A finite element model for the vibration of spatial rods under various applied loads

Int. J. Mech. Sci. Vol. 34, No. 1, pp. 41 51, 1992 Printed in Great Britain.

0020-7403/92 $5.00+ .00 © 1992PergamonPresspie

A FINITE ELEMENT MODEL FOR THE VIBRATION OF SPATIAL RODS UNDER VARIOUS APPLIED LOADS Y. XIONG and B. TABARROK Department of Mechanical Engineering, University of Victoria, Victoria, B.C., C a n a d a VSW 3P6 (Received 21 November 1990; and in revised form 24 July 1991)

A l ~ t r a e t - - I n this study, a comprehensive energy formulation is outlined for the vibration analysis of spatially curved and twisted rods under various applied loads. In addition to the initial axial force, this formulation takes into account the initial moments and shear forces as well as initial deformations. The latter is accomplished through the updating of the curvature and twist of the rod under increasing loads. Based on this formulation, a spatially curved and twisted rod finite element is developed and used for analysis of several examples. The accuracy of the results obtained indicates that a small number of such elements can yield satisfactory results.

1. INTRODUCTION Spatially curved and twisted rods are used in many mechanical and aeronautical engineering components. For purposes of linear stress analysis and natural vibration, such rods are often modelled by an assemblage of prismatic beam elements. For the vibration of spatial rods under the action of various applied loads, however, such a representation has a number of drawbacks since a geometric stiffness matrix is required. The most important of these is that in the standard geometric stiffness matrix, for prismatic beam elements, initial deformations are ignored and only the initial axial load is taken into account. This is acceptable for such systems as pretwisted rods and deep arches which exhibit insignificant changes in geometry under the action of prebuckled loads and where the initial axial load has a dominant effect on the vibration behaviour of the system. On the other hand, for such systems as helices which, under most loading conditions, exhibit appreciable changes in geometry and have other initial forces-bending moments and torque for instance, the use of the usual geometric stiffness matrix for beam elements will lead to erroneous results. It should also be noted that the representation of curved and twisted rods by prismatic beam elements introduces an added error into the analysis; namely the geometric error. Reduction of this error requires the use of a large number of elements and results in large size system matrices. This incease in the size of the system matrices is generally not of major concern for linear stress analysis, governed by linear algebraic equations. However, for vibration analysis with applied loads, governed by eigenvalue equations, the increase in the size of system matrices is of importance from the computational perspective. Until recently, spatially curved and twisted rod elements had received little attention. However, there is extensive literature on the elements for purely curved and purely pretwisted rods, see, e.g., [1-5]. For curved and pretwisted rods of circular cross-section an exact stiffness matrix accounting for shear as well as flexural and torsional deformations was derived by Mottershead [6]. Mottershead [7] also derived a consistent mass matrix and through the addition of some nonlinear strain~lisplacement terms which neglected the shear deformations, he investigated the dynamic stability of helical springs. A constant strain curved and twisted element was derived by Tabarrok et al. [8, 12] for the static and dynamic analysis of spatial rods. In this study a comprehensive analysis is carried out and element matrices are derived for the vibration analysis of spatial rods. The formulation takes into account the initial moments and shear forces as well as the initial axial force. The initial deformations are also considered through the updating of the curvature and twist of the rod under increasing loads. The performance of the element developed is then assessed through a comparison of the results obtained for several examples, including a pretwisted rod under an axial load, a 41

42

Y. XIONG and B. TABARROK

deep arch subjected to uniformly distributed (dead and follower) forces and end couples, and a helical spring under an end load. 2. B A S I C E Q U A T I O N S

In T a b a r r o k and Xiong [9], a set of general equations was derived for the buckling analysis of spatial rods. By incorporating inertia forces in these equations, the governing equations for the vibration of spatial rods, under applied loads, expressed in the principal coordinate system, may be given as follows: Equilibrium equations

Z'Q2 + K'Q3 = pAfi~,

(1)

q~ + zql -- ~¢*Q3 + z*Qa = p A f i ~ ,

(2)

q'3 -- tcql -- x*Q~ + x * O 2 = pAil's,

(3)

q'l -- zq2 +

~cq3

--

m'l - z m 2 + xm3 -- q2 - z ' M 2 m tE q- z m 1 -Jr- qx m 3'-

+ r'M1

K m 1 --

+ K'M3

2 0 *1, = P A 'J1

(4)

