UNI VERSITETET I TRONDHEIM
DEPARTMENT OF MARINE TECHNOLOGY - THE NORWEGIAN INSTITUTE OF TECHNOLOGY THE UNIVERSITY OF TRONDHEIM
MARINTEKNISK SENTER
MARINE TECHNOLOGY CENTRE TRONDHEIM. NORWAY- b
tab. y.
Schee.psbouwkunde
Technische HO9SChOOI UR-86-52
e
Force measurements in
oscillating flow on ship
sections and circular
cylinders in a U-tube
water tank
BY
BJØRN SORTLAND
ON SHIP SECTIONS ANO CIRCULAR CYLINDERS IN A U-TUBE WATER TANK
by
Bjern Sortland
Trondheim, October 1986
Division of Marine Hydrodynamics The Norwegian Institute of Technology The University of Trondheim.
ABSTRACT.
A U-tube water tank with a drive mechanism to produce stable oscillations of the water has been designed. A force measuring system capable of measuring in-line and transverse forces simultaneously on a section of rigid cylin-ders of arbitrary shape has been built up, and tested by force measurement on circular cylinders. The force measuring system has been used to measure sway damping on a midship section. The measuring section was placed upside down on a false bottom simulating the free surface.
Flow visualization with hydrogen bubble techniques and polystyrene particle has been tested and found useful. A special light device has been designed for this purpose.
CON TENTS
Page Nomenclature
Introduction 4
1.0 Experimental technique and equipment 6
1.1 The U-tube water tank 7
1.2 Driving mechanism and wave probe 11
1.3 Characteristics of the tank 13
1.4 Force measuring system 16
1.5 Force transducers 19
1.6 Dynamic response of the force transducers 22
1.7 Blockage effects 26
1,8 Data acquisition system 27
2.0 Force coefficients 30
2.1 In-line force coefficients 30
2.2 Transverse force coefficients 32
2.3 Governing parameters 33
3.0 Data logging
3.1 Calibration of forces 35
3.2 Calibration of the wave probe 38
3.3 Location of oscillating cycles 38
3.4 Maximum fluid velocity 39
3.5 Calculation of force coefficients 41 3.6 Output from the calculations 43 3.7 Testing of the computer program 43
Page
4.0 Force measurements on smooth circular cylinders 45
4.1 Circular cylinders 45
4.2 Force measurements 46
4.3 Inline force coefficients for smooth
circular cylinders 53
4.3.1 Inertia and drag coefficients 57
4.3.2 The rms coefficients of the in-line force 64
4.3.3 The maximum in-line force coefficient 64
4.3.4 Fourier components of the in-line force 65
4.4 Lift force for smooth circular cylinders 66 4.5 Effect of air bubbles on the measuring cylinder 71
4.6 Summary of the results for smooth circular cylinder 72
5.0 Measurements of sway damping on Ship sections 74
5.1 Experimental set up 75
5.2 Measurement on a midship cross-section 77
5.3 Inertia and damping coefficients 78
6.0 Flow visualization 82
6.1 Hydrogen bubble technique 82 6.2 Visualization with particles 83 6.3 Visualization with fluorescent additive 85
7.0
Conclusion
86Acknowledgement 88
References 89
B beam of the ship section
Ca added-mass coefficient, CM - 1
C0 drag coefficient from Fourier analysis (eq. 8), for
cir-cular cylinders based on the diameter 0, and for ship sec-tions based on the draft D, see eq. 7
Cols drag coefficient from least square method (eq. 9)
CFmaX maximum in-line force coefficient (eq. 10)
CFrmS root-mean-square in-line force coefficient (eq. 11)
CL lift force coefficient used in a quasi steady model eq. 22(
CLmax maximum lift force coefficient (eq. 12)
0Lrms root-mean-square lift force coefficient (eq. 13)
CM inertia coefficient, CM = Ca + 1, from Fourier analysis,
for circular cylinder based on the volume of the cylinder, and for ship section based on the volume given by B D L,
see eq. 7
s inertia coefficient from least square method (eq. 9) D diameter of the cylinder
D draft of the ship section frequency
filter cut-off frequency
F in-line force
F0 maximum drag force given from C0
FL lift force (transverse force(
FLm measured lift force
FM maximum inertia force given from CM Fm measured in-line force
Q acceleration due to gravity k structurai stiffness
k roughness height, average grain size k1,k2,k3,k4 force calibration factors, (eq. 17) KC Keuleqan Carpenter number, Um.T/0
L length of the measuring section
m mass of model
ma added mass of the model Reynold number, U5D/v
S Strouhal number, fLO/Um
T period of oscillation (natural period of the water in the
U-tube)
Um amplitude of the free-stream velocity u instantaneous free-stream velocity
W width or height of the test section
x instantaneous deflection
a phase angle between the zero crossing and the first sampled point of the displacement signal in each period found from the Fourier transform
frequency parameter, D2/uT
phase shift of each n-th harmonic introduced by the analog filter
phase angle introduced by Che discrete Fourier transform
(eq. 18)
e
2t/T
V kinematic viscosity coefficient p density of water
phase angle (eq. 22)
INTRODUCTION
To predict the slow-drift oscillation of moored structures in irregular seas, more information about the hydrodynamic damping is needed. Based on the results from theoretical models and the correlation with experimental data, attempts are made to establish a procedure for estimating slow-drift damping of different ship forms. This will be combined with an established procedure for calculating the slow-drift excitation forces in irregular waves. The results will be given as slow-drift response of a moored ship in irregular seas.
