Int. J. Mech. Sci. Vol. 22, pp. 267-283 Pergamon Press Ltd., 1980. Printed in Great Britain

FINITE ELEMENTS FOR DYNAMICAL ANALYSIS OF HELICAL RODS

JOHN E. MOTTERSHEAD Lucas Research Centre, Dog Kennel Lane, Shirley Solihull, West Midlands, England

(Received 16 November 1979; in revised form 15 January 1980)

Summary--Finite elements are presented for dynamical analysis of helical rods. The element stiffness and mass matrices are based on the exact differential equations governing static behaviour of an infinitesimal element. Natural frequencies obtained by use of the element, which allows for both shear deformation and rotary inertia, are compared to the frequency spectra of helical compression springs. The element performance is compared with that of other finite elements.

NOTATION A cross sectional area c~ constant of integration E Young's modulus G Shear modulus I second moment of area ] radius of gyration

[k] ~ element stiffness matrix I element length measured along helical path

M bending moment [m] ~ element mass matrix [N] transformation array

P force {P}0" vector of element nodal forces

[Q] transformation array {Q} vector of applied loads

r mean coil radius s length measured along helix T kinetic energy t time

U strain energy u, v, w co-ordinate directions

{u} vector of displacements {u} e vector of displacements within element

{u}0 e vector of element nodal displacements {ti} e vector of velocities {ti} e vector of accelerations

:c helix angle y shear coefficient A displacement 0 slope/rotation K curvature v Poisson's ratio p mass density "r torsion

angle measured about axis of helix ~o circular frequency

INTRODUCTION The good p e r f o r m a n c e of helical c o m p r e s s i o n spr ings is crucial to the in tegr i ty of an ex tens ive range of au tomobi l e c o m p o n e n t s . This is par t icu lar ly t rue of spr ings opera t ing in dynamica l e n v i r o n m e n t s and where the t r end is to min ia tur iza t ion .

Al though the gove rn ing differential equa t ions for helical spr ing d y n a m i c s were publ i shed in 1899 by L o v e [ l ] the p rob lems of spr ing surging ( resonance) r ema ined unso lved ; the equa t ions were too difficult to solve by hand. In c o n s e q u e n c e the " s i m p l e " theories[2--4] for spr ings were deve loped . These can be thought of as t rea t ing the spr ing as a h o m o g e n e o u s rod hav ing modified elast ic and mass proper t ies ; the coupl ing effects re la t ing ex tens ion , to rs ion an d bend ing are however , prohibi ted.

267 MS Vol. 22, No. 5--A

268 J .E . MOTTERSHEAD

Phi l l ips and Cos te i io [5] e x t e n d e d wha t is e s sen t i a l l y the s imple t he o ry , coup l ing e x t e n s i o n and to rs ion , to fo rm a set of n o n q i n e a r equa t i ons gove rn ing large s t ra in b e h a v i o u r of spr ings unde r the ac t ion of axia l f o r c e s and m o m e n t s . This a p p r o a c h has been fu r the r e x t e n d e d by Cos te l lo [6 ] in an inves t iga t ion of rad ia l e x p a n s i o n due to axia l impac t and a numer i ca l e x e r c i s e c o m p a r i n g l inear and non- l inea r theor ie s has been p e r f o r m e d by S inha and Cos te l lo [7].

Wi th the i n t r o d u c t i o n of the digi tal c o m p u t e r in the 1950s c a m e r e n e w e d in te res t in L o v e ' s equa t i ons and t hey we re e x t e n d e d in 1952 b y Y o s h i m u r a and M u r a t a [8] and again in 1966 by Wi t t r i ck[9] . The first of these c on t r i bu t i ons i n t r o d u c e d to r s iona l iner t ia and the la t te r c o n s i d e r e d ro t a ry iner t ia and T i m o s h e n k o [10] shea r d e f o r m a t i o n effects .