-- K * M 3

' 2 0 *2, = P A J2

(5)

K'M1 +

tc~M 2 =p

J3A"20"3,

(6)

where q~ and m~ are the force and moment components, respectively, in the perturbed state, while Qi and Mi are the initial force and moment components; u~ and 0~ are the displacement and rotation components; p is the mass density, A the cross-sectional area andj~ the radii of gyration; x and z are the curvature and twist of the rod in the current state and the changes in the curvature components may be expressed in terms of incremental displacements about the current state (u*, u~, u~) and the axial rotation 0". These are given in Refs [9, 14]; x* = u * " - 2zu*' + ~:u~' - z2u *,

(7)

r* = 0~' + xu~' + ~:~ul',

(8)

x* = - u*" - 2zu*' + zZu'~ - zxu'~ + xO*.

(9)

F o r c e - d i s p l a c e m e n t relations

qx _ e* = u*' - zu~ + xu* - 0",

(10)

q2

(11)

yGA

_ e~ = U*' + ZU* + 0 " ,

7GA q3 = g~ = U*' -- KU~,

(12)

EA ml - k* = 0 " - zO~ + gO*, EIa

(13)

m2 -- k~ = 0 " + zO*, EI2

(14)

m3 -- k~ = 03 ' - - to0*,

(15)

GIa

where y E G A I1,12 /3 k*, k*, k]' e*, e~', e~

= = = = = = = =

shear coefficient, Young's modulus, shear modulus, area of cross-section, second moments of area, torsional constant, incremental curvature quantities associated with ml, m2, m3, incremental strain quantities associated with ql, q2, q3.

Vibration of spatial rods 3. V A R I A T I O N A L

43

FORMULATION

For the purpose of developing the element matrices it is useful to construct a variational functional, the extremum conditions of which would yield the equilibrium equations derived above, and the related boundary conditions. This may be accomplished by writing the inner product of the equilibrium equations, when expressed in terms of the kinematic quantities, with the appropriate virtual displacements. It is useful to integrate such a virtual work expression by parts. Such an integration has the following desirable effects: (i) the associated boundary conditions become identified, (ii) the energy terms emerge, and (iii) derivatives of lower order will appear in the functional. The latter effect is of importance for inter-element compatibility requirements. On carrying out this integration the following energy terms may be obtained:

Strain energy

,;2;i

U=~

,

{?,GAe.2 + 7GAe .2 + EAe~ 2 + Ell k .2 + EIzk72 + GI3k~E}dsdt

= 2

,

{ ) , 6 A ( u * ' - ru7 + ~u~ - 0 7 ) 2 + ?,GA(u*' + ru* + 0 2 ) 2

+ EAIu*' - Ku*) 2 + E l l ( O * ' - - tO* + ~ 0 ~ ) 2

+ EI2(O*'+ TO*)2 + 6 1 3 ( 0 " - hO*) 2} dsdt.

(16)

Complementary kinetic energy ~ l ft ~fo pA{fi,2 + fi,2 + ti,2 + s ;2,,2 +j20,2 + j 2 0 , 2 } d s d t .

T= }

(17)

Virtual work of initial forces OW =

;J fi{ l

-- ~ ~ 3 ( ~ [ ( u * ' - - 7.u 7 -t- KU~) 2 ~- (U7 t q- "CU*)2]

1

"+" [ Q 2 0 ~ ( U * ' -- TU7 -}" KU~ -

* *' "~- 3u* -4- 0 " ) ] 07) -- QlO3(~(U2

"k- [ Q I ( U * ' "-- TU* q- KU*) -'b Q2(U*' + TU*)]b(U~'-- KU*)

+ [Q,O* + Q2O*]ao'; -k- [ -- M 3 0 * ~ ( O * ' - - zO* .-1- KO*) + M 3 0 " 6 ( 0 "

+ 30*)]

+ [MIO* -- M20"36(0~'- ~cO*) + [MEO*6(O*'-- TO* + KO*)- M10"6(07'+ 30*)] ~dsdt. )

