INITIAL INVESTIGATIONS INTO THE DAMPING 

CHARACTERISTICS OF WIRE ROPE 

VIBRATION ISOLATORS 

M. A. Cutchins 

J. E. Cochran, Jr. 

K. Kumar 

and 

N. G. Fitz-Coy and M. L. Tinker 

Aerospace Engineering Department 

Auburn University, Alabama 

(SASA-CR- 180648) III'IIAL lNVES3.XGATIONS 187-20569 

X%Ta IHE DAPIEING CBABZCkER3S3lCS CP YXBB 

EWE Beport YIBBATIOL ISCLATCGS Einal pecbnical (Auburn lioiv,) 89 F CSCZ 20K Unclas 

G3/39 45403 
Auburn Univ er s it y 

Aerospace Engineering 

Technical Report 87-1 

FINAL REPORT 

INITIAL INVESTIGATIONS INTO THE DAMPING CHARACTERISTICS 

OF WIRE ROPE VIBRATION ISOLATORS 

M. A. Cutchins 

J. E. Cochran, Jr. 

K. Kumar 

M. L. Tinker 

Ne G. FitZ-Coy 

Engineering Experiment Station 

Auburn University, Alabama 36849 

Prepared under 

NASA Research Grant NAG8-532 

George C. Marshall Space Flight Center 

National Aeronautics and Space Administration 

Auburn Univ er s it y 

January 1987 
ABSTRACT 

Passive dampers composed of coils of multi-strand wire rope are 

investigated. Analytical results range from those produced Ey complex 

NASTRAN models to those of a Coulomb damping model with variable 

friction force. The latter agrees well with experiment. The Coulomb 

model is also utilized to generate hysteresis loops. Various other 

models related to early experimental investigations are described. 

Significant closed-form static solutions for physical properties of 

single-and multi-strand wire ropes are developed for certain specific 

geometries and loading conditions. NASTRAN models concentrate on model 

generation and mode shapes of 2-strand and 7-strand straight wire ropes 

with interfacial forces. 

i 
TABLE OF CONTENTS 

Section Title Page 

1 . INTRODUCTION 1 

2 . DAMPING CONSIDERATIONS ................ 3 

Literature Discussion ............... 3 

Summary of Techniques ............... 4 

3 . STATIC MODELS .................... 9 

Background ..................... 9 

Summary ...................... 10 

Effective Modulus of Rigidity (E) ......... 15 

Effective Torsional Modulus of Rigidity (E) . 16 

Application to Dynamic Investigations ....... 17 

4 . SIMPLEST DYNAMIC MODELS ............... 19 

One-Dimensional Models ............... 19 

NASTRAN Single Loop Model . . . 22 

Single Loop Wire Rope Tests ............ 23 

Isolator-Supported Momentum Wheel Equations . . 24 

5 . ADVANCED DYNAMIC MODELS . 0 0 e 28 

Basic Finite Element Models ............ 28 

Equations of Motion ................ 29 

Discussion ..................... 33 

6 . CONCLUSIONS ..................... 35 

7 . REFERENCES ...................... 36 

8 . BIBLIOGRAPHY ..................... 38 

ii 
TABLE OF CONTENTS (CONTINUED) 

Section - Title Page 

9. FIGURES . 41 

APPENDIX A. ISOLATOR SHAKER TESTS . . A-1 

APPENDIX Be WIRE ROPE PROPERTIES . . A-2 

iii 
LIST OF FIGURES 

Figure 1.1 Helical isolators. 

Figure 3.1 

Figure 3.2 Plots comparing the analytical and numerical values 

Rope geometry with m2 = 6 and mi = 1. 

of effective modulus of rigidity of wire rope for 

the practical range of a. 

Figure 3.3 Modulus of Rigidity as affected by the helix angle. 

Figure 3.4 Torsional stiffness as affected by the helix angle. 

Figure 4.1 Wire rope vibration isolator. 

Figure 4.2 Undamped single-degree-of-freedom model. 

Figure 4.3 Rigidly connected Coulomb damper model. 

Figure 4.4 Elastically connected Coulomb damper model. 

Figure 4.5 Isolator force versus dfsplacement curve. 

Figure 4.6 Listing of ACSL program for Coulomb friction model. 

Figure 4.7 Theory (undamped model) vs. experiment. 

Figure 4.8 Theory (Coulomb model with constant friction force) 

vs. experiment. 

Figure 4.9 Theory (Coulomb model with variable friction 

force) VS. experiment. 

Figure 4.10 ACSL-generated 60 Hz hysteresis loop. 

Figure 4.10 (Cont.) ACSL-generated 40 Hz hysteresis loop. 

Figure 4.10 (Cont.) ACSL-generated 100 Hz hysteresis loop. 

Figure 4.11 Single loop wire rope experimental arrangement. 

Figure 4.12 Momentum wheel with three isolator support pairs. 

Figure 4.13. ACSL model definition arrangement - wire rope 

isolated GMC momentum wheel. 

iV 
LIST OF FIGURES (CONTINUED) 

Figure 4.14 

Figure 5.1 

Figure 5.2 

Figure 5.3 

Figure 5.4 

Figure 5.5 

Figure 5.6 

Figure 5.7 

Figure 5.8 

Figure 5.9 

Figure 5.10 

Figure A-l 

Figure A-2 

Figure A-3 

Figure A-4 

Planned output arrangement for isolator-supported 

investigation. 

NASTRAN model of two-strand wire rope. 

Cross section of two-strand wire rope showing 

finite elements. 

PENTA MSC/NASTRAN solid element. 

Cross section of seven-strand wire rope showing 

finite elements. 

NASTRAN model of seven-strand wire rope without 

end mass. 

NASTRAN model of seven-strand wire rope with 

end mass. 

Two-strand wire rope. 

Typical mode shape showing end separation 

(no end mass). 

Typical mode 'shape showing end separation 

(with end mass). 

Typical mode shape showing slipping between strands. 

Mass and base accelerations avg. 

Mass and base displacements avg . 

Isolator transmissibility. 

Isolator effectiveness. 

V 
LIST OF TABLES 

TABLE I. ISOLATOR VIBRATION TEST DATA. 

TABLE 11. ONE POSTULATED VARIATION OF FRICTION FORCE WITH 

FREQUENCY 

TABLE 111. NASTRAN SINGLE LOOP MODEL. 

vi 
1. INTRODUCTION 

Space activities have revitalized the importance of damping 

investigations. There has been a significant increase in the number of 

papers on the topic in the literature and a number of research 

investigations have been initiated. Long of interest to many, damping 

is receiving increased attention due to a variety of actual and 

potential applications; among them: large space structures, space 

structures with stringent pointing requirements, computer controlled 

flexible structures, flexible manufacturing systems, composite 

materials, swept-forward wings, SDI structural dynamics, and others. 

The primary interest of this investigation centers on vibration 

isolators constructed of wire rope. Such isolators have been known to 

have good characteristics for some time, but have not been analyzed to a 

thorough enough degree to be yet used in space applications. This lack 

of analysis is at least partly due to the difficulty in modeling the 

dynamics of wire rope isolators. 

Wire rope is, from the basic point of view, simply several stiands 

of wire twisted, or wound, together. 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 (Silverman, 1985; Gilbert, 1976). The 

structure of wire rope provides many interfaces at which a portion of 

the relative motion of strands of wire is converted 6y friction into 
2 

heat, thereby dissipating vibrational energy (Harris and Crede, 1976). 

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 are fixed in clamps (see Fig. 1.1) 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, (Silverman, 1985) 

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 devices (Silverman, 1985; Jewell, 1984). 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 

damping rates and dynamic response must be very accurately known to 

prevent resonance and control interaction problems. 
2 DAMPING CONSIDERATIONS 

Literature Discussion 

Aside from the rather obvious use of isolators as passive damping 

devices for equipment mounting and/or isolation of rotating devices, it 

has been concluded (Rogers, et al., 1986, and others) that the use of 

passive plus active damping devices results in a reduced number of 

active control components and reduced energy and power requirements. A 

combination of the two types of damping can lead to more robust and 

reliable systems, and less expensive systems. 

An examination of the literature concerning damping indicates the 

widely differing approaches to its investigation. The most significant 

paper in this regard which relates to wire rope damping is that by 

Pivovarov and Vinogradov (1985). Their study, however, was limited to 

excitation of straight wire rope which suspended a mass in pendulum 

fashion. Experimental hysteresis loops (shapes only) are given and 

several single nonlinear differential equations are solved in an attempt 

to simulate the phenomena. 

The only other two items of literature found which discuss wire 

rope damping directly are LeKuch and Silverman (1983) in a designer 

notebook setting--a guide to isolator selection, and Kerley (1984) who 

just mentions wire rope "complex cable" arrangements as one type of 

passive damping device. 

Several of the other literature papers, while not addressing wire 

rope damping at all, are very related to some of the tasks undertaken 

3 
4 

during this research. 

separating and determining combined dampings from free vibrations. 

Another is the paper by Vakakis (1985) on unidirectional isolators 

(multiple mass, many degrees of freedom models). 

