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36.3 Examples

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Four examples are given in this section. The first example illustrates the roles of the normal and the

tangential components of the drag force in the static configuration of a towing cable. The second example

shows how the equation of motion of an articulated tower can be formulated in the presence of

surrounding fluid. The third example shows how to choose a single significant wave height to represent a

certain condition from significant wave height data over a long period of time. The final example shows

how to reconstruct time series data from a given spectrum.

36.3.1 Static Configuration of a Towing Cable

For the purpose of ocean surveillance, oceanographic or geographic measurements, or ocean exploration,

marine cables with instrument packages or Remotely Operated Vehicles are often towed behind ships or

submarines. For example, the goal of the VENTS program by the National Oceanic and Atmospheric

Administration (NOAA) is to conduct research on the impacts and consequences of submarine volcanoes

and hydrothermal venting on the global ocean. In attempts to locate and map the distributions of

hydrothermal plumes in the Mid-Ocean Ridge system, an instrument package called a CTD (Conductivity,

Temperature and Depth Sensors) is towed behind a ship.

Fluid-Induced Vibration 36-23

Let us consider a cable and a body towed behind

a ship at a constant velocity with no current as

shown in Figure 36.17. What kind of shape will the

cable take? What will be the distance between the

ship and the towed body?

We immediately recognize that this is equivalent

to having a stationary ship with a steady current in

the opposite direction. The equation of motion is

given by

F~ ¼ ma~ðs; tÞ ¼ 0~ ¼

›

›s ðT~t Þ þ f nn~ þf t~t þ mg~k

where m is the mass of the cable per unit length,

a~ðs; tÞ is the acceleration of the cable, s is the

coordinate along the cable, T is the tension which

is a function of s, ð~t; n~; b~Þ is the set of unit vectors of

the curvilinear coordinate system, ~k is the unit

vector downward in the direction of gravity, g is the gravitational acceleration, f n is the normal drag

force, and f t is the tangential drag force. The added mass and the inertial terms are zero because the fluid

acceleration and the cable acceleration are zero. The normal and tangential drag forces are given in

Equation 36.27 and Equation 36.29. In our case, they are given by

f n ¼ CDr

U 2 cos2u; f n ¼ 2CTr

U2 sin2u

The corresponding scalar equations are given by

2 CTr

U 2 sin2u 2 mg cos u ¼ 0

2 T

ds þ CDr

U 2 cos2u 2 mg sin u ¼ 0

ð36:33Þ

where u is the angle that the tangential vector makes with the vertical and measured positive clockwise.

Note that we have used ›~t=›s ¼ ð2›u=›sÞ~n and ~k ¼ 2cos u~t2sin un~: Equation 36.33 shows that the

tangential components of the external forces act to increase the tension, while the normal components

cause the towline to bend. Because the normal component of the drag force is much larger than the

tangential component, most of the fluid force is used to turn the cable.

From the force diagram (in Figure 36.17), the angle that the cable makes with the vertical where it is

connected to the towed body is given by

Tð0Þcos uð0Þ ¼ W ; Tð0Þsin uð0Þ ¼ Drag

Once we know the weight and the drag force on the towed body, the tension and the angle at s ¼ 0 can be

found. If the drag is negligible compared with the weight, then the cable must be near vertical and the

tension must be equal to the weight of the towed body at s ¼ 0 :

Tð0Þ < W and uð0Þ < 0

For now, let us assume that this is the case. Then, with these initial conditions, the system of ordinary

differential equations (Equation 36.33) can be solved numerically for TðsÞ and uðsÞ: For example,

Cable

Ship

Tension

Drag

Weight

Instrument package

q n

FIGURE 36.17 Towed system in equilibrium and the

forces acting on the towed body.

36-24 Vibration and Shock Handbook

even very simple finite difference equations will work. A set of equations

Tiþ1 ¼ Ti þ mg cos ui 2 CTr

U 2 sin2ui

􀀏 􀀐

Ds ð36:34Þ

uiþ1 ¼ ui 2 mg sin ui þ CDr

U 2 cos2ui

􀀏 􀀐

Ds=Ti ð36:35Þ

where Ti ¼ TðiDsÞ; are used here, and it works very well for Ds ¼ 0:05:

The Cartesian coordinates, x and y, are related to u by

ds ¼ sin u and

ds ¼ cos u

and can also be obtained by integrating them numerically.