(18)

It is seen that while the expressions for the strain energy and the complementary kinetic energy are straightforward, the expression for the virtual work of initial forces needs some explanations. The terms in the round brackets, in the first term of this integral, are the slopes of the centreline. Hence the first term describes the work of the axial force Q3 as it moves through a distance equal to the difference between the original length of the centreline (in the current state) and the centreline for the bent and twisted perturbed state, projected onto the centreline in the current state. That is, this term is the generalization, to curved and twisted rods, of the familiar potential energy of the axial force for prismatic rods. Now while in the principal coordinates of the critical state the shear forces Q1 and Q2 have no components in the binormal and the normal directions respectively, they will have components in the directions of the binormal and the normal of the perturbed state. These components, given by Q10~' and QzO~, will have virtual work contributions associated with the shear strains. These virtual work terms appear in the second term of the above integral. In a similar manner the components of Q1 and Q2 will have a virtual work term associated with the axial strain in the perturbed state. This work is given in the third term of the above integral. The expressions Q ~0* and Q20~',in the fourth term, are the contributions of Q x and

44

Y. XIONGand B. TABARROK

Q2 to the torque per unit length and hence this term describes the virtual work by these incremental torques associated with the rotation 0~. The fifth term accounts for the work of the components of twisting torque M 3, in the perturbed state, associated with changes of curvature in the normal and the binormal directions. The sixth term takes account of the work done by the components of M1 and M2, in the perturbed state, associated with the changes of centreline rotation. Finally, the seventh term accounts for the work of the components of M1 and M2, in the perturbed state, associated with changes of curvature in the normal and the binormal directions. With the energy terms at hand, one can write the expression for Hamilton's principle as 6H = 6T-

(19)

6U + 3W.

By carrying out the variations and utilizing the equilibrium relations for the initial force quantities, one can obtain the following governing equations and boundary conditions:

,~1-1 =

{,~u~[pA~'~ - ~,aA(u~' - ~u3 + ,~u'~ - 03)' 1

+ "rvGA(u*' + zu* + 0") - ~cEA(u'~' - xu*)

+ Q~(O*' + ~u*' + ~:~u*) - Q 3 ( u * " - 2ru*' + Ku*' - r~u~)] + b u * [ p A f i 3 - 7 G A ( u 3 ' + zu~ + 0")' - zTGA(u*' - ru* + ~cu* - 0")

+ Q3( - u2*" -- 23uI" + 32u3 - 3~:u~ + ~0~) -- Ol(0~' + Ku*' + Kzu*)] + 6u*[pAfi'~ - EA(u'~' - ~:u*)' + xTGA(u*' -- zu3 + tcu'~ -- 0") z , tt + ~dl~ul - 2zu*' + xu*' - 32u *)

- Q~( - u3" - 2~u*' + r~u3 - 3Ku~ + ~ 0 " ) ] + 30*[pAj2"O'( -- E I , ( O * ' - - 303 + xO'~)' + 3 E I 2 ( 0 " + tO*) - KGI3(O*' -- ~cO*) + zTGA(u 3' + zu* + 0';) + M 2 ( O * ' - - KO*) -- m 3 ( o 3' + 30*)] + 303[pAj2"O * - EI2(O*' + 30*)' - z E I , ( O * ' -

zO 3 + KO*)

- 7GA(u*' - zu* + ~cu~ - 0") - MI(O'~' - ~0") + M3(O*'

-

-

"cO3 +

1<0")]

+ 303* [ p A j 3"20"'* 3 -- GI3(O ~' - t¢0"1)' + xEII(O*' - tO* + xO'~) + M1(03' + TO*)-- M2(O*' - 303 + x O ~ ) ] } d s d t {~u,* [~GA(u~*' - ru3 + ~cu~ - 03) + Q3(u,*' - r u 3 ' + Ku~) - 0 ~ 0 3 ]

+

f,21 + ~u3[~aA(u3' + ru'f + O'f) + Q3(u3' + ~u~) + ( L 0 ~ ] + ~u~[EA(u~'-- ~cu~*) - QlIU~" - ru3 + ~ u ~ ) + ~0~[el~(0~'

~(u3'

