Analytical and experimental study of the dynamic response of segmented cable systems

Journal ofSound and Vibration (1971) 18 (3), 311-324

ANALYTICAL DYNAMIC

AND EXPERIMENTAL

RESPONSE

OF SEGMENTED

STUDY

OF THE

CABLE SYSTEMS-f

J. E. G~ELLER Applied Mechanics Group, Hydvoballistics Division, Naval Ordnance Laboratory, White Oak, Maryland, U.S.A. AND

P. A. LAURA$ Department of Engineering, Universidad National de1 Sur, Bahia Blanca, Argentina (Received 15 March 1971)

The present paper is a review of experimental and analytical investigations dealing with the dynamic response of vertically hanging segmented cable systems where the upper portion is a stranded steel cable and the lower segment is a nylon rope. Such systems are frequently used in oceanographic applications. A generalized distributed mass model has previously been developed by the authors for a segmented cable made up of two viscoelastic materials including internal damping and linear external damping of the payload and cable. In that model the viscoelastic behavior is simulated by a two-parameter Voigt model and external linear viscous damping is used for the payload and the cable, but an approximate method takes into account non-linear damping elfects due to the solid-fluid interaction. In general, good agreement is obtained between that mathematical model and the experimental results. It is also shown that the agreement is better if one approximates the viscoelastic behavior of the nylon rope by a lumped three-parameter system. The behavior of single nylon cables is also studied analytically and experimentally under dynamic conditions in air and in water media. The problem of “snap loading” is also studied and several conclusions of practical interest from a designer’s point of view are drawn.

1. INTRODUCTION

Nylon rope and stranded steel cable are frequently used in oceanographic cable systems. Nylon is highly desirable for buoy systems, mooring systems, and other oceanographic applications because of its light weight, relatively good strength, and ease of handling. However, in cases where fishbite is a problem [I], steel stranded cable is used in the upper portion of the cable while nylon is used in the lower portion. Many applications involve the suspension of a payload by a segmented cable of steel and nylon where the upper end is excited by longitudinal oscillation due to ocean wave motion. This problem was studied analytically by the authors [2] by approximating the wave motion as a sinusoidal displacement function. The cable segments were considered as distributed masses with viscoelastic behavior including internal damping and also external damping due to fluid viscosity. The purpose of this paper is to compare experimental data with theoretical prediction using this model and a t Paper sponsored by the Society of Naval Architects and Marine Engineers for presentation at the Second Offshore Technology Conference, Houston, Texas, U.S.A., 22 to 24 April 1970. $ Also Research Scientist, Naval Ship Research and Development Center, Washington, D.C. 20034, U.S.A. 311

312

3.E.

GOELLERANDP.A.LAUR.4

mathematical lumped parameter model which approximates viscoelastic behavior of nylon rope by a three-parameter model frequency referred to as the standard linear solid. Eumped parameter models were also developed to predict the cable force during a snap condition caused by slack in the cable. second

2. REVIEW OF THE MATHEMATICXL 2.1. MODELNO.

MODELS

I-DI.C~XIBUT~DMASSV~IGTMODEL

The following is a summary of the pertinent characteristics of the tkst model. For a detailed discussion and derivation of equations, the reader is referred to [2]. The model is shown schematically in Figure 1. The model considers the cable as being made up of two segments of different viscoelastic materials. A two-parameter Voigt model is used for each segment. A continuum approach was used as opposed to the frequently used discrete parameter model.

dx3

Figure 1. Two material cable system.

The viscous damping of the water was considered for both the cable and payload by assuming the damping to be linearly dependent on the local velocity. Damping forces are, in general, proportional to velocity squared, but an approximate linear damping coefficient was obtained by equating the frictional energy dissipated by linear damping to that dissipated by damping which varies with the square of the velocity. In order to have a generalized theoretical model, the payload was considered to be attached to a rigid foundation by an elastic spring and dashpot. This model is an approximation to the salvage recovery problem where the mass is originally settled in the ocean sediments. If the spring constant is set equal to zero, the model corresponds to a floating buoy or ship with a payload suspended from the bottom of the cable.

