Bridge

Vibration Control by Tuned Mass Damper and Tuned Liquid Damper Bridge Mellichamp Department of Physics and Mathematics Colby College 7154 Mayflower Hill Waterville, ME 04901-8871 6021 Jocelyn Hollow Rd. Nashville, TN 37205-3209 [email protected] Mentors: Dr.Yahui Zhang and Miss Liu Xiao Hui State Key Laboratory of Structural Analysis for Industrial Equipment Dalian University of Technology Dalian, China 116024

Abstract This paper investigates the feasibility of improving and implementing passive dampers onto jacket platforms in the Bohai Bay as a means to mitigate ice-induced vibrations. Tuned Mass Dampers (TMDs) are currently the most widely used devices for vibration control of the Bohai Sea’s jacket platforms. The parameters of the TMD are optimized using MATLAB. The optimal TMD applied to the JZ20-2 MUQ oil platform in the Bohai Bay has parameters of tuning frequency ratio, f = 0.985 and the TMD coefficient of damping, ςd = 0.065, which corresponds to a stiffness of, kd = 329633.4 N/m, and a damping of, cd =7.91*10-7 Ns/m. The Tuned Liquid Damper (TLD) is a second type of passive vibration mitigation. The theory of applying a TLD is relatively new and is still under development. Unlike the TMD’s simple linear force, the TLD responds in a complicated fashion, making it difficult to create a mathematical model to predict accurately the TLD’s behavior. Testing was done to calculate the sloshing force as well as to compare the experimental data to predicted standard theory. While the theory and the experimental data differ significantly on the extremes of the applied frequency range (0.94 to 1.04Hz), both have a maximum force in the vicinity of 0.98 to 0.99Hz. Currently, the TMD is the preferred choice of damper for ice-induced vibration mitigation, though it can be continually improved as the research of TLD application is simultaneously furthered.

Introduction The world’s need for resources continues to grow at a rapid rate, resulting in a diminished supply of petroleum resources. As technology develops alternative means of energy production, it is imperative to improve simultaneously the assessment of current reserves, focusing on efficiency, safety, and environmental protection. It is believed that the Bohai Bay contains approximately 20.5 billion tons or 1.5 billion barrels of offshore

oil reserves, which will allow China to rely predominantly on domestically produced energy – one of its primary goals. However, access and extraction of oil in the Bohai Sea is complicated by the presence of sea ice. The Bohai Sea, the southernmost sea in the Northern Hemisphere, contains substantial amounts of sea ice every winter. Sea ice, moved by ocean currents, pushes against the offshore oil platforms causing structural vibration that results in discomfort for human occupants, equipment malfunction, and potential structural failure. To limit damage, conical shells are placed around the structure’s legs to reduce the force imparted by the breaking sea ice. While the imparted force is lessened, it is periodic, which results in another problem. The general problems associated with the exertion of ice forces on structures has been subject of scientific interest for a number of years, beginning when Korzhavin initially addressed the issue in a comprehensive study of ice strength under dynamic loading (Nadreau 1987). A significant portion of the concern exists because the ice failure frequency is nearly identical to the resonant structural frequency of the jacket platform, which amplifies the effect of ice-induced vibrations and increases the risk of structural failure (Yue & Liu 2003 and Wang 2002). Various techniques of vibration mitigation have been implemented worldwide to decrease the effects of wind, seismic, and ice-induced vibrations. There are various types of dampers – passive, semi-active, and active. Tuned Mass Dampers and Tuned Liquid Dampers are both types of passive auxiliary damping devices, which depend on adding an auxiliary mass to increase the level of damping and require no external source of energy. Auxiliary dampers are often used when inherent damping is insufficient. They are particularly useful because they provide a predictable, adjustable, and reliable method of damping that can be adjusted or added after the structure is complete, often necessary as the actual structural damping is unknown until construction of the structure is complete (Kareem 1999). Similarly, passive damping systems are advantageous due to their simplicity and cost effectiveness. They are adjusted easily, can be retrofitted, and require little maintenance. TMDs and TLDs add only a small amount of mass to the structure and can reduce their acceleration responses from ½ to ⅓ of the undamped response (Maebayashi, 1993). The Bohai Sea platforms are different from other platforms that are also subject to ice-induced vibrations. Unlike the oil fields along the Alaskan coast, the Bohai Sea is considered a marginal oil field, making the efficiency with which oil is extracted highly important because there only a small amount of oil can be extracted. In the Bohai Sea, platforms have been inadequately constructed because the presence of ice was not considered, resulting in the collapse of several platforms. To increase the cost efficiency of exploiting oil, mitigate structural fatigue, and reduce the possibility of failure, it is vital to develop and implement an optimal method to mitigate the effects of ice-induced vibrations, most poignantly by adding auxiliary passive dampers such as Tuned Mass Dampers or Tuned Liquid Dampers to the jacket platforms.

