckling configuration. Further investigation 
must determine the more "exact" limits of the two phenomena. 

Approach: The solution was approached by two techniques; both of 
which assumed the existence of a buckling configuration, and then by different 
approaches derived their basic equations. In the first a differential equation 
approach was formulated through the use of the elastic equations for bending and 
twisting of simple beam configurations. This resulted in three equations, two of 
which were coupled in their first-order effects. The eigen-values of this coupled 
pair of equations, subject to the correct boundary conditions, provides the 
critical load. Unfortunately because the resultant equation was highly non-linear 
its solution was considered beyond the scope of this exploratory activity. 

In the second approach an integral-equation formulation was obtained 
through the use of energy techniques. Although exact solutions are again difficult, 
this approach lends itself to approximate solutions. The assumption of one 
variable, subject to all the displacement boundary conditions, allows the complete 
integration and a reduction to an algebraic form; from which the critical load 
can be obtained. Although this is simpler than the differential equation 

approach, it is quite laborious. 


5.2 











A third alternative present* itself for the solution of this problem; 
that of hounding the correct solution by others which are presently known. 
Although this technique is limited if precise values of the critical "g" loading 
are required, as would bo necessitated in optimum design; it is of real value 
in the solution to the present problem. I‘he bounding problems utilized were 
the uniformly-loaded and end-loaded cantilever beam, and the uniformly-loaded 
simply-supported and clamped beams. 

In addition to the limitations imposed by the "bounding" solutions 
another limitation present s itself, and must be considered in all problems of 
this type; plastic deformation of the beams. It is found that this often provides a 
lower limit on the "g" loading, over and beyond the possibility of lateral buckling; 
unless the beam is optimum designed to avoid this type of limitation. 

I he accompanying figure presents a curve of "n" (ng's) which the 
spring c u'» withstand as a function of beam length; all other dimensions are as 
specified on the c ros s - sec l ion. From this it is seen that plastic flow isa real 
limitation in this problem, and that the lateral buckling rigidity is more than 
adequate; even though only a lower bound on this rigidity is calculated. In the 
interpretation of this limit, based on the initiation of plastic flow, it is important 
to note that a yield stress of 200 x 10^ to 250 x 10^ psi was assumed; which may 
be beyond that which the material can develop. In the latter event the "n" curve 


5.3 













would be suitably lowered. An optimum design of this beam would raise the 
plastic curve with little, if any, drop of the buckling curve, subject to little 
or no weight increase; thereby substantially increasing the load carrying capa¬ 
city of the beam-like member. Further work in these directions could be ex¬ 
pected to result in a definitive evaluation of beam hair spring design. 


5.5 









5.2 KH UT 01 A WhDGK HhTAlMR ON HAIR SPRING 

Statement Ins eat Igalr the dlleili upon the bending characteristics 
of the idlin' Scam a* described tn Problem #1, of ait imposed curvature at 

the fixed end. 

Background The function of the beam la to perform a* a apnng 
resisting an applied turouc a> tllualrated bclo* 



The bending ha r act c rist ic s of the above spring are influenced by 
.m initial cur .atore imposed on a portion of the spring during the fabrication of 
the above assembly. The extent of the deviation of these bending characteristics 


from those of tie hitherto employed idealisation (i.e. , a flat strip) has been 












questioned. This discussion is, therefore, directed toward discerning that 
extent. 

Remarks : The following approach indicates that the presence of the 
wedge retainer (in that it deforms the hair spring) greatly changes the bending 
characteristics of the hair spring from those of an assumed "flat" hair spring. 

A more "exact" evaluation of these changes awaits further investigation and will 
be undertaken under the scope of the present proposed task . 

Approach : 


Phase a: Extent of Deformation at Fixed End 


Assuming that the insertion of the brass pin and the spring 
into the steel ring causes no practical change in the curvature of the steel ring, 
then the surface of the fuze spring in contact with the steel ring will take on the 



The maximum relative transverse deflection (sag) thus 
imposed on the spring is equal to the relative deflection between the edges of 
the spring and the center of the spring. 