One is the work of Badrakan (1985) dealing with 

Summary of Techniques 

Perhaps the best source for damping considerations is the new book 

by Nashif, Jones, and Henderson (1985), although again, no mention of 

wire rope damping is made therein. 

of freedom hysteretic, i.e., k(l+iq), damping. A summary of their 

suggested methods of determining damping follows : 

Their emphasis is on single degree 

a. Half-Power Bandwidth: While probably not appropriate for high 

damping devices such as the isolators of interest in this work, the 

technique involves determining Aw and ures where Aw is the bandwidth 

q-wl, between two amplitudes usually taken as 0.707 Amax. 

response peak at wres. For viscous damping, this results in: 

hax is the 

while for hysteretic damping: 

w 

res 

Note that the right-hand side of Eq. (2.2) equals 0 for q<<1. 

b. Resonant Response Amplitude: In this method, an amplitude A is 

defined, Eq. (2.3), and determined experimentally. 
5 

(2 -3) ('p'res - amplitude of response at resonance 

F/k static displacement A= 

For viscous damping, Eq. (2.3) becomes 

- for <<<I -25 2 <m 

1 A= 

while for hysteretic damping: 

(2.4) 

c. Use of a Nyquist Diagram which is equivalent to a, above. 

d. Hysteresis Loops: The area contained within a force versus 

deflection plot over one cycle of motion equals the energy dissipated 

per cycle, Ds. The simplest example of this is that of base excitation 

of a single degree of freedom system which can be shown to yield, for 

viscous damping : 

Ds = IT c w Wr2 

and for hysteretic damping: 

In Eqs. (2.6) and (2.7), Wr is the relative displacement between the 

single mass and the base, o is the forcing frequency, while c, k, and n 

are the properties of the viscous, c, and hysteretic, k( l+in), dampers. 

Expressions which are alternatives to Eqs. (2.6) and (2.7) are 

derived by Nashif, et al., and are 

IT n k C4 Wo2 

'D = 

S (1+)2 + 112 

(2.6a) 
6 

and 

(2.7a) 

In Eqs. (2.6a) and (2.7a), two new terms appear. They are: 

Wo = amplitude of harmonic steady state base motion 

While this technique seems to hold the greatest promise for application 

to the isolators of interest, it should be noted that all equations 

(2.6-7a) are for single degree of freedom models with constant 

properties. 

e. Quadrature Bandwidth: In this method, the imaginary part of 

the response, is plotted versus frequency and two points on 

1 opposite sides of the peak are located by - 

points is then found. 

The Aw between these 

n a~* Q 

For n = fi, the relationship can be shown to be: 

A “Q = y/1 + 0.6436~ - 41 - 0.6436~ 

w 

res 

(2.8) 

If n<<l, then using the binomial theorem, Eq. (2.8) becomes 

(2 -9) -= 

w A‘ (1 + 0.3218~) - (1 - 0.321817) + ... = 0.643617 

res 

Hence, 

AwQ 17 z.i 1.554 - 

w 

res 

(2.10) 

f. Dynamic Stiffness: For a single degree of freedom hysteretic 

model, the complex ratio of displacement excitation gives the form:] 

kraig, R. R., Structural Dynamics, Wiley, 1981, p. 101. 
7 

1 

k - muz + ikq 

w* 

F 

a=L= (2.11) 

which can also be written in the form 

a = [alei$ = a,, + iaQ (2.12) 

If la1 and 

difficulty (though not impossible) of determining k and rl by solving the 

four equations for Ial, $, aD, and aQ.2 

the dynamic stiffness as K = l/a. Therefore 

are measured experimentally, Nashif, et al., note the 

Alternatively, one may define 

where KD is called the direct dynamic stiffness and 

KQ is the quadrature dynamic stiffness. 

Equating real and imaginary parts of Eq. (2.13) yields 

Plotting KD + mu2 versus frequency and KQ/(KD + mu2) versus 

frequency yields 

k = KD + mu2 

(2.13) 

(2 .i4) 

(2.15) 

(2.16) 

(2.17) 

Nashif, et al., note that this method is very helpful when q is high. 

Turning from single degree of freedom considerations, techniques 

for damping analysis vary widely. The authors contend that ultimately 

2Nashif, et al., Vibration Damping, Wiley, 1985, pp 142, 145. 
8 

the use of finite element software probably holds the best promise for 

solving very complicated geometric arrangements of wire rope. 

et al., discuss the merits and pitfalls of such an approach. 

of damping phenomena by this method is far from trivial, however, and 

greater insight into the basic mechanisms has been the intent of this 

Nashif, 

Treatment 

study 

Attempts to date using finite element techniques for this research 

period are discussed in Sections 4 and 5. 
3. STATIC MODELS 

Background 

The uses of stranded cables in conventional engineering 

applications are well recognized. For rope-selection and handling, 

practicing engineers have long depended upon extensive experimental 

results, such as those compiled by Scoble (1920-1928) that began to 

appear in the early 1900s. More recently, the IJire Rope Board and 

federal agencies (1980) have utilized the available empirical data to 

provide general guidelines for rope selection. Attempts to correlate 

the experimental data have met with some success (Druckers and Tachau, 

1945; Huang, 1975), yet these fall short of an in-depth understanding of 

the effects of various wire-rope parameters on their static and dynamic 

behavior. In view of the recently proposed applications of wire ropes 

for augmentation of structural damping in large space structures, the 

importance of such theoretical static and dynamic investigations has 

increased considerably. 

Of the various investigations, special significance is attached to 

the work of Costello and Phillips (1973, 1974) who adopted a more basic 

approach to study the static behavior of the cables. Here the cables 

were treated as groups of separate curved rods (Love, 1944) in the form 

of helices. This static analysis leads to a set of nonlinear algebraic 

equations in several variables. These equations are solved 

simultaneously using the Newton-Raphson algorithm. However, the utility 

of the analysis is severely limited as it depends on computational 

9 
10 

results obtained for particular cases of cable data. For design, it 

would be far more desirable to have the solution in analytical form that 

can provide an insight into the influence of the various wire rope 

parameters on its deformation characteristics. Such analytical results 

may be of even greater significance for the future investigations of 

wire rope dynamics. 

Summary 

To obtain the analytical solutions, we separately consider two 

different models of cables (Fig. 3.1). In the first, the cable is 

assumed to be a single strand constituted by a single layer of an I 

arbitrary number of thin helical wires. The core is assumed to be 

fibrous so that it does not contribute to the strength of the rope 

directly. In the second model, the cable has a metallic wire core that 

is wrapped around by an arbitrary number of layers of helical wires, 

each having its own direction and magnitude of lay. In either of the 

two cases, we assume that an axial force, F, and an axial or torsional 

moment, M, when applied to the cable cause it to undergo elongation as 

well as "twist." This results in elongation, changes in the lay angles 

and helix radii of helical wires in each of the layers. An assumption 

of small changes in the wire helix angle and helix radius is utilized to 

develop simple governing linear relations for the applied forces and 

moments in terms of the linear and rotational cable strains. 

A summary of static analyses follows: 

a. Cables with Fibrous Cores. 

The results characterizing the deformation characteristics can be 

summarized as follows : 
11 

(I) The longitudinal strain, €,, in the helical wire and the overall 

cable strain, E, are related by 

(= E- Aacot a 

where 

a = lay angle of helical wires when cable is not loaded 

Aa = increment in helix angle due to the applied load 

(ii) The expression for rotational strain, B, is given by 

$ = (l+v) E cot a - (Aa/sin2a)[l - (f-v) cos2a] 

where 

u = Poisson's ratio of wire material 

(iii) The effective Modulus of Rigidity has been obtained for two 

different end conditions: 

$ ='O: When the free cable end is not permitted to rotate under 

end moment constraints, i.e., when B=O, the expression for the effective 

modulus of rigidity can be written as 

(E)B=o = g sin a (3 *3) 

F where E = - AE E 

A = metallic area of the cross-section 

E = modulus of rigidity of the wire material 

g = Ii - (l+v)cos2a - (l+v){cos2(m/m) - v) cos4aI 
12 

M = 0: When the axial (torsional) moment M=O, the expression for 

- 

the modulus of rigidity, E, takes the form: 

{ 4h cos2a + sin4a sin2( n/m)( ( l+u)sin2a cos2a 

where 

h = [l+{cos2(n/m)-u)cosa2 + (1-2u)cos4a] 

Figure 3.2 shows a comparison of the values of effective modulus of 

rigidity for cables obtained using the above analytical results with the 

corresponding numerical results of Costello and Phillips (1976) over the 

entire feasible range of a. A generally excellent agreement between the 

two is noted, thus establishing the validity of the approximate 

analytical approach adopted here. 

b. Cables with metallic core 

The above approximate analytical approach is now applied to 

investigate the various deformation characteristics in a more general 

cable model. Here, we consider that the core wire is metallic and large 

enough to prevent helical wires in a layer from touching each other, 

although these wires remain in contact with those in the adjacent 

layers. We assume the cable to be made up of n successive layers, 

counting from the innermost core which is taken as the first layer. The 

results characterizing the deformation properties of such a cable are 

summarized below: 
13 

(i) The longitudinal strain in helical wires in the ith layer 

denoted by si and the overall longitudinal cable strain, E, are related 

by 

where 

ai = original lay angle of the helical wires in the ith layer. 

