The nonlinear steady-state response of cable networks

The nonlinear steady-state response of cable networks

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 9 (1976) 191-201 0 NORTH-HOLLAND PUBLISHING COMPANY THE NONLINEAR STEADY-STATE RESPONSE OF CABL...

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 9 (1976) 191-201 0 NORTH-HOLLAND PUBLISHING COMPANY

THE NONLINEAR STEADY-STATE RESPONSE OF CABLE NETWORKS D. HITCHINGS and P. WARD Department of Aeronautics, Imperial College of Science and Technology, Prince Consort Road, London SW72BY, U.K. Received 19 January 1976 The nonlinear steady-state response of a pianar cable net under periodic excitation is investigated. The nonlinearity is attributed to changes in the cable tension resulting from finite displacements. A lumped mass finite element formulation is employed to derive the governing nonlinear differential equations. These equations are solved by assuming a single mode expansion reducing the governing equations to the single-degree-of-freedom Duffing’s equation, from which an amplitude-frequency relationship is derived and compared with results previously obtained by direct integration of the equations of motion.

Introduction The steady-state response of flexible cable nets undergoing finite dispIacements is an important probIem which has received much attention recently. Morris has employed direct integration techniques [ 1, 2 ] with varying amounts of success. However, when the governing equations are solved by direct integration methods, the computer time required is often quite prohibitive. In this paper a combined numerical and analytical technique is used to determine the response of cable nets to harmonic excitation. Emphasis is placed on the case when the frequency of excitation is near a natural frequency. This is shown to be a very efficient method for obtaining the steady-state response. The nonlinear problem is approached by representing the deflection curve or surface by an expansion in terms of the linear free oscillation modes or some other “mode shape” decided upon by physical arguments, e.g. the linear steady-state response. The deflection is then dete~ined in two stages. Firstly, the linear modes (eigenvectors) and natural frequencies (eigenvaIues) are obtained from a finite element model of the structure. Secondly, the time-dependent coefficients are determined from a set of coupled, nonlinear, ordinary, second order differential equations - the linear “modes” being used to determine the coefficients in these equations. In this paper a “one-mode” expansion will be considered. In this case the resulting equation is shown to be of the form of Duffing’s equation. Similar techniques have been applied by several authors to the problem of a pinned beam, and a survey of this literature is available in a paper by Nayfeh et al. 131.

1. mathematical

formulation

The problem of interest here is the vibration of cable nets in which the nonlinear effect is pro-

D. Hitchings and P. Ward, The nonlinear steady-state response of cable networks

192

Fig. 1. Element of cable net.

duced by the change in tension caused by finite displacements. The deformations are assumed to be sufficiently small for nonlinear material effects to be ignored. The cable net is idealized as a series of axial load-carrying elements that are considered to be straight between nodal points (fig. 1). The strain displacement relationship for a typical element may be written as

or

=?E+i((gy+(f)‘] y;z ax yz = L(u) +

$N(u)

(1)

,

(2)

where L is a linear spatial operator, N is a nonlinear spatial operator, R, D, w are measured in the local element axis system, and u is the generalized displacement vector. To formulate the strain energy of the structure for finite amplitudes, assume

u = e. + &

+

t2a2 + ...

(3)

)

7 = 80 + ~L(r.4,) + .p ;N(u,).

The strain energy U of the structure may be written as u=t

j-a’ydV,

(4)

V

where the integration is taken over the volume of the entire structure, Substituting (3) into (4) gives

u= a J{(T)+Eu1 + E2u2)’

(co + tL(u,)

V

Writing I, = L(u,) and collecting terms gives

+ ;~2N(u1))

dV.

D. Hitchings and P. Ward, The nonlinear steady-stateresponse of cable networks

+.$3[fu~N(ul)+u~lll

+~4[;u;N(~1)1) dV.

193

(5)

We assume that the reciprocity relation (u’)‘y2 = (dyyl

)

is valid, where 1 and 2 are arbitrary states of stress and strain. Then we may write t Ql&O

‘Ufol,

)

(6)

+1

=ufiy2

=

;u’, N(u,) .