Work with numerical models always creates the need for more basic hydro-dynamical data which may be found from experiments. Free decay tests with a moored ship have been performed, but it was shown by Faltinsen, Dahie and Sortland [i] that it is difficult to obtain slow-drift damping terms from that kind of tests, particularly for sway and yaw motion. A U-tube
oscillating water tank has been found to be more convenient for the experi-ments needed here. The tank can be used to study two-dimensional ship sec-tions by measuring damping coefficients and visualizing the flow. Flow
visualization is an important basis for physical explanation of flow pherlo-mena and for support of the numerical calculations.
The work reported here describes the design and construction of the U-tube oscillating water tank, the drive and control mechanism and the force mea-suring system. A major part in the design has been to produce harmonic
fluid motion which is capable of staying stable with a Constant amplitude for a long period of time. The force measuring system is designed for mea-suring in-line and transverse forces simultaneously on fixed rigid
cylin-ders. The diameter and length of the measuring sections can be changed. The force measuring system is fixed outside the tank and can easily be mo-ved down to the bottom of the tank. In this position, forces on two-dimen-sional ship sections in transverse oscillating flow, corresponding to sway motion, can be measured.
Force measurements on smooth circular cylinders have been compared with published results from other U-tube water tanks. This was done to test the force measuring system and the computer program evaluating the force
Different kinds of flow visualization techniques have been tested. Visuali-zation with hydrogen bubble technique and white polystyrene particles seems to work well in an oscillating water tank. Fluorescent additives have also been tested and are useful for special cases with small velocities.
This report is the main part of a dring. thesis which also include ref.
[iJ
and [31] and a short introduction and a summary in Norwegian. Thisreport is also an expanded version of ref. [2] hich is completely con-tained in in this report.
1.0 EXPERIMENTAL TECHNIQUE AND EQUIPMENT
When studying forces on bodies in oscillating flow there are two main types of experimental rigs used in laboratory investigations. In first one the body under test is made to move in some predetermined manner in a fluid otherwise at rest, and in the other the body is kept stationary and the fluid is made to move. Our requirement for the experimental rig is that it should be capable of producing harmonic flow or movement with a wide range of stable amplitudes. Flow visualization and force measurements must be
possible.
Mathematically, there is no difference between oscillating flow about a body at rest, and oscillating the body in a fluid otherwise at rest. However, one has to remember that the inertial coefficient CM for the fluid
accelerat-ing about a body at rest is equal to CM = i + Ca, where Ca is the added
mass coefficient.
Although both these types of experimental rigs may give the needed results, there are on the other hand significant differences between the difficulties encountered in the two situations. Both methods have been
tested and reported and the main advantages and disadvantages of the two types of rigs are mentioned below.
a) Oscillating the body in a fluid otherwise at rest.
The main advantages of this system is that the amplitude and frequency may be varied independently. Then the effects of Reynolds number or of Keulegan-Carpenter number may be studied separately. Usually a higher Reynolds number may be achieved than when the fluid is oscillated in a U-tube os-cillating water tunnel. When oscillating the body, however, the inertial force due to the mass of the body must be subtracted from the total force giving the added-mass coefficient. This is usually not so difficult to do, but special care has to be taken. The main disadvantage of the system is the vibration problem, particularly at low frequencies. The mechanical complexity of such a system also has to be taken into account. Forced os-cillations of the body are usually done in an open-water tank. Then the pos-sibility of free-surface effects on the measurements, which are difficult to assess, also have to be taken into account.
b) Oscillating the fluid about a body at rest.
The main disadvantages of this system are that the oscillations are usually at the natural frequency of the water in the system, and thus the frequency of oscillation is fixed. Consequently the Reynolds number and the Keulegan-Carpenter number cannot be varied independently. Forced oscillations, off the resonant frequency, could be performed. The power required to do so would, however, be much greater compared to when the fluid is oscillating
at its natural frequency, and the complexity of the system will increase. The main advantages of a free oscillating system like a U-tube water tank is that there is less mechanically generated noise. This kind of measuring rig is simpler and thus easier to maintain, and gives easier access to the measuring section.