Aai = increment in the lay angle of the helical wires in the 

ith layer when load is applied. 

(ii) The expression for Aai can be written as 

i- 1 

where 

rj = helix radius of the helical wires in the jth layer; 

j = 1,2,..., i,...,n. 

Rj = radius of the helical wires in the jth layer; 

j = 1,2 ,..., 2,...,n. 

Vj = v(Rj/rj) COS~CI~; j=1,2 ,..., i, ... n. 

R = radius of the cable. 
14 

(iii) The expressions for the applied force, F, and applied moment, 

M, can be stated in the following convenient linear form: 

where 

= F/(AE) 

E = M/(ER3) 

A = metallic area of cross-section; C(miRi2) 

i 

E = modulus of rigidity of the cable material 

R = cable-radius 

mi = number of helical wires in the ith layer 

Ri = radius of the helical wires in the ith layer 

FE = l[miRi2 sin a.(sin ai-v cos2 ai)l/l[miRi21 

i 1 i 

M = 7r1[mi?ii3 cos ai{(ri/Ri)(sin2 ai - u cos2 “i) 

i E: 

M = 7r1[migi4 sin ai{(ri/Ri)2 cos2 a i 

@i 

+ (1/4)(l+vf sin4 ai cos 2ai)}1 
15 

Next, we demonstrate the usefulness of the analytical results by 

applying these to study some important extensional and torsional 

stiffness characteristics of wire ropes. 

Effective Modulus of Rigidity (E) 

(a) No end rotation; i.e., f3=0: Here the expression for the 

effective modulus of rigidity can be written as 

(E)6=o = F = l[miRi2sin ai(sin2 ai - U COS2 ai)l/l[miRi21 (3.9) 

i i E 

(b) Cable ends free to rotate; i.e., M=O: Here, we obtain 

- i l[miEi2(ri/R) sin2 ai cos a,] i l[migi3 cos a i 

1[miEi4 sin ai{(r./RiI2 1 cos2 ai 

i 

+ (1/4)(l+vf sin4 aicos2ai)}]] . (3.10) 
16 

Effective Torsional Modulus of Rigidity (E) 

Effective torsional modulus of Rigidity (E) is given by 

- 

G = ~[l migi4 sin ai{(ri/Ri)2 cos2 “i + (1/4)(l+vf sin4 ai c0s2ai)} 

i 

c[Mili3(ri/Ri) sin2 ai cos ail/l[miEi2 Sin a.(sin2 ai 

i .i 1 

- v cos2 ai)]] (3.11) 

To illustrate the ease of application, we consider a specific cable 

with the following data: 

- First layer (or metallic core): 

ml = 1, a1 = ~/2, R1 = a 

- Second layer: 

m2 = 6, a2 = a, R2 = a 

- Third layer: 

m3 = 12, a3 = r-a, R3 = a 

- - - 

Evidently , the above three characteristics , (E) B=o , (E)M=o , and G, 

would be dependent on the value of a chosen. Figure 3.3 shows the 

effect of varying this lay angle a on the stiffness properties of the 

cable. In all cases, effective extensional modulus of rigidity 

significantly decreases as a decreases. The stiffness remains higher, 

however, for the case of “fixed ends” when the rotational strain is 

zero. 
17 

Figure 3.4 presents the effect of varying a on the torsional 

stiffness. Here the trend is reversed as increasing a leads to rapid 

fall in the torsional rigidity of the cables. Needless to say, such 

plots could be readily generated for any arbitrary situation, thus 

facilitating the choice of geometric parameters of the cable. 

Application to Dynamic Investigations 

To illustrate the usefulness of the analytical results in dynamic 

studies, consider a vertically hanging cable clamped at the upper end 

and carrying a weight at its free end. The differential equations 

governing the cable "extension-twist" oscillations (assuming the weight 

of the cable itself to be negligible) can be written as 

(d2B/dt2) = - [M E + MB B] 111 E 

(3.12) 

(3.13) 

where 

W = weight of the hanging mass 

I = moment of inertia of the mass (related to cable twist) 

R = length of the cable . 

It is now easy to determine the frequency of the coupled oscillations. 

The expression for the natural frequencies (w) of the coupled 

oscillations can be written as 
18 

The dimensionless analytical force and torsional moment relations 

developed here are rather general. The explicit form of results is 

found to be useful in predicting the rope stiffness against elongation 

as well as rotation. The examples considered here demonstrate the 

computational ease and effectiveness with which the closed-form solution 

can be utilized in various studies. The results of the static analyses 

would be quite useful in dealing with any general problem of wire rope 

dynamics. Finally, the valuable insight thus gained into the relative 

deformation behavior of wire strands is likely to facilitate modeling of 

energy dissipation in the rope through internal rubbing action. 
4. SIMPLEST DYNAMIC MODELS 

One-Dimensional Models 

It is possible to postulate several spring-mass-damper onedimensional isolator models. These models include an undamped model, a 

rigidly connected Coulomb damper model, and an elastically connected 

Coulomb damper model. In Fig. 4.1 is shown the wire rope vibrator 

isolator being modeled, and in Figs. 4.2 through 4.4 are shown the 

simplest mathematical models studied. 

For the isolator shown in Fig. 4.1, an experimental study has been 

performed to determine the frequency response characteristics. The 

experimental arrangement consisted of exciting the base vertically. 

Most of the isolator frequency response data is presented in Appendix A. 

In Table I are shown the numerical values of acceleration and 

displacement for the isolator base and center mass shaker test. A plot 

of this experimental data appears with analytical curves discussed and 

ref erred to below. 

For use in an analytical comparison, an experimental stiffness 

curve has been obtained. The arrangement used for the stiffness 

measurements was as follows. The isolator was mounted vertically; 

upside-down from the position shown in Fig. 4.1. A horizontal metal 

strip was attached to the center mass of the isolator with an LVDT 

arranged at its opposite end to measure mass displacement. The 

procedure used in the measurements was to apply known static loads to 

the isolator center mass and monitor the downward displacement by use of 

19 
20 

the LVDT and a voltmeter. The stiffness curve is shown in Fig. 4.5 and 

is seen to be nonlinear. The average stiffness is approximately 100 

lblin. 

The governing equation of motion for the model shown in Fig. 4.2 is 

.. 

Mx + K(x-u) = 0 (4.1) 

where x and u are positive as shown in the figure and M and K are the 

mass and stiffness, respectively. The weight of the center mass for the 

isolator is 0.1359 lb, and the stiffness used (an average value) is 

97.656 lblin. Equation (4.1) was solved using the Advanced Continuous 

Simulation Language (ACSL) in preparation for more involved model 

solutions. A listing of the program is shown in Fig. 4.6. In Fig. 4.7 

is shown a comparison of the frequency response for the experimental 

arrangement, previously described, and the undamped model of Fig. 4.2 as 

solved by ACSL. The mass displacement for the undamped model is higher 

than the mass displacement for the real isolator at all frequencies as 

was expected. However, Fig. 4.7 does show that the undamped onedimensional model displays the same basic behavior as that of the real 

isolator. Of course, the response of the undamped model increases 

without bound at the natural frequency of the model. The natural 

frequency of the undamped model is approximately 84 Hz, while the 

resonant frequency of the real isolator is approximately 75 Hz. Figure 

4.7 is significant because it shows that one-dimensional models hold 

promise for modeling the behavior of, at least, the uni-axial motion of 

the isolator. From these results, it would be expected that by 

incorporating damping into the model, good results would be obtained. 

This was the next step taken in the analyttcal study. 
21 

It has been surmlzed that the damping in wire rope isolator systems 

is dry friction, or Coulomb, damping. By incorporating a friction force 

in the equation of motion, a mathematical model of the system shown in 

Fig. 4.3 is obtained. The equation of motion is 

Mi + K(x-u) + sgn(i) Ff = 0 (4 *2> 

where Ff is the Coulomb friction force which always opposes the motion 

of the mass. As a first approximation, it was assumed that the friction 

force is constant for all frequencies. Using a constant force of 2.25 

lb, the mass displacement response shown in Fig. 4.8 was obtained, again 

using ACSL. It is seen that the model is overdamped for most of the 

frequency range. 

dependence of the friction force Ff. It was found that for low 

frequencies (below 60 Hz) the analytical mass response is not as 

sensitive to changes in the friction force Ff as it is at higher 

frequencies. Using the values of Ff shown in Table 11, the frequency 

response shown in Fig. 4.9 was obtained. It is seen that this model is 

in much better agreement with experiment than the model which assumes no 

frequency dependence of the friction force. It is interesting to note 

that for the Coulomb-damped models, the natural frequency is about 80 

Hz, slightly lower than the resonant frequency of the undamped model. 

'l%ese results led to the investigation of frequency 

The insensitivity of the Coulomb-damped model to changes in Ff at 

low frequencies suggests that the isolator might best be modeled as an 

elastically connected Coulomb damper system as shown in Fig. 4.4. 
22 

To date, good results as described above have been obtained for the 

displacement response of the Eq. (4.2) model. An analytical hysteresis 

loop, generated by the ACSL model with variable friction force, is shown 

in Figure 4.10. The acceleration response of the mathematical model, 

however, has not agreed well with experiment. This needs further 

study. 