+ ru~*)]

- r03 + K0~) + M303 - M~O~]

+ ~03[E1~I03' + 30~) - M30* + M,O';] + ~0~ [ ~ I ~ ( 0 " - ~ 0 r ) - M , 03 + M~ 0 r ] }1'0 d~ + f l p A { f i * b u * + U"* .2 ", , + J 3-20 3",6 0 3 }, d s l t , . t2 26U * 2 -[- fi*3u~ + J l.2", O l ( ~ O 1, --[-J202302

(20)

It is not difficult to see that if the displacement quantities are not varied at times t~ and t2, then the Euler-Lagrange equations that emerge from the above variations yield the correct

Vibration of spatial rods

45

equilibrium equations in translation and the correct boundary conditions, for incremental quantities. For equilibrium equations in rotation one finds that the curvature changes of the centreline, namely x*, 3" and x*, which appear in the equilibrium Eqns (4)-(6), are replaced by the curvature changes associated with the constitutive relations, as given in Eqns (13)-(15). This is not unexpected since in reality the moments and the twist act on finite size cross-sections and not on the centreline. For slender rods for which shear strains may be neglected the two forms of curvature changes become identical, as one can readily verify by solving for 02* and 0* in Eqns (10) and (11) and eliminating the same from Eqns (13)-(15). The result found for changes of curvature will be identical to the expressions given in Eqns (7)-(9). 4. A FINITE ELEMENT MODEL With a variational statement at hand we are now in a position to develop a finite element model for the vibration analysis of spatial rods. Because of the ease with which polynomials are differentiated and integrated they are almost always used for the development of finite element models. However, in some cases, such as that at hand, polynomials cannot describe the rigid body modes, see Ref. [8], and are therefore not suitable for the development of element matrices. Evidently exact solutions of the governing equations provide the best choice for the shape functions. For the problem at hand the exact solutions will be functions of the vibration frequencies and applied forces and will result in a nonlinear eigenvalue system equation. The solutions of the governing equations in the absence of initial (internal) and external forces provide a good compromise. This solution may be obtained from the equilibrium equations: {q}' + [Z] {q} = {0},

(21)

{u*'} - [z]T{u *} -- ED]-X{q} = {0}

(22)

and the constitutive relations

where {q}T=[qa q2 q3 rnl mE m3], {u*}T=[ u* U2* U~ 0* 0* 02*] and [D] is a diagonal matrix containing the stiffness parameters 7GA, ?GA, EA, EI1, EI z and GI 3 and

Ix]=

[c] [J] [0] [c]J'l

(23)

where ,r

[c] = -k

0 0

0 ,

[J] =

0

1

0

0

0

.

Because of the similarity of the form of these equations to that of the homogeneous linear equilibrium equations, the expressions for {q} and {u*} have the same form as that given by Mottershead 1-6]. Use of these expressions will yield an exact stiffness matrix and an approximate geometric stiffness matrix in the system eigenvalue equation. Equations (21) and (22) are best solved by first obtaining a solution for {q} from Eqn (21) which, upon substitution into Eqn (22), will allow one to obtain a solution for {u*}. The functions obtained appear as trigonometric and products of polynomials and trigonometric terms and they involve 12 integration constants. These constants may be related to six nodal variables (three in translation and three in rotation) at each end of the element. The transformation from the integration constants to the nodal variables may be incorporated into the integrals in Eqn (20). That is, one may express the functions {q(s)} and {u*(s)} in terms of the nodal displacement variables {u*}e, i.e. {q(s)} = [A(s)] {u*}e,

(24)

{u*(s)} = rB(s)l {u*}e.