313

RESPONSE OF SEGMENTED CABLE SYSTEMS TABLE 1

Summary description of mathematical models Model schematic

Applicable equations

Description Distributed mass Voigt model (no. 1)

Equations

2 viscoelastic cable

2

segments with internal damping Cable external linear damping Payload linear damping Elastic base support Virtual mass of water included

3 Parameter model (720.2)

of motion

_ p. $7

I

a3uj = l a2uj

,

ax=at Co&titutive

a:

at2

relation

ax. au+

a au. 2 at ax3

u~j=Ej--i+pj-

Boundary conditions u3(L3,t) = x0 sin wt u&, t) = UJ(O,t)

( *1

viscoelastic

Freely suspended payload Internal damping dependent on frequency Stiffness dependent on frequency Payload damping proportional to V Virtual mass of water included

Analog simulation for snap condition (no. 3)

Payload damping proportional to V2 Determines snap load after slack condition Virtual mass of water included Freely suspended payload

Equation of motion M, z& + C, & = G(w) (x1 - x2)

Complex stiffness G(w) = K(w) [l + iS(w)] Constitutive eauation

Real part of complex stiffness K(w) =

K1 K; + uz $(K,

+ Kz)

Kz + UJ’pL2

Equation of motion M.~,+~~C~AI~*~R~+K~(X~-X~)=O if F, = T,,,, + K,(x, - XI) < 0

then M,R,

+ -$pC,, Al&l Rz - Z-stat= 0

where T,,,, = W, - W, Boundary conditions x2(0) = 0 n,(o) = 0

Slit model (no. 4) 2 Parameter

lumped model 2 Viscoelastic materials Lumped cable external damping proportional to vz Payload damping proportional to Vz Determines snap load after slack condition Virtual mass of water included

21

- K3(x3 - xl) - C’&

Cable forces Pz = Kz(x2 - xs) + C&h P3 = K&Q - .Y,) + C&p

Slack conditions ifPz
- Rs) - 2,)

P,=Pa=O

- R,) + W, - WCS

314

J. E. GOELLER AND P. A.

For each segment (j = 2, 3), the material constitutive relationship of the form

LAURA

is considered

to behave

in accordance

with a

oxj=E,dtr,+p .a 55 i3Xj

1

UJat i axj 3

where Ej is the elastic modulus and poj is the internal viscosity coefficient. Considering external damping on the wetted cable, the equation of motion for each segment is

where Tj = in the cable to equation part of the

linear

pnj/‘Ej, time constant of the viscoelastic model?, oj = (Ejlpj)i12, sound velocity material, fij = fjpfj/Aj Ej, viscous damping coefficient of the fluid. The solution (2) for a sinusoidal displacement function x = x0 sin wt is given by the imaginary following equation : uj(xj, t) = Xj(xj) eimf. (3)

The solution involves evaluation of four complex constants by using the following boundary conditions which are illustrated in Table 1. The first boundary condition is the prescribed displacement at the upper end. The second and third require compatibility of displacement and force at the joint between the cable segments. The last boundary condition is obtained from a force balance at the payload. The results obtained can be expressed in terms of the “natural frequency w,,” of a massless cable system using the equivalent spring constant K =

K2 &W2

(4)

+ 41,

%e = (K/~-Y*,

(5)

where K2 and K3 are the spring constants corresponding to each segment. The “‘effective mass M,” includes the payload mass and the virtual mass of water. The details of the solution are straightforward [2] but quite lengthy and will not be repeated here. Equations are developed in [2] for force in each segment of the cable which are valid provided the cable does become slack during the oscillatory motion. If slack initiates, an impact force is developed when the cable becomes taut. This phenomenon is usually called “snap”. The details of a simple but quite accurate analysis of the problem are given in reference [3]. A summary of the characteristics of the models used to analyze this condition are given in Table 1. The payload damping coefficient C, and the cable external damping coefficient /I were computed by equating the energy dissipated for linear damping to that dissipated for velocity squared damping. The details of the method are given in [4]. This method allows use of wellknown drag coefficients for cables [5] based on damping coefficients which are functions of the square of the velocity, i.e.