5.7 









V 



Actually this "sag", d . can vary with the tolerance* on the 
ring and on the fuze spring itself; a representative value, however, was found 
to be quite large. 


Phase b: Effect of an Assumed Distribution of the Deformation 

In the initial stages of this investigation, it was assumed that 
the angle was small. This can be shown to lead to incorrect results and the 
analysis must be revised to account for being large. This revision is reflected 
in the reported Phase a above, and Phase c below. For Phase b only . tho 
original assumption of email, will still be assumed to hold, thus providing 
a lower bound on the effects of the curvature (i.e. , the small o( assumption 
yields a smaller value for the curvature than the value subsequently found). 

Under the above condition, and assuming St. Venant's 
principal concerning the extent of the influence of an edge disturbance, it was 
theorized that a parameter describing the effect of the deformation might be the 
number of widths thru which the deformation might be equivalently assumed to 
hold in an undiminished state. 

A number of curves are presented based on that investigation. 


5.8 









These curves show the deflection of the end of the spring for a load. £p (at that 
end) considering spring lengths of .3" to .45". It was assumed that the actual 
deflection of the spring would fall somewhere between that of the spring with an 
undiminished cylindrical deformation extending one width (b) and the spring with 
an undiminished cylindrical deformation extending three widths (3b). (Note: 
the remainder of the spring was assumed flat). 














In view of the shortcomings of the assumption upon which 
these curves arc based, they are presented merely for an appreciation of the 
possible magnitude of the deviation of bending characteristics of the deformed 
spring from those of the flat spring idealization. As was pointed out previously, 
the actual deviation would be expected to be greater than that shown,, 

Phase c : Determination of the actual extent of the deforma¬ 
tion caused by an initially imposed curvature at the fixed end of the spring. 

The revised maximum deflection (Phase a) was found to be 
non-trivial. Since this deflection falls into the large deflection category (i. e. , 
& > h), no conclusion can be drawn as to the local character of the deformation 

and, for this reason, an estimate of the actual extent of the deformation was 
undertaken. 


The spring was treated as a flat plate subjected to loading 
conditions so distributed as to bend the plate to the required configuration; but 


to have no resultant force transverse to the plate. 











T 


Based on this an infinite series expansion was found for the 
deflection of the spring in the % direction. The following curves show this 
deformation when cither one or two terms of the series are employed. The 
series converges so rapidly that addition of the third term would not be warranted. 

A number of factors arc evident from inspection of the curves: 

(1) The spring retains some portion of the initially 
imposed curvature over its entire length. 

(2) The curvature in the direction gives rise to a 
curvature in the X direction. 

It appears that the former assumption of a local deformation 
associated with the end effect requires much further justification. The determin¬ 
ation of the bending characteristics of the spring in its presently conceived 
configuration (as shown in the curves using first and second terms of the series) 

is complex. The procedure would hold only if the yield stress of the spring were 
not exceeded. The local relaxation resulting from such excessive stresses, would 
result in an attenuation of the cylindrical end effect, i.e. , the spring would act 
more like a flat plate or beam. 


5. 11 











£/</u»rr M40JJ ?*h»f9//D 

J'Z’ o£‘ jz ' aZ ’ jr/> or ' 











\ 

6. TORSION HAIR SPRING 

Torsion hair springs are often subject to the same phenomena associ¬ 
ated with rotating shafts, i.e. , resonance at critical speeds. This reson¬ 
ance consists of the coincidence of the spin rate of the projectile (and there¬ 
fore, of the hair spring) with one of the modes of transverse bending vibration 
of the hatr spring. Thus the values of those spin rates which correspond to 
critical speeds of the torsion hair spring, are dependent upon the properties 
and configuration of that hair spring. 

The ultimate purpose of the hair spring, which is concerned with its 
time keeping characteristics, is to oscillate in its torsional mode. For this 
reason, the design of the spring is based upon the objective of obtaining a 
specific natural frequency of the spring acting in concert with a rotary inertia. 
The attainment of this natural frequency is dependent upon the same properties 
and configuration considerations as are the values of the critical speed spin 
rates. 