Using (6) in (5) gives

u=; $I +,

+ ~[2ubZ,l + ~2[u;N(ur)

+u~fJ

+ t3[u; N(u,)l + ~4[;+V(u1)11 dV,

(7)

V

s

i!! = {‘(,l, +&J~N(~l)+u;lll

at

+~3[u;N(ur)]}dV,

+;t2[+'(u1)1

(8)

v

where the integration is taken over the volume of the entire structure. Element displacements for the gth element are defined in a local (element) axis system as a column vector

The element displacements are related to the global displacement vector r by jig=

Tag’,

(9)

where T is an ax.is transformation matrix, and as is a Boolean connectivity matrix [4]. Eq. (8) can be expressed in matrix form for the gth element by assuming a linear interpolation of displacement in each element. Then (1-v)

0 0

0

(l-77) 0

0

rl0

0

0 rl 0

0

9 = Ill , 1

(l-77) 0 0 7)

i$'

(10)

D. Hitching and P. Ward, The nonlinear steady-stateresponse of cablenetworks

194

=aps,

1, =g

a= l-1 00 1001/t.

Also, e, = El,, where E is Young’s modulus. For a harmonic excitation of the form l’sin S2t we assume a solution of the form

in which .$is an undetermined function of time, and X, is a “mode shape” which describes the spatial variation of the assumed solution. The “mode shape”, although classically an eigenvector of the equiv~ent linear system, may also be an assumed shape based on physical arguments, e.g. the linear steady-state response. Then (13) where ks is the element elastic stiffness in the local coordinate system. Eq. (14) may be written as

It can be shown (see appendix) that fvukZ, dV = 0 for a structure initially in equilib~um. Further, in this case, N(u,)=

(g,’+(g,’

)

which may be expressed in matrix form, using (lo), as

where 0

0

00

0

1

00-l

0

10

0

0

10

xg=-

0 0 0

-1 O-10

00

0

0 0

O-l 0

0

0

10 0

1

Also, for the gth element we may write

D. Hitchings and P. Ward, The nonlinear steady-stateresponse of cable networks

Pop, the initial tension ,

*o CM? =

s

195

4 *1 dAe = Pig,

.f

the increase in tension linearly proportional

,

to the amplitude

Ae

s ‘3 dAe Ae

=

Pzg, the increase in tension quadratically proportional

to the amplitude

,

where the integration is taken over the area of the element. Finally, eq. (8) may be expressed in matrix form as = x; c {~[POgTtxgT+

g

8

T’EEgT] + ;p[PigT’xgT]

+ p[P2pgTI)X1

.

(19

The kinetic energy of the structure is given by T=

;i'Mi.

(16)

A consistent or lumped mass formulation may be used in deriving M. Using (12) in (16), T=fX;MX,i2.

The dissipation function for linear viscous damping is given by D= pci

,

which, using (12), may be expressed as

The work done by the applied forces is given by

Q=XiPsinW. The change in element tension is a function of the current local displacements. The functional relationship may be derived [S] by means of the Principle of Minimum Potential Energy, and is found to be P-

XA

=y

(U,-uU1)+f(U2-uJ2+ (

which may be expressed as CA = PlgE + 4,E’

*

(17)

In the case af a structure possessing symmetry about an axis perpendicular response it is found that Pig = 0, Applying Lagrange’s equation in the form

to the main

+E

at =Q

(18)

gives after re-arranging (19) i.e. a Duff&g’s type equation with damping in which

cm

For the case of r = 0, ( 19) becomes

(21) i.e. Duffing’s equation with viscous damping. Stoker [6) presents a method for obtaining approximate harmonic solutions to (21). The displacement and impressed force can be expected to be out of phase, and in the solution it is more convenient to fix the phase of the solution and leave the phase of the impressed force to be determined. Hence, ~+c~faf+8~3=Rcosof+Ssinwt, in which p = JRis considered fixed. As a first approximation to the solution we take t=AsinS’2t. After some manipulation

(23) and noting that sin3Stt = z sin Clt - 4 sin 3Rt, we get

(ol - fi2)A +‘[email protected] = S

(24) ACQ Squaring and adding gives

197

D. Hitch&s and P. Ward, The nonlinear steady-state response of cable networks [((11-fi2)A

t$&43]2

+c2A2fi2

z&)2

)

(25)

which gives a cubic equation in A’. This cubic may be solved by a variety of numerical methods. In this paper one root was found by Newton’s method and then eliminated leaving a quadratic to be solved by completing the square.

2. Numerical results A computer program has been developed to perform the calculations outlined in the previous section for the case of proportional damping of the form C=b,M+b,K,

in which b, and

b,

are constants.

16

1

Fig. 2, A planar cable net.