A similar drag coefficient

can be obtained

2.2. MODEL NO. 2-THREE-PARAMETER

for the payload

depending

on its shape and size.

DISCRETE MODEL

A three-parameter model shown as model no. 2 in Table 1 will also be examined. The characteristic behavior of this model is useful in explaining the experimental behavior observed in nylon ropes. For example, the constitutive relation in terms of force-strain is

t These parameters are evaluated in reference [4] from free oscillation tests.

315

RESPONSE OF SEGMENTED CABLE SYSTEMS

From the theory of linear viscoelasticity, the substitution of (iw)” for P/W is permissible This yields 0 L K,K:+p * u2(KI + K2) + ioK: E A [ --=

= G(w).

K$+/_L*CJ

(8)

I

If a harmonic force of the form P = P,,,, + APcos wt

(9)

is applied, then a plot of P vs. E can be made [6] as shown in Figure 2 by using equation (7). The slope of the diagonal A-B can be interpreted as an apparent dynamic spring constant. Thus, the apparent spring constant is frequency-dependent. Moreover, for low frequency it approaches the static spring constant, and a high frequency it approaches (K, + K2). The “apparent spring constant K,” at resonance can be estimated from natural frequency measurements by K, = Me ute. This result is higher than the static spring constant. This was found to be true in the experiments where the static spring constant of 73 ft (22 m) of 0.25 in (6.35 mm) diameter nylon rope was found to be about 2.9 lb/in (4.32 kg/m) compared to the estimated apparent spring constant (K,) of 3.4 lb/in (5.05 kg/m). Thus in dynamic computations on nylon rope, the use of a static spring constant can lead to misleading results.

Figure 2. Hysteresis loop for harmonic cycling of the three-parameter

model.

The details of the derivation for the dynamic response of this model to a sinusoidal displacement are given in [7]. The amplitude magnification ration is give by the expression X

-=

(LP

+

my

+

(2Qm8,,,)*

fI*+z*

x0

(10)

I ’

where H=Q2+m2_w2+m*4u

+m*)

l+ccm* z

=

25,Q(Q*+ m24 (1 + m*)+ l+ctm*

2mQ8

Illax*

The corresponding maximum cable force is P max = Ke Xo[(~/~ne)~ + (25,

~GJ~I~'*4x0.

(11)

Several parameters in these equations require discussion. The “natural frequency wne” is measured experimentally. The spring constant K, is computed from K, = M,wz,. The damping term a,,, is approximately l/n times the logarithmic decrement from a free oscillation test [4]. This is also equivalent to TW,, which is used in model no. 1. Free oscillation tests on 0*25-in. (6.35 mm) nylon rope indicated 6,,, to be O-158.The term m is the ratio of transition frequency to the natural frequency. The “transition frequency We”is defined as that frequency

316

J. E. GOELLER AMD P.

A. LAURA

at which the damping 6(w) is a maximum. In the calculations performed, it was assumed that the maximum damping occurred at resonance [4,7]. The term a is the ratio of the high to the low spring constant; namely (Ki + &)/Ki. Knowing a,,,, u can be computed from the equation [4, 71 6 max= (a - 1)/G. Thus all the parameters

required

to compute

(12)

the response

can be determined.

