1. Vibration and Stochastic Wave Response of a Tension Leg
Platform
Written By: Joshua Harris, August 26, 2015
Abstract
The equation of motion for a single tendon of a tension leg platform is
presented. The equation is uncoupled, linear and non-homogenous. The force on the
tension leg platform is modeled as a random harmonic load, which is interpreted to
be the waves hitting the tension leg platform at random. The model is a compliant
structure that allows for small deformations and displacements. The tension leg
platform is modeled as a rod connected to a torsional spring with a mass at the end.
There is viscous damping that accounts for the drag that is caused by the seawater.
The first method of analysis involves finding the input spectral density and
multiplying it by the transfer function to receive the output spectral density plot.
The second method involves solving the equation of motion to find the motion of the
tension leg platform as a function of time.
1. Introduction
A tension leg platform (TLP) is a vertically moored floating structure that is
usually used for the offshore production of oil and gas, however the idea has been
considered for wind turbines. There are best suited for use in water that is 1000 feet
in depth. The usual method for installing TLPs start with foundation piles being
lowered into the seabed and hammered into the bottom of the ocean floor. The
platform itself is moored by tethers or tendons, which are connected to the
foundation piles. A group of tethers is known as a tension leg. A tendon support
buoy will next be inserted on top of the topmost tendons. The giant TLP hull is then
bought in by boats and attached to the tendons.
There are typically two types of TLP structures: fixed and compliant. Fixed
structures are rigid and do not allow for any motion in the tethers. Compliant
structures, which are more commonly used, allow for small deformations and
displacements to occur, thus making it easier to design and account for the waves
that are hitting the TLP. The deeper the water depth, the better it is to use compliant
structures.
2. Formulation of Spectral Density Plots
The most direct process in acquiring the spectral density plot of a system is
to find the corresponding autocorrelation function and then take the Fourier
transform of it. This would give the input spectral density plot. To get the output

2. 2
spectral density plot, which is of importance here, the transfer function must be
found and multiplied by the input spectral density plot.
2.1 Physical Representation of the Model and Nomenclature
Figure 1, shows the physical representation of the model. The nomenclature
is found in Table 1. Some of these values are borrowed from [5], while others were
calculated, in order to create a more realistic model
Figure 1 – Schematic Diagram of Tether

3. 3
Table 1- Nomenclature
Symbol Meaning Value
L Length 260 m
ϕ Random Variable [0, ∞)
π Pi 3.1415
τ Time constant Seconds
t Time [0, ∞) seconds
k Torsional stiffness 100 kN•m
c Viscous Damping 3.036 x 1010 kg•m/s
ω Frequency [range of values] rad/s
ωn Natural Frequency [range of values] rad/s
ωo Initial Frequency (when
n=0)
[range of values] rad/s
m Mass 169,353 kg
A Amplitude 2000 N
θ Angle (of motion) Radians
N Upper Limit ∞
Boldfaced values are borrowed from [5]
2.2 Identification of the Input Forces
It is necessary to derive the input force on the system. The autocorrelation
function of this force can than be found. The ocean waves can be modeled as a
stochastic, harmonic force that acts on the system. The different natural frequencies
need to be added together because they contribute to the overall force. A random
variable generator can be used to create the random variable that is within the same
range as the natural frequency. It does not matter if the random variable is added or
subtracted.
(1)
Although the cosine function is used here, the sine function can be used as well. The
upper limit can be adjusted based on how many frequency values need to be
evaluated. It should be noted that the natural frequency is defined as:
(2)
Where T is a fixed time period, but n if from the range of 0, 1, 2,..N.

4. 4
2.3 Derivation of Input and Output Spectral Densities
A major assumption that is made is that the entire process is stationary. In a physical
sense, it is to be assumed that the function and the process will enter steady state
after a long period of time. There are no sudden impulse forces acting on the system.
Statistically this means that the joint probability density function at a distinct set of
times will equal the joint probability density function and an entirely different set of
times. Thus, this slightly changes the definition of the autocorrelation function:
(3)
This is the autocorrelation function of a stationary random process where F(t) is the
input force as a function of time. When the force is plugged into equation 3,
(4)
The two expectations are first found separately and then multiplied together. After
further evaluation and simplification and factored out coefficients, the final result is
an autocorrelation function that is a function of the time constant.
(5)
The detailed process of finding these expectations and simplifying terms is found in
the appendix. The input spectral density is found by taking the Fourier transform of
the autocorrelation function. By using a table of Fourier transforms, the Input
spectral density is equal to:
(6)
δ represents the Dirac delta function evaluated at the value within the parenthesis.
The next step is to find the transfer function, H (iω), and multiply the transfer
function by the input spectral density to get the output spectral density.
(7)