The planar cable net of fig. 2 has been analysed and the results are now presented. The initial coordinates of the nodes of the net are given in table 1 in metres, while the initial tensions and Ed values of its members are given in table 2 in newtons. As a prelude to the analysis proper the linear mode shapes and natural frequencies for the net were obtained with the aid of Householder reduction and the QL algorithm. The effect of gravity load was neglected in the computation of the cable static tensions. The structure is that previously analysed by Morris [ 1, 21 and, as in his work, two mass distributions were considered for the eigenvalue analysis. Firstly, the structure was analysed for the case in which 1 kg is placed on the lower node points. Table 3 gives the first five transverse mode shapes and their corresponding frequencies. In this case the frequency spectrum was found to be 5.70-1558.78 Hz. A second eigenvalue analysis was performed for a modified mass distribution. A uniform distribution of mass was employed with 0.5 kg at each node. The frequency spectrum narrowed to 5.80-382.15 Hz. The significant reduction in the high frequencies is to be expected since the

D. Hitchings and P. Ward, TYle nonlinear

198

steady-stateresponse of cabienetworks

Table I Initial coordinates of nodes

-

Table 2 Tensions and EA values of members

Node

X

Y

Member

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.243 0.250 0.499 0.499 0.751 0.753 I.008 I.009 I.259 t.259 1.513 1.524 1.762 1.777 2.016

-0.391 -0.059 -0.343 -0.108 -0*313 -0.133 -0.304 -0.145 -0.3 15 -0,133 -0.343 -0.108 -0.387 -0.063 -0.447

1 2 3 4,5 6 7 8 9 fff I1 12,13 14 15 16 17-23

16 17 18

2.010 0.009 0.0

-0.005 -0.446 -0.0

Table 3 Transverse modes and frequencies (toad = 1 kg) Node

1 2 3 4 5 6 7 8 9 10 li 12 13 14

Frequency in hertz

EA

294 290.77 288.12 286.06 288.71 290.47 293.12 294 290.47 288.12 283.06 288,71 289.59 296.35 22.05 ,_,..._-I___-

190 000 190 000 190 000 190 000 190 go0 190 000 r90000 190000 ~9~~~ 190 000 ~9~~0~ 190000 I90 000 190 000 103 000

Table 4 Transverse modes and frequencies (load = 0.5 kg) Node

----

5.70

8.06

10.45 --I--

Il.95

12.93

-0.673 -0.668 -0.987 -0.988 -0.702 -0.699 0.016 O.OlB 0,717 0.718 0.997 1.000 0.678 0.695

-0.527 -0.524 -0.332 -0.333 0.500 0.501 0.998 1.ooo 0.586 0.587 -0.316 -0.317 -0.61 I -0.628

0.972 0.970 0.265 0.266 -0.644 -0.645 0.003 0.003 0.996 1.000 0.293 0.294 -0.665 -0.686

-0.303 -0.302 0.733 0.137 0.997 LOOQ -0.392 -0.394 0.434 0.437 0.945 0.952 -0.048 -0.052

0.998 1.000 -0.214 -0.2 16 0.650 0.654 0.753 0.757 O.f03 0.104 0.139 0.141 0.911 0.939

-

Tension

1 2 3 4 5 6 7 8 9 IO 11 12 13 14

Frequency in hertz 5.80

8.19

10.62

12.15

13.17

-0.679 -0.672 -0.990 -0.990 -0.704 -0.700 0.020 0.022 0.723 0.723 0.998 1.000 0.672 0.687

-0.530 -0.525 -0.326 -0.326 0.512 0.513 1.000 1 .OOO 0.576 0.576 -0.327 -0.328 -0.608 -0.622

0.977 0.968 0.261 0.260 -0.628 -0.627

-0.298 -0.294 0.744 0.744 1.000 0.998 -0.396 -0.397 0.446 0.447 0.955 0.957 -0.05 1 -0.055

1.OOO 0.992 -0.264 -0,265 0.665 0.666 0.752 0.75 1 0.046 0.046 0.185 0.185 0.906 0.924

0.032 0.032 1 .OOO 1.000 0.273 0.272 -0.659 -0.675

higher modes predominantly represent the oscillation of individual nodal masses. The first five transverse mode shapes and their frequencies are listed in table 4, Morris analyses the modified structure under harmonic excitation P sin @t applied in the y-direction at node 12 where P = 3.8 N. As the time step used may be at most one quarter of the minimum period of the system, the modified mass distribution offers the advantage of admitting a maximum time step of 6.54 X 1W4 set compared with 1.6 X 1W4 set for the original structure. Damping for the system was defined by b, = OS, b, = 0.002.

