COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 9 (1976) 191201 0 NORTHHOLLAND PUBLISHING COMPANY
THE NONLINEAR STEADYSTATE 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 steadystate 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 singledegreeoffreedom Duffing’s equation, from which an amplitudefrequency relationship is derived and compared with results previously obtained by direct integration of the equations of motion.
Introduction The steadystate 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 steadystate 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 steadystate 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 timedependent 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 “onemode” 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 steadystate 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 loadcarrying 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
ja’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 steadystateresponse 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=; $I +,
+ ~[2ubZ,l + ~2[u;N(ur)
+u~fJ
+ t3[u; N(u,)l + ~4[;+V(u1)11 dV,
(7)
V
s
i!! = {‘(,l, +&J~N(~l)+u;lll
at
+~3[u;N(ur)]}dV,
+;t2[+'(u1)1
(8)
v
where the integration is taken over the volume of the entire structure. Element displacements for the gth element are defined in a local (element) axis system as a column vector
The element displacements are related to the global displacement vector r by jig=
Tag’,
(9)
where T is an ax.is transformation matrix, and as is a Boolean connectivity matrix [4]. Eq. (8) can be expressed in matrix form for the gth element by assuming a linear interpolation of displacement in each element. Then (1v)
0 0
0
(l77) 0
0
rl0
0
0 rl 0
0
9 = Ill , 1
(l77) 0 0 7)
i$'
(10)
D. Hitching and P. Ward, The nonlinear steadystateresponse of cablenetworks
194
=aps,
1, =g
a= l1 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 steadystate 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
00l
0
10
0
0
10
xg=
0 0 0
1 O10
00
0
0 0
Ol 0
0
0
10 0
1
Also, for the gth element we may write
D. Hitchings and P. Ward, The nonlinear steadystateresponse 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(U2uJ2+ (
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 rearranging (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 steadystate response of cable networks [((11fi2)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 5.701558.78 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 5.80382.15 Hz. The significant reduction in the high frequencies is to be expected since the
D. Hitchings and P. Ward, TYle nonlinear
198
steadystateresponse 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 1723
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 ydirection 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 steadystate response of net.
Frequency
Fig. 4. Nonlinear steadystate response of net when assumed “mode shape” is 1st mode of &ear system.
The linear steadystate response of the net was obtained by means of a modal analysis, and a frequencyamplitude curve is given in fig. 3, These results compare favourably with the direct integration results of [ 11. In this paper the nonlinear steadystate 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 steadystate 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 amplitudefrequency 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 steadystate
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 steadystate solution expressed as complex
vector.
Secondly, some difficulty was encountered in obtaining the amplitudefrequency curve in the case of the linear steadystate 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. Steadystate 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 steadystate 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 steadystateresponse 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) 20912108. [2] N.F. Morris, Modal analysis of cable networks, J. Strut. Div., ASCE 101 (1975) 97108. [ 31 A.H. Nayfeh, D.T. Mook and D.W. Lobitz, Numericalperturbation method for the nonlinear analysis of structural vibrations, AIAA J. 12 (1974) 12221228. [4] J.H. Argyris, Recent advances in matrix methods of structural analysis (Pergamon Press, 1964). [S] J.T. Oden, Finite elements of nonlinear continua (McGrawHi& New York, 1972). [6] J.J. Stoker, Nonlinear vibrations in mechanical and electrical systems (Interscience Pub., New York, 1950).