Background

There are multiple designs of Tuned Mass Dampers. Usually, the TMD system contains a mass, a spring, and a damper. The TMD is applied to the structure at the location of maximum dynamic response, typically at the top. The spring and damping components are tuned to a specific frequency, resulting in the TMD’s control being limited to a single structural mode. A spring and a damping mechanism cause the energy to dissipate resulting in the reduction of the structure’s dynamic response through which the TMD transfers inertial force to the building. To effectively absorb energy, the TMD’s natural frequency is tuned to the structure’s and operates based on a phase shift. There are two categories of Tuned Liquid Dampers - Tuned Liquid Column Dampers (TLCD) and Tuned Sloshing Dampers (TSD). Tuned Sloshing Dampers can be further sub-divided into two categories based on the depth of water contained in the tank. TSDs rely on the amplitude of fluid motion and wave breaking pattern to provide damping, while gravity provides the restoring force. The experiments use a shallow water TSD, which dissipates energy through viscous action and wave breaking that occurs primarily at the boundary. The natural frequency of a TLD can be adjusted easily since the depth of the water and the dimensions of the container determine it. When practical limitations result in an inadequate natural frequency of the TLD, it is possible to apply a spring mechanism that will adjust the frequency of the sloshing motion if the building experiences a change in dynamic characteristics (Shimitzu 1994). TLDs are investigated as a possible alternative or addition to the implementation of TMDs, to mitigate the dynamic response of the jacket platforms. They are considered because they require no extra energy from an actuator and, furthermore, there is no mechanical friction. The resulting system is effective enough to control the slightest vibrations, the period is easy to adjust, and the system is easy and inexpensive to maintain (Noji 1991). However, research on Tuned Liquid Dampers is complicated because unlike the TMDs, which have a linear response, that of TLDs is highly nonlinear due to the fluid motion. This characterization of TLDs is very important both analytically and experimentally. The majority of research focuses on a better understanding of the water’s motion within a tank through physical experimentation, although some mathematical models have been developed.

Procedure TMD Parameter Optimization Tuned Mass Dampers respond to the dynamic motion of the structure in a linear manner. A single degree of freedom (SDOF) structure and the absence of damping in the primary structure were assumed to simplify the structure to facilitate parameter optimization.

md

u + ud kd 2

kd 2

cd

P(t )

u m k 2

k 2

c

Figure 1: Singe Degree of Freedom System with TMD. Labels with a subscript “d” correspond to the TMD and those with no subscript correspond to the structure. The variables presented can be found in Table 1.

Table 1: Definition of Variables for TMD Optimization

m : mass of the platform c : damping of the platform

(kg) ⎛ Ns ⎞ ⎜ ⎟ ⎝m⎠ ⎛N⎞ ⎜ ⎟ ⎝ m⎠

k : stiffness of the platform ⎛ rad ⎞ k ⎟ ⎜ ω= m ; natural frequency ⎝ s ⎠ of the platform c ζ= 2mω ; damping ratio of the platform u : displacement of the platform

Ω: load frequency f =

ωd ω ; Tuning frequency ratio

(m) ⎛ rad ⎞ ⎟ ⎜ ⎝ s ⎠

m d : mass of the TMD

c d : damping of the TMD kd : stiffness of the TMD kd ωd = md ; natural frequency