The w o r k of W i t t r i c k d e s c r i b e d an e igenva lue p r o b l e m for semi- inf in i te he l ica l spr ings p rov id ing a so lu t ion for cha rac t e r i s t i c phase and g roup ve loc i t ies . In con t r a s t , the w o r k p r e s e n t e d here enab le s na tu ra l f r e q u e n c i e s and m o d e shapes to be o b t a i n e d f rom W i t t r i c k ' s d i f ferent ia l equa t ions . D i s p l a c e m e n t func t ions are o b t a i n e d by in- tegra t ing the d i f ferent ia l equa t ions . This is an unusua l f o r m u l a t i o n but not an un- k n o w n one , and the e l e m e n t may be c o n s i d e r e d to be of the same f ami ly as t hose p r e s e n t e d by Davis , H e n s h a l l and W a r b u r t o n [11-13]. F o r s ta t ic p r o b l e m s the e l e m e n t p r o v i d e s e x a c t so lu t ions whi l s t in d y n a m i c s the na tura l f r e q u e n c i e s a re una f f ec t ed by mesh dens i t y p r o v i d e d the mas t e r deg ree s of f r e e d o m [ 1 4 ] are unchanged .

DIFFERENTIAL EQUATIONS The co-ordinate reference frame u, v, w (principal normal, binormal and tangent to the helix) is used in

formulation of the equations, and illustrated in Fig. I. The length of the wire along it's centre line is denoted by

s(4,) = C[(K2 + r:)l (1)

where

COS 2 ~x

r

r = r t a n a

(= curvature) (2)

( = torsion) (3)

and :~ = helix angle. To clarify matters, if the origin of the reference frame moves at unit velocity along the central line of

the wire the curvature, K, is given by the angular velocity of the principal normal about the binormal and the torsion, r, is given by the angular velocity of the principal normal about the tangent.

If ti, 0, ~v are co-ordinate unit vectors the forces and displacements may be resolved at any point on the wire centre line, "

0 = tio. + SO,, + ¢vO. (4)

A = tiA + f~A + ~A., (5)

M = tiM. + ~,M, + ~M.. (6)

P = tiP. + ~,P,. + ~,P... (7)

Then assuming plane sections remain plane,

dO tiM. +@M" +#Mw d~ = I . E I , E J~G (8)

and if bending curvatures are small,

(9)

Equations (8) and (9) are specialised forms of Hooke's law. Considering bending equilibrium about an infinitesimal element 6s and assuming small slopes and

negligible stretching of the wire centre line,

dM ds tiP" + ~P" = 0 (10)

Finite elements for dynamical analysis of helical rods

,. 573'.

w-tangent

t ~ r ! I I

269

FIG. 1. Helical rod.

and equilibrium of forces (if point loads only are allowed)

dP - - = 0 . (11) ds

Equations (4)-(7) may be substituted into equations (8)-(11) and differentiated with the aid of the formulas of Frenet which may be found in the book by Kreyszig[15]. On separation into co-ordinate directions Wittrick's equations for statics emerge,

0v = ~s~ - cA~ + KAw P" GAy (12)

Pv O. = - - ~ - ' : A . 4 GAy (13)

dd• P w O= --KA. AE (14)

2M. = d0___x~ _ ¢0~ + K0w (15) IE ds

2My = ~ s + tO. (16) IE

Mw = d0_._~. _ K0. (17) IG ds

dM. . _

P~= ds - r M ° + x M ~ (18)

-dMo P" ds "rM, (19)

dMw 0 = ~ - ,,M, (20)

270 J. E. MOTTERSHEAD

= ~ - - 7P,, + KP., (21) 0

0 = dP,, + zP. (22) ds

0 = dP.. _ rP , (23) ds

where I = Jw = 21. = 2/~ for circular sections and y is the shear coefficient [16].

DISPLACEMENT FUNCTIONS The displacement functions are obtained by integrating equations (12)-(23). Firstly, considering equations (21)-(23). Differentiating (21) with respect to s and substituting for

(dPdds ) and (dPw/ds) from equations (22) and (23) the following is obtained,

0 = ~ + "r2Pu + r2Pu

or

d2P, 0 = - ~ s 2 + a2Pu (24)

where

a = X/(r: + K-').

Recognising equation (24) as the harmonic equation, then the solution will be of the form,

P. = c~a cos (as) + c2a sin (as).

Eliminating P . from equations (22) and (23) gives on integrating,

Po = - c t r sin (as) + c27 cos (as) + c3

P., = CIK sin (as) - c , r cos (as) + "r c3 K

(25)

(26)

(27)

(28)

where ct, c2 and c3 are constants of integration. In similar fashion expressions for M.M~M.. are obtained from equations (18)-(20), 0ug,0. from 0 5 ) - 0 7 )

and A.AvA~ from (12)-(14) in terms of 12-constants of integration. These can be written in matrix notation,

{P(s)} = [A(s)]{C} (29)

{u(s)} = [B(s)]{C} (30)

where

{P} = Pu

M~ M,, M~

{u} = Au A r Aw O~ Ov Ow

{C} = "

The contents of matrices [A] and [B] are given in full in Appendix 1.