NASTRAN Single Loop Model 

A NASTRAN bar element model in the form of an ellipse has been 

generated. Table 111 is a program listing. There are 100 grid points 

symmetrically arranged to match geometrically the single wire rope loop 

test to be described. Those grid points in the region of the clamped 

portion of the ellipse are fixed against any movement; those in the 

region of the sliding block (to be described) are allowed only 

translation in the direction perpendicular to the plane of the loop. 

All other grid points are allowed six degrees of freedom. A restricted 

parameter study has been carried out with the following as parameters: 

E, G, A, I,, Iy, and J. 

several of these. 

Appendix B summarizes the computation of 

A tentative conclusion at this point is that varying E and G can 

result in a stiffness value equal to that obtained experimentally; 

however, questions still remain concerning the validity of the 

parameter values used. In general, E and G values seem to have to be 

lowered an order of magnitude more than those reasonably expected in 

order to agree with experiment. This geometric (looped) case has not 

yet been solved analytically like the other simpler cases in SECTION 3, 

however. 
23 

Single-Loop Wire Rope Tests 

A single wire rope loop has been mounted in a special device which 

has the capability to clamp one "side" of the elliptical loop. The 

opposite "side," though free to move perpendicularly to the plane of the 

wire rope loop, prevents the wire rope from experiencing any degrees of 

freedom other than this one. In this manner, one "loop" of a typical 

wire rope isolator is simulated. The means by which such a restrictive 

motion is accomplished is through the use of two low friction linear 

bearings which fit closely on a close tolerance shaft. The block which 

houses the bearings also has the means to clamp the moving "side" of the 

wire rope loop. Figure 4.11 is a photograph of this particular test 

arrangement. Interestingly, if only one half of the wire rope loop is 

attached, the bearings and their housing "roll" about the shaft as the 

loop is moved perpendicular to its plane. No such "roll" is observed 

with a symmetrical loop in place. 

No attempt was made to match the size of wire rope tested in the 

aforementioned device with the actual size used for the candidate 

isolator. This is primarily due to the impractical small bearinglshaf t 

dimensions required to do so. Instead, it was decided that a larger 

loop would exhibit a phenomenon similar to a smaller loop and our 

research would center on exploration of such at present. 

To date, tests have been performed primarily to ascertain wire rope 

stiffness values for comparison with the NASTRAN model already 

described. The arrangement should be more significant when the static 

models in Section 3 are extended to the combined bendingltwisting case, 
24 

and/or the bending/twisting/extension case which should more 

realistically describe the isolator application. 

Another test being planned is a dynamic test to determine the 

hysteresis loop for a single wire rope loop. It is very likely that 

this will have to involve - two loops (both in the same plane, 

symmetrically arranged to each side of the bearing/shaf t movement) and a 

special two-point shaker excitation scheme with at least two dynamic 

force gages. 

Using traditional terminology of a 2a x 2b elliptical loop of wire 

rope, loops with large values of a/b are less stiff than those with a/b 

values approaching unity. The table below gives specif Ic experimental 

values : 

2a 2b Lgth of Cable StifEness 

a/b (inches) (inches) (inches) (lbsjin.) 

1 .ll 2.5 2.25 7.461 5.138 

1.22 2.75 2.25 7.854 3.292 

1.44 3.25 2.25 8.639 2.144 

Experimental force versus deflection curves, while not shown, are 

reasonably linear. 

Isolator-Supported Momentum Wheel Equations 

Conslder a momentum wheel of mass M supported by three 

symmetrically spaced uniaxial isolators. The momentum wheel, while 

symmetrical, has an unbalanced mass m offset by a distance e from its 

spin axis. Figure 4.12 shows both a side view and a top view of the 

disc, axes, etc. For the spin axis 2, and an imaginary fixed plane 

which contains m and e located by the coordinate 9 from x, the modified 

Euler equations are (Meriam, 1975) : 
25 

(4 -3) 

0 - - 

cMx - IXX nx 

= I i - (Izz-Ixx)Rznx - 

YY Y 

cMz - IZZ AZ+I ZZ i 

Using Fig. 4.12 and assuming that the eccentric mass m is a distance h 

above point 0, the left-hand sides of Eqs. (4.3) can be found to be 

CMx = - mg e sin+ + RIRcos al - Ro R cosa2 + R R sina3 

02 

+ me(nz + $)h cos+ - me(Rz2 + $ )h sin $ 

CM = mge cos$ - RIR sin a1 + R R cosa3 - RoR sin a2 (4.4) Y 2 

.2 

+ me(Qz2 + $ )h cos$ + + $)h sin$ 

.. ZM = - rneZ(;= + 4) + M (t) - ~RR, z z 

IT Rzt - b 3 4- where a = Rt + 1~12 - 120' = Rt + IT - - IT = 1 6 

3 n *2 = 2a - - 'TI - Rt = - - 

9 = Rt + n - 2 (7 lr) = nzt - 7 

2 2 RZt 

2 71 

and Rt = tangential reactions at each isolator pair. 

The simplest form of the support forces would be in the form of k 

times deflection. These are 
26 

R1 = kl(Rexcos al - Re Y 

R2 = k2(R9, sin a3 + R9 

Ro = - 

Rt = kt R$ 

Sin all 

cos 0,) Y 

ko(RBx @os a2 + R8 sin a,) Y 

Actually the k's shown in Eq. (4.5) will be modified to reflect a more 

realistic Coulomb friction support action. 

The mass moment of inertia of the disc referred to point 0, 

ignoring the eccentria mass m (m<<M), is 

0 

- 

0 

Also needed are the relationships between the 8's and R's which are 

9, = &dt 9,(0) = oo 

BZ = %dt eJ0) = 0' 

The set of equations ((4.3) through (4.7)) can be solved for the 

highest ordered derivatives and integrated numerically. A scheme to do 

this with ACSL is shown in Fig. 4.13 with the additional capability of 

incorporating the successful model-withvariable-friction-force into the 
27 

simulation as a multiply-invoked MACRO. Vector integrators can be 

conveniently used to handle the 8 and Q relationships. 

operations are shown in Fig. 4.13 by the { } brackets. 

which would be a measure of successful isolation is postulated as 

Vector 

A parameter 

emax equals zero would be perfect isolation. It should be noted that 

the three variables in Eq. (4.8) vary with time, in general, so that 

peak-to-peak or rms values would be most meaningful for the general 

case. Actual parametric runs on the computer exceed the scope of this 

phase of the research. An anticipated form of the output is shown in 

Fig. 4.14. 
5. ADVANCED DYNAMIC MODELS 

Basic Finite Element Models 

For the purpose of preliminary analysis, three NASTRAN finite 

element models have been developed. Two of these are complete models 

(i.e., including all seven strands--six wound and one maiden) of a line 

segment of the cable. The other is an incomplete model of the same line 

segment, which has been developed as a geometric check for the complete 

models. 

The incomplete model, hereafter referred to as "the two strand 

model," was the first to have been developed. This model consists of a 

maiden strand around which a single strand is wound (see Fig. 5.1). 

Both the maiden and the wound strands are subdivided into 276 elements; 

each strand requiring 1932 grid points for definition. The hexagonal 

cross section of each strand consists of six triangular segments (see 

Fig. 5.2). Each triangular segment represents the upper (or lower) 

surface of solid "pie-shaped'' PENTA elements (solid triangular elements 

used in MSC-NASTRAN). Figure 5.3 shows a typical PENTA element. The 

thickness of each element (0.002 inches) is twice the length of each 

element. This results in the maiden strand being 0.09 inches in length 

and 0.002 inches in diameter. That is, the line segment modeled has a 

L/D ratio of 45. 

The wound strand forms a helix angle of 72 degrees. Both strands 

are connected by scalar elastic elements (springs), which were 

incorporated to model the normal forces acting on each strand. Figure 

5.2 shows the arrangement of the scalar elastic elements. 

28 
29 

The complete models are similar to the two strand model with one 

exception; that is, the complete models consist of a maiden strand and 

six wound strands. Hence, the name, "seven strand model." The wound 

strands are connected to the maiden strand and to adjoining wound 

strands by springs. A typical cross section of a seven strand model is 

shown in Fig. 5.4. Also shown in this figure is the arrangement of the 

spring elements. 

The difference in the two seven-strand models occurs in their 

boundary conditions. Both models are suspended at one end. The first 

model has the other end free, thus forming a fixed-free line segment of 

the wire rope (see Fig. 5.5). The second model has a mass attached to 

the other end of the wire rope (see Fig. 5.6). This model has been 

designated as simply "the pendulum model". The tip mass, M, is one and 

one-half times as heavy as the wire rope. 

Equations of Motion 

The deformation of the cable was modeled using the assumed modes 

method. Using this method, the position vector of a generic mass 

element (see Fig. 5.7), dm, can be expressed as 

m " 

where are the mode shape vectors, and q are the generalized 

coordinates. The mode shape vectors, 4 are functions of the 

undeformed coordinates, xo, yo, and zO. 

NASTRAN finite element models described above. 

j j 

j' 

They were obtained from the MSC 
30 

The equations of motion were obtained via a Lagrangian approach. 