(kg) ⎛ Ns ⎞ ⎜ ⎟ ⎝m⎠ ⎛N⎞ ⎜ ⎟ ⎝ m⎠ ⎛ rad ⎞ ⎟ ⎜ ⎝ s ⎠

of the TMD cd ζd = 2 md ω d ; damping ratio of the TMD ud : relative displacement of the (m) TMD to that of the platform m m= d m

ρ=

Ω

ω

The goal of the optimization is to determine the values of kd and cd that result in the minimal amplitude of the TMD and platform system while keeping the parameters of the TMD (m, c, and k) constant, as well as md, which is defined. Based on the harmonic force given in Eq.1, the equations of motion follow

P (t) = P?e iΩt

(1)

mu&& + cu& + ku − c d u& d − k d u d = P md u&&d + c d u& d + k d u d − md u&&&d = 0,

(2)

(3) where Eq. 2 gives the motion of the system of the TMD and the Platform and Eq. 3 refers to that of the TMD alone. The displacements can be calculated using Cramer’s rule to find that the displacement of the platform and the relative displacement of the TMD to the platform given respectively by P?(−md Ω2 + ic d Ω + kd ) , (−mΩ2 + icΩ + k)(−md Ω2 + ic d Ω + kd ) − md Ω2 (ic d Ω + k d ) P?md . u?d = 2 2 (−mΩ + icΩ + k)(−md Ω + ic d Ω + kd ) − md Ω2 (ic d Ω + k d )

u? =

(4) (5)

The program relied upon nested loops to perform multiple iterations as ρ, ςd, and f were varied. Within the innermost loop, Eq. 5 was solved for and the maximum value of ρ’s range was calculated for all possible combinations of the TMD’s damper ratio and the tuning frequency ratio on the intervals specified in Table 2. The method used to determine the optimal parameters is the min-max method, which consists of minimizing the maximum dynamic response. Table 2: Specifications for Optimization of TMD Applied to Jacket Platform

ω = 8.164 rad/s k = 13330179.2 N/m c = 65312 Ns/m P? = 133301.792 0.5 ≤ ρ ≤ 2.0

m = 200,000.0 kg ζ = 0.02 m = 0.01 0.5 ≤ f ≤ 1.5 rad/s 0.0 ≤ ς d ≤ 1.0

TLD Experimental Setup & Procedure

The difficulty in developing mathematical models capable of representing the complex motions of the liquid and force of the Tuned Liquid Damper necessitates physical experimentation. Experimentation is relied upon heavily to characterize the TLD’s behavior and investigate its ability to mitigate structural vibrations. The experiments strive to quantify the behavior of the TLD and establish the accuracy of mathematical models. The purpose of this experiment is to directly measure the sloshing force of the TLD. Determining the sloshing force will allow the size of the liquid tank to be optimized without the presence of a sliding table capable of handling a scale model. Previous experiments conducted at Dalian University of Technology have determined the fundamental natural frequency of the water sloshing motion through a swept experiment to be 1.00Hz. These tests measured the response of the TLD under harmonic excitation and utilized the experimentally determined dynamic response of the

structure to determine the accuracy of mathematical models. Table 3: Definition of Variables for TLD

m = mass of the structure, including (kg) the empty tank mi = mass of the solid (kg) F1 = force applied using Setup I (N) L = length of container (m) w = width of container (m) μ = coefficient of friction

m d = mass of the water in the tank

(kg)

u(t) = the displacement of the table F2 = force applied using Setup II ht = height of the container hw = depth of the liquid in container

(m) (N) (m)

f =

ωd ω : resonance frequency

The width of the TLD container was selected to ensure the moving table in the laboratory could accommodate the TLD and the height of the container was chosen to ensure that the water was not lost due to sloshing out of the tank. Based on the principle of resonance, it is known that the frequency of the TLD as well as the natural frequency of the structure should be approximately equal, meaning that the tuning frequency ratio should be equal to 1.0, which can be seen graphically in Fig. 2, and is defined as fw =

1 2π

g

ω L

tanh

hwπ , L

(6)

where g is gravity and the remaining variables are defined above in Table 3.

Figure 2. Graphical Representation of Eq. 6. The graph shows the relationship of how the frequency changes as the TLD’s parameters- water depth and tank length- vary.

When the frequency ratio is valued at 1.0, Eq. 6 simplifies to a relationship of the two remaining unknowns- the depth of the water in the tank and its length, which can be seen

graphically in Fig. 3.