Cl C2 C3 C4 C5 C~ C7 C8 C9 Cio CII Ci2

FORMULATION OF STIFFNESS AND MASS MATRICES

The finite element shown in Fig. 2, with the convention for generalised forces and displacements positive in the directions indicated by the local co-ordinate axes, may be obtained from equations (29) and (30) and the appropriate co-ordinate transformation arrays. It has twelve degrees of freedom; three rotations and three translations at each node.

The local co-ordinates are chosen such that 'y ' coincides with the spring axis and is positive in the direction l ~ 2 . ' x ' is in the plane perpendicular to the spring axis and connects node 1 to the projected position of node 2 and is positive I ~ 2 . ' z ' completes the right handed set.

Finite elements for dynamical analysis of helical rods 271

z

/['--.z FIG. 2. Local co-ordinates.

The local co-ordinates, whilst not appearing obvious at first sight, are conventional for beam-type elements of the same family and thus comply with a uniform approach which is helpful to the finite element user.

Entering s = -+ 1/2 at the nodes and introducing transformation arrays [N] and [Q] we have that,

o r

and

o r

{P}oe = [ N ] [ ~ ]{C} (31)

{P}o e = [ Y]{C} (32)

{u}g = [ Q l [ ~ ]{C} (33)

{U}o e = [X]{C}. (34)

The contents of matrices IN] and [Q] are given in Appendix I. According to D'Alembert 's principle dynamics problems may be converted into static ones by taking

inertia forces into account so that the principle of virtual work may be employed. Integrating with respect to time between two instants tt and t2 when the displacements are prescribed, Hamiltons principle is obtained,

6fi~Ldt=O (35)

where L, the Lagrangian, is given by,

L = T - U - lq (36)

and T is kinetic energy, U is strain energy, ~ is potential energy of the applied loads. Considering the strain energy,

U" = ~u}~r{p}o e (37)

272

but

J. E. MOTTERSHEAD

{P}o" = [k]'{U}o"

... U" - l {u}:r[k]'{U}oe.

Also by inspection of equations (32) and (34) and eliminating {C}.

{P}o" = [Yl[X]-'{u}o'.

S o the e l e m e n t st i f fness matrix is given by,

[k] ~ = [Y][X]-'.

Cons ider ing the Kinet ic energy ,

w h e r e

T , = ~ p A f #2 J-t/2 {a(s)}'r [G]{'i(s)}" ds

° ° ° ° i] 1 0 0 0 0 1 0 j212r

[G]= 0 0 j212 0 0 j212r 0 ]212 o o o o Y_I.

Combining equations (30) and (34) and eliminating {C},

{u(s)}" = [B][X]-'{U}o"

... {~i(s)}" = [Bl[XF'{zi}o"

then substituting in equation (42) g ives ,

¢ 1 f(#2 T = ~ PA{{li}°'r .I-|#2 [X]-r[B]T[G][B][X]-I ds{ii}o'}

or

/

T • = 2 {zi}er [m]'{i~}°"

w h e r e the element mass matrix is given by,

f l l2

= [XI-r[Blr[GI[BI[XI -I ds. [m]" pA J-~n

Forming global stiffness and mass matrices the energy expressions may be written down as,

= ~ {~i}rlM}{ ~i} T

= 1 {u}riK]{u} U

f~ = - {uV{O}.

So,

o r

f2 1 . r . 1 r r 8[~, {~{u} [M]{u}-~{u} [K]{u}+{u} {Q}}dt]=O

~ '~ {{Sti}rtMl{u} - {Su}r[Kl{u} + {Su}r {Q}} dt = 0 i

since both [M] and [K] are symmetric. Integrating the first term by parts,

1st term = [{Su}r [M]{li}]~ - f~i 2 {Su }r [m]{12} dt.

(38)

(39)

(4o)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

Finite elements for dynamical analysis of helical rods

But {,Su} is only an admissible variation provided {~u} = 0 at t~ and t2; thus

f '-" {<~u}T{ lM] {a} + [ K l { u } - {Q} } a t = o J,|

but {6u} is arbitrary so,

273

[M]{ti} + [Kl{u} = {Q}. (54)

If harmonic motion is assumed equation (54) becomes,

IlK] - w-'[M]}{u} = 0 (55)

the eigenvalue equation for structural vibration.