Figure 36.18 shows the results when mg ¼ 1:5 N/m, CDrDU2=2 ¼ 10 N/m, CTrDU2=2 ¼ 0:1 N/m,

W ¼ 100 N, and the cable is 100 m long. Care is taken so that the ship is located at x ¼ 0 and y ¼ 0:

It is interesting to note that u approaches a critical value, and the shape gradually becomes linear

toward the ship. Mathematically, du=ds becomes zero. This is when the drag force is completely balanced

by the normal component of the cable weight. The angle at which this occurs, ucr; can be obtained from

the second governing equation and

mg sin ucr ¼ 2f n;

sin ucr

cos ucr ¼ CDr

U 2 1

In our case, ucr ¼ 1:184 rad, and this value agrees with Figure 36.18.

−90 −80

−50

−40

−30

−20

−10

−70 −60 −50 −40

Horizontal coordinate, x (m)

Water depth, y (m)

−30 −20 −10 0

0 10

0.5

1.5

20 30 40 50

Coordinate along the cable, s (m)

q (rad)

60 70 80 90 100

FIGURE 36.18 The equilibrium configuration of a towed cable and the angle that the cable makes with the vertical

when mg ¼ 1:5 N/m, CDrDU2 =2 ¼ 10 N/m, CTrDU2 =2 ¼ 0:1 N/m, and W ¼ 100 N.

Fluid-Induced Vibration 36-25

36.3.2 Fluid Forces on an Articulated Tower

Offshore structures are used in the oil industry as

exploratory, production, oil storage, and oil

landing facilities. They are designed to be selfsupporting

and sufficiently stable for offshore

activities such as drilling and production of oil.

An articulated tower as seen in Figure 36.19 is an

example of an offshore platform that consists of a

base, shaft, universal joint that connects the base

and the shaft, ballast chamber, buoyancy

chamber, and deck. The ballast chambers provide

the extra weight so that the tower’s bottom stays

on the ocean floor, and the buoyancy chamber

adds the necessary buoyancy so that the tower

does not fall.

An articulated tower can be effectively modeled

as a rigid inverted pendulum, where the deck is

modeled as a point mass, the shaft as a uniform

rigid bar, and the buoyancy chamber by a point

buoyancy. In two dimensions, motion of the tower

can be described with a single DoF (Chakrabarti

and Cottor, 1979; Bar-Avi, 1996). The equation of

motion in terms of the tower’s deflection angle is

obtained by summing the moment about the point

O in Figure 36.20 and is given by

d2u

dt2 ¼

MO ¼ mg

sin u

þ MgL sin u 2 Bl sin u þ

ðL

f nx dx

where I is the mass moment of inertia about the

point O given by I ¼ mL2=3 þ ML2; m is the mass

of the shaft, g is the gravitational acceleration, L is

the length of the shaft, M is the point mass at the

top, B is the buoyancy provided by the buoyancy

chamber, l is its moment arm, f n is the normal

fluid force per unit length, and x is the coordinate

along the shaft from O.

The fluid force per unit length in the normal direction is given by

f n ¼ CDr

2 ðwn 2 vnÞlwn 2 vnl þ CMrp

w_ n 2 CArp

where the last term is the force in the normal direction due to the added mass. vn and an are the velocity

and the acceleration of the body in the normal direction and are given by

vn ¼ x

and an ¼ x

d2u

dt2

Deck

Buoyancy chamber

Ballast chamber

Base

Shaft

Universal joint

FIGURE 36.19 Schematic of an articulated tower.

L/2

f n

FIGURE 36.20 Free-body diagram.

36-26 Vibration and Shock Handbook

If we assume that the surrounding fluid is stationary, then the normal velocity and the acceleration of the

fluid (w and w_ ) are zeros. Thus, the moment due to the fluid force is given by

ðL

f nx dx ¼

ðL

2CDr

x2 du

􀀏 􀀐2

sign

􀀏 􀀐

þ CArp

d2u

dt2

􀁻 !

x dx

¼ 2CDr

􀀏 􀀐2

sign

􀀏 􀀐

þ CArp

d2u

dt2

and the equation of motion is given by

3 þ ML2 þ CArp

􀁻 !

d2u

dt2 ¼ mg

2 þ MgL 2 Blb

􀀏 􀀐

sin u 2 CDr

sign

􀀏 􀀐

􀀏 􀀐2

Note that the normal fluid drag force adds directly to the restoring moment in the case of a rigid bar.

The equation of motion can be solved numerically once the initial conditions (u½0􀀉 and du=dt½0􀀉) are

given.