Based on these advantages and disadvantages for the two types of experi-mental rigs and the fact that the University had started building a small U-tube oscillating water tank, it was decided to use this facility. The tank had to be equipped with systems for steady oscillations force
measurements and flow visualization. The design and construction of this U-tube water tank is reported in the next sections.
1.1 The U-tube water tank
Important factors of the design of the U-tube water tank to be considered are the geometry of the corners, the length of the working section and the height of the upright arms. The overall size of the tank, and in particular the size of the working section, are also very important. From an energy con-sideration it can be shown that the natural period of oscillation of the
liquid in a U-tube of arbitrary shape is given by
T = 2ir 1/
ii±a
2gpwhere the parameters mentioned are given in figure 1.1 below. (1)
Figure 1.1
Geometry of a U-tube water tank.
From this equation it is shown that if the tank size is increased by a factor of k (geosim), the natural period will increase withViz. A
corres-ponding increase in amplitude of oscillation and measuring section diameter gives a increase in Reynolds number with a factor of k3/2.
The size and shape of the different parts are also important. If the bends are too tight, separation can occur, which will introduce disturbance n
the flow. The height of the upright arms has to be greater than the double oscillating amplitude such that when the water is at its lowest operating level, it is still above the entry to the corners.
The most important part of the tank is nevertheless the working section. A
wider working section enables a larger body to be tested, and hence a higher Reynolds number to be achieved. The length of the working section has to be larger than twice the actual amplitude of oscillation to ensure fully developed uniform flow at the test section. The height p of the working section could be made less than the width h of the upright arms, to obtain the actual amplitude or velocity of oscillation higher than that of the free surface. See figure 1.1. Such a construction would on the other hand give a higher risk of separation occurring in the bends.
26TO
Figure 1.2
Main dimension of the U-tube water tank The tank cross section is 400 mm square inside. It is made of $ mm water resistant aluminium alloy, with three 20 mm thick plexiglass windows in the working section
and one in the right upright arm.
The main dimension of the U-tube water tank is given in figure 1.2. The tank cross-section is 400 mm square inside. The tank is built in five modular sections. Each of them was made of 8 mm thick water resistant aluminium alloy, welded together, and with 8 nm flanges for assembly. The modules are bolted together using a rubber seal between adjacent flanges. This construction allows the tank to be dismounted and moved easily. On the other hand missalignment between adjacent sections has to be checked care-fully because discontinuity of the inner surface will cause separation and thus affect the velocity-field substantially.
The tank lies on a wooden bed to reduce vibrations from the surroundings being transferred to the rig. A wood stiffener on each upright arm has later been glued on to the aluminium to stop these faces from flexing. Measurements have shown that those stiffeners have helped to reduce struc-tural low frequency fluctuations.
Measurements of structural vibrations have shown that there are no need for more stiffeners. The main part of the vibrations left is introduced from the driving fan with frequencies above 35 Hz. Those high freouency vibra-tions are far from the frequency of interest for the force measurements and will therefore be taken away with filters.
One of the upright arms has a small plexiglass window about the mean water level. This window is used for inspecting the water surface for waves and ripples when it is oscillating. lt also helps in calibrating the con-ductance wave probe, which is used for measuring the instantaneous water surface level and the oscillation amplitude.
The working section has three 20 mm thick plexiglass windows, one on each side and one at the top. The top window is removed when changing models or when cleaning the tank. All the windows are constructed such that they are flush with inside of the tank. The windows have a rubber seal preventing leakage and are bolted on. Removing or installing the windows take a few minutes. The windows make it possible to do force measurements and flow visualization simultaneously.
To prevent corrosion and maintain a uniform inner surface, the inside of the working section has been painted. The black colour also gives a good contrast for flow visualization. The roughness of this surface is estima-ted to k=0,02 mm, average grain size, from a table given by Hoerner [3]
L
o
o
0O
I
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1.2 Orivinq mechanism and wave probe
A conductance wave probe is both used to measure the instantaneous water level and to control the U-tube drive system. This probe consists of two 3 mm stainless steel rods connected to an amoliuier. The change of conduc-tance between the two rods is measured when the water level is changing. The wave probe is fixed in one of the upright arms of the tank, see figure
1.3.