IMPLEMENTATION The computation of the stiffness and mass matrices for the element involves many trivial but tedious

arithmetic manipulations. Development and coding of the element is thus highly prone to human error and in consequence the aim has been to minimise the likelihood of such, even at the cost of program efficiency.

In coding the stiffness matrix the approach has been to set up matrices [X] and [Y] from [A], [B], [N] and [Q]; matrix [K] being obtained by matrix manipulation.

For the mass matrix the coding is simplified if the integral ft_l~/: [B]T[G][B] ds is expressed as follows,

L,.,,.,.,,ds+L,.2,.-,.2,<,s tn rln j,-

+ fi//2 {R3}r{R3} ds + J-,n 2 {R4}r{R4} ds

F-r" f": + ~- {R~}Z{R~} ds + j2{R6}r{R~} ds ll2 112

where Rt, R2 . . . . Rn are rows of matrix B. Indefinite integrals have been found using Dwight's Tables of Integrals [18], the coding provides the limits.

Despite this relaxed attitude to program efficiency the element remains most inexpensive to use and it should be remembered that for most applications very coarse meshes will provide good results.

Since stiffness and mass matrices are symmetrical a useful check is automatically provided on the accuracy of the computer coding.

RESULTS The element has been developed because of the lack of a suitable analytical method which will allow the

calculation of spring natural frequencies by hand; by the same token the task of proving the helical element is made difficult.

To begin, the static performance of the helical element will be examined. A spring has been modelled, having one end axially fixed and the other subject to unit axial load. A further constraint at both ends of the mesh has been applied in order to prevent rotation in the plane containing the spring axis and the end node. The coils have been represented using meshes with varying degrees of refinement and the end displace- ments have been compared, in Table 1, with the 'strength of materials' theory for springs whose helix angle is significant.

Clearly, the use of the element for this type of application is not recommended, it is not necessary; it does, however, serve to illustrate the power of the formulation. Since the error remains constant with the mesh refinement an exact solution of Wittrick's equation is confirmed.

An analytical theory due to Timoshenko[17] for the flexural vibrations of circular rings provides a further test for the helical element. By setting torsion to zero (z = 0) the element degenerates to allow solutions for circular beams and arches; in this mode the element is identical to some earl ier formulations[12, 13]. For completeness, the element natural frequencies are compared to the Timoshenko theory in Table 2.

Further discussion of this analysis would be to reiterate the work of others so an investigation of springs whose spectral characteristics have been established by physical testing will now be presented.

TABLE 1. STATIC PERFORMANCE OF THE HELICAL ELEMENT

Mesh density % (coils/element) Error

1 0.085

1 0"085 6 0-085

274 J. E. MOTTERSHEAD

TABLE 2. FLEXURAL VIBRATIONS OF A CIRCULAR RING

1 2 - E l e m e n t s 24-Master DOF

Mode % Error

1 0 2 -0 .2 3 0.35 4 0.8 5 1.6

Frequency searches have been conducted on two springs with clamped/clamped terminals. The natural frequencies obtained from the tested springs are compared to the finite element simulation (employing 30-master degrees of freedom) in Table 3. The mode types have tended to fall into three major categories,

1. Predominantly "snaking" (mode type "S" in the table),. 2. Predominantly torsional (mode type "T"). 3. Predominantly longitudinal (mode type "L").

The details of the two springs are as follows,

(a) Spring 1 Overall axial length, 32.5 ram. Mean coil diameter, 12 mm. Wire diameter, 1 mm. Number of active coils, 5.

(b) Spring 2 Overall axial length, 36 mm. Mean coil diameter, l0 mm. Wire diameter, I ram. Number of active coils, 7.6.

Under test the springs were shaken in longitudinal and transverse directions (coil motion being detected by an inductive proximity transducer) and this is thought to explain the failure to detect the second torsional mode in both springs.

The simulation of the second spring indicates for the fundamental and second predominantly snaking modes the existance of doublets. On close examination of the mode shapes it was found that the motion occurs in different planes; one twin vibrating in a plane about 90 ° removed from the plane of vibration of the other. This effect might be expected since the spring terminals are clamped at discrete points on the helix (not on the spring axis) and hence the flexural stiffness will depend upon the plane of bending. Two predominantly longitudinal third modes have also occured but their shapes are quite different.