Figure 3. Relationship Between Tank Length and Water Depth for a Frequency of 1 Hz. The relationship is based on Eq. 6 when fw = 1.0. Shown on the graph is the water depth, 0.2m and tank length, 0.6m used in the experiment.

Using Eq. 6 and practical limitations, the container’s length and corresponding water depth were selected to maintain nearly resonant frequencies as outlined in Table 4. Table 4: Specifications of TLD Experiment

mi = 0.06 kg w = 0.5 m

m d = 0.06 kg

L = 0.6 m

hw = 0.2 m fw = 1.007 Hz

ht = 0.5 m

The experimental apparatus consists of a moving table, dSPACE control system, hydraulic actuator, sensors, and a mass system. The mass system can either be the TLD constructed and filled with water to specifications given in Table 4 or a mass block with mass equal to the mass of water contained in the TLD. A harmonic force is converted into its corresponding displacement to the system since it is potentially dangerous to apply a raw force when its repercussions are unknown. The displacement is applied to the moving table by sending an electronic signal via the hydraulic actuator. The displacement used during these experiments was x = 0.002sin(ωt), where ω is the frequency and t is the time. The harmonic displacement was applied with different frequencies, ranging from 0.94 to 1.04 Hz. The frequencies in this range that were used were 0.94, 0.97, 0.98, 0.99, 1.00, 1.01, and 1.04 Hz. The sensors are placed on the mass system and can be set to read displacement, velocity or acceleration. The downfall of using a single sensor to read three types of data is that they are generally less effective than a sensor designed to read only one type of data. During the experiment, the sensors are set to either acceleration or displacement. An Analogue Digital Converter (ADC) converts the response signals collected by the sensors from analogue to digital signals. After they are converted, they are input into the dSPACE computer program where the data is read and then stored in MATLAB. Data is collected for two different scenarios. The first is shown in Fig. 4, where the

TLD is placed on top of the moving table. The second case is shown in Fig. 5, where a block with equal mass to the water contained in the TLD is placed on top of the moving table.

Figure 4. Tuned Liquid Damper Experimental Setup I. The apparatus above is shown with the tank of liquid, which may be substituted for a solid of equal mass as seen in Fig. 5. Variables are defined in Table 3.

Figure 5. Tuned Liquid Damper Experimental Setup II. The apparatus above is shown with the solid mass in the place of the TLD. Variables are defined in Table 3.

In order to calculate either the damping or sloshing force of the TLD, it is necessary to perform preliminary experimental tests to accurately determine and calibrate the equipment used in the experiment. The force of friction cannot be assumed as negligible due to the small size of the sloshing force. For this reason, the experimental set up shown in Fig. 5 is used and the harmonic force is applied to the table. The equation of motion governing this stage of the experiment is (m + mi )u&& = F2 − μ (m + mi ) g.

(7)

Secondly, the solid mass is replaced with the TLD according to the specifications listed in Table 4. The same forces were applied to the moving table and the data was collected. Similarly, the equation of motion governing this stage of the experiment is

mu&&1 = F1 − μ (m + md ) g − FTLD .

(8)

Eqs. 7 and 8 can be manipulated to determine the force of the TLD, which is given as

FTLD = F1 − F2 + md u&&, (9) such that the sloshing force is F1 - F2. The force of the TLD consists of two forces. The most dominant is the sloshing force, which was calculated experimentally. The second is the inertial force of the TLD, given as the third term of the governing equation, Eq. 9.

Results and Discussion TMD Parameter Optimization

The optimal parameters of the TMD based on specifications found in Table 2 were calculated using a MATLAB program. The program performed many iterations using small step sizes. However, for the purpose of presentation, the small step size will not be used. Instead, a larger step size is shown in the subsequent figures, neglecting many of the data sets produced by the program. Initially, the tuning frequency ratios were held constant to compute the relationship between the maximum displacement and the damping ratio of the TMD. When tuning frequencies that varied by a very small step were considered, they appeared to have a maximum displacement that did not vary with the TMD’s damping ratio. However, when the step size was increased so the effect of the tuning frequency could be seen, it became evident that, many of the curves did have a clear-cut minimum as opposed to a function similar to the inverse of the natural log. For tuning frequency ratios slightly above or below 1.00, a minimum was seen clearly. From Fig. 6 below, it is easy to see that the minimum of the maximum displacement occurs around f =1.00 and ςd = 0.05.