Figure 1.3 U-tube water tank
In order to achieve continuous oscillations with stable amølitudes of the water in the tank, a fan mounted on the other upright arm is used, see figure 1.3. The fan is run periodically to counteract the damping. It is
switched on and off by a fan-controller connected to the wave probe ampli-fier, see fig. 1.4. The fan-controller is trigged by negative-going zero crossing (i.e. negative gradient) of the wave probe signal, (because the wave probe is in the opposite upright arm of the tank). After triggering,
Wave probe
Main body in aluminium Wood stiffener
Plexiglass window Force measuring system Plexiglass window Honeycomb section Driving fan Wood bed
Amplifier Fan -controUer Driving fon W.V tnaginq point waiting runl-nng me time
Figure 1.4
U-tube driving system
the fan-controller waits a predetermined (adjustable) period of time for
the wave to rise. The fan is then switched on at full power and runs for
another predetermined adjustable period of time. The amplitude of
oscilla-tion is adjusted by the running time, and the adjustable waiting time is to
get the fan running when the water is aoout at its top level.
The fan itself is of a very low power rating and takes some time to
acce-lerate to full speed.
When running the tank at small amolitudes of
oscil-lations the fan runs at low speed.
The fan is mounted on a cover plate.
When switched off, the air escapes between the blades of the fan, which
then is free to rotate in either directions.
This method increases the damping considerably. To account for this, a
seoa-rate air escape opening with a Flap was made.
The flap was opened when the
fan was switched off, but this system did not seem to have any noticable
effect on the oscillations of the water.
Below the fan a honeycombed section is mounted, see figure 1.3, to reduce
the velocity fluctuations in the air-jet from the fan. The water surface
below the fan has been observed when the tank is run on its maximum
ampli-tude, Only negligible ripples on the water surface have been found. Waves
generated on this water surface would have a considerable effect on the
velocity in the tank and have to be prevented.
water surface at the wave probe
t me
water surface
at the fon
1.3 Characteristics of the tank
In all the experiments the still water level chosen was 0.30 m from the top of the tank. At this level the oscillations have a period of about 2,86 seconds. From equation (1) the natural period of the water in a U-tube of constant cross-sections is given by
/B+2H 2m P T = 2m
n 2g - g
where B and H is given by figure 1.1 and L is the overall length of the tank. By measuring L along the centerline of the tank this equation gives a theoretical value of 2,86 seconds. The period of the tank is very close to
the theoretical natural period.
The maximum amplitude of oscillation of the water in the tank is about 240 mol, but for big cylinders it would be a little less because of the larger damping. In those cases a separate air escaoe opening as mentioned above could help with getting larger amplitudes.
The minimum amplitude of oscillation depends on the wave probe amplifier and the fan controller. The fan controller needs a minimum signal of ± i Volt to work, but with an extra amplification of the wave probe signal it has been possible to run the tank with an amplitude of oscillation less
than 10 mm.
The damping of the oscillation of water in the tank is shown in figure 1.5. This trace is taken from the moment the fan was turned off, and shows that it takes about 15 cycles for the amplitude to fall to half its maximum value. Obviously the damping arises from the drag on the model and fric-tion at the walls, and is therefore not constant, and will be smaller at the lower amplitudes. In this particular case there was no model inside the working section.
't -C
-9
d CE
Time [sec] Fig. 1.5Damping of the oscillation of the water
In order to examine the stability of the oscillation of the water, the dis-placement signal (calibrated wave probe signal) has been Fourier trans-formed. As shown later no significant high order term has been found, leaving a pure harmonic oscillation of the water.
However, the displacement signal shows an offset in the mean water level between when the tank is at rest and when it is running. This offset is due to the one-sided pressure introduced from the driving fan. It is believed that this offset does not affect the velocity in the working section of the
tank.
The breathing (vibration) of the tank walls when the water is oscillating has been measured, and found to be very small.
The uniformity of the velocity distribution at the test section has been checked by Kjeldsen [] who has used the tank for calibration of an ultra-sonic current meter. -le found that the vertical velocity profile in the center of the test section was almost constant with a standard deviation of
about 1 from 8 different measurement-heigths.
The uniformity of the velocity has also been checked by use of the hydrogen bubble technique. Pictures have been taken of bubbles generated from a vertical wire mounted in the center of the test section without any model
present. From those pictures the velocity seems to be uniform except for a
very small region close to the wall, i.e. the boundary layer. The influence of this appeared to be in a region less than 10 mm from the wall.
A theoretical estimate of the influence of viscosity in a periodic 2-dimen-sional flow can be found from an exact solution of the Navier-Stokes
equ-ations. From Schlichting [5], the depth of penetration, which is defined by one wave length (Or a phase lag of 2ml of the motion into the fluid. is
written
/ 2v
y = 2m
V
-w
n
for laminar flow. Here y is the kinematic viscosity and w the frequency of oscillation. In our case, this equation gives a depth of penetration of about 6 nm. At this distance off the wall, the velocity reduction is about 2 0/00. A boundary layer defined by i velocity reduction will be about 4
mm.