The equation of motion can be simplified if we assume that the angle of rotation u is small. More

specifically, if we assume that u 2 is negligible when compared with 1, then we find that2

sin u < u

The equation of motion can be simplified to

3 þ ML2 þ CArp

􀁻 !

d2u

dt2 2 mg

2 þ MgL 2 Blb

􀀏 􀀐

u þ CDr

sign

􀀏 􀀐

􀀏 􀀐2

¼ 0

which resembles the equation for a linear oscillator with a nonlinear damping term. Note that the system

becomes unstable when the stiffness term (the coefficient of u) becomes negative. This occurs when the

buoyancy is not sufficient or

B ,

2 þ MgL

􀀏 􀀐

36.3.3 Distribution of Significant Wave Heights — Weibull and Gumbel

Distributions

The National Buoy Data Center (NBDC) run by NOAA collects ocean data such as wind, current, wave,

pressure, and temperature data in various locations and the records are made public. Let us say that we

are to design an articulated tower (in Section 36.3.2) in one of these locations where the data are

available. The first task is to characterize the environment. Using all of the information that is collected

is inefficient and impractical. Instead, we are interested in choosing a single number that can

represent typical and extreme situations such as 10- and 50-year storms. For now, let us only consider

random waves. We are then interested in finding the significant wave heights representing 10- and

50-year storms.

From NBDC data for a buoy outside Monterey Bay, the number of occurrences for ranges of

significant wave heights is constructed in Table 36.2. The measurements were taken every hour

for about 12 years. We first construct the corresponding Weibull distribution using the method

described in Section 36.1.6. We first guess g so that a pair of lnð2ln{1 2 FðhÞ}Þ and lnðh 2 gÞ form a

2This is called the small angle assumption.

Fluid-Induced Vibration 36-27

straight line. Figure 36.21 shows that the pair

yields nearly a straight line when g < 0:84: The

slope and the y intercept of this line are 1.6 and

2 0.78, respectively. The Weibull parameters are

then m ¼ 1:6 and b ¼ 1:6:

Similarly, we can find the corresponding

Gumbel probability density function by plotting

pairs of ðh; lnð2ln{FðhÞ}ÞÞ to form a line. For the

data shown in Table 36.2, the line has a slope of

2 1.52 and y intercept of 2.84 so that a ¼ 21:52

and b ¼ 1:87:

Figure 36.22 shows the Weibull probability

density and the cumulative distribution (Equation

36.17) in solid lines, the Gumbel probability

density and the cumulative distribution in dotted

lines (Equation 36.18), and the discrete probability

density and the cumulative distribution derived

from Table 36.2 in symbols.

TABLE 36.2 Number of Occurrences of Various Sea States

Significant Wave Height, h (m) Number of Occurrences Sum

,1 2,367 2,367

1 – 2 46,353 48,720

2 – 3 3,4285 83,005

3 – 4 1,3181 96,186

4 – 5 3,813 99,999

5 – 6 716 100,715

6 – 7 145 100,860

7 – 8 32 100,892

8 – 9 8 100,900

9 – 10 2 100,902

Total 100,902

−5

−4

−2

−3 −1 1

0.99

0.5 0.84 0.95

γ = 0.2

ln(h−γ)

ln(−ln{1−F(h)})

FIGURE 36.21 Plots of ðlnðh 2 gÞ; ln½2ln{1 2 FðhÞ}􀀉Þ

for various values of g.

FGumbel (h)

FWeilbull (h)

fWeilbull (h)

0.8

0.6

0.4

0.2

0 2 4 6 8 10

Significant wave height, h (m)

Probability density and

cumulative distribution

FIGURE 36.22 Weibull approximations of the probability density and cumulative distribution of significant wave

heights measured in the outer Monterey Bay area. The symbols are the values given in Table 36.2.

36-28 Vibration and Shock Handbook

The next step is to find a significant wave height that can represent an N-year storm, hN : The

probability that we will not have an N-year storm in any given year is 1 2 1=N and is equivalent to the

probability that the significant wave height will not exceed hN in the same year. The probability that

h , hN in a single measurement is FðhN Þ; and the probability that h , hN in every measurement taken in

a year is FðhN Þ24£365: Then, we have

1 2

N ¼ FðhN Þ24£365

Table 36.3 shows significant wave heights that represent 5-, 10-, and 50-year storms obtained using the

Weibull and Gumbel distributions.

The Gumbel probability distribution gives higher significant wave heights. For this particular set of

data, the Weibull distribution seems to fit the data better (Figure 36.22), and the Weibull distribution is

the most often used distribution in the offshore industry.

36.3.4 Reconstructing Time Series for a Given Significant Wave Height

Previously, we found significant wave heights that could represent 5-, 10-, and 25-year storms for a given

site. Recall that the significant wave height can entirely characterize the Pierson –Moskowitz spectra.