Figure 6: Maximum Displacement vs. TMD’s Damping Ratio. The figure shows the relationship of the maximum displacement as a function of the damping ratio of the TMD for five different values of the tuning frequency ratio. The tuning frequency ratios shown on this graph vary from 0.50 to 1.50 in increments of 0.25.

To verify that the loop was executed correctly, the maximum displacement was plotted as a function of the tuning frequency ratio for various values of ςd. The value ςd = 0.00 is included in the simulation to further ensure that the TMD’s behavior is predictable and is similar to previously created dynamic response curves. As Fig. 7 shows, when ςd = 0.00, there is no minimization of the maximum displacement other than that due to the structural damping. Negative damping was not considered in this case and, therefore, it is confirmed that the absence of damping is the worst scenario in dynamic response mitigation.

Maximum Dispacement (m)

Maximum Displacement vs. Tuning Frequency Ratio 0.5 Vd =0.00

0.4

Vd =0.05

0.3

Vd =0.20

0.2

Vd =0.30 Vd =0.50

0.1

Vd =0.10

0 0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Tuning Frequency Ratio

Figure 7: Maximum Displacement vs. Tuning Frequency Ratio. The figure shows the relationship of the maximum displacement as a function of the tuning frequency ratio various values of ςd on the range of 0.00 to 0.50. Please note that in the legend, Vd refers to ςd.

The maximum displacement is minimized when the TMD efficiently counteracts a force with a frequency slightly less than the structure’s natural frequency. For this reason, the optimal design of a TMD is only possible when the frequency of the induced force is known. When the excitation frequency is unknown, it is impossible to design an effective, let alone optimal, TMD to control the vibrations. Fig. 7 confirms what Fig. 6 found: the optimal parameters for the specified TMD are in the vicinity of f =1.00 and ςd = 0.05. Upon further iteration with increasingly smaller step size as the vicinity of f and ςd was focused in on, the optimal parameters for a Tuned Mass Damper were found to be f = 0.985 and ςd = 0.065. Manipulating the equations in Table 1, the optimally designed TMD was found to have stiffness, kd = 329633.4 N/m, and damping cd = 7.91*10-7 Ns/m.

Structural Displacement (m)

Displacement of the Structure 0.3 0.25 0.2 With TMD

0.15

Without TMD

0.1 0.05 0 0.5

1

1.5

2

f , theload - natural structure frequency ratio

Figure 8: Maximum Displacement of the Structure with and without an Optimally Designed TMD as a Function of f, the Load to Natural Structure Frequency Ratio Structural Displacement vs. Time 0.4 0.3 0.2 0.1 0 -0.1

0

5

10

15

20

with TMD Without TMD

-0.2 -0.3 -0.4 Time (s)

Figure 9. Maximum Displacement as a Function of Time when the Load Frequency, Ω = 8 rad/s.

The dynamic response of the structure with and without the TLD is shown above to demonstrate the relative effectiveness of the TMD.

Figs. 8 and 9 graphically show the effectiveness of the TMD when applied to reduce the effects of a harmonic ice load. Using the data shown in Fig. 8, the absolute smallest maximum displacement the structure undergoes was determined as well as the corresponding ice load frequency. These were found to be when ρ ≈ 1.0, which corresponds to Ω = 8.0 rad/s. A TMD optimally designed to counteract a force with a frequency of 8.0 rad/s will offer the maximum amount of dynamic response mitigation. The effects of an 8.0 rad/s ice-load frequency on the structure’s displacement with and without an optimally designed TMD are shown in Fig. 9. TLD Experiment

This TLD experiment and data analysis deals with the very initial stages of the research. The sloshing force of the TLD was calculated by manipulating Eqs. 7, 8, and 9. At the current time, the primary goal is to observe the variances in the sloshing force as the frequency of the induced force varies, which is shown in Figs. 10 to 16. Experimental Sloshing Force vs. Ti applied frequency of 0.94Hz.

