The usefulness of wire rope in shock and vibration isolation is briefly reviewed and its modeling for the purpose of vibration analysis is addressed. A model of a nominally straight segment of wire rope is described in which the rope structure is represented by a maiden, or central, strand of wire with one (or more) strand(s) wrapped around it in a helix (helices). The individual strands are modeled using finite elements and MSC NASTRAN. Small linear segments of each wire are modeled mathematically by dividing them lengthwise into triangular prisms representing each prism by a solid NASTRAN element. To model pretensioning and allow for extraction of internal force information from the NASTRAN model, the wound strands are connected to the maiden strand and each other using spring (scalar elastic) elements. Mode shapes for a length of wire rope with one and fixed to a moving base and the other attached to a point mass, are presented. The use of the NASTRAN derived mode shapes to approximate internal normal forces in equations of motion for vibration analyses is considered

Cochran, J. E., Jr.

Cutchins, M. A.

Fitz-Coy, N. G.

English

NASA Technical Reports Server

N87-22748
FINITE ELEMENT MODELS OF WIRE ROPE
FOR VIBRATION ANALYSIS
J. E. Cochran, Jr., N. G. Fltz-Coy and M. A. Cutchlns
Auburn University, Alabama
Workshop on Structural Dynamics and Control
Interaction of Flexible Structures
Marshall Space Flight Center
Huntsville, Alabama
Aprll 22-24, 1986
k
PRECEDING PAGE BLANK NOT FILMED
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https://ntrs.nasa.gov/search.jsp?R=19870013315 2020-03-20T10:32:59+00:00Z
FINITE ELEMENT MODELS OF WIRE ROPE
FOR VIBRATION ANALYSIS
J. E. Cochran, Jr.,* N. G. Fitz-Coyt and M. A. Cutchins*
Auburn University, Alabama
Abstract
The usefulness of wire rope in shock and vibration isolation is briefly
reviewed and its modeling for the purpose of vibration analysis is addressed.
A model of a nominally straight segment of wire rope is described in which the
rope structure is represented by a maiden, or central, strand of wire with one
(or more) strand(s) wrapped around it in a helix (helices). The individual
strands are modeled using finite elements and MSC NASTRAN. Small linear
segments of each wire are modeled mathematically by dividing them lengthwise
into triangular prisms (thick "pieces of pie") representing each prism by a
solid NASTRAN element. To model pretensioning and allow for extraction of
internal force information from the NASTRAN model, the "wound" strands are
connected to the maiden strand and each other using "spring" (scalar elastic)
elements. Mode shapes for a length of wire rope with one end fixed to a
moving base and the other attached to a "point" mass, are presented. The use
of the NASTRAN derived mode shapes to approximate internal normal forces in
equations of motion for vibration analyses is considered.
*Professor, Aerospace Engineering
tGraduate Research Assistant
1252
Introduction
Wire rope _ is, from the basic point of view, simply several strands of
wire twisted, or wound, together (see Fig. I). Some types are commonly called
"cable" and are used to carry electricity, support bridges and "cable cars,"
raise and lower heavy loads and in many other practical ways. A less obvious,
but equally important, use of wire rope is in shock and vibration isolation
devices.2, 3 The structure of wire rope provides many interfaces at which a
portion of the relative motion of strands of wire is converted by friction
into heat, thereby dissipating vibrational energy. 4 Furthermore, the
stiffness of wire rope structures can be tailored to provide support and
restoring forces. Stiffness and damping are adjusted by varying wire
diameter, the number of strands, pretensioning and the arrangement of lengths
of the wire rope. Commonly, helical coils of ropes 2,3 are fixed in clamps
(see Fig. 2) to form individual shock and/or vibration isolators. The
isolators are used to support and isolate communications equipment in vehicles
which are subjected to large magnitude, short-term accelerations; i.e.,
"shocks." In addition to absorbing shock, the internal, or system,
damping,_, 5 of the wire rope devices provides vibrational isolation over wide
ranges of frequencies and amplitudes.
The damping characteristics of wire rope and vibration isolators made
from it are not well understood from the theoretical standpoint. Apparently,
the design of individual isolators is accomplished by experimentation by
engineers with considerable experience in applications of these devlces. 2
Realistic mathematical models of wire rope isolators would be useful in the
design process and perhaps would allow the achievement of the confidence
levels in isolator characteristics needed for more applications in which
1253
ORIGINAL p_7. _.3
pOOR QUALri'Y
Fig. I. Wire rope.
1254
O4
0
0
m
i-i
0
m
u
I-I
_4
1255
damping rates and dynamic response must be very accurately known to prevent