Once the spectral density is determined, a sample time history of the wave profile, hðtÞ; can be

determined using either Borgman’s or Shinozuka’s method (Section 36.1.4). Here, Shinozuka’s method is

used to generate the random wave elevations.

Let us first find the random frequencies distributed according to So

hh ðvÞ=s2h

: The P–M spectrum in

terms of the significant wave height is given by Equation 36.8.

hh ðvÞ ¼ 0:7795v25 exp 2

3:118

v24

􀀏 􀀐

The variance is given by

s2h

ð1

hh ðvÞdv ¼

The probability density and the cumulative distribution functions are given by

f ðvÞ ¼ So

hh ðvÞ=s2h

12:472

v25 exp 2

3:118

v24

􀀏 􀀐

; FðvÞ ¼ 1 2 exp 2

3:118

v24

􀀏 􀀐

The inverse of the cumulative distribution function is given by

F21ðxÞ ¼ 2

3:118

lnð1 2 xÞ

􀁻 !21=4

The random frequencies distributed according to f ðvÞ can be obtained from uniformly distributed

random numbers x from 0 and 1. Table 36.4 shows uniform random numbers between

TABLE 36.3 Comparison of Representative Significant Wave

Heights for Long-Term Predictions from Gumbel and Weibull

Distributions

5-Year (m) 10-Year (m) 50-Year (m)

Weibull 7.84 8.15 8.79

Gumbel 8.83 9.33 10.4

Fluid-Induced Vibration 36-29

0 and 1 and the random frequencies distributed according to f ðvÞ3. The significant wave height

of 7.84 m is used.

We can obtain 100 in this way, and the wave elevation is also obtained using Equation 36.13. The

random phase wi is obtained by multiplying uniform random numbers (different from the ones used to

generate the random frequencies) by 2p.

Figure 36.23 shows the surface elevation as a function of time, the corresponding wave velocities at the

water surface (Section 36.1.7) as functions of time, and the wave velocities at t ¼ 0 as functions of the

water depth. Note that the wave velocities decay with depth.

TABLE 36.4 Generation of Random Frequencies Distributed According to f ðvÞ from Uniform

Random Numbers

Uniform Random Numbers 0 , x ,1 Random Frequencies v Distributed According to f ðvÞ

0.950 (2 19.713 ln[1 2 0.950])21/4 ¼ 0.360

0.231 (2 19.713 ln[1 2 0.231])21/4 ¼ 0.662

0.606 (2 19.713 ln[1 2 0.606])21/4 ¼ 0.483

.. .

−10

−20

Wave elevation (m)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

−5

−10

Wave velocities (m/s)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

−1

−2

−3

Wave velocities (m/s)

0 50 100 150 200 250 300 350 400 450 500

Water Depth (m)

FIGURE 36.23 Wave elevation and velocities.

3The uniform random numbers can be generated by the MATLAB rand function.

36-30 Vibration and Shock Handbook

36.3.5 Available Numerical Codes

Many numerical codes are available for modeling the dynamics of slender structures such as risers, tether,

umbilicals, and mooring lines. The first example in this section was solved by a numerical code, WHOI

Cable, developed at Woods Hole Oceanographic Institution. WHOI Cable is a time-domain program

that can be used for analyzing the dynamics of towed and moored cable systems in both two and three

dimensions. It takes into account bending and torsion as well as extension.

Comparative studies investigating flexible risers were carried out by ISSC Committee V7 from

computer programs developed by 11 different institutions in the period between 1988 and 1991, and the

results were reported by Larsen (1992). More recently, Brown and Mavrakos (1999) conducted a

comparative study on the dynamic analysis of suspended wire and chain mooring lines and reported

results from 15 different numerical codes. The participants included engineering consultancies, and

academic and research institutions involved in marine technology. Some of the time-domain programs

that were included in the comparative study are MODEX by Chalmers University of Technology,

FLEXAN-C by Institute Francais du Petrole, DYWFLX95 by MARIN, R.FLEX by MARINTEK,

CABLEDYN by National Technical University of Athens, DMOOR by Noble Denton Consultancy

Services Ltd, V.ORCAFLEX by Orcina Ltd Consulting Engineers, ANFLEX by Petrobras SA, TDMOORDYN

by University College London, FLEXRISER by Zentech International. Some of these programs are

available to academic institutions and government laboratories at no cost.

Acknowledgments

The author wishes to express gratitude for the funding from the Woods Hole Oceanographic Institution

and the Department of Mechanical Engineering at Texas Tech University.

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