$

o

Node

3

I

tnttgntron

Node 3

e

Nods

12

n

Intcgratlon

Node

12

143

2 “a “_

E

a

Frequency

Fig, 3. Linear steady-state response of net.

Frequency

Fig. 4. Nonlinear steady-state response of net when assumed “mode shape” is 1st mode of &ear system.

The linear steady-state response of the net was obtained by means of a modal analysis, and a frequency-amplitude curve is given in fig. 3, These results compare favourably with the direct integration results of [ 11. In this paper the nonlinear steady-state solutions were obtained in two ways. Firstly, the assumed ‘“mode” shape was taken to be the first mode of the linear system. fn this case eq. (25) is a cubic in A2 with real coefficients. Only real solutions for A are retained; thus there can be one or three roots for a given frequency. A frequency-~plitude curve for this approximation is given in fig. 4. Secondly, the assumed “mode shape”’ is taken to be the Iinear steady-state solution expressed as a complex vector. In this case the cubic of (25) has complex coefficients giving, in general, three complex roots for A. The admissibility of these solutions should strictly be tested by a stability analysis. However, in this paper the stable solutions were obtained by inspection from amplitude-frequency diagrams. The curves obtained are given in fig. 5. Morris obtained amplitudes at four frequencies, and these results are included in fig. 5 for comparison. It is found that the method outlined in this paper reproduces the general features of the response weII, al though detailed comparison is difficult for a number of reasons. Firstly, as only a small number of points on the curve were obtained by direct integration, a large part of the response is open to speculation. It is a feature of numerical integration techniques that the curve around the nonlinear jump point is difficult to reproduce as this solution should always converge to the upper or lower stable branch and the connecting transition curve cannot be obtained.

D. Hitchings and P.

200

Ward,The nonlinear sfe~~s~u~e response ofcable networks

0

Approx.

0

Integration

20 c

No&

3

Node

12

l 08

9

B

a0 O0

0

0

l

00

%

as”” Of 4

5

Fig. 5.

Nonhear steady-state

c

(yoQ

[email protected]@

5%

6

Frequency

1

7

0

0 l

0

0

00 I

0°% 5

6

7

Frcqurncy

responseof net when assumed “mode shape’”is linear steady-state solution expressed as complex

vector.

Secondly, some difficulty was encountered in obtaining the amplitude-frequency curve in the case of the linear steady-state approximation as visual inspection became ambiguous. This difficulty will be surmounted when a stability analysis of these solutions is undertaken. Emphasis must be placed on the saving in computer time which is a feature of this approach. Steady-state amplitudes for a given frequency of excitation were obtained on the CDC 5400 in less than 1 set, whereas the corresponding time for a direct integration solution is of the order of 9000 sec. Also it must be noted that the presence of high frequencies in the frequency spectrum is not a problem with regard to computer time in this method.

3. Conclusions A combined numerical and analytical technique for obtaining the steady-state response of cable nets undergoing moderately large displacements has been outlined in this paper. The present technique offers large savings in computer time and shows encouraging a~eement with results obtained by direct integration techniques. The technique described is in the process of further devetopment including considera~on of alternative “mode” shapes and mult~~e solutions.

201

D. Hitchings and P. Ward, The nonlinear steady-stateresponse of cable networks

Appendix The integral J+bl,

dV may be written as

(A.11 Since Pogut is (pg&, which is the element load vector at state 0, then (A. 1) can be written as

F r’uj { 7”
References [ 1] N.F. Morris, Dynamic response of cable networks, J. Strut. Div., ASCE 100 (1974) 2091-2108. [2] N.F. Morris, Modal analysis of cable networks, J. Strut. Div., ASCE 101 (1975) 97-108. [ 31 A.H. Nayfeh, D.T. Mook and D.W. Lobitz, Numerical-perturbation method for the nonlinear analysis of structural vibrations, AIAA J. 12 (1974) 1222-1228. [4] J.H. Argyris, Recent advances in matrix methods of structural analysis (Pergamon Press, 1964). [S] J.T. Oden, Finite elements of nonlinear continua (McGraw-Hi& New York, 1972). [6] J.J. Stoker, Nonlinear vibrations in mechanical and electrical systems (Interscience Pub., New York, 1950).