200

200

100

100

0

0

Experimental Sloshing Force vs. Time for Applied Frequency of 0.97Hz.

-100

-100

-200

-200 0

2

4

6

8

0

10 12 14 16

2

4

Figure 10: Experimental Sloshing Force vs. Time for an Applied Frequency of 0.94 Hz.

100 0 -100 -200

4

10

Experimental Sloshing Force vs. Time for an Applied Frequency of 0.99Hz.

Experimental Sloshing Force

Experimental Sloshing Force

200

2

8

Figure 11: Experimental Sloshing Force vs. Time for an Applied Frequency of 0.97 Hz.

Experimental Sloshing Force vs. Time for an Applied Frequency of 0.98Hz.

0

6

time (s)

time (s

6

8

10

12

time (s)

Figure 12: Experimental Sloshing Force vs.

200 100 0 -100 -200

0

2

4

6

8

10 12 14 16

time (s)

Figure 13: Experimental Sloshing Force vs.

Time for an Applied Frequency of 0.98 Hz.

Time for an Applied Frequency of 0.99 Hz. Experimental Sloshing Force vs. Time for an Applied Frequency of 1.01Hz. Experimental Sloshing Force

Experimental Sloshing Force

Experimental Sloshing Force vs. Time for an Applied Frequency of1.00Hz. 200 100 0 -100 -200

0

2

4

6

8

10 12 14 16 18

time (s)

Figure 14: Experimental Sloshing Force vs. Time for an Applied Frequency of 1.00 Hz.

200 100 0 -100 -200 0

2

4

6

8 10 12 14 16 18 time (s)

Figure 15: Experimental Sloshing Force vs. Time for an Applied Frequency of 1.01 Hz.

Experimental Sloshing Force

Experimental Sloshing Force vs. Time for an Applied Frequency of 1.04Hz.

200 100 0 -100 -200

0

2

4

6

8 10 12 14 16 18 time (s)

Figure 16: Experimental Sloshing Force vs. Time for an Applied Frequency of 1.04 Hz.

From the graphs, it is clear that the sloshing force is harmonic and varies with the frequency of the applied force. Both the amplitude and the period of the sloshing force are different for each given case. Presently, the greatest concern is whether the TLD can produce a force large enough to mitigate the dynamic response of the structure. The largest theoretical force produced occurs when the frequency is set to 0.99Hz. The maximum amplitude calculated for the sloshing force is 191.406N. It is important to note that the sloshing force is not symmetrical directionally. The case with the most symmetry is when the frequency is 1.01Hz. In the future, this will be an important factor to consider when determining how to optimally design and implement the TLD. Ultimately, the goal of the TLD, as is the case with any damper, is to consistently counteract as opposed to augment the dynamic response of the structure. Modeling a fluid motion is a complex task.

This task is further complicated by the

introduction of boundaries, such as the sides of the TLD and, as a result, it is difficult to determine precisely how the contained fluid will respond. It is essential to investigate how the fluid will respond in order to design a TLD that will effectively mitigate the dynamic response of the structure. Current theory maintains that the force of the TLD is given by

(10)

&&(t ) + w w(t ) = − &x&(t ) , w

where the variables and relationships are defined in Table 5. Table 5: Definition of Relationships and Variables the Theoretical TLD Force

σ n = (2n − 1)π , an =

8

σ

2 n

t = time (s) ρ = 1000kg / m 3 , density of water

x(t) = 0.002sin(ωt) , TLD displacement(m) fn =

L

tanh(

hσ n ), L

hσ n M = ρLBh , mass of the water (kg)

L= 0.6m, length of the tank

B = 0.5m, width of the tank h = 0.2m, depth of the water &&(t ) + w w(t ) = − &x&(t ) , defining equation for ω = 0.94 – 1.04, frequency of applied w harmonic displacement displacement w n = 0.985 Hz The theory governing the force of the TLD is based on an infinite number of iterations, as n is varied from 1 to infinity. However, in calculating the force of the TLD in these experiments, only the maximum is considered. For this reason, the only case considered is n = 1. Using the relationships in Table 5 and Eq. 10, the maximum force of the TLD was calculated for the frequencies of 0.94 to 1.04Hz, which were increased in 0.01Hz increments. Based on the theoretical model, the maximum force of the TLD is 352.9263N, which occurs when the frequency is 0.99Hz. The peak appears to occur when the frequency is in the vicinity of 0.98 to 0.99Hz. Similarly, the experimental peak occurs at a frequency of 0.98 and 0.99Hz, when the maximum TLD force range is 328.125N. This can be seen plotted in Fig. 17. It should be noted that the experimentally calculated force is the sloshing force, which lacks the inertial force component, which differentiates it from the TLD force.