A hybrid snaking]longitudinal mode is displayed by the first spring, both in simulation and under test. Non-dispersive waves such as the extensional and torsional waves in uniform bars are characterised by

natural frequencies which are multiples of the fundamental frequency. The interval separating the natural frequencies of each modal family of the tested springs is surprisingly regular and so tends to support the use of the simple theories referred to in the introduction. The mode shapes of the two springs have been plotted using the computer and are presented in Figs. 3 and 4. Photographs of the first spring operating in the first and second longitudinal modes have been taken and are shown in Figs. 5 and 6.

TABLE 3. COMPARISON OF FINITE ELEMENT SIMULATION TO SPRINGS UNDER TEST

Spring 1 Spring 2

Natural frequency Error Mode Natural frequency Error Mode Hz % type Hz % type

Helical Test Helical Test FE FE

436 431 - 1-2 L 396 478 474 - 0"8 S 397 497 490 - 1.4 SI L 469 506 509 0"6 T 532 856 844 - 1.1 L 887 937 932 - 0"5 S 900 964 960 - 0"4 S[L 937 987 T 1067

1173. 1145 - 2.4 L 1348 1282 1302 I "5 S 1409

-1 .3 S 391 - 1.3 S 459 - 2.2 L 528 - 0'4 T

- 1-0 S 878

-2 .5 S 906 - 3.4 L

T 1282 -5 .1 L 1386 - 1.7 L

Finite elements for dynamical analysis of helical rods

G L

HYBRID

SNAKING FAMILY

TORSIONAL FAMILY

FIG. 3. Mode shapes--spring 1.

275

It is important to note that snaking type deflections are present for all modes, a characteristic which is thought to explain side wear marks on the coils of fuel injector nozzle springs. Such transverse motions in predominantly longitudinal and torsional modes cannot be detected using the simple theories.

A fine helical mesh of Timoshenko beam elements[l l] should provide natural frequencies which converge upon those obtained from the helical element. For the straight beam elements, helical coupling is provided by virtue of the nodal connections, so with further mesh refinement greater accuracy is expected.

For helical springs having twelve coils and clamped terminals the spectral data is summarised in Table 4. Notice that when the number of master degrees of freedom are increased the natural frequencies converge from above. If however, the mesh density of the helical elements is altered without changing the masters the natural frequencies are unchanged. Again, this is to be expected since the displacement functions satisfy Wittrick's equations exactly and so a criterion is provided by which the accuracy of the coding may be judged.

In so far as comparing the performance of the two elements is concerned the most striking points are threefold: (1) The natural frequencies converge to the same value for both elements. (2) For a mesh density ratio of 24:1 in favour of the Timoshenko beam element, the helical element performs best. (3) The helical element having a mesh density of 6-coils/element and 6-master degrees of freedom performs only marginally worse over the first four eigenvalues than does the Timoshenko beam element whose mesh density is 72 times greater and which has 138 extra master degrees of freedom.

In view of the accuracy obtained in the analysis of the two physically tested springs the assumptions employed in the development of the differential equations (notably those concerning small strains) seem to be justified, For the more severe dynamical case, namely that of coil-clashing, the assumptions must be reviewed and the problem, perhaps, can only be treated satisfactorily if geometrical non linearity is accepted. In-any-case coil contact problems must be approached using a time stepping technique and consequently the computation time will be much increased.

276 J . E . MOTFERSHEAD

LONGITUDINAL FAMILY

SNAKING FAMILY

TORSIONAL FAMILY

FIG. 4. Mode shapes--spring 2.

TABLE 4. NATURAL FREQUENCIES (Hz) FOR A HELICAL MESH HAVING TWELVE COILS AND CLAMPED TERMINALS

Timoshenko Element beam

type Helical element element

No. of elements 144 12 6 12 2 144

Mesh density 12 EIs/coil 1 El/coil 2 Coils/El I El/coil 6 Coils/El 12 Eis/coil

No. of master DOF 66 66 30 6 6 66

142 142 142 145 145 144 143 143 143 146 146 145 297 297 300 328 328 310 337 337 340 369 369 351 362 362 366 505 505 370 364 364 367 513 513 374 592 592 612 616 645 645 658 661

Finite elements for dynamical analysis of helical rods 277

Flc. 5. Fundamental longitudinal mode--spring 1.

278 J . E . MOTrERSHEAD

FIG. 6. Second longitudinal mode--spring 1.

Finite elements for dynamical analysis of helical rods 279

CONCLUSIONS Finite elements have been presented for the dynamical analysis of helical rods and

springs. The formulation satisfies Wittrick's equations exactly and has been shown to produce natural frequencies which compare well with springs under test. Extremely coarse meshes (six coils per element) have been shown to provide surprisingly accurate results.