We have a painted aluminium surface where the roughness is estimated to k=0,02 mm average grain size. For design purposes, Jonsson [6] suggests RE=105 as a laminar to turbulent transition value for oscillating flow along a smooth surface like this. RE is the amplitude Reynolds number defined as
2
w a
RE n 1m
V
where aim is the free stream particle amplitude. In our case for a maximum amplitude of 0,24 m we get RE=1,1.105. From this point of view we can pro-bably have a turbulent boundary layer for the large amplitudes.
For oscillating Smooth turbulent flow, the boundary layer thickness is given from Jonsson as
0,0465 - 1O}/ aim
which gives a value of 3,5 mm for the maximum amplitude of oscillation. Here 46 corresponds to the depth of penetration y defined by Schlichting.
1.4 Force neasurinq system
A complete force measuring system has been designed and constructed. The system used at Imperial College, reported by Singh [7], has been studied closely. The experience from this system and the requirement for damping coefficients on two-dimensional ship sections, have formed the basis for
the design. Both Sarpkaya [8] and Singh [fl measure either in-line or transverse force alone, although it would be of great interest to measure both forces at the same time.
The basic requirement for this system was a system which could measure in-line (drag) and transverse (lift) forces simultaneously, although this would make the system more complex.It was believed to be possible to
achieve this on account of the experience with force transducers at MARINTEK. The measurement should be performed on either side of fixed
rigid cylinders on which the diameter and shape could be changed.
It was decided to fix the force measuring system outside the tank for two
reasons. The breathing of the tank walls, when the water is oscillating, would then not affect the measurements, and it would be possible to do flow visualization through plexiglass windows during the measurements. If the
force measuring system had been fixed to the windows, experience has shown that it would be necessary to change the plexiglass to reinforced alumi-nium to stop the faces from flexing.
The final design of the force measuring system is shown in figure 1.6 with a circular cylinder mounted. A dummy cylinder on each side of the
measuring cylinder goes through the plexiglass window and is fixed outside
the tank. This supporting structure is very strong so as not to allow any noticeable deflection to take place. It is bolted onto the U-tube on the underside of the measuring section. By use of different spacers, the height of the measuring system above the bottom of the U-tube can be adjusted. The size of the two-dimensional measuring section is adjustable from a circular cylinder with minimum diameter of 20 mm (the dimension of the force transducer) and up to a circular cylinder of 100 mm. The cylinder could be given any shape within those boundaries.
The force transducers are mounted inside the dummy cylinders where only a thin pointed finger with 7 mm square cross section protudes. The force
transducers are made in one piece ending in a steel cone. This cone fits into a conical hole in the measuring section with a slightly different angle to prevent any moment from beeing transferred from the measuring section to the force transducers. The system is identical on each side of the measuring section, which then is kept in position by a light pressure from the steel cones. The only forces transferred to the force transducers are then the in-line and transverse forces.
The design chosen also gives the opportunity to change the length of the measuring sections. This is interesting for studying the correlation length of the transverse force. On the other hand the gap between the measuring-and the dummy-section has to be extremely small in order not to affect the flow around the cylinder especially for short measuring sections.
The deflection of the measuring section is only due to the deflection of the two fingers on the force transducers and is measured to be less than
0,1 mm. The velocity introduced from this movement of the section, even for the lift force where the frequency of oscillation is higher than the particle frequency, has only negligible effects on the measurements. These
imperceptibly small movements are not believed to give any lock-in effect
either.
The gap between the dummy and measuring section is adjusted by nuts on the force transducers. The gap has to be large enough to allow the deflection of the measuring section take place; on the other hand it has to be so small that no effects on the velocity distribution around the cylinder are introduced. It is found that a gap adjusted to about 0,1 mm will not affect the measurements on account of the boundary layer thickness calculated in the previous section.
The length of the measuring section can be changed by sliding the dummy back and forth. When changing shape or diameter on the Section, the force transducer is removed to the new section, and a new ring around the Section through the plexiglass window is nade. The measuring sections have been made of both stainless steel and plexiglass and are usually filled with
water.
In an early version of the system the rotating moment was also transferred to the force transducer by the means of a steel pin through one of the
+
0
LS
o
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/
&1oI
'A
r
p.u.
Q
o
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Figure 1.6 Force measuring system1. Fixing clamp 9. Spacer
2. Support 10. Spacer
3. Plexiglass sleeve (seal) 11. Shimring
4. Disk 12. 0-ring
5. Nut 13. 0-ring
6. Dummy cylinder 14, Plexiglass nindoc
7. Nut 15. Force transducer
steel cones. Those moments on the circular sections tested were however too small to be measured by the force transducers.