Maximum Sloshing Force vs. Frequency 400 350 300 250 Theory Experiment

200 150 100 50 0 0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Frequency (Hz)

Figure 17: Maximum Sloshing Force vs. Frequency. The solid line shows the maximum theoretical TLD force. The maximum sloshing force representing the experimental data is represented by the dotted line, and represents the range of the experimentally calculated sloshing force to account for the lack of symmetry.

Fig. 17 shows that the maximums of the experimentally obtained sloshing force and the theoretical TLD force are similar, in general shape only on the region of frequency from 0.97 to 1.00 Hz, which is only a small portion of the region considered. The shape is substantially different for the outlying data, with the experimental data consistently larger than the theory predicts. The explanation is unknown, but can most likely be attributed to equipment that picks up a significant amount of noise and is sufficiently sensitive. The high amplitude liquid impact or “slamming phenomena” may play a role in damping as well, but require further investigation since they were not considered in the experimentation (Yalla & Kareem 1999). Another possible cause for the difference is that, in the case of a TLD with small dimensions like the one used in this experiment, the frictional force is much larger than that of the slosh, making it difficult to determine accurately the force and motion of the water sloshing within the TLD.

Conclusions Research on the most effective and practical methods of dynamic response mitigation remains a work in progress. In the area of ice-induced vibrations, a significant area remains unexplored, although a good starting point is past research based on wind-induced and seismic vibration mitigation. The Tuned Mass Damper has proven effective at mitigating the dynamic response of structures, including the JZ20-2 MUQ oil platform in Bohai Bay. Numerical analysis was performed in MATLAB to better design the TMD to mitigate vibrations for this specific oil platform. The optimally designed

TMD will have parameters: tuning frequency ratio, f = 0.985 and the TMD coefficient of damping, ςd = 0.065, which relates to a stiffness, kd = 329633.4 N/m, and a damping of, cd =7.91*10-7 Ns/m. An alternative to the TMD or an option to supplement the TMD is the Tuned Liquid Damper, which uses the sloshing motion of a liquid against boundaries to provide a force capable of counteracting that of the induced force. Should a scale structure not be present to test the effects of the TLD on dynamic response mitigation, the sloshing force can be directly calculated using only the TLD. The maximum force of the TLD is found when the applied force has a frequency of 0.98 to 0.99Hz. The data found experimentally for the maximum sloshing force only roughly corresponds to that predicted by the theory, and further investigation is required. Overall, the TLD is a viable option for dynamic response mitigation, though research must first be furthered in order to reduce accurately the effects of ice-induced vibrations.

Future Research Future research will use the data collected in the TLD experimentation to determine the efficiency of the TLD by a method that has yet to be developed. One of the criterions is the TLD’s ability to diminish as opposed to augment the dynamic response of the structure. In addition, random ice loads and a time history of actual ice forces taken in the field will be applied and tested. The goal of this research is to ultimately determine an optimal design of a TLD to mitigate ice-induced vibrations as an alternative or in conjunction with TMDs. Research was simultaneously conducted with the TMD using a half simulation and half experimental approach to investigate the TMDs ability to reduce the effects of an ice-induced vibration on the jacket platforms.

Acknowledgements The author would expresses appreciation to Professor Zhang Yahui for his assistance and support during this project, as well as graduate students Liu Xiao Hui and Zhang Li for their assistance and allowing her to be a part of their research project. The author also thanks Dr. Hayley Shen and Dr. Hung Tao Shen for coordinating this research experience. This material is based on work supported by the National Science Foundation under Grant No. OISE-0229657. Any opinions, findings, and conclusions expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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