To formulate the Lagrangian, the potential and kinetic energies of the 

wire rope segment are required. The former is the sum of the elastic 

and gravitational potential energies. For the contribution due to the 

elasticity of the system, we can write 

where gs is the system's stiffness matrix and q = (ql q2...q )T. - - n 

The following assumptions were made in order to formulate the 

gravitational potential energy : 

i. length of segment = L 

ii. mass per unit length = ci 

iii. small displacements in the xz-plane 

With these assumptions, the expression for the gravitational potential 

energy becomes 

where K is the gravitation stiffness matrix. The matrix, K is given =g =g 

by 

N-1 L z 

K = -1/2 1 / u. / o[g']T[z']d< dzo 

=g i=o 0 =o 

(5.4) 

and 2 - = (,jl ~~...k). In the case of the d where 4' = (100)- (9) 

pendulum model, the gravitational stiffness matrix also has the 

following term added to it: 

- dz 
31 

L 

0 

1/2 M I [$'lT[p'1 dzo 

With the cable fixed at the upper end, the kinetic energy of the 

pendulum system can be expressed as 

*-' OT -T 

i=O m. 

T = 1/2 1 1 zdm :dm dm + 1/2 M & 

1 

where fdm and 

tip mass, respectively. N represents the number of strands included in 

the model. The strands are numbered such that the maiden strand is i=O, 

and the others are numbered in a counterclockwise manner beginning with 

i-1. Using Eq. (5.1), the expression for the kinetic energy can be 

rewritten as 

are the velocities of a generic mass element, and the 

7Il 

where - is the generalized mass matrix given by 

where the bracketed Q terms are evaluated at the lower end. For the 

seven-strand model without the weight, the second term in Eq. (5.7) is 

neglected (i .e. , M=O) . The nonconservative forces acting on dm are assumed to be due to 

viscosity and Coulomb friction between the strands. 

damping forces are modeled as 

Generalized viscous 
32 

where [ _D ] is a diagonal matrix of structural damping coefficients. 

The generalized Coulomb damping forces are obtained from 

- 

T N- 1 

QC i=O m = 1 1 2 dmidm 

i 

(5.9) 

where fdmi denotes the force on an element dm of the ith strand. If p 

denotes the coefficient of friction between the strands, and fij denotes 

the magnitude of the normal force between the ith and jth strands, then 

for the maiden strand, fdm becomes 

0 

(5.10) 

A A 

t. and b. are the unit tangent and binormal vectors, respectively, of 

dmi. For the other strands, we have 

-1 -1 

with 

(5.11) 

(5.12) 
33 

The unit tangent and binormal vector are given by 

A A A A 

= cos a[- sin ei & + cos e jl + sin a k, si iand A A A A 

bi = sin a[sin Bi 1 - cos Bi j] + cos a k - 

- 

where ei(zo) is the angle of the center line of the ith strand at zoo 

The equations of motion are then obtained from 

& {:I-% ab- + 

where = T - Vs-Vg. The resulting equations of motion are 

(5.13) 

where Hj, dj , and K. are the generalized mass, damping coefficient and 

s tiffaess coefficient, respective:y, for the jth vibrational mode. 

K 

J 

is the jth row of gg, and Q, is the jth element of 9,. 

j -g j 

Discussion 

Complete normal mode analyses have been performed on the finite 

element models previously discussed. The cases of primary interest are 

the two seven-strand models. The information acquired from these 

analyses are twofold. First, the mode shape vectors, 4. are necessary 

to define the deformation of the structure in the analytical equations. 

Figures 5.8-5.9 are typical mode shape plots. Second, the contact 

forces normal to the strands, fij , have also been obtained from these 

analyses. 

-J 
34 

It has been observed that for several of the vibrational modes, the 

lower end of the cable, whether free or weighted, has a tendency to 

separate. When this occurs, the contact forces normal to the strands 

become negative (i.e., the forces in the springs (scalar elastic 

elements) become tensile instead of compressive). For these cases, the 

negative forces are neglected since they would not contribute to Coulomb 

damping. Another tendency displayed in these analyses is the slipping 

between strands. Figure 5.10 shows a typical case. 

A simulation code has been adopted to perform the "analytical" 

calculations. The inputs for this code are the mode shape vectors and 

the contact forces. Results from this code are expected in the next 

phase of this research. 
6. CONCLUSIONS 

The phenomena associated with wire rope damping is quite 

complicated. A macro approximation of it using a uniaxial Coulomb 

damper with a variable friction force has yielded very good results of 

isolator response. Various NASTRAN models have been developed which are 

beginning to provide the kind of analytical capability and insight 

required for modeling of arbitrary complicated wire rope arrangements. 

Several other dynamic models have been explored. Beginning experimental 

studies have shed some light on appropriate wire rope characteristics 

related to static properties, damping, and dynamic response. 

Significant progress in deriving closed form solutions for wire rope 

static properties has been accomplished and their extension to dynamic 

appiications has begun. 

35 
7- REFERENCES 

Badrakan, "Separation and Determination of Combined Dampings 

from Free Vibrations," Journal of Sound and Vibration, 

Vol. 100, No. 2, 1985, pps. 243-255. 

Costello, G. A,, and Phillips, J. W., "Contact Stresses in Thin 

Twisted Rods," Journal of Applied Mechanics, Transactions ASME, 

Series E, Vol. 40, 1973, pps. 629-630. 

Costello, G. A., and Phillips, J. W., "Contact Stresses in Thin 

Twisted Rods," Journal of Applied Mechanics Division, ASCE, Vol. 

100, 1974, pps. 1096-1099. 

* 

Costello, G. A., and Phillips, J. W., "Effective Modulus of 

Twisted Wire Cables," Journal of Engineering Mechanics 

Division, ASCE, Vol. 102, 1976, pps. 171-181. 

Craig, R. R., Structural Dynamics, John Wiley and Sons, New 

York, 1981. 

Druckers, D. C., and H. Tachau, "A New Design Criteria for 

Wire Rope," Journal of Applied Mechanics, Transactions ASME, 

Vol. 67, 1945, pps. A33-A38. 

G i 1 be r t /C ommo nw e a 1 t h , T ran sm i s s-.pn Lln e Rgfzr 

Wind-In'duced Conductor Motion, Electric Power Research 

Institute, Palo Alto, CA, 1976. 

Book , 

Huang, N. C., "Finite Extension of an Elastic Strand with 

a Central Core," Journal of Applied Mechanics, Transactions 

- ASME, Vol. 45, 1978, pps. 852-857. 

Jewell, R., NASA/MSFC, Private Communication, December 19, 

1984. 

Kerley, J., "Concepts and Effects of Damping in Isolators," 

paper in Rogers (1984), pps. L-1 through L-15. 

LeKuch, H. and Silverman, G., "Selecting Shock and Vibration 

Isolators," Astronautics & Aeronautics, pps. 30-32, 

September 1983. 

Love, A. E. H., A Treatise on the Theory of Elasticity, Dover 

Publications, Inc., New York, 1944, Chapters 18 and 19. 

Meriam, J. L., Dynamics (2nd Ed.), John Wiley, New York, 1975. 

Nashif, A. D., Jones, D. I. G., and Henderson, J. P., 

Vibration Damping, John Wiley & Sons, 1985. 

36 

I 
37 

Phillips, J. W., and G. A. Costello, "Contact Stresses in 

Twisted Wire Cables," Journal of the Engineering Mechanics 

Division, ASCE, Vol. 99, 1973, pps. 331-341. 

Pivovarov, I. and 0. G. Vinogradov, "The Phenomenon of 

Damping in Stranded. Cables, " Proceedings of the AIAA 26th 

Structures, Structural Dynamics, and Materials Conference - 

Part 11, April 15-17, 1985, pps. 232-237. 

Rogers, L. (Ed.), Vibration Damping 1984 Workshop Proceedings, 

AFWAL-TR-84-3064, November 1984 (Held at Long Beach, CA, 

February 1984). 

Rogers, L. (Ed.), Damping 1986 Proceedings, AFWAL-TR-86-3059, 

Vols. 1 and 2, May 1986 (Held at Las Vegas, Nevada, March 

1986 1. 

Silverman, J., Aeroflex Lab, Plainview, N. Y., Private Communication, March 22, 1985. 

Scoble, W. A., "First Report of the Wire Rope Research Committee," Rochester Institute of Mechanical Engineers, Vol. 

115, 1920, pps 835-868; Second Report, Vol. 119, 1924, pps. 

1193-1290; Third Report, Vol. 123, 1928, pps. 353-404; 

Fourth Report, Vol. 130, 1935, pps. 373-478'. 

Vakakis, A. F. ar?d S. A. Paipetis, "Transient Xespmse cf 

Unidirectional Vibration Isolators with Many Degrees of 

Freedom," Journal of Sound and Vibration, Vol. 99, No. 4, 

1985, pps. 557-562. 
a. - BIBLIOGRAPHY 

Bert, C. W., and R. A. Stein, "Stress Analysis of Wire Rope in Tension 

and Torsion," Wire and Wire Products, Vol. 37, 1962, pp. 769-770, 

7 72-81 6. 