(25)

46

Y. XIONC,, and B. TABARROK

Then on using Eqns (24) and (25) in Eqn (19) and following the conventional assembly procedure, one can obtain the following system matrix eigenvalue equation: ( [ K ] - J,[Kg.])lt.l* } = ¢o2[ ~'/] 't/t* {,

(26)

where the system stiffness matrix [ K ] and mass matrix [M] are symmetric while the geometric stiffness matrix [Kg] is nonsymmetric, and [M] is the mass matrix; 2 and (,)2 are the load and frequency factors, respectively. 5. IM P L I ! M E N T A T I ( ) N

The finite element model developed here is based on the updated Lagrangian equilibrium equations, that is, the initial displacements are taken into account by updating the geometric configurations. More precisely, Eqn (26) may be written as: ~ [K(tc, r)] -

2[Kg(~, r ) ] )

{u* } =

t,)2[M(~c, r)]

{u* },

(27)

where to, ~ are the curvature and twist, respectively, in the current state, which are the summations of the initial curvatures and the curvature changes, as expressed in Eqns (7) and (8). This element can thus be used for the vibration analysis of spatial rods of different configurations, including the cases where the initial displacements have a significant effect on the rod's geometry, prior to buckling. For these cases, the solutions are obtained iteratively by solving Eqn (27) and Eqns (7) and (8). An effective procedure is as follows: Step I. In the initial undeformed state, when the load factor 2o = 0, form the stiffness matrix [K(~c o, 3o)] and the mass matrix [ M 0 , , r ) ] , and conduct a natural vibration analysis; Step 2. Apply a load )k and conduct a static analysis; Step 3. Update the change of geometry by computing t,, z with load 2~; Step 4. Form system matrices with the new geometry and conduct a vibration analysis with load 2k; Step 5. Judge if the first frequency is zero, 2 k is the critical load for static buckling; otherwise Step 6. Now increase the load factor to 2k 4 ~ = )) + A2~ and go to Step 2. For some spatial rods, such as pretwisted rods and deep arches for instance, which do not suffer significant prebuckling deformations, one does not need to distinguish between the configuration of underformed state and that for the current deformed state. In these cases the third step in the above procedure may be ignored.

6. I L L U S T R A T I V E

EXAMPLES

As a first example consider a pretwisted cantilevered rod subjected to an axial load P. The data used for this example are as follows E=68.13GPa,

G--25.61GPa,

A = 1.1718cm 2,

1=45cm,

p=2882kgm

~"o=0,

3

r0=4.7124.

The first two vibration frequencies of the rod under different axial load, along with the experimental natural frequencies and the analytical static buckling load, are shown in Table 1. It is seen that for natural vibration frequencies only one or two elements give very good results. As a second example, the vibration of a quarter-circular arch with rectangular crosssection was analyzed under uniform external pressure. The boundary conditions were pinned, allowing the arch to rotate freely about its normal and binormal, at its two ends. However, rotations about the centreline were set equal to zero. In order to show the full capability of the present element, both the in-plane and out-of-plane vibration analysis of the arch under both dead load and follower forces were carried out. In the latter case, with follower forces, an extra virtual work term (see the Appendixl must be appended to the

Vibration of spatial rods

47

TABLE 1. THE FIRST TWO VIBRATION FREQUENCIES (IN HERTZ) OF A CANTILEVERED PRETWISTED ROD WITH AXIAL LOAD

Axial load P (N) No. of elements

0

50

100

150

200

250

300

350

Critical load

1

157.98 340.69

147.20 135.39 122.23 107.21 89.35 66.34 27.18 332.59 324.23 315.60 306.67 297.43 287.84 277.90

359.94

2

157.79 337.87

147.08 135.33 122.22 107.20 89.29 66.11 25.98 330.08 322.01 313.64 304.95 295.90 286.48 276.66

358.99

3

157.78 337.57

147.07 135.33 122.21 107.19 89.27 66.07 25.83 329.81 321.77 313.44 304.77 295.76 286.37 276.57

358.87

experim.

158 339.3

Analytical Pc,

358.83

variational formulation in view of the nonconservative form of the follower force. The data used for this example are as follows: E=68.13GPa, R=30.5cm,