resonance and control interaction problems.
The purposes of thls paper are (i) to present the results of work on the
problem of obtaining accurate mathematical models of wire rope isolators using
MSC NASTRAN and (2) to present some preliminary results of work toward
obtaining equations of motion of a wire rope "pendulum" for use in correlating
experimental and theoretical results.
Wire Rope Model
Basic Model
A sketch of a "wire rope" segment of length L, composed of a maiden wire
strand and a single "wound" strand, is shown in Fig. 3. It is assumed that
when the rope Is in its undeformed state the angle of the helix formed by the
centerline of the wound strand, a, Is constant over the length L. For the
finite element analysis, the maiden strand is divided into 372 elements
such that each cross section of the strand looks like a hexagon cut into tri-
angular pieces which have thicknesses equal to the radius of the strand. The
other strands are modeled in a similar manner. In addition, scalar elastic
elements ("springs") are used to connect the wound and maiden strands. These
springs are incorporated so that the distribution of the normal force between
the two strands can be determined for each mode shape computed (see Fig. 5).
NASTRAN produced figures showing two-strand and seven-strand wire rope are
given as Figures 6 and 7, respectively.
Boundary Conditions
The boundary conditions are chosen so that the "top" of the segment is
fixed. Two choices for the boundary conditions on the other end are
considered here. The first is to use a "free" end condition, (Case I). The
1256
O X
• /
B
Fig. 3. Two-strand wire rope.
1257
e,
MAIDEN STRAND WOUND STRAND
SECTION B-B
Fig. 4. Cross section of two-strand wire rope showing finite elements,
1258
N1259
oRIGINaL PAr_ |_
OF POOR Qu/_t.n'x
Fig. 6. NASTRAN model of two-strand wire rope.
1260
ORIGINAL PAGE IS
POOR QUALITY
Fig. 7. NASTRAN model of seven-strand wire rope.
1261
second choice (Case 2) is to attach to the "bottom" end a mass which is
relatively large compared to the mass of the rope segment.
The fourth mode shape for the seven-strand rope alone (Case 1) is shown
in Fig. 8. The corresponding mode shape for the Case 2 "pendulum" model is
given in Fig. 9.
Equations of Motion
Our stated objective is to obtain "analytical" models of wire rope
vibration isolation. Here, "analytical" implies that mode shapes from NASTRAN
models without damping will be used rather than analytlcal mode shapes such as
the classical ones for beams. The mode shapes are to be used to approximate
the displacements of the wire rope and attach "rigid" bodies when linear
damping is present and when nonlinear nonconservative and conservative forces
act on the system. The wire rope "pendulum" is a simple system which we
hope to use to test this method for modeling these effects.
Except for the determination of nonconservative internal forces, the most
attractive way of deriving equations of motion for the pendulum is to use the
Lagrangian formulation.
The Lasran_ian
To obtain the Lagrangian _, we must write the kinetic and potential
energies of the pendulum. The support point, A, is assumed to move in a
prescribed manner so that the vector _A from the fixed point 0 is known. The
kinetic energy, T, of the pendulum, assuming the mass M is a "point mass," is
Ni
T-_ I f (VA + "-rdm)'(VA + -_dm)dm
i=l m i
+ 1/2 M (VA + _M)-(V A + __M) (i)
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ORIGINAL P;C_ 15
OF POOR QUALiTy
Fig. 8 Mode shape for wlre rope model, Case i.
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GR|G_._L PAGE IS
OF pOOR QUALITY
Fig. 9 Mode shape for wire rope model, Case 2.
1264
where _A = _A ' the velocity of A, _dm is the velocity of a generic mass
element dm, N is the number of strands (see Fig. 3), and mi is the mass of the
m
ith strand. We approximate _dm by [ _j qj, where the _j are mode shapes,
jffil
the qj are generalized coordinates, m is the number of modes used and
m
_M = j=l[ _j (0 0 L)qj. The _j are functions of the coordinates Xo' Yo and zo
of dm in the undeformed "structure" and the qj are, of course, functions of
time to be determined.
In matrix form, Eq. (i) is
I T I _T#T_L_MT = _M T VA2 + M Vi _L _ +_ ffi=
N N
T I
+ vA I f a am + I _ f _T_T#ffi= _ am (2)
i=i m i iffii m_1
where MT is the total mass of the system, q ffi(qlq 2...qm )T
and _ = (_I _2"''_m ) and ffi#Lis _ffievaluated at xO ffiYo ffi0, z° = L.
As expected, we may write
i .T T
T =_-q M _ + a _ +_b , (3)
where
N
M ffi l f ffi#T_ dm +M_T
= = =L _=L (4a)
i=l mi
N
T T T
a ffi MV_A_L + VA I f _ dm (4b)
i=I mi
and
i
_b MT.!A2 (4c)
1265
the potential energy, V, is due to the elastic stiffness of the system and to
gravity. For the former, write
Vs = 1/2 T _s _ (5)
where K is the stiffness matrix.
=s
The potential energy due to the location elements of mass in the rope and
the mass M in a uniform gravity field is obtained by assuming that M is a