The steel pin was then removed and the section kept from rotating by use of a tape between the dummy and measuring section inside the cylinder. The tape, however, introduced a small effect on the force measurements which was shown when calibrating. The use of a steel pin through the cone of the force transducer however, gave no effect when calibrating, but when running an experiment the cylinder flipped back and forth )rotated a small angle), because there had to be a spacing between the steel pin and the
corresponding fixing system on the measuring section. Different methods have been tried to stop this rotation, but with no success. The best system for stopping this rotation seem to be by the friction between the
steel cone and the hole in the measuring section, but this friction seems to be too small for large KC values.
1.5 Force transducers
The design of the force transducers must incorporate such features as high natural frequency with the model connected, linearity, imperceptibly small deflections, sensitivity and mechanical strength for any expected load. The requirement of sensitivity depends on maximum and minimum load. Because different size of cross-sections are to be used, it has proved to be somewhat difficult to achieve a reasonable sensitivity for all loads. The
measurement of rotating moment on the measuring sections was also
attempted, but it was later shown that these moments on circular cylinders were far below the sensitivity of the force transducers which it was possible to achieve.
Another problem was how to protect the strain gauges in the force trans-ducers from water. When doing flow visualization with hydrogen bubble technique, acid was put into the water. The standard coating used then proved to be not good enough for more than a short period of time. The in-line and transverse force transferred to the force transducer are measured by means of Strain gauges. The strain gauges are mounted on a membrane at the neutral axis in each of the two holes in the thin pointed
finger of the force transducer, see figure 1.7. The two holes which are identical are used for measuring forces in each directions. This system is called the shear force transducer system.
Figure 1.7 Shear force transducer
Another aesign of the thin pointed finger has also been used, called the moment force transducer system and shown in figure 1.8.
Figure 1.8 Moment force transducer
This system measures the bending moment from forces in the two directions. The strain gauges are mounted on either side of the two membranes.
The Strain gauges used for the force transducers were a Micro Measurement product EA-13-062TW-120 with a resistance of normally 120 Ohms and K=1.995. After bonding to their respective surfaces using a Micro Measurement adhe-sive called M-Bound 610, they were first coated with a layer of Araldit epoxy and next a 3M 1706 protective coating. Upon this standard coating the whole finger of the force transducer were coated with Hottinger sealing
compound. Although this procedure is used to protect the strain gauges from water they, should be removed from the water when not in use, espe-cially when acid is added to the water.
The Strain gauges were then connected to an Hottinger Baldwin Messtechnik carrier frequency amplifier of type KWS 3073 to form a full bridge. The output from the amplifier is ± 10 Volt.
The first set of shear force transducers, made of aluminium, were found not to give the desired sensitivity for the actual load. The dimensions of the
transducers were already very small and it was not easy to make such a transducer with smaller dimensions. For this reason the moment force transducers were made with a much greater sensitivity, but then the deflec-tion of the measuring model also became much greater than when using the shear force transducers. The moment force transducers were also found to have some more cross-torque (a false force reading in horizontal direction when they were loaded in vertical direction and vice versa) than the shear force transducers, and both transducers suffered from some kind of drifting-in after they had been loaded or unloaded. (When a load was removed, the output from the amplifier did not go quite back to the initial value at once. The last part of about 30 nV it drifted-in over a period of atout 10 seconds.) Because of that, the response for a sudden change in the force was not so good, and it also affected the linearity of the
trans-ducers.
To get rid of the small but unwanted errors introduced by the force trans-ducers, a new set of shear force transducers was nade. Ideally, the deflection of the force transducer should be almost zero to keep the trans-ducers linear, but then a very high amplification is needed. The trans-ducers were made of a steel specially intended for strain gauges measure-ments, and the dimensions were kept to an absolute minimum. The strain gauges were only coated by 3M 1706 protective coating by a special proce-dure because the Araldit epoxy was found to give the unwanted drifting-in
problem. To protect the strain gauges from water, the hole in the force transducer where they were mounted, was filled with Hottinger sealing
com-pound. This sealing compound was thought to give some drifting-in, but only a negligible effect was observed.
The new shear force transducers were calibrated very carefully to check both linearity and cross-torque. The linearity for the actual loads was found to be perfect, and the cross-torque was about S mV in the opposite direction when the transducer was loaded to full output, 10 V, in one
direction. These new shear force transducers were somewhat more sensitive than the old shear force transducers and gave full output (10 Volts) from the amplifier, on maximum amplification, for a load of 1 Newton.