G=25.61GPa, b=1.89cm,

p=2882kgm

-3

t=0.62cm,

where R is the mean radius, b the width, and t the height of the arch. Iterative computations show that the prebuckling deformations are not appreciable. Therefore the third step of the iteration was not carried out and the vibration frequencies and buckling loads obtained for the two load cases, along with the buckling loads obtained analytically [10, 11], are shown in Tables 2a and b, respectively. It is seen that a small number of elements gives quite good results. It is interesting to note that in the follower force case the first and second vibration modes cross over when the load increases from 12,000 to 13,500 N m - 1 . That is, although the first natural vibration mode is out-of-plane symmetric (OPS), the first static buckling mode is inplane antisymmetric (IPA). This is pointed out in Table 2b. As a final example let us consider the vibrations of a helical spring with overall axial length 36 mm, mean coil diameter 10 mm, wire diameter 1 mm, and number of active coils 7.6. The data used are E = 206.84 GPa, G = 79.29 G P a and p = 7866 kg m - 3. First the free vibration frequencies of the spring, with clamped ends, were calculated and compared with those obtained by Mottershead [6] through calculations and experiments. These are shown in Table 3. The nearly pure bending modes in the x and y directions are characterized by close frequencies, as one can expect from the near symmetry of the spring about the x and y axes. The torsional bending modes are characterized by distinct properties. Next the vibration frequencies of this spring under the action of an axial load P with one end rigidly clamped and the other clamped for rotations only were determined. In this case the rod has noticeable deformation. That is, the geometry of the rod under increasing loads must be updated. Table 4 shows the four lowest frequencies under different loads along with the static buckling load Per, obtained by the present element. It is seen that the vibration frequencies decrease as the applied load increases and the static buckling load calculated is very close to that obtained from Haringx's formula, which is well known to be valid for springs with small pitch angles. 7. CONCLUDING COMMENTS In the foregoing we have derived a general element for the vibration analysis of spatially curved and twisted rods under various applied loads. The element takes account of the initial bending moments and shear forces as well as axial loads. Furthermore, the element

48

Y. XIONG and B. TABARROK

TABLE 2a. THE FOUR LOWESTFREQUENCIES OF ARCH WITH UNIFORMLYDISTRIBUTEDDEAD LOAD (IN HERTZ) Distributed load P (N m - 1) No. of elements

10

12

14

16

Mode*

0

1000

2000

3000

4000

5000

6000

I(OPS) 2(IPA) 3(IPS) ~OPA)

494.5 1287 3025 3471

454.4 1242 2979 3449

410.3 1195 2932 3427

360.9 1147 2885 3405

303.5 1097 2837 3383

232.4 1044 2788 3361

126.0 987.9 2738 3339

1 2 3 4

494.5 1286 3020 3469

454.1 1241 2973 3447

409.6 1194 2926 3425

359.7 1145 2878 3403

301.7 1094 2830 3381

229.4 1041 2780 3358

119.2 984.1 2729 3335

1 2 3 4

494.5 1286 3018 3469

453.9 1241 2971 3447

409.3 1194 2924 3424

359.2 1144 2876 3402

300.8 1093 2827 3380

228.0 1039 2777 3357

116.0 982.3 2726 3334

1 2 3 4

494.5 1286 3017 3468

453.8 1241 2971 3446

409.1 1193 2923 3424

358.9 1144 2875 3402

300.3 1093 2826 3379

227.2 1038 2775 3356

114.1 981.4 2725 3333

1 2 3 4

494.5 1286 3017 3468

453.8 1240 2970 3446

409.0 1193 2923 3424

358.7 1144 2874 3401

300.1 1092 2825 3379

226.7 1038 2775 3356

113.0 980.9 2724 3333

1 2 3 4

494.5 1286 3017 3468

453.8 1240 2970 3446

409.0 1193 2922 3424

358.6 1144 2874 3401

299.9 1092 2825 3379

226.4 1038 2774 3356

112.3 980.5 2723 3333

Buckling load (N)

6416

6370

6349

6337

6331

6326

Analytical solution P~r = 6316 *OPS---out-of-plane symmetric IPA--in-plane antisymmetric

OPA--out-of-plane antisymmetric IPS--in-plane symmetric

Vibration of spatial rods

49

TABLE 2b. THE FOUR LOWESTFREQUENCIES OF ARCH WITH UNIFORMLY-DISTRIBUTEDFOLLOWER FORCE (IN HERTZ) Distributed follower force P (N m - ' ) No. of elements

6

8

10

12

14

16

Mode

0

3000

6000

9000

12000

13500

I(OPS) 2(IPA) 3(IPS) 4(OPA)