point mass at the end of the length of rope and that the rope has mass per
unit length O(Zo). Furthermore, the motion is restricted to the xz-plane and
dx/dz o = (I00) (d_/dz o H) is assumed small enough for terms of order q_ and
higher to be neglected. Then, the potential energy contribution due to
gravity can first be written as
L
Vg = f Mg(l - cos O)dz °
o
L zo
+ f o[l - f cos[8(_)]dE]dz
o 0 o
(6)
i (dx/dzo) 2 we may writeBy noting that cosO = i - _
L
1 dx 2
Vg = z M f (d--_-o) dz °
o
Z°.dx I ]2+ ½ ° f [d-'z-'l d_ dz °
0 0 OI =
IZo
(7)
Obviously, Vg is of the form,
Vg-_ qT =Kg q (io)
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The nonconservative forces acting on dm are assumed to be due to
viscosity and coulomb friction between the strands. Viscous forces can be
included in the equations by using some form of modal damping. For the coulomb
friction, we need to consider the rope's structure.
Let the strands be renumbered so that the maiden strand is i=O, and at
any value of Zo, the other strands are numbered 1 through N-I in a
counterclockwise manner about the zo axis. Also, let _dmi denote the force on
denote the force on the element dm i of the ith strand and let _ denote the
coefficient of friction. Furthermore, let fij denote the magnitude of the
normal force between the ith and the jth strands. Then, for dmo, we have the
coulomb friction force,
N-1
where ti and bl are the tangent and bi-normal unit vectors, respectively, of
dmi. For dmj, J > O, we may write.
_d.j "+ .% "g_E(id.o-_d_j)'(_J_J+ij_j)
- .qk sgnt(idmj- i_.k).(_jij +_j_j)
3
with J, k and A in the followlng triplets:
[!][I[il[!][i]If• = , , , , ... , (13)L.-2j
In Eqs. (12) we have also used 6
1267
__j " COS a [- sin e j i + cos Oj_] + sin a _I_ (14a)
and
_j - sin _[,in ej -;"- cosOj_] + cos_,_ (14b)
where 8j(Zo) is the angle of the centerline of the jth strand at zo.
Because _dmi = RA + _(Xo,Yo,Zo) H, where (Xo,Yo,Zo) is a point in the
undeformed i th strand, 6_dmi= _ (Xo,Yo,Zo)6 H. The generalized force matrix is
therefore 7
N-I
" f _T fdml,i_0 mi
(15)
where the integral indicates a summation of the quantities within it.
The equations of motion then follow from
d _£ ___' _IT
_ _s - (v._)T,
= T-Vs-Vg, viz.,
(16)
where - _ _ is the generalized viscous damping force. For normal modes of the
undamped structure, we have (with _ a diagonal matrix),
(17)
where Mj is the generalized mass and kj the generalized stiffness,
respectively, for the jth mode _gjKT is the jth row of _g, Qj is the jth element
of _ , dj the diagonal element of _ in the jth row and aj is the jth element
of a.
1268
The motion in qj is coupled in two ways. First, through the gravity
terms and, second, through the Qj since up to m of the qk appear in Qj to
determine its sign.
The next steps to be taken are (I) to obtain experimental data for the
position of M and (2) to match it with
which can be determined when _A is specified and the qi are known.
(18)
Summary
The use of wire rope in shock and vibration isolation devices has been
reviewed briefly. Finite element models of a nominally straight length of
wire rope have been described. These models were developed as part of an
effort to construct mathematical models of wire rope for use in the analysis
of vibration isolation devices which are constructed from wire rope.
Equations of motion for one of the finite element models, a seven-strand
rope suspended from one end and with amass attached to the other end to form
a "pendulum," were derived. The use of these equations to simulate the motion
of the pendulum was discussed.
Further work needs to be done with the current models and new models of
helical isolators should be developed.
Acknowledgment
The support of NASA, Marshall Space Flight Center, Huntsville, Alabama,
under Grant NAG8-532 is gratefully acknowledged. The COTR for this effort is
Mr. Start Guest.
1269
References
lo
o
o
o
o
o
o
, American Wire Rope, American Steel and Wire Company, New York,
, "Helical Isolators Protect Communications Gear," National
Defense, April 1986, pp. 72-73.
, "Shock-Free Flight Into the Future," Isolation News, Vol. 14,
No. i, Fall 1984, Newsletter published by Aeroflex International, Inc.,
Plainvlew, Long Island, NY.
Gobdman, L. E., "Material Damping and Slip Damping," in Shock and
Vibration Handbook (C. M. Harris and C. E. Crede, editors), McGraw-Hill,
New York, 1976, 36, pp. 1-28.
Badrakln, F., "Separation of and Determination of Combined Damplngs from
Free Vibrations," Journal of Sound and Vibration, Vol. i00, No. 2, 1985,
pp. 243-255.
Miele, A., Flight Mechanics, Vol. I, Addison-Wesley Publishing Company,
Inc., Reading, Mass., 1962, pp. 15-16.
Whittaker, E. G., A Treatise on the Analytical Dynamics of Particles and
Rigid Bodies, Cambridge University Press, London 1936, reprinted 1970,
pp. 34-38.
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Finite element models of wire rope for vibration analysis

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