Attempts have been made to calibrate all the force transducers dynamically (not in water) by use of a rotating force and a spectrum analyser, but no phase shift could be found.
1.6 Dynamic response of the force transducers
Prior to a set of experiments on a model, the force transducers were care-fully mounted into the force measuring section, and calibrated without water in the tank. The gap between the dummy and the measuring section was
adjusted to about 0.1 mm. Special care was taken to avoid contact between the two sections. The calibration of the force transducers was then done by hanging loads on the cylinder and recording the output from the force transducer. From this output, calibration factors were calculated to give the component of the in-line and vertical forces on each end of the cylinder.
A dynamic calibration in water of the force measuring system is not easy to
achieve, Instead, a theoretical estimate of the response of such a system has been calculated.
The force measuring system with the force transducer and the dummy- and measuring cylinder is shown schematically in figure 1.9. The idealised system shown is a mass-spring system. The total mass of the system is the maas of the measuring section and the added mass due to water around the
model. The spring constant k (structural stiffness) is found by hanging loads on the measuring cylinder and measuring the deflection.
I
measuring
section
k
4 m
k2k: k1. k2
Figure 1.9The force transducer and measuring section may be regarded as a mass-spring system. For this system the natural frequency is given by
ik
w =v
n m+m a
if the damping is neglected. a and ma are the mass and added mass of the
cylinder.
In practice however, damping will be present, thus modifying the system's
response. The equation of motion of a damped mechanical system is given by
mx + B x + kx = F (3)
s
where a: mass of the system, B5: structural damping, k: structural
stiffneSs, F: external forces, and x is the displacement of the model. In
the present case, the external forces may be obtained from a small modifica-tian of Morison's equation. The external forces F on a circular cylinder of length L and diameter D is then given by:
2 F = pCDDL(u_) (u-k) I
4 M a
+p
LCu-m x
where u and are the fluid velocity and acceleration, respectively of the incident flow. The def1ecton x is very small, such that
> «
uSubstituting equation (4) into equation (3) and neglecting terms of higher order gives:
(m+m )i+ (B pC DLI ul )x + kx = 1pC DL u4 uJ
+ p
LC u (5)a S D 2 D 4 M
To simplify this equation, some assumptions will be made according to Singh
[7J.
The structural damping Bs is believed to be small and can be ignored in comparison with the hydrodynamic damping. If we let the damping be constant over a cycle, then C0 is the drag coefficient averaged over a cycle and ul may be replaced by Um, the amplitude of the fluid velocity.The right hand side of the equation (S) may be written as F0 sin wt Equation (S) may then be written
(m+m
(X +
pC DLU + kx = F sin wt (5)a D a o
(4) (2)
The steady State solution of this equation giving the dynamic response of the measuring Section in in-line direction may be written as
i.here
static deflection
dynamic magnification factor
=arctg
/1
2 w 2--) (2Ç)
ww2
n n PCDDL Um 2(m+m)wj
k C,)=Y
n m+m a a 788 g 2Ç-itO2= p-7--
= 565 g C0 = 1,5 U5 = 0,42 rn/S damping coefficient phase lag natural frequencyBefore the response of the system can be determined, the structural
stiff-ness k for the desired force transducer has to be found experimentally. The added mass coefficient is found from potential flow and the damping coefficient is based on the steady flow drag coefficient. This gives only
a rough estimate of the response of the system.
We consider a 200 mm long plexiglass cylinder of 60 rna diameter which is a typical size of a test section. This cylinder has a weight of 423 g and is filled with 365 g water. We assume a Keulegan Carpenter number of 20 giving a large velocity and amplitude of oscillation. For this cylinder we
then have: X X A sin
(wt
-F o X s k 1 A-The structural stiffness for the shear force transducer made of aluminium is found to be
k = 4 1O
m
The shear force transducer, together with the above mentioned cylinder, has the following properties:
= 544 rad/sec => f = 87 Hz Ç = 0,0051
The structural stiffness of the moment force transducer is found to be
k = 2 lO4
The moment force transducer together with the same cylinder then gives the following properties
= 122 rad/sec => f = 19,4 Hz
n n
Ç = 0,023
The oscillations of the water in the tank have a period of 2,86 seconds giving a frequency of 0,35 Hz. As shown there is almost no dynamic effect ori those two transducers for this particular cylinder, even for a excitation frequency 10 times this oscillation frequency. From Fourier
ana-lysis of the force signal it is shown that there is almost no contribution to the force above 3 Hz. The moment force transducer has a 3 effect for the dynamic magnification factor for this frequency. This transducer is on the other hand designed for small forces and will be used for smaller cylinders giving a higher natural frequency and then a lower dynamic magni-fication factor. The new shear force transducer made of steel has a structural stiffness a little less than for the old one made of
alumi-nium, and consequently the properties are not so far away from those found for this aluminium force transducer.
f[Hz] A
[0]
0,35 2,18 1.000 0,002 3,5 22 1.002 0,24 f [Hz] rad sec A[0]
0,35 2,18 1.000 0,047 3,5 22 1.03 0,5This analysis is obviously not exact, but it provides the worst possible response of the system. It shows that there is no need for an accurate dynamic calibration of the force measuring system.