494.5 1287 3025 3471

466.1 1138 2881 3431

435.7 967.6 2729 3390

603.1 759.2 2567 3348

367.7 465.5 2395 3306

191.7" 348.6 t 2304 3285

1 2 3 4

494.5 1286 3020 3469

465.2 1137 2874 3428

433.8 963.6 2720 3387

400.0 752.0 2556 3344

363.1 449.7 2382 3302

143.7 343.1 2289 3280

1 2 3 4

494.5 1286 3018 3469

464.8 1136 2871 3427

432.9 961.8 2716 3385

398.6 748.7 2552 3343

361.0 442.5 2376 3300

115.8 340.6 2283 3278

1 2 3 4

494.5 1286 3017 3468

464.5 1135 2870 3427

432.4 960.9 2715 3385

397.8 746.9 2550 3342

359.8 438.6 2373 3299

97.70 339.2 2320 3277

1 2 3 4

494.5 1286 3017 3468

464.4 1135 2870 3427

432.2 960.4 2714 3384

397.3 745.9 2549 3342

359.1 436.2 2405 3298

85.12 338.4 2310 3276

1 2 3 4

494.5 1286 3017 3468

464.3 1135 2869 3426

432.0 960.0 2713 3384

397.0 745.2 2578 3341

358.6 434.7 2398 3298

75.92 337.8 2303 3276

Buckling load (N)

13806

13670

13610

13578

13559

13547

Analytical solution Per = 13520 *IPA tOPS

MS 34:1-D

50

Y. X1ONG and B. TABARROK TABLE 3. NATURALFREQUENCIESOF HELICALSPRING (IN HERTZ} Present element

Mottershead

Mode

30 d.o.f.

42 d.o.f.

54 d.o.f.

30 d.o.f.