5. 5
Equation 7 is the fundamental result for linear, stationary systems. The most
reliable way to find the transfer function is to take the Fourier transform of the
equation of motion. Because there are imaginary numbers in the transfer function,
the magnitude of it is taken, before being multiplied by the input spectral density,
because the output spectral density plot must be real and only a function of the
frequency. Thus, it is necessary to find the equation of motion for the system.
The equation of motion is found by applying Newton’s second law to the system and
then equating it to the force (1) previously described. As noticed in Figure 1, there is
a mass, viscous damper, and torsional stiffness that are associated with the system.
Moreover, the tether’s motion is modeled as an angular motion that rotates in
radians. The equation of motion is represented below, with a single over-dot
representing the first derivative and two over-dots representing a second
derivative.
(8)
Again F (t) is denoted by equation 1. This same equation of motion will be used in
solving the position as a function of time. Once F (t) is plugged into the equation, the
Fourier transform can be taken to get the transfer function. However, in order to
make it simpler to take the Fourier transform, a simple substitution to convert
equation 8 as a function of θ to a function of x.
This can be applied to any rotational system. Because the tether is fixed about a
torsional spring, it can be used here as well. It is to be noted that here r is equivalent
to L. The Fourier transform of the right side of the equation of motion gives the
input function F(ω) and the transform of the left side of the equation of motion
gives the output function X(iω). Dividing the latter by the former is the transfer
function.
The Fourier transfer is taken and then substituted into the formal definition of the
transfer function.

6. 6
As aforementioned from equation 7, the magnitude of H (iω) must be taken.
(9)
All of the unknowns in equation 7 are found. After factoring the appropriate
constants, the output spectral density is obtained. Note that ω can equal ωn.
(10)
The complete, detailed mathematical solution for each step is found in the appendix.
2.4 Plots of Power Spectra
By definition of the Dirac delta function, it is zero everywhere except when the term
evaluated inside the parenthesis is equivalent to zero. Also, its area must equal one.
So in this case, the Dirac delta function is only zero when either ωn=-ω0 for the first
term of the summation or when ωn=ω0 for the second term of the summation. As
Figure 2 shows, the following result will be spikes at the aforementioned values of
ω. The spectral density plot can be used to measure the amount of energy in a
stochastic process. Higher amplitudes in graphical results indicated that there is
more energy at those particular frequencies. Therefore, the physical significance of
these graphical results lies in the fact that the energy of the system is highly
concentrated at two different frequencies. The coefficient in front of the summation
of equation 10 is what determines the amount of area under the spike. Again, the
area under the Dirac delta function must equal one. Thus, the coefficient can
increase or decrease the amount of area underneath the spike.

7. 7
Figure 2 - General Graph of Equation 10
The area underneath the power spectrum is equivalent to the mean-value squared,
which also can be expressed as the variance subtracted by the mean. Thus, it can be
used to calculate the variance if the mean is known or vice-versa. Statistically, this
means that the spectral density plot is a distribution of the variance according to the
frequency.
X(t) is simply the random variable as a function of time. In this system, it is the
position of the tether as a function of time.
By using the nomenclature from Table 1, a specific coefficient, as a function of ω, in
equation 10 was found. Specific frequencies were plugged in, evaluated, and
graphed. Figure 3 shows the various graphs at various frequencies. The area
underneath each spike becomes more visible as the frequencies become higher.

8. 8
a. The frequency ω equals 0. The value of
266.87 is multiplied by δ (0).
b. The frequency ω equals 10 rad/s. The
value of 2.9 x 10-11 is multiplied by δ (0).
c. Frequency of ω equals 100 rad/s. The value of 2.9 x 10-13 is multiplied by δ (0).
Figure 3 – Spectral Density Plots at Different Frequencies
3. Formulation of the Position Function and Its Mean-Value Squared
The process behind finding the position as a function of time requires solving
the equation of motion. This position function can be used as a “random variable”
and be used to find the mean-value and the mean-value squared. Equation 8 shows
the equation of motion in terms of θ, but by using the relationship of between θ and
x, the equation of motion can be a function of x. However, this will be done after the
equation is solved in terms of θ first.
3.1 Solving the Differential Equation
As previously mentioned, equation 8 is a second-order, linear, uncoupled,
nonhomogeneous differential equation. The solution is therefore the homogeneous
solution added to the nonhomogeneous solution. To find the homogeneous solution
to the equation, the characteristic equation is solved and its solution is used as
exponents of e, multiplied by t.

9. 9
(11)
The coefficients c1 and c2, are found by the initial conditions that can be uniquely
formulated based on the specific conditions surrounding the system. It is not to be
confused with the c in the exponent, which is the viscous damping caused by the
water.
To find the nonhomogeneous solution, the method of undetermined coefficients can
be used, especially since the force is a harmonic function of cosine.
(12)
The next step is to add equations 11 and 12 together to get the general solution. It is
still a function of θ.
(13)
The final step is to relate θ to x. As previously mentioned, r is equal to L. Therefore, L
is in the final solution in place of r.
(14)
The complete, detailed mathematical solution for each step is found in the appendix.
3.2 Finding the Mean-Value Squared
There is a relatively simply way to find the mean-valued squared of a
continuous random variable that involves only calculus.