Chi, M., "Analysis of Multi-Wire Strands in Tension and Combined Tension 

and Torsion," Developments in Theoretical and Applied Mechanics, 

Proceedings of the Seventh Southeastern Conference on Theoretical 

and Applied Mechanics, Vol. 7, 1974, pp. 599-639. 

Cochran, J. E., Jr., N. G. Fitz-Coy, and M. A. Cuthins, "Finite Element 

Xodels of Wire Rope for Vibration Analysis," presented at the 

Workshop on Structural Dynamics and Control Interaction of 

Flexible Structures, Marshall Space Flight Center, Huntsvil le, 

Alabama, April 22-24, 1986. 

Costello, G. A., "Analytical Investigation of Wire Rope, " Applied 

Mechanics Reviews, pp. 897-900, July 1978. (This was the feature 

survey article which included 30 references; none address wire 

rope damping, however. ) 

Cutchins, M. A., J. E. Cochran, Jr., S. Guest, N. G. Fitz-Coy, and 31. L. 

Tinker, "An Investigation of the Damping Phenomena in Wire Rope 

Isolators, accepted for ASME Vibrations Conference to be held -in 

Boston, Massachusetts, September 27-30, 1987. 

Fabunmi, J. A., "Extended Damping Models for Vibration Data Analysis," 

Journal of Sound and Vibratlon, Vol. 101, No. 2, 1985, pp. 

181-192. . 

Ferri, A. A., and E. H. Dowell, "The Behavior of a Linear, Damped Modal 

System with a Non-Linear Spring-Mass-Dry Friction Damper System 

Attached, Part I," Journal of Sound and Vibration, 1984. 

Ferri, A. A., and E. H. Dowell, The Behavior of a Linear, Damped Model 

System with a Non-Linear Spring-Mass-Dry Friction Damper System 

Attached, Part 11," Journal of Sound and Vibration, Vol. 101, No. 

1, 1985, pp. 55-74. 

Hall, H. M., "Stresses in Small Wire Ropes," Wire and Wire Products, 

Vol. 26, 1951, pp. 228, 257-259. 

Harris, C. M., and C. E. Crede, Eds., Shock and Vibration Handbook, 2nd 

Ed., McGraw-Hill Book Company, Inc., New York, 1976, pp. - 32-23 

through - 32-44 and pp. 36-36 through - 36-46. 

3a 
39 

Hruska, F. H., "Calculations of Stresses in Wire Rope," Wire and Wire 

Products, Vol. 26, 1951, pp. 766-767, 799-801. 

Hruska, F. H., "Radial Forces in Wire Ropes," Wire and Wire Products, 

Vole 27, 1952, ppo 459-463. 

Hruska, F. H., "Tangential Forces in Wire Ropes," Wire and Wire 

Products, Vol. 28, 1953, pp. 455-460. 

Johnson, C. D., and D. A. Kienholz, "Finite Element Prediction of 

Damping in Structures with Constrained Viscoelastic Layers," AIM 

Journal, Vol. 20, No. 9, September 1982, pp. 1284-1290. 

Kumar, K., and J. E. Cochran, Jr., "Analytical Solutions for Static 

Elastic Deformations of Wire Ropes," accepted for the April 1987 

28th AIAA SDM Conference in Monterey, California (AIAA Paper 

No. 87-0720-CP). 

Lazan, Benjamin J., Damping of Materials and Members in Structural 

Mechanics, Pergamon Press, New York, 1968, pp. 22-23. 

Leissa, A. W., "Contact Stresses in Wire Ropes," Wire and Wire Products, 

Vole. 34, 1959, pp. 307-316, 372-373. 

Machida, S., and A. J. Durelli, "Response of a Strand to Axial and 

Torsional Displacements ,- Journal of Mechanical Engineering 

Science, Vol. 15, 1973, pp. 241-251. 

Phillips, J. W., and G. A. Costello, "Analysis of Wire Ropes with 

Internal-Wire-Rope Cores," Journal of Applied Mechanics, Vol. 52, 

No. 3, 1985, pp. 510-516. 

Rogers, L. C., and K. E. Richards, Jr., "PACOSS Program Overview and 

Status," pp. 85-109 of Wright (1986). 

Ruzicka, J. E., and T. F. Derby, Influence of Damping in Vibration 

Isolation, the Shock and Vibration Information Center, Naval 

Research Laboratory, Washington, D. C., 1971. 

Santini, P., A. Casteilani and A. Nappi, "An Introduction to the Problem 

of Dynamic Structural Damping," AGARD Report No. 663, January 

1978. 

Sayers, D. D., and M. C. Potter, "The Effects of Curvature and Torsion 

on the Temperature Distribution in a Helix," Journal of Applied 

Mechanics, Vol. 52, No. 3, 1985, pp. 529-532. 

Starkey, W. L., and H. A. Cress, "An Analysis of Critical Stresses and 

Mode of Failure of a Wire Rope," Journal of Engineering for 

Industry, Transactions, ASME, Vol. 81, 1959, pp. 307-316. 
40 

Vinogradov, 0. G., and I. S. Atatekin, "Internal Friction Due to Wire 

Twist in Bent Cable," Journal of Engineering Mechanics, Vol. 112, 

NO. 9, pp. 859-873, 1986. 

Vinoeradov. " 0. G., and I. S. Atatekin, "Structural Damping of Stranded 

Cable," Mathematical Modeling in Service and Technology, Pergamon 

Press, Avula, X., et al. (Eds.), pp. 537-541, 1984. 

Wright, R. L. (Ed.), NASA/DOD Control/Structures Interaction Technology, 

NASA CP 2447, Part I, Proceedings of a conference held in 

Norfolk, VA, November 15-21, 1986. 

, American Wire Rope, American Steel and Wire Company, New York, 

1913. 

, Vibration Damping Short Course Notes, University of Dayton 

Research Institute, June 1983. 

I - 

b W 

tF 

iM 

t 

SECTION X-X 

NOTES: SECTION X-X 

FOR ROPE WITH FIBROUS CORE 

HELICAL WIRES CONTACT 

a=r b=R 

FOR ROPE WITH METALLIC CORE 

HELICAL WIRES DO - NOT CONTACT (SHOWN) 

a = ri b = Ri' 

Fig. 3.1 Rope geometry with m2 = 6 and mi = 1. 
I .o 

0.8 

0.6 

0.4 

0.2 

0 

60 

Numerical (Costello and Phillips, 1976) 

70 80 

Original Helix Angle, a' 

90 

Fig. 3.2 Plots comparing the analytical and numerical values of 

effective modulus of rigidity of wire rope for the practical 

range of a. 
I .o 

0.5 

C 

Fig. 3.3 

0 

/ 

p=0 

------ M=O 

I 1 I 

3 60 70 80 

a' 

0 

Modulus of Rigidity as affected by the helix angle. 
0.4 

0.2 

0 - 

50 60 70 80 90 

Fig. 3.4 Torsional Stiffness as affected by the helix angle. 
7 

Attached to isolator housingJ 

(a) Partial isometric 

v 

i' 

7 

1 

Figure 4.1 Wire 

Wire Rope Loops I ir Wire (All four 

Rope 

sides) 

LOOP a 

777 

(b) Side view 

rope vibration isolator. 

t +" 

Figure 4.2 Undamped single-degree-of-freedom model. 

//I// 

-Y 

777 

0 

0 
F' I 

/////////////////////' 

Figure 4.3 Rigidly connected Coulomb damper model. 

Figure 4.4 Elastically connected Coulomb damper model. 
CI 4 Y"4 11.0 

/ U 

Q) 

P 

0 

.- 

Z. 4.0-- 

Q) 3.0-- 

0 

I .o I I I I I I I I I I I I I I I I 1 I I I + 

0.0 .01 .02 .03 .04 .05 .06 .07 .08 .09 .I 0 .I I 

0 

isolator displacement (in.) 

Fig. 4.5 Isolator force versus displacement curve. 
PROGRAM ISOL 

INITIAL 

CONSTANT MASS=.1359 , G=386.4 , SLOPE=97.656 

CONSTANT N=O .O , OMEGA=40.0 , UZERO=.0840 

CONSTANT TDEL=O .O , PI-3.1416 , PHASE=O.O 

CONSTANT TSTOP- 6.0 , XO=O.O , XDO=O.O 

TABLE FF, 1, 14 

/ 10.0, 20.0, 40.0, 60.0, 70.0 ... , 75.0, 80.0, 100.0, 120.0, 140.0 ..e 

, 160.0, 180.0, 200.0, 250.0 ... 

, 0.007, 0.007, 0.004, 0.004 / 

, 0.75, 0.85, 1.25, 0.15, 0.10 ... , 0.10, 2.25, 1.45, 0.50, 0.005 ... CINTERVAL CINT=O .002 

END $ 'OF INITIAL' 

DYNAMIC 

iJ = 2.*PIkeiviEGA 

DERIVATIVE 

F1 = UZERO*HARM(TDEL,W,PHSE) 

XDD = -G/MASS*(SLOPE*(X-F1) + FF(OMEGA)*SIGN(l.O,XD)) 

XD = INTEG(XDD,XDO) 

X = INTEG(XD,XO) 

END $ 'OF DERIVATIVE' 

TERMT(T.GT.TSTOP) 

END $ 'OF DYNAMIC' 

END $ 'OF PROGRAM' 

Fig. 4.6 Listing of ACSL program for Coulomb friction model 

with variable friction force. 
TABLE I. ISOLATOR VIBRATION TEST DATA 

Frequency 

(H,) 

10 

20 

40 

60 

70 

75 

80 

84 

100 

120 

140 

160 

180 

2 00 

250 

300 

Base Peak 

Accel. 