The element is easy to implement and has been designed to fulfil industry's urgent need for a reliable design aid in the complex area of spring dynamics.

Acknowledgements--The author would like to thank the Directors of Lucas Industries Limited for permission to publish this paper. The work forms part of an MPhil thesis which is registered with the CNAA at the Polytechnic, Wolverhampton and the advice of Dr E. W. Parker, of the Department of Mechanical Engineering, regarding spring testing, is gratefully acknowledged. Thanks are also due to fellow members of the Engineering Computing Section, in particular Dr. P. R. Wilson, and Mr H. C. Grigg of Lucas CAV Limited.

REFERENCES !. A. E. H. LOVE, The propagation of waves of elastic displacement along a helical wire. Trans. Camb.

Phil. Soc. (1899). 2. J. P. S. CURRAN, Further considerations in injector design for high specific output diesel engines. I.

Mech. E. Symp. Critical factors in the application of diesel engines, Southampton University (1970). 3. S. C. GROSS, Coil spring performance. Automobile Engng (1959). 4. J. DICK, Shock-waves in helical springs. The Engineer (1957). 5. J. W. PHILLIPS and G. A. COSTELLO, Large deflections of impacted helical springs. J. Acoust. Soc. Am.

51,967-973 0972). 6. G. A. COSTELLO, Radial expansion of impacted helical springs. J. Appl. Mech. Trans. A S M E 42, 789-792

(1975). 7. S. K. SINHA and G. A. COSTELLO, The numerical solution of the dynamic response of helical springs.

IJNME 12, 949-961 (1978). 8. Y. YOSnIMURA and Y. MURATA, On the elastic waves propagated along coil springs. Inst. Sci. and Tech.

6(1), 27-35. Tokyo University (1952). 9. W. H. WIYrRICK, On elastic wave propagation in helical springs. Int. J. Mech. Sci. 8, 25-47 (1965).

10. S. P. TIMOSHENKO, On the correction for shear of the differential equations for transverse vibrations of primsmatic bars. Philisophical Mag. 41,744-746 (1921).

I 1. R. DAVIS, R. D. HENSHALL and G. B. WARaURTON, A Timoshenko beam element. J. Sound Vibration 2, 475--487 (1972).

12. R. DAVIS, R. D. HENSHALL and G. B. WARBURTON, Constant curvature beam finite elements for in-plane vibration. J. Sound Vibration 25, 561-576 (1972).

13. R. DAVIS, R. D. HENSnALL and G. B. WARaURTON, Curved beam finite elements for coupled bending and torsional vibration. Earthquake Engng Struct. Dynam. 1~ 165-175 (1972).

14. B. M. IRONS, Structural eigenvalue problem: elimination of unwanted variables. A I A A J 3, 961-962 (1965).

15. E. KREYSZIG, Advanced Engineering Mathematics, 3rd Edn. Wiley, New York (1972). 16. G. R. COWPER, The shear coefficient in Timoshenko's beam theory. J. Appl. Mech. 33, 335-340 (1966). 17. S. TIMOSHENKO, Vibration problems in Engineering, 3rd Edn. Van Nostrand, New York (1955). 18. H. B. DWIGHT, Tables o f Integrals and Other Mathematical Data, 4th Edn. Macmillan, New York

(1961).

Matrix A aj., = a cos (as)

al.2 = a sin (as)

all = -- ~- sin (as)

a2,2 = r cos (as)

a2. 3 = l

a3,j = K sin (as)

a3.z = - x cos (as)

a3.3 = "IlK

a4.1 = - rs sin (as)

a4.2 = rs cos (as)

a4.4 = a cos (as)

a4.~ = a sin (as)

APPENDIX 1

280

K2 T2S COS a s . l = - ~ s in(as)- a (as)

K2 ,i.2 a5.2 = ~'~ Cos (as) - a s sin (as)

aL4 = - r sin (as)

a~,5 = r cos (as)

a5.6 = 1

K ' I . K ' I -- - - sin (as) + - - s cos (as) a6.t - - - - a 2 a

KT K ' I a6.2 = - ~ COS (as) + - - S sin (as) a

a6,3 = l i t

a6.4 = K sin (as)

a6.5 = - K COS (as)

a6.6 = "ILK.