Thi.e, one finds an interrelationship which exists between the primary, 
time keeping, characteristic of the torsion hair spring and the secondary, 
critical speed, characteristic which constitutes possible degradation of the 
functioning capability of the time keeping system. 

In order to illustrate this relationship, the following specific case will 
be investigated. 


6.1 









a 


spun 


A* * spsin^ • 
% 


i 


* 


2oizz 


r> 

'effectiveSt/mce 
di«**iata+ 


J 


(effective 

pz,;n 

2^fo<e«cl* of 

two musses • 


*"*f 


k 


Imh'ice. motto* 

* ad frepueacf'f 
Cs«perp»seJ opoh 

proj&c ti/o apt") 


6.1 TORSIONAL FREQUENCY ( f ) 


The balance oscillates in a torsional vibration about the centerline, 
superposed on the overall projectile spin, & . The frequency, f , of this 
oscillation can be shown to be given by: 



for a steel wire (modulus of elasticity, £■ 5 30x10^ psi and Poisson's ratio, 
P’s . 3), where 

f- natural frequency in torsional mode (cps) 
d - spring diameter, circular section (in) 

M- concentrated mass ^ 1/2 total balance mass 
effective diameter between masses, M. (>*) 
l - effective length of hair spring (irt) 


6.2 









6.2 'WHIRL FREQUENCY ) 


Likewise, It can be shown Hint (or the special case where the rotary 
Inertia of the end Is constrained so that the shaft can be considered fixed- 
fixed, the critical speed, , Is given by 



4^/00 


i*r 


C4^cJ 

l t 


(or steel again, where 

(/ £4 t • • • • discrete constants, the value of which depends on 

the mode shape (of the bending vibration). 

' 22.0 

</, • 61.7 

o } . 121.0 


6.3 










Combining these two equations, one finds 


M : & 


AR 


R (£<60 


For small bending deformations of a full coll -* LA , this can be linearised 
as: 


M'-EI 


Thus the strain energy of the full coil is given by 

U ' fa 

U*ir£T jf. 


The work done by centrifugal forces in moving the coll clement through 

iven by 

£(v*irdi£) ^ dr 


the displacement, U , is given by 

fi* 

w- 

J 


o J 


* ^ 




d ) 




d ( for* C ) 


which for a full coil becomes, 7 *? 77 

d (^ork) 

w= zrreo 1 ^ [R l « r flu 1 . ] 

Again, for small deformations, a , this can be linearized as: 


A. 2 









j, •/ 


APPENDIX 



. 125 
. .SO 
. 17 b 
. 200 
.225 
.250 
.275 
. 300 
. 325 
. 350 

27 


li: I D<S*T ITERATION KOR SECTION I 


* / * M "* 




" 7 . ' 


'l 


l 

1.2 

1.4 
1.6 
1.8 
2.0 
2.2 

2.4 
2.6 
2.8 


1 *' 

*.* 



'ix j? J 

K *• 

K 

-k’" 

- 

- 

• 

- 

- 

. * 

- 4.65 

-.142 

1.44 

1.728 

. .44 

.76 

-8.04 

- 7.28 

-.222 

1.98 

2.7. 

.98 

2.68 

-12.7 

-10. 

-.306 

2.56 

4.09 

1.56 

6.37 

-19.0 

-12.36 

-.378 

3.24 

5.82 

2.24 

13. 

- 27. 1 

-14.1 

-.431 


8.0 

3.00 

24. 

-37.2 

-13.2 

-.404 

4.84 

10.6 

3.84 

40.80 

-49.3 

-8.5 

—. 260 

5.78 

13.9 

4.78 

66.50 

-64 • 6 

1.9 

.058 

6.78 

17.6 

5.78 

101.80 

-81.9 

19.9 

.61 

7.82 

21.9 

6.82 

149.00 

-102. 

47. 

1.44 


86.38 

. 


404.91 



O **' 


• 4 


", 




.x[+!t2LJ 

L 97 s 9 J 


- - v^r /r 


<- * 

<■— 4 4 


awf rtnc<4 7 ^ 


?<?*