(g's) 

1.1 

2.9 

6.1 

8.3 

7.6 

7.6 

10.2 

12.3 

12.45 

12.6 

12.7 

12.6 

12.2 

11.8 

11.1 

Run 2 Vertical 

Base Disp. Mass Peak 

(in.) 

0.302 

0.170 

0.084 

0.051 

0.033 

0.028 

0.034 

= 0.032 

0.024 

0.016 

0.012 

0.008 

0.006 

0.004 

0.0025 

- 

Accel. 

(g's) 

1.1 

3 .O 

7.7 

18.5 

40.1 

50.6 

31.3 

15.5 

11.45 

9.6 

8.2 

7.1 

6.3 

5 -6 

4.3 

Mass Disp. Relative Disp. 

(in.) 

0.309 

0.180 

0.103 

0.113 

0.181 

0.191 

0.105 

0.030 

0.014 

0.012 

0.004 

0.002 

0.003 

0.0003 

- 

(in.) 

.0070 

.0100 

.0190 

.0620 

.1480 

.1630 

.07 10 

.0060 

-.002 

-.004 

-.004 

-.004 

-.001 

.0005 

- 
3. 

+-+;. 

I I I I I c 00 59.00 108 @@ 157 00 206 00 2555 00 3 

FREQUENCY OF BASE MGTION 

Fig. 4.7 Theory (undamped model) vs. experiment. 
- 

0) I I 1 I I I 

I EXPER 

e ACSL MDEL 

r 

I 

i - 

I 

iI) 

.= 

3 

* 

3 

0 

0 

0 

\ 

\ 

I 

I 

t 

5b 90 0' 10 00 

ACSL MODEL WITH 

CONSTANT FKICTLON i 

I 

FORCE I 

I 

! 

L 

I 

I 

h Y _I- -, .a 

90.00 ijo 90 170 00 zio 50 ~CCI 

FREQUENCY OF BASE MOTION 

Fig. 4.8 Theory (Coulomb model with constant friction force) vs. 

experiment. 
TABLE 11. ONE POSTULATED VARIATION OF FRICTION FORCE WITH FREQUENCY. 

Frequency (Hz) Ff (lb) 

10 0.75 

20 0.85 

40 1.25 

60 0.15 

70 

75 

80 

100 

120 

140 

i 60 

180 

200 

250 

0.10 

0.10 

2.25 

1.45 

0 .so 

0.005 

0.007 

0 -007 

0.004 

0.004 
I 

i- 

! I I I I + 

50 00 90 CO 130 OC! 170 00 ?io 53 153 

FREQUENCY OF BASE MOTION 

Fig. 4.9 Theory (Coulomb model with variable friction force) vs. 

experiment . 
m 

I I I I I I m 

wl+ t 

In 

I t 

0+ I 

- 

L u T 

m 

rt 

'J 

0 

+ A 

+ iiYST1 

+ HYSTZ 

I 

1 

t r I 

m I 

CT, *c 

I 

I 

(0 A' 

dl 

A c 

'9 m 

, 1 I , , v 

'-0 10 -0 07 -0 33 0 00 0 93 0 07 0 10 

MASS DI SPL ACE ME NT 

Fig. 4.10 ACSL-generated 60 Hz hysteresis loop. 
t 

i 

I 

j 

i 

j 

!- 

I I - I I 

I i1 -0 07 -0 04 0 OC 0 04 0 08 0 11 

UTSPL ACEMENFig. 4.10 (Cont.) ACSL-generated 40 Hz hysteresis loop. 
- HYSTlO 

-,+ HYSTll 

+ HYSTlP 

Fig. 4.10 (Cont.) ACSL-generated 100 Hz hysteresis loop. 
TABLE 111. NASTRAN SINGLE LOOP MODEL 

ID WIRE ROPE ISOLATOR SINGLE LOOP LINE ELEMENT HODEL 

TIME 100 

SOL 24 

CEND 

TITLE=WIRE ROPE ISOLATOR SINGLE LOOP LINE ELEMENT MODEL 

SUBTITLE= 1.25” MAJOR AXIS AND 1.125” MINOR AXIS 

LABEL= STATIC LOADING MODEL 

ECHO=BOTH 

SET 10 =1,76,51 

DISPLACEMENT=lO 

SUBCASE 1 

LABEL= ,302 LB STATIC LOADING 

LOAD= 101 

SUBCASE 2 

LABEL= ,652 LE STATIC LOADING 

LOAD= 102 

SUBCASE 3 

LABEL= ,852 LB STATIC LOADING 

LOAD= 103 

SUBCASE 4 

LABEL= 1,052 LB STATIC LOADING 

LOAD= 104 

!§OUTPUT (PLOT) 

$PLOTTER NAST 

$SET 1 =ALL 

$VIEW O., O., 0. 

$AXES Z,X,Y 

$FIND SCALE ORIGIN 0, SET 1 

$PTITLE=WIRE ROPE ISOLATOR SINGLE LOOP LINE ELEMENT MODEL 

$PLOT SET 1, ORIGIN 0, SET 1 

BEGIN BULK 

GRID, 1, ,-1.25,O. 0,O. 0, , GRID,2, ,-1.20, .315,0.0, , GRID,3,,-1.15,.440908,0.0,, 

GRID, 4, ,-1.10, .5343454,0.0, , GRID,5,, -1.05,. 6104097,O. 0, , GRID,6,,-1.00, .675,0.0,, 

GRID,7,,-.95, .7311635,0.0,, 

GRID,8, ,-.9,. 7807208,O. 0, , GRID, 9,, -. 85, .8248636,0,0, , GRID, lo,, -. 80,. 8644218,O. 0, , GRID, 11, , -. 75,. 9,O. 0, , GRID, 12,, -. 70,. 9320542,O. 0, , GRID, 13, ,-.65,. 960937,O. 0, , GRID, 14,, -. 60, -986827,O. 0, , GRID, 15, ,-.55,1. 010248,O. 0,, 

GRID, 16, ,-.50,1.031080,0,,, 

GRID, 17, , -. 45,l. 049571,O. 0, , GRID, 18, , -, 40,l. 065845,O. 0, , 123456 

GRID, 19, , -. 35,l. 08,O. 0, , 123456 
GRID, 20, , -. 30,l. 092120,O. 0, , 123456 

GRID,21, ,-.25,1.102270,0.0, ,123456 

GRID, 22,, -. 20,1.110507,0.0, ,123456 

GRID,23, ,-. 15,l. 116871,0.0, , 123456 

GRID,24, ,-. 10, 1.12139,O. 0, ,123456 

GRID, 25,, -. 05,l. 124100,O. 0, , 123456 

GRID, 26, , 0.0,l. 125,O. 0, ,123456 

GRID,51, ,1.25,0.0,0.0,, 

GRID,50, , 1.20, ,315,O. 0, , GRID, 49,, 1.15, ,440908, 0.0, , GRID, 48,, 1.10, .5343454,0.0, , GRID, 47,, 1.05,. 6104097,O. 0, , GRID,46,, 1. 00, ,575,O. 0, , GRID, 45,, .95, .7311635,0.0, , GRID,44,, .9, .7807208,0. 0,, 

GRID, 43, , .85, .8248636,0.0, , GRID, 42,, .80, ,8644218,O. 0, , GRID,41,, .75,. 9,O. 0, , GRID, 40,, .70, .9320542,0.0, , GRID,39,, .65,. 960937,O. 0,, 

GRID, 38,, ,60,. 986827,O. 0,, 

GRID,37,, .55,1.010248,0.0, , GRID, 36,, .50,1.031080,0.0, , GRID,35,, .45,1.049571,0.0, , GRID,34, , . 40,l. 065845,O. 0, ,123456 

GRID, 33,, .35,1.08,0., , 123456 

GRID, 32, , .30,1.092120,0,0, , 123456 

GRID,31, , .25,1.102270,0.0, , 123356 

GRID, 30, , .20,1.110507,O.O, , 123456 

GRID,29,, . 15,l. 116971,O. 0, , 123456 

GRID,28,, . 10,l. 12139, 0. 0,, 123456 

GRID,27,, .05,1.124100,0.0, , 123456 

GRID, 52,, 1.20, -. 315,O. 0, , GRID,53,, 1.15,-.440903, 0.0, , GRID,54,, 1.10,-.534345,0.0,, 

GRID, 55, , 1.05, -, 610410, 0.0, , GRID,56,, 1.00, -. 675,O. 0, , GRID,57,, .95, -. 731164,O. 0, , GRID,58,, .9,-. 780721,O. 0,, 

GRID, 59, , .85, -. 824864,O. 0, , GRID, 60,, -80, -. 864422,O. 0, , GRID,61, , .75, -. 9,O. 0, , GRID, 62,, .70, -.932054,0.0, , GRID, 63, , .65, -. 960937,O. 0, , GRID,64,, .60,-.986827,0.0,, 