3. EXPERIMENTAL SET-UP The tests were performed at the Naval Ordnance Laboratory using the hydroballistics tank shown in Plate 1. The tank is approximately 35 ft (10.7 m) wide by 100 ft (30.5 m) long with a water level ofup to 45 ft (19.8 m). The driving mechanis,m for oscillating the cable was located cm the top deck with the cable inserted through a porthole in the deck. A portion of the cable

0

/

/

!O

20

30

/

/

40

50

SO

Extension (in = 25.4 mm)

A

Figure 3. Force-extension data for 0.25~in (6.35 mm) nylon rope [initial length 73 ft (22.25 m)]. (first test); A-A, B (bt teSt),

o __

3,

length was in air while the major portion was immersed in the water. The payload could be photographed through ports in the sides of the tank. The driving mechanism for oscillating the cable is shown in Plate 2. It is essentially a cranktype device with a drive rod attached to the flywheel. The prime mover consisted of a d.c. shunt electric motor with a reduction gear for reducing the speed and increasing the torque capability. The motor had adjustable speed, but at a given setting the speed was insensitive to changes in the applied torque.

‘late 1. Interior of hydroballistics facility. Water tank: 100 ft (30.48 m) long; 75 ft (22.86 m) deep; 3: (lor .7 m) wide. Water depth: 65 ft (19-8 m) maximum. Construction: stainless steel lined tank, supported reirIforced concrete.

(facingp.

316)

RESPONSEOFSEGMENTED

CABLESYSTEMS

317

The force in the cable specimens was measured by load cells located at top and bottom of the cable. The load cell used was model A 8293 as manufactured by Schaevity with a maximum load capability of 1000 lb (454 kg). The voltage output was fed to a Beckman type R oscillograph where a trace of force versus time was obtained. Prior to each test, the instrument was calibrated by applying known weights to the load cells. The payload was a sphere of 8 in (20.32 cm) diameter which was made of solid aluminum with a weight of 26.9 lb (12.65 kg) including attachment fittings. The weight of the sphere in water was 17.8 lb (8.53 kg) excluding the weight of the attached cable. The virtual mass or added mass of the displaced water was 0.15 slug (4.85 lb). Tests were performed on segmented cables of steel and nylon, and cables of nylon alone. Experiments on wire rope have been reported in [3]. The steel can best be described as aircrafttype cables constructed of carbon steel according to MIL-W-1511. The diameter was O-1875-in (4.83 mm) with seven wires per strand and 19 strands per cable. The nylon rope was 0*25-in (6.35 mm) diameter and can best be described as a braided rope with a breaking strength of 1150 lb (521.6 kg). Experimental static force-elongation data for a 73-ft (22.25 m) length of nylon rope is shown in Figure 3. The data shown are for the loading phase. The nylon rope exhibited appreciable hysteresis during unloading. The force elongation data also depends on previous history. For example, curve A is for the first test. Curve B is for the last of the three succeeding tests. The rope exhibited an increase in stiffness for the latter test. Dynamic force-elongation data on 1*2-in (30.5 mm) nylon rope from reference [8] also indicated a hysteresis loop. The data also show that the apparent spring constant (slope of the principal axis of the hysteresis loop) increases with increasing mean load. It was also observed that the apparent spring constant under dynamic cyclic load is higher than the static spring constant. It was also shown that K, is significantly lower in water compared to air. 4. EXI’ERIMENTAL RESULTS 4.1. NYLONR~PBTESTS INAIR(SYSTEMNO. 1) Forced oscillation tests were performed on O-25-in (6.35 mm) nylon rope 73 ft (22.25 m) long with the S-in (20.32 cm) diameter spherical payload. The tests were performed in both air and water. The tests were performed in air for several reasons. The first was to estimate the internal damping of the cable free of any external damping. Second, it was desired to observe the pulse shapes in air so that non-linear effects of water damping could be observed. Third, it is a known fact that the elastic properties of nylon change when exposed to water. This then would have an effect on the natural frequencies and the cable force. The system was oscillated at the top at displacement amplitudes of l-3 in (2.54-7.62 cm). The range of forcing frequency was O-3 Hz. The force response was found to be very nearly sinusoidal and was reproducible. In order to effectively get a comparison between experiment and the theory formulated by mathematical models (no. 1) and (no. 2), the experimental data were non-dimensionalized by plotting P/Kc X,, versus w/w,, where w,, is the damped natural frequency which is measured experimentally. The dynamic component P was obtained by subtracting the static force LGpstat”from the maximum cable force. Verification was obtained from the free oscillation tests. For the nylon rope system no. 1, it was found to range from 1.06 to 1.18 Hz. The internal damping factor of rw,, = 0.158 was obtained from the force-time response of a free oscillation test. With these values, the computed dimensionless cable force was compared to experimental results as shown in Figure 4. The comparison is reasonably good for frequency ratios up to resonance. The experimental results are slightly lower at the low frequency range, but this is probably due to the non-linear characteristics of the cable. The steeper rise near