J. E. MO'VI'ERSHEAD

Matrix B All elements of [B] given below are to be multiplied by 1/AE, i.e. b 0 = bJAE

T2 K2 3 T2K21~ 2 cos(as)+ s sin(as)

+{~a { l+ j_~ ( l+~ '2v~+ (l+v)(rZ+aZ)~s \ - ~ - } J y-d j cos (as)

b,,2 = - 6 - ~ (as) - - - ~ s z cos (as)

x 2 2 ( l+r2v~+( l+v) ( ' I2+a2) s in (a s ) + {~-~ { 1 + ~ -~ a 2 / j 3"a s

r 4 ( l + v ) . 2(1+ v ) _ 1 } b,.3 = ~ { ~ ± 3'

"i (2a2+ K2V) r x2v b,.4- ~ ~ s2s in(as)-] i~-~ sc°s(as)

T K211 r (2a 2 + K2V) S 2 COS (as) -- ~ ~ s sin (as) bl.5 = ~ 2a z

bl.6 = ~ (a2 + 2v7"2) a 4

2"i • bl.7 = - 7 s sm (as)

b,.s = - ~ s cos (as)

b,.,o = a cos (as)

hi . , = a sin (as)

T3 K2V 3

"i f3CK2v (2z2-K2v) l }sZ cos (as) + ~ / ~ - - ' ~ + 2a z

r [ K 2 f4"rZv 5 ,'1 , ( l + v ) ( r 2 + a 2 ) . K21 -~-~ ~ ~ - - h ~ - ~ v - ,~-,- Y + - ~ - j ~ s s i n (as)

• IK 2 a3 {~2-~ J4"I2v--5 - , } - ( l ; v ) + ~ } c o s ( a s ) [ a 2 2 v

,1.3 1£21" 3

a 4 2 a 2

+___z[K z ~4"i2v 5 } ( l + u ) ( ' i 2 + a 2) K2} a Z t ] r ~ I - - d ~ - - - ~ v - 1 ~ + 2 scos(as)

Finite elements for dynamical analysis of helical rods

1 ,rK2f_~T~.a~._~__~v_l~_l ~4,r2v 5 .I (l+v) + )sin(as

T 2 (2a2~a~2V) s2 cos (as) . - ~ { 2 z'(r2- 2x2)ls K 2 b2' = j2 ~ j s in(as)

K2 { v ( r2 - 2K2)/ f a ~ 2 ~ j c o s ( a s )

r2(2a2+K2v) 2 • K2 b23 = ~ ~ T s s l n ( a s ) + ~ - ~ { 2 v(r2-2x2)ls 2a 2 j cos (as)

K2 {2 v ( rz -2x2) / j2a3 ~ j sin (as)

2 T t 2

b 27 ~ 2x ~ . 2,7 = - ~'~s cos (as) - - ~ sin (as)

2~ -2 . 2K 2 b2.s = - ~-~ $ sin (as) + ~T~ cos (as)

b2.1o = - r sin ( a s )

b~,H = r cos (as) b2,~2 = 1

KT2 K21~ 3 "

xr 2 / 3 K21~ 2 cos(as)

+ ~ f 1 I'. 2 2- 4r~x2~) ~2 ~,)(ae+~.e))ssin(as) a~[~a-~[[ a + T ) + T ] + T + ( I + y '

+ x _ _ _ _(a2+r2){l_(l~_v)}}cos(as)

KT2 K21~ 3

f a ~ . + ~ - - ~ } s s in(as)

- --a2X {~a~ f~(a 2 + ,r ) + T ~ + T . + 2 4"/'2K21" / K2 (1 + ~-')(__d 2 + 'r2)1 s ' ) I ) COS (as)

+ x--- ~ l-L { (a2 + r2) + ~ } - (a2 + r2){1- ~ - ~ } } sin (as 3 [j2a z

K'r (2a 2 + r2v) s2 cos (as) b3.4 = j2 2a 3

KT 2 3 2 " j 2 ~ (2a +~K //')S Sln(as)

U \ .~ /

=Kr(2a2+~2v) 2. I b3.5 j2 2a 3 s sin ~asp

+~_~K~" (2a 2 +~3 x2~,)s cos (as)

i2a~ 2a2+~x2v s in(as)

2K , 2 b3.~=-f~(a + 2~2)s

2Kr . . 2xr b3.7 -- ~ s cos tas~ - ~ sin (as)

2~:~" . . 2K1" b3.8 = ~r~ s sm ~as) + ~ cos (as)

