This document describes an experiment to understand hydrodynamic forces on a floating body. The author manufactured an experimental rig and selected transducers to measure displacement, acceleration, pressure, and create vibrations. Tests were conducted with and without water, and results compared to theoretical values using multiple scale analysis and the Duffing equation. The rig design achieved the desired results but could be improved for accuracy and real-world applications.

1. NONLINEAR HYDRODYNAMIC
FORCES
MANUFACTURE AND TRANSDUCER
SELECTION
By Sachindra Max
2010

2. Overview of the Project
• This experiment is to
understand the fluid
hydrodynamic forces acting
on a floating body.
• This is practically applicable
for borehole oil drilling in
offshore and onshore.
• Properties of Fluids inside
the bore will be able to
measure from this equipment.
2

3. Overview of the Project
• My main objective was to manufacture the experiment rig
and select transducers to it.
• Apart form that I compared the experimental results in
different conditions with a theoretically obtained values.
• Methods of multiple scale analysis is used to evaluate the
theoretical values.
Experimental Theoretical
Results Values
3

4. About bore hole drilling
• Bore holes may construct for may different purposes.
• In the Oil and Gas industry bore hole drilling is very important.
• Drilling is divided mainly in to 2 categories.
1. Onshore Drilling 2. Offshore Drilling
4

5. About bore hole Drilling cont..
• This technology has a history of hundreds years.
• Drilling methods have been developed over the time for
different purposes and environments to increase its efficiency.
• Few main drilling types,
I. Auger drilling
II. Air core drilling
III. Cable tool drilling
IV. Diamond core drilling
V. Hydraulic-rotary drilling
VI. Sonic (vibratory) drilling / Resonance enhanced drilling
5

6. Manufacturing of apparatus
6

7. Experimental rig – Manufacturing
• The rig is adjusted couple of times to achieve the desired
experimental results.
7

8. Transducer Selection

9. Transducer Selection – Linear variable
displacement transducer
• Linear variable displacement transducer (LVDT) is used to
measure the amplitude of the oscillation.
• Change in position vertically of the actuator rod measures the
amplitude of the oscillation.
• This device uses the eddy current principle to get the
measurement.
• Accuracy of the LVDT used is sufficient enough for the
experiment.
9

10. Transducer Selection – Accelerometer
• Acceleration of the inner cylinder is measured by using this
device.
• Device is connected to a oscilloscope or to a computer
software to get the experimental results.
• Fixing this unit to the experimental rig is easy and accuracy is
high enough for the experiment.
10

11. Transducer Selection – Pressure
Transducer
• Pressure transducer is necessary to fix to the bottom of the
large cylinder to measure the variable fluid pressure due to
the impact of the actuator.
• This can be connected to a oscilloscope or to a computer
software just like the Accelerometer or LVDT.
• Similarly fluid level can be measured by using the basic
hydrostatic pressure equation.
P = hρg
• Suitable transducer for this experiment has a pressure range
of 0 – 6 bar
11

12. Transducer Selection – Vibro impactor
• This device creates the vertical sinusoidal force to vibrate the
system.
• The metal bar is placed in a solenoid which creates a magnetic
flux according to the current supplied.
• This is a very simple design and it’s the most appropriate for the
experiment.
12

13. Transducer Selection – Motor selection
• Initially experiment rig was supposed to design with a electric
motor instead of a vibro impactor.
• Setup of the motor could be done as shown above.
• Main disadvantage of connecting a motor to the experiment is
that it can create vibration NOT only vertically but also
horizontally.
• Therefore it can create unnecessary friction to the vibrating
system.
13

14. Arrangement of the experimental rig
14

15. Experimental rig – Signal generation /
inputs
Signal Vibrating
Transformer
Generator system
• A Sinusoidal current is generated from a signal generator and
sent it to the vibro impactor, solenoid of the impactor creates
a sinusoidal magnetic flux in the same frequency as the
current. Therefore the metal bar will vibrates at the same
frequency as the current.
• Amplitude of the sinusoidal motion of the metal bar is varied
by varying the supplied voltage from the transformer.
• Best results of the experiment can be achieved by setting the
vibrating frequency to 6Hz and voltage to 12v.
15

16. Experimental rig – Results / data
outputs
Oscilloscope Vibrating
Transducers
/ Computer system
• Output signals from all the transducers are connected either
to a oscilloscope or to a computer.
• Graphical representation of the data can be filtered well only
in the computer software.
• Oscilloscope used to show the general idea of the motion
graphically.
• Experimental data is taken at different conditions such as with
water and without.
• Data is recorded as a function of time from the computer
software and then plotted as shown in next slides. 16

17. Data acquisition for various cycles
17

18. Water damped system with the LVDT :
Displacement Vs. Time
18

19. Water damped system with the
Accelerometer : Acceleration Vs. Time
19

20. Mathematical model for the
experimental results
• Experiment rig can be simplified to a simple spring damper
system as shown above.
• Springs of the rig will act same and water of the system
behave as a damper. 20

21. Transient motion of the system
without water
• Suspended system is lifted up n released manually with a
magnetic system and let the system vibrate freely with no
external forced applied.
• Equation of the acceleration Vs. time graph of the system can
assumed as;
1
• Oscillating Velocity of the system can be obtained by
integrating the equation 1
2
21

22. • Graphical representation of the equation 2
Velocity Vs. time – without water
( in a time interval of 0 – 2 seconds )
22

23. • Equation 2 can be integrated once and obtained the displacement
Vs. time equation
3
Displacement Vs. time – without water
( In a time interval of 0 – 2 seconds)
23

24. Theoretical model for the system
24

25. Method of multiple scales
• This is a perturbation method.
• Duffing equation is used to evaluate the theoretical values.
• This equation can be used to this experiment since it’s
amplitude is low and it can be used for water damping
systems.
• Duffing equation :-
• This equation does not contain a damping term. There for its
modified with an additional damping term.
• Duffing equation with damping term :-
25

26. Duffing equation
• Substitutions;
• After the substitution coefficients of each power of ϵ is
separated;
• Solutions for u0 , u1 , u2 obtained.
• u is obtained by substituting u0 , u1 , u2
• Where;
26

27. Comparison
• Data obtained experimentally and numerically needs to
compared.
• Amplitude of the oscillation will be lower in experiment
values due to the friction acting at contacting points.
• The values can be compared as shown in the next slide.
27

28. Graphical comparison
Experimental results Theoretical values 28

29. Conclusion
• Properties of water such as the damping term can be
evaluated by comparing the two methods.
• Accuracy of the system can be increased buy remanufacturing
some parts. And by using proper rigid connections.
• Experiment rig needs to be modified further for real life
applications.
29

30. Thank you.
Any Questions ?
30