J.

318

E.

GOELLER AND P. A. LA’,‘RA

resonance is very close to a jump condition. The good agreement was proba because of the relatively small elongation of the cable resulting in near linear behavior.

At

73

50

50 e $ k?

4.0

p e $ B ii

30

2.0

!O

_-I 0

05

I.3

i.5

Frequency

ratio,

20

2,5

3.0

w/w,,

Figure 4. Dimensionless force at top of the cable vs. frequency ratio showing comparison of experimental air tests with analytical model (no. 1). -, Theory-distributed mass Voigt model (no. 1); ----, experiment x0 = I; -----, experiment x0 = 2; -----, experiment x0 = 3.

I

05

I

,o

Frequency

/

I

2.0

25

1

I.5 ratio,

u/wne

Figure 5. Dimensionless force at top of the cable vs. frequency ratio showing comparison of experimental air tests with analytical models. ----, Theory-distributed mass with Voigt model (no. I); --, Tbeoryaverage experimental . three-parameter viscoelastic model (no. 2); o ----0,

higher frequencies, the experimental results are higher than those obtained using the distributed mass model. Also, the experimental results show a dependency QM excitation amplitude, whereas the theoretical force ratio P/K,X0 plots as a line independent of excitation amplitudes. The dependency on amplitude was also observed in the water tests. It was hoped that better agreement between theory and experiment could be obtained by a

319

RESPONSEOFSEGMENTEDCABLESYSTEMS

mathematical model which accounts for increase in stiffness with frequency. The threeparameter model (no, 2) has this characteristic plus the characteristics of damping dependency on frequency. The internal damping reaches a maximum at the “transition frequency Ok”, then continues to decrease. Figure 5 shows a plot of equation (11) compared to the distributed mass-Voigt model (no. 1) and the average of the experimental data. The three-parameter model (no. 2) shows slight improvement at the higher frequency. It is believed that better agreement could be obtained if the material properties are obtained in a dynamic test as a function of frequency and these properties are used in combination with the distributed mass model. 4.2. NYLONROPETESTS

INWATER(SYSTEMNO.

1)

Forced oscillation tests were performed in water on O-25-in (6.35 mm) rope 73 ft (22.25 m) long with the 8 in (20.32 cm) spherical payload. The system was oscillated at the top at displacement amplitudes of l-4 in (2.54-10.16 cm). Typical force-time response plots are shown in Figure 6. At low frequency, a non-linear response was observed which was not observed in the air tests. The non-linearity is probably caused by the damping of the water. At higher frequency the effect of non-linearity disappears and the response is very nearly sinusoidal.

(bj /&= 4ot 20 o~~k,

24 lbCO.896 kg)

I lb 03.07kg)

w (cl

Figure 6. Experimental force response at top of 025-in (6.35 mm) nylon rope in water (system no. 1) at various forcing frequencies [x0 = 3 in (76.2 mm)].