GRID,65, , .55, -1.01025,O. 0, , GRID, 66, , $50, -1.03108,O. 0, , GRID, 67, , -45, -1.04957,O. 0, , GRID, 68,, .40, -1.06585,O. 0, ,12456 

GRID,69,, .35,-1.08,O. 0,, 12456 

GRID, 70,, .30,-1.09212,O. 0, ,12456 
GRID, 71,, .25, -1.10227,O. 0, , 12456 

GRID,72,, .20, -1. 11051, 0.0, ,12456 

GRID,73,, . 15,-1.11687,O. 0, ,12456 

GRID,74,, . 10,-1.12139,0.0, ,12456 

GRID, 75,, .05, -1.12410,O. 0, , 12456 

GRID,76,, 0.0,-1.125,O. 0, , 12456 

GRID, 100, ,-1.20,-.315,0. O,, 

GRID, 99, ,-1. 15, -. 440908, 0. 0, , GRID,98, ,-1.10,-.534345,0.0, , GRID,97, ,-la 05,-.61041,0.0, , GRID,96, (-1. 00,-, 675,O. 0, , GRID,95, ,-.95,-.731164,0.0, , GRID, 94,, -. 9, -. 780721,O. 0, , GRID,93,,-.85,-,824864,0.0,, 

GRID, 92,, -. 80, -. 864422,O. 0, 

GRID,91,, -.75,-.9,0.0, , GRID,90, ,-.70,-.932054, 0. 0, , GRID,89, ,-.65,-.960937,0.,, 

GRID,88,,-.60,-.986827,0.0,, 

GRID,87, ,-.55, -1.01025,O. 0, , GRID, 86,, -.50, -1.03108,O. 0, , GRID, 85, , -. 45, -1. 04957, 0. 0, , GRID, 84,, -. 40, -1.06585,O. 0, , 12456 

GRID,83, ,-.35,-1. 08,O. 0, , 1245G 

GRID,82, ,-.30,-1. 09212,O. 0, , 12456 

GRID,81, ,-.25,-1,10227,0.0, , 12456 

GRID,80,, -.20,-1.11051,O. 0, , 12456 

GRID,79, ,-. 15,-1.11687,O. 0, , 12456 

GRID,78,, -. 10, -1. 12139, 0. 0,, 12456 

GRID, 77, ,-* 05, -1.12410,O. 0, , 12456 

FORCE, 101,76,, .302, 0. 0, 0. 0, 1. 0 

FORCE, 102,76, , .652,0. 0,O. 0,l. 0 

FORCE, 103,76,, .852, 0.0,O. 0,l. 0 

FORCE, 104,76,, 1.052,O. 0,O. 0,l. 0 

CBAR, 1,35,1,2,0.0,1.0,0.0 

=,*(1>,=,*<1>,*(1>,== 

=97 

CBAR, 100,35,100,1, 0. 0,l. 0,O. 0 

PBAR, 35,75,0.013829,1.95312-5,l. 95312-5,3.90648-5 

MATl, 75,30. +5,11.54+5, ,7.324-4 

PARAM, GRDPXT, 1 

PARAM, WTMASS, 0,0025907 

ENDDATA 

1 
i 

A 

i 

BH = Bearing Housing SS = Shaft Support 

DOCM = Direction of Constrained UC = Upper Clamp 

LC = Lower Clamp 

S = Shaft (fixed in both SS's) 

Motion (of BH & UC) WRL = Wire Rope Loop 

Pig. 4.11 Single loop wire rope experimental arrangement. 
I 

to x 

Ref. 0' 

axis 

Z 

I % 

Figure 4.12 Momentum wheel with three isolator support pairs. 
PROGRAM 

MACRO ISOPAIR( N) 

"ONE-DIMENSIONAL COULOMB FRICTION MODEL W/VARIABLE FRICTION" 

"e.g., MOST OF FIG. 4.6 CODING SLIGHTLY MODIFIED" 

LMACRO END 

ARRAY I( 3) 

CONSTANT "INPUT OF M,e,R,K's,h,m,INITIAL CONDS." 

7 

INITIAL 

"COMPUTATION OF IXX,IYY,IZZ, AND m x e" 

"CONVERSION OF IC's TO RADIANS AND RAD./SEC." 

END 

DYNAMIC 

DERIVATIVE r "ROTATIONAL EQS . FOR MOMENTUM WHEEL W/3 SYMM .-SPACED, PAIRED 

ISOLATORS" 

"FUNCTION OF MASS, e, I~S, a's, R'S, its, i, $, R AND 3 ISOLATORS" 

MZ = f(T) 

{Q} = /{i?}dt 

{e) = jjQ)dt 

& = /$ dt 

I$= / idt 

ISOPAIR(1) 

ISOPAIR(2) INVOCATION OF MACRO'S 

ISOPAIR(3) 

THEW = SQRT(BX**~ + ey**2 + 1$**2) $ **A MEASURE OF ISOLATION" 

END $ "OF DYNAM" 

END $ "OF PROGRAM" 

Figure 4.13. ACSL model definition arrangement - 

wire rope isolated momentum wheel 

([t S +] denotes sorted sections). 
0 

me 0.5 

RPM MR= MR 

3000 0 

Figure 4.14 Planned output arrangement for isolator-supported 

investigation. 
Fig. 5.1. NASTRAN model of two-strand wire rope. 
SPRING 

MAIDEN STRAND WOUND STRAND 

SECTION B-B 

Fig. 5.2 Cross section of two-strand wire rope showing finite elements. 
r( 

0 

m 

Fig. 5.5. NASTRAN model of seven-strand wire rope without 

end mass. 
Fig. 5.6. NASTRAN model of seven-strand wire rope 

with end mass. 

0 

0 
0 X 

L I 

Fig, 5.7, Wo-strand wtre wpe. 
Fig. 5.8. Typical mode shape showing end separation 

(no end mass). 
Fig. 5.9. Typical mode shape showing end separation 

(with end mass). 
a 

0 

Fig. 5.10. Typical mode shape showing slipping between 

strands. 

0 
APPENDIX A. ISOLATOR SHAKER TESTS 

A comparison of the accelerations of the isolator base and the 

center mass is shown in Fig. A-1 and the displacements of the mass and 

the base are shown in Fig. A-2. Isolator transmissibility and 

effectiveness as functions of frequency are shown in Figs. A-3 and A-4 

respectively. Transmissibility is defined as the ratio of mass 

displacement to base displacement, and isolator effectiveness is defined 

as one minus transmissibility. While the behavior shown in these 

figures may not seem atypical to that of other vibration isolation 

systems, Fig. A-2 shows a displacement much like that of materials with 

high loss factors (see pp. 133-134 of Nashif, -- et al. (1985)) or 

beneficial damping (see pp. DDD-3 and DDD-16 of Rogers (1984). 

A- 1 
Figure A-l 

MASS AND BASE ACCELERATIONS A'VG 

I 1 I I I 

F + iSd88..\A 

& lS'dBV.4.4 

1.30 56.33 102.00 i48.30 194.30 24c, ss 

FREQUENCY (HZ) 

0 

AUSilRN ilN 1 'JERS i T v 
Figure A-2 

MASS AND BASE DISPLACEMENTS AVG 

I I I I I 

+ iS'JBSD.4 

.+ 15VBY34 

'h 

I 

u C 

1.00 412.90 74.03 158.30 i7C.33 

FREQUENCY (HZ) 

AUBURN i)NiVERSiTY 
Figure A-3 

ISOLATOR TRANSMISSIBILITY 

0 

.- 

co 

h 

01 

,.p 

00 42 - 90 74.00 106.30 i3F3.50 i73.33 

FREQUENCY (HZ) 

I 

00 42 - 90 

\ 
Figure A-4 

ISOLATOR EFFECTIVENESS 

I I I 

1.00 42 00 74.30 106.30 138.90 i70.30 c 

FREQUENCY (HZ) 

- 

Q 

L 

e 

z 

LL 

AUBURY UNI'lEwSiTY 
APPENDIX B. WIRE ROPE PROPERTIES 

The area moments of inertia used in this NASTRAN model are found by 

using I for a circle of radius r for each wire making up the wire rope 

in combination with the parallel axis theorem. For six strands plus a 

core strand, 

(B -1 1 I = 7 I + [nr2(2r)2]2 + [lrr2(r2)]4 = 4 55lT r2 

X C 

It can be shown that any orientation of the axes yields the same result. 

For the cable used in the experimental arrangement (one with a 

non-fibrous core), Eq. (B.1) yields 

= I = 1.04293 x in4 (B.2) IX Y 

where r = D/6 and D = 0.133 in. The polar moment of inertia is given 

by 

J = Ix + I = 2.085 x in4 (B.3) Y 

Various areas were calculated, based on different assumptions, but were 

found not to affect the stiffness calculated from the NASTRAN output. 

Increased confidence in the static results from the model is 

- 

anticipated prior to extensive dynamic runs of the model; however, 

various schemes for including damping in the NASTRAN model and for 

producing the resulting hysteresis loops have been initiated. 

B-1