281

282

2

b3.,o = K s i n ( a s )

b3.,, = - K cos (as)

b3,12 =

r / x2v\ 2 • 3"rK2v b4., = - ~ k2 + - ~ - ) s sm ( a s ) - - ~ T s cos (as)

b,~ ~ ( ~ + ~ ) s ~ c o s 3 ~ • = r ( a s ) - ~ T s s in (a s )

2 ( l + v ) b4.3 = -- j2 0 2

(2a2+ K2v) s cos (as) b4A = ajZ

b - (2a2+ K2v~) s in (a s ) 4,5 -- a j 2 S

2v.r b4.6 = j2 a 2

2 b4,7 = j~ a c o s (as)

2 b4,s = ~ a sin (as)

T 2 b,, = - 2 ~ (2 + r-~-2 ~s2cos (as, + 5 ~ s sin(as)

-, a ~ z l a

2 K2 (1 ~ 5 L ~ ) cos (as)

~ 2 ( K2 t 5 ~2K2v bL2 = - 2 ~ 2 + ~ v s z sin ( a s ) - ~ - - ~ s cos (as)

2 K 2 (1 + 5- 7-2v~ s in (as ) + ~ ' ~ 4 a z ]

b 2 ( l + v ) ~.3 = j-~ z a---- T - s

bs4 (2a~2~2 2V) s s in (a s ) 7-r2v • = - ~ - ~ c o s (as)

(2a2+ K2v) rK~V bs.5 = 7- j2a2 s cos (as) - ~ sin (as)

_ ( a 2 + ~r 2) b5.6 = 2 ~ s

2 b~.7 = - ]~ 7- sin (as)

2 b~s = ~ 7- cos (as)

bL9 =

KT- ( -~2 ) 2K'rV [ 5/< 2 \ b6., = 2 ~ 2 + s2 COS ( a s ) + ' ~ - T (as)

- ~ - ~ ) } cos (as)

K T K2 )scos(os. = t a s ) - - ~ [ l - ~

51( 2 + j~ 0-"3

2r2 (I b6.3 = j2a2 K + v ) s

• KP ( 2 a 2 + K 2 v ) s l n ( a s ) - ~ ( a ' + 7 - 2 ) c o s ( a s ) b6,4 = K ~ S

J . E . MOTTERSHEAD

Fin i te e l e m e n t s for d y n a m i c a l ana lys i s o f helical rods

( 2 a 2 + l c 2 v ) , , K u b6~ = - x j2 - - - - -~y~a s co s t a s ) - - ~ (a 2 + ~.z) sift ( a s )

2 ~" {1 ~r2 6..-~ ~ - +~-)~ b

b~,7 = ~ x sin ( a s )

2 b6,8 = - :~ K cos ( a s )

I"

2 r b~,9 = j-~ ~ .

2 8 3

M a t r i x N

n l , ] ~-- n 4 . 4 = n 7 , 7 = n l O . J 0 --- - s in 4)

/11.2 = n4 ,5 ~-- - 117,8 = - - I ' l [0 , l l = s in a c o s 4'

111,3 = n 4 , 6 ~--- - - 117.9 = - - / 110 ,12 ~--- - - COS O~ COS (~

112,2 = 11'[5,5 = -- 118.8 = -- 1111,11 ~'~ - - C O S Of

112,3 ~--" /'/5.6 = -- 118,9 = -- 111132 = -- sin Of

113,1 ~ 116,4 = -- n9,7 = -- 1112,10 = cos (~

113.z = n~,~ = n9.s = n~2.. = sin a sin ~b

n3.3 = 116,6 ~ 119,9 ~ 1112,12 = - - C O S Of s in 4,

M a t r i x Q

q l , I = q4 ,4 = - - q 7 . 7 = - - q l 0 . 1 0 = s i n

q l . 2 = q 4 , 5 = qT .S = q l0 .11 = - s i n t~ c o s 4,

q l . 3 = ( /4,6 = q 7 . 9 = q ] o A 2 = C O S Ot C O S ( ~

q2,2 = q5,5 = qa,s = q l I . N = COS a

q2.3 = q~ .~= qs,9 = qll ,Z2 = s i n a

q3,1 = q~.4 = q 9 . 7 --- q12,10 "~ - c o s

q 3 . 2 = q6 ,5 = - - q 9 , s = - - q l Z I I = - - s i n a s i n ~b

q3.3 = q6,6 = - q9.9 = - ql2.~2 = co s a s in 4)