Free oscillation tests were also performed in water to estimate the damped natural frequency. It was found that the damped natural frequency in water was about 0.72 Hz compared to an average 1.10 Hz in air. This is due to a significant lower spring constant in water [4] and the differences in values of external damping. The experimental data were non-dimensionalized by estimating the spring constant from measurements of the natural frequency. Figure 7 shows the comparison between experiment and theory using the distributed mass model (no. 1). The tangential drag coefficient per unit length of cable, CDc, was estimated to be 0.010 from Figure 5-15 of [5]. The plot of CD, is relatively constant beyond a Reynolds’ number of 500. The Reynolds’ number for 0.25 in (6.35 mm) nylon rope at a velocity of 1 ftjs (0.31 m/s) is about 2000 and is therefore greater than 500 over the major portion of a cycle of oscillation. The assumption of constant drag coefficient then appears reasonable. In general, the comparison between theory and experiment is reasonably good for frequencies up through the resonant frequency. The results tend to diverge at the higher frequencies in the same way as the air tests.

320

J. E. GOELLERAND P. A. LAURA

0

I .o

0.5

Frequency

I.5 ratio,

20

25

3.0

u/w,

Figure 7. Dimensionless force at the top of the cable vs. frequencyratio showing comparison water tests with analytical model (no. 1). (a) x0 = 1 in (25.4 mm); (b) x0 = 2 in (50~8mm). --, model no. 1; -----, experimental results.

ofexperimental Theory from

Comparing Figure 4 with Figure 7, it will be seen that water has a significant effect in lowering the cable force [4]. For absolute values of force, the difference in spring constant between air and water must be considered.

4.3. SEGMENTEDCABLESYSTEMSBNWATER Forced oscillation tests were performed on segmented cables consisting of stranded steel in the upper portion and nylon in the lower portion. Two basic configurations were studied; namely, system no. 2 consisting of 34.5 ft (10.5 m) of 0.25-in (6.35 mm) nylon joined to 36.5 ft

(11-l m) of 0.1875~in (4.83 mm) steel cable, and system no. 3 consisting of 6 ft (1.83 m) of O-25-in (6.35 mm) nylon joined to the bottom of 62 ft (18.9 m) of 0*0933-in (2.39 mm) steel cable. This latter configuration was tested to illustrate the dynamic response during a snap condition viz condition following initiation of slack in a cable. Figure 8 shows the comparison between experimental and the theoretical force at the top and bottom of the cable in system no. 3 using the distributed mass model (no, 1). The results were non-dimensionalized based on a measured natural frequency of 1.05 Hz. As in the case of pure nylon cables, the comparison is good up through resonance. At higher frequencies, there is a difference which is believed to be caused primari!y by the nylon characteristics. As mentioned previously, the nylon stiffness and damping appears to be sensitive to frequency in the high-frequency range. Actually, if the frequency had been increased to cover the second resonant frequency, the cable force at the top is predicted to be somewhat higher than at the first resonant frequency, This is illustrated in reference 121.At resonance, the cable force is a maximum at the top of the cable, but in the anti-resonance range, the force is higher at the bottom of the cable than at the top. This can be seen both by theory and experiment from Figure 8. Therefore at the higher “modes”, the location of maximum cable force can occur anywhere along the cable length depending on the forcing frequency.

RESPONSE OF SEGMENTED CABLE SYSTEMS

321

During the oscillation tests of stranded steel cable systems, it was observed that there is a combination of wave amplitude and forcing frequency that causes slack in the cable.? The cable subsequently becomes taut and experiences a severe impact load. The mathematical models (no. 1) and (no. 2) are not applicable to this condition. In reference [3] the snap phenomenon has been modeled on the analog computer assuming elastic cables and by a digital computer program assuming cables of two viscoelastic materials. Pertinent characteristics are summarized in Table 1 as models (no. 3) and (no. 4). The results of the tests on system no. 3 are presented to show typical behavior. Figure 9 shows the maximum cable force as a function of forcing frequency for a system no. 3 oscillated at 2 in (5.08 cm) amplitude. 5.0 40 3.0 2.0 I-O *(" e O 2 5G 4-o 3.0 20 IO

.?