10. 10
In this case, n=2. However, a problem arises because there is no function f(x) and it
cannot be derived or found with ease. Therefore, the assumption of ergodicity has to
be made. Formally, a stationary random process is ergodic if the time average of an
event at a single time period is equal to the ensemble average. In other words, the
average is constant. With ergodicity, an average over a long period of time can be
taken, instead of numerous averages at many different time periods. Also, an ergodic
process is always stationary. Since TLPs are designed to last over long periods of
time, the assumption of ergodicity can be made. It is safe to say that the average
over a long period of time is the same as over a very long period of time. For
example, the average over a six-month period will not be too different from an
average over a two-year period. Mathematically, as the time approaches infinity, the
exponentials will approach zero; it is the exponential part that would provide the
most change and discrepancy in averages. With this new ergodic assumption, the
definition of the mean-value squared changes, and it does not require a function of x.
(15)
The coefficients that precede the sine and cosine terms are now constant, and
throughout the integration process, can be factored out in front of the integral. For
ease of calculation, the coefficients are renamed as A and B.
As seen, the coefficients of A and B are not functions of T, so they are constant
always. These same coefficients are used and represented in the final answer.
(16)

11. 11
As aforementioned, the mean-valued squared can be compared to the spectral
density plot and can be used to find and evaluate the variance. The complete,
detailed mathematical solution for each step is found in the appendix.
4. Summary and Conclusions
A model of a tendon in a Tension Leg Platform (TLP) is modeled as a rotating
beam about a torsional spring. It has viscous damping, torsional stiffness, and a
random harmonic load. The process is assumed to be ergodic. Taking the Fourier
transform of the autocorrelation function provided the function of the power
spectrum, specifically the input spectral density. After using a Fourier Transform for
the equation of motion to get the transfer function of the system, the output spectral
density is found.
The second method of analysis involved solving the equation of motion. The result
was used to find the mean-value squared of the system, which is quite useful in
finding other statistically properties.
The many results stemming from such analyses can be used for the design process
of TLPs. The spectral density plots can show where all of the energy is concentrated.
When testing, such frequencies can be focused on. If the motion of a tether needs to
be limited or even made more movable, values of viscous damping and torsional
spring constants can be used in the position function, to see which parameter affects
the motion of the tendon the most. Different frequencies, lengths, and masses can
be easily substituted to see how the system reacts to changes in these parameters.
By using the variance and mean-value squared, the changes in the position can be
evaluated, analyzed, and accounted for during the design process. A designer can
recognize if the average position would lead to failure or instability. Although the
forces due to the ocean waves are stochastic, they can still be predicted using
probabilistic models.
5. Appendix
The appendix provides all of the detailed calculations done in this work in
their original form. It includes the key assumptions that were made in order to
apply certain formulas and carry out the mathematics.
References
1. Benaroya, Haym, and Mangala M. Gadagi. "Dynamic Response of an Axially Loaded
Tendon of a Tension Leg Platform." Journal of Sound and Vibration 293
(2005): 38-58. Elsevier. Web. 26 Aug. 2015.
2. Benaroya, Haym, and Ron Adrezin. "Response of a Tension Leg Platform to
Stochastic Wave Forces." Probabilistic Engineering Mechanics 14 (1999): 3-
17. Elsevier. Web. 26 Aug. 2015.

12. 12
3. Benaroya, Haym, and S. M. Han. "Non-Linear Coupled Transverse and Axial
Vibration of a Compliant Structure Part 1: Formulation and Free Vibration."
Journal of Sound and Vibration 237.5 (200): 837-73. Ideal Library. Web. 26
Aug. 2015.
4. Han, Seon Mi, and Haym Benaroya. "Comparison of Linear and Non-linear
Responses of a Compliant Tower to Random Wave Forces." Chaos Solitons
and Fractals 14 (2001): 269-91. Elsevier. Web. 26 Aug. 2015.
5. Mathisen, Jan, Oddrun Steinkjer, Inge Lotsburg, and Oistein Hagen. Guideline for
Offshore Structural Reliability Analysis - Examples for Tension Leg Platforms.
Tech. no. 95-3198. Ed. Vigliek Hansen. N.p.: n.p., n.d. Joint Industry Project.
Det Norske Veritas, 27 Sept. 1996. Web. 26 Aug. 2015.
6. Benaroya, Haym, and Mark L. Nagurka. Mechanical Vibration: Analysis,
Uncertainties, and Control. 3rd ed. Boca Raton, FL: CRC/Taylor & Francis,
2010. Print.