Experiment

1 2 3 4 5 6 7 8 9 10

396 399 469 534 886 898 947 1087 1417 1433

395 398 465 528 870 884 925 1049 1333 1387

395 398 464 528 868 881 919 1043 1321 1373

396 397 469 532 887 900 937 1067 1348 1409

3917 391'? 459 528 878? 878'? 906 n/a 1282 1386

TABLE 4. THE FOUR LOWESTFREQUENCIESOF HETICALSPRING WITH DIFFERENTLOADP (IN HERTZ) No. of elements

10

13

14

15

16

Buckling load

Axial load P(N) Mode 0

2

4

6

8

10

11

1 2 3 4

114.8 115,2 233,1 512.7

106,9 107.3 232.6 511.2

97,59 98.10 232.2 509.6

86.32 86.96 231.7 508.1

72.05 72.88 231.3 506.8

52.17 53.42 230.9 505.7

37.44 39.23 230,8 505.7

l 2 3 4

! 14.8 115,2 233.0 512.8

106.8 107.2 232.5 510.9

97.32 97.83 232,1 509,0

85.86 86,49 231.6 507.1

71.31 72.11 231.2 505,2

50.86 52.04 230.9 503,5

35.38 37,10 230.7 502.7

1 2 3 4

114.8 115.2 232.9 512.4

106.8 107.2 232.5 510.5

97.27 97.78 232.0 508.5

85.77 86.40 231.6 506.4

71.16 71.97 231.2 504.5

50.60 51.79 230.8 502.6

34.95 36.70 230.6 501.8

l 2 3 4

114.8 115.2 232.9 512.3

106.8 107.2 232.4 510.3

97.25 97.76 232.0 508.3

85.73 86.36 231,5 506,2

71.09 71.91 231.1 504.2

50.48 51.68 230.8 502.2

34.75 36.52 230.6 501.3

1 2 3 4

t 14.8 115.2 232.9 512.2

106.7 107.2 232.4 510.2

97.22 97.74 231,9 508,1

85.69 86,33 231.5 506.0

71.03 71.85 231.1 503.9

50.37 51.57 230.7 501.9

34.58 36.36 230.5 501.0

1 2 3 4

114.8 115.2 232.8 512.1

106.7 107.2 232.4 510,1

97.22 97.74 231.9 508.0

85.68 86.32 231.5 505.90

71.02 71.83 231.1 503,8

50.34 51.54 23(/.7 501.8

34.34 36.30 23.05 500.8

1 2 3 4

114.8 115,2 232.8 512,1

106.7 107.2 232.4 510,1

97.21 97.73 231.9 508.(I

85.68 86.31 231,5 505,9

71.01 71.82 231.1 503.8

50.32 51,53 230.7 50t.7

34.49 36.28 230.5 500.8

1 2 3 4

114.8 115.2 232.8 512.1

106.7 107.2 232.4 510.1

97,21 97.73 231.9 508.0

85,67 86.31 231.5 505.9

71.00 71.81 231.1 503.7

50.30 51.51 230.7 501.7

34.47 36.26 230.5 500.7

12.01

I 1.89

11.87

11.86

11.85

I 1.85

11.85

11.85

Haringx [13] P+, - t 1.99

Vibration of spatial rods

51

can be used for the vibration analysis of rods subjected to conservative and nonconservative follower-type forces. For some curved and twisted rods initial deformations become significant and affect the natural frequencies. The major contributions for such initial deformations come from finite rotations and result in changes in the rod's curvature and twist. A procedure has been outlined to incrementally account for such changes of geometry. The illustrative examples given verify the formulation and demonstrate the rather good performance of the element developed. REFERENCES 1. B. A. H. ABBAS,Simple finite elements for dynamic analysis of thick pretwisted blades. Aeronaut. J. 83, 450 (1979). 2. J. TOMAS and E. DOKUMACI, Simple finite elements for pretwisted blading vibration. Aeronaut. Q. 25, 109 (1974). 3. T. M. WANG, A. J. LASHEYand M. F. AHMAD, Natural frequencies for out-of-plane vibrations of continuous curved beams considering shear and rotary inertia. Int. J. Solids Structures 20, 257 (1984). 4. Y. YAMADAand Y. EZAWA, On curved finite element for the analysis of circular arches. Int. J. Num. Meth. Engng 11, 1635 (1977). 5. T. Q. YE and R. H. GALLAGHER,Instability analysis of pressure-loaded thin arches of arbitrary shape. J. Appl. Mech. 50, 315 (1983). 6. J. E. MOTTERSHEAD, Finite elements for dynamical analysis of helical rods. Int. J. Mech. Sci. 22, 267 (1980). 7. J. E. MOTTERSI-IEAD,The large displacements and dynamic stability of springs using helical finite elements. Int. J. Mech. Sci. 24, 547 (1982). 8. B. TABARROK,M. FARSHADand H. YI, Finite element formulation of spatially curved and twisted rods. Comp. Meth. Appl. Mech. Engng 70, 275 (1988). 9. B. TABARROKand Y. XIONG, On the buckling equaton for spatial rods. Int. J. Mech. Sci. 51, 179 (1989). 10. S. TIMOSHENKO and J. GERE, Theory of Elastic Stability. McGraw-Hill, New York (1961). 1l. G. A. WEMPNER, Mechanics of Solids with Applications to Thin Bodies. McGraw-Hill, New York (1973). 12. B. TABARROK,A. N. SINCLAIR, M. FARSHADand H. Y1, On the dynamics of spatially curved and twisted rods-a finite element formulation. J. Sound Vibr. 123, 315 (1988). 13. A. M. WAHL, Mechanical Springs, 2nd edn. McGraw-Hill, New York (1963). 14. Y. XIONG and B. TAaARROK, An energy formulation for buckling analysis of spatially curved and twisted rods. Presented at the Second World Congress on Computational Mechanics, Stuttgart, Germany.

APPENDIX For the rod with follower distributed loads, an extra virtual work term, accounting for the contributions of the follower forces, must be appended to the functional. This term may b¢ written as follows: ,5 wd =

fo

{pl [ O ~ 6 u , - (u*' - ~u* + , , u ~ ) 6 u * ]

+ p 2 [ - O~,Su* - ( u , ' + ~ u * ) 6 u * ]

+ p~[(u*' -

~u* + ,,u~)Ou* + ( u * ' +

~u*)6u~]

+ t2[0"60" -- 0*60*] + t3[0~60" - 0*607]}

ds,

(AI)

where Pl, P2, Pa, tl, t2, t a are the distributed force and moment parameters. The various terms in the above integral may be interpreted in a manner similar to that used in Eqn (18).