I.0 Frequency ratio,

2.0

3.0

w/wne

Figure 8. Dimensionlessforce at top (a) and bottom (b) of systemno. 2 vs. frequencyratio showingcomparison of experimentwith analyticalmodel (no. 1). --, Theory from model no. 1; o -0, experimental results. 7~~~= 0.152,une= 1.05Mz,X0= 1 in (25.4mm). The experimental results compare well with those predicted by the mathematical model. Note that the cable force increases gradually with forcing frequency until the snap condition is experienced at about 1.3 Hz. The force then increases drastically from about 35-87 lb (15.88-39.46 kg) for a very slight increase in frequency. It should be noted that the snap condition initiated well below the natural frequency (or resonant frequency) of the system which was estimated to be about 2.28 Hz. It is interesting to note that the cable force reaches a maximum during the first snap condition (1.3 Hz), followed by a minimum and then a rapid increase as a second snap condition was experienced. The second snap condition occurred very near the natural frequency of the cable system. Tests were performed on the same cable system except the 6 ft (1.83 m) piece of nylon rope was removed leaving 62 ft (1X.9 m) of stranded steel cable. The snap load reached 170 lb (77 kg) at 1.6 Hz and still had not yet reached the maximum. Theoretically the force should reach a peak of 300 lb (136 kg) at a frequency of 1.65 Hz, then drop off slightly. This force level was not achieved in the experiment because of fear of damaging the apparatus. Compared to the previous test, the results showed that a small length of nylon is extremely effective in mitigating the snap load. Without the nylon, the maximum cable force was estimated to t This phenomenon is basicallya case of dynamic stability [9],

322

J. E. GOELLER AND P. A. LAURA

Forcing frequency, Hz

Figure 9. Maximnm cable force vs frequency for specimen no. 3 in water showing comparison of experiment with mathematical model (no. 4). CD, = 0.5, CD, = 0.01, w,, = 2.28 Hz, X0 = 2 in (50.8 mm). --, Theory from model no. 4 ; o -o , experiment.

be 300 lb (136 kg) whereas with the nylon the maximum force was 87 lb (39.46 kg). Furthermore, in designing cable systems, a factor of safety of six to eight is frequently applied to the static wet weight of the payload to compensate for dynamic effects and fatigue. In tests performed on stranded steel cables, cable forces greater than nine times the static wet weight of the payload were encountered due to snap loads. Hence it is concluded that the safety factor can be purely fictitious if adequate provisions are not taken for the snap condition.

5. CONCLUSIONS

Based on the experimental

and theoretical

results the following

conclusions

are made.

(i) The force response of nylon rope in air for small elongation is fairly linear. The distributed mass Voigt model, which is based on linear theory, yields good predictions up through resonance. At high frequencies, the dimensionless force P/K,X,, is dependent on excitation amplitude, whereas the theoretical dimensionless force is not. Better agreement in the high frequency range can be obtained by a three-parameter viscoelastic model no. 2 with stiffness and internal damping dependent on forcing frequency. The apparent spring constant computed from the natural frequency must be used in preference to the static spring constant. (ii) The internal damping measured by a logarithmic decrement approach can be applied to forced oscillation problems with good accuracy. The damping in @25-in (6.35 mm) nylon rope tested is significant (TW,, = 0.158). (iii) Water damping has a significant effect on reducing the cable force in the vicinity of resonance. At low frequency, damping appears to actually increase the force slightly. The approximate linear damping coefficient for the payload obtained from velocity squared damping based on equivalence of frictional energy yields results from the distributed mass Voigt model no. 1, which compare well with experiment. The drag coefficients were based on steady flow conditions in a range practically insensitive to Reynolds’ number. (iv) The apparent spring constant of nylon braided rope is significantly lower in water than in air. A factor as high as two was computed based on the change in natural frequency between air and water.

RESPBNSEOF SEGMENTEDCABLESYSTEMS

323

(v) The distributed mass Voigt model no. 1 for segmented cables yields good comparisons with experiments in water over the fundamental mode. The experiments verified the theoretical predictions that the force is a maximum at the top of the cable up through the first resonance frequency. The force is a maximum at the bottom of the cable in the vicinity of antiresonance. This indicates that the location of maximum stress in the cable must be examined closely since it can occur anywhere along the length of the cable, depending on the forcing frequency. (vi) A snap condition usually develops in steel cables before resonance is achieved. This snap condition results in an impact load in the cable following initiation of slack in the cable. This impact load, which is maximum at the top of the cable, can be of sufficient magnitude to cause premature failure even in cable systems with a high safety factor based on payload static weight “TStat”. Maximum cable forces nine times the static payload weight in water were observed in the tests. It was also found that significant mitigation of the impact load can be achieved by a shock absorber such as a highly deformable element. It was then possible to get through the snap condition by increasing the frequency in much the same way that it is possible to get through

a resonant

condition. ACKNOWLEDGMENT

The authors wish to express their gratitude to Drs A. Seigel and V. C. D. Dawson of the Naval Ordnance Laboratory for their cooperation in making various test facilities available. The manuscript was typed by Miss Dierdre Aukward. REFERENCES 1. W. S. RICHARDSON1967 Transactions 2nd Infernational Buoy Technology Symposium, Washington, D.C. pp. 15-18. Buoy mooring cables, past, present and future. 2. J. E. GOELLERand P. A. LAURA 1969 Journal of the Acoustical Society of America 46, 284-292. Dynamic stress and displacements in a two-material cable system subjected to longitudinal excitation. 3. J. E. GOELLERand P. A. LAURA1970 Proceedings of the 5th Southeastern Conference on Theoretical and Applied Mechanics, North Carolina University and Duke University. A theoretical and experimental investigation of impact loads in stranded steel cables during longitudinal excitation. 4. J. E. GOELLER1969 Ph.D. Dissertation, Department of Mechanical Engineering, The Catholic University of America, Washington, D.C. Theoretical and experimental investigation of a flexible cable system subjected to longitudinal excitation. 5. B. W. WILSON1960 Texas A &M. AMProject 204. Technical Report No. 204-l. Characteristics of anchor cables in uniform ocean currents. 6. R. 0. REID 1968 Texas A & M. AMProject 204. Technical Report No. 68-l 1F. Dynamics of deep sea mooring lines. 7. J. C. SNOWDON1968 Vibration and Shock in Damped Mechanical Systems. New York: Wiley 8~ Sons. 8. R. G. PAQUETTEand B. HENDERSON1965 Genera.? Motors Corporation. The dynamics of simple deep sea buoy mooring. 9. P. A. LALIRAand J. E. GOELLER 1971 RILEM International Symposium, Buenos Aires. On some considerations on the behavior of mechanical cable systems from a dynamic stability viewpoint. APPENDIX NOMENCLATURE A, A a C c CR

projected area of spherical payload A, = rr/4D2 effective cross-section area of homogeneous cable A = 0.404d2 acoustic sound speed in elastic cable damping coefficient critical damping ratio Cc, = 2 k, M,

3. E. WELLER

AND P. A. LAURa4

drag coefficient of payload drag coefficient of cable diameter of payload mass diameter of cable modulus of elasticity drag force on cable damping constant of fluid complex modulus of three-parameter viscoelastic model gravitational constant spring constant length of cable frequency ratio m = wJu,, effective mass = MD + M,, lumped mass of cable mass of payload virtual mass of displaced water axial force in cable static force in cable due to payload W, - W, time axial displacement tangential velocity of cable maximum displacement of payload buoyancy force due to displaced payload weight of payload displacement function spatial variable (see Figure 1) maximum amplitude of displacement function ratio of spring constants Y = kz/ks ratio of high to low spring constant viscous damping coefficient p = fpjAE damping factor depending on frequency viscosity strain au/ax density axial stress of cable time constant of viscoelastic model natural frequency of simple spring mass system; w,, = K,/h& circular frequency of forcing function damping ratio = C/Cc, frequency ratio w/w,,

Subscripts

p c e v j x stat

payload cable effective viscoelastic model identifies segment of cable j = 2, 3 longitudinal or axial direction static