1.7 Blockage effects.
To achieve high Reynolds number in the U-tube the diameter of the body has to be increased. This gives rise to wall-interference effects which influ-ence all kinds of measurements. The formulae used for blockage corrections in steady flow cannot be applied to oscillating flow, and there is no unique blockage correction for the entire period of the flow. This is evi-dent from the fact that within a given cycle the fluid undergoes varying accelerations and velocities in the wake width, and the wake pressure changes accordingly.
Sarpkaya [a] has made a set of experiments to determine the role of blockage in oscillating flow. From pressure measurement around a circular cylinder in a U-tube he concluded that blockage effect in harmonic flow is
negligible for 01W ratios less than 0.18. Here D is the diameter of the cylinder and W is the width of the test section. On the other hand if the flow had been steady with a velocity equal to the maximum velocity used in the oscillating flow, a 6 Correction in the drag coefficient should have been applied for D/W = 0.18, according to Sarpkaya.
Sarpkayas result was obtained from experiments with 13 = 8370 and Rn > 81O. It is believed that the blockage effects are a function of Reynolds' number and KC-number. On the other hand it is not obvious that Sarpkaya's result Can be fully achieved in this U-tube having another 13 value for the same blockage ratio.
The largest cylinder tested in this study has a diameter of 60 mm giving a blockage ratio of 0.15 and 13 = 1200. As shown later, the measured forces on
this cylinder seem to suffer from some kind of blockage effect for KC greater than 10.
1.8 Data acquisition system.
The force and displacement signals from the force and the wave probe amplifier respectively, in all 5 channels, are connected to the Ana-log to Digital converter (AID-converter) on a VAX 730 computer. The signals are in the range of ±10 Volt and the signal cables used is about 10 s
long. Just before entering the A/D-converter the signals are filtered using 4 pole lowpass Sutterworth filters. The use of filters are mainly to take away high frequency electrical noise introduced in the amplifiers and cables. To prevent aliasing or folding when digitising the signals, the sampling frequency has to be more than two times the highest frequency-component of the signal.
When the force signals are passed through the filters, higher frequency components will be shifted in relation to the wave probe signal according to figure 1.10. For this reason high frequency settings of the filters are to be preferred, but this will give a high sampling frequency. Then more space and computer time are needed for storing and doing calculations on the data. On the other hand higher frequency components do not contribute to the drag and inertia coefficient. It is only the maximum and root mean square of the Forces that can be affected. lt will be shown later that this phase shift can be accounted for, but it seems not to affect the results noticeably for the actual filter frequency.
Figure 1.10
Phase shift through a 4 pole lowpass Butterworth filter For different cut off frequencies The input signal Frequency f is nondimensional by the water oscillating
Usually the filters were set to 20 Hz cut-off frequency running the AID-converter on a sampling frequency of 100 Hz. (The filter- and sampling frequencies have to be the same for all channels.) This sampling frequency results in about 286 points per cycle. As the data consist of fairly low frequency inforniation, at first sight it would seem that using this samp-ling frequency gives an over-representation of the data and is too high.
However, it will be shown later that all those points on a cycle are needed particularly for the displacement signal giving the phase angle of integra-tion of the force coefficients.
When using 20 Hz filter frequency the amplitude of the main electrical noise at 50 Hz is attenuated to 0.7
0/00
of the input signal according to figure 1.11. From Fourier analysis of the force signal, it is shown that there is no contribution to the force signal above 4.5 Hz. For this reason no components of the force signal are attenuated or cut off by this filter setting.Figure 1.11.
Characteristic of a 4 pole lowpass Butterworth filter. tnput signal frequency f and cut off frequency
When running a registration, the data are just digitised and stored on the computer-disc. It is then possible to run very long registrations, because only the disc-capacity limits the registration length.
It is also possible to have analog outputs of the displacement and force signals. A light pen recorder can be connected to the wave probe and force transducer giving a simultaneous recording of the forces versus elevation.
2.0 FORCE COEFFICIENTS
The prediction of force and flow characteristics of a periodic and sepa-rated flow around a circular cylinder is a difficult task. One of the dif-ficulties arises in the description of the history of the motion and the effect of vortices. One way around these difficulties is to assume that the total time - dependent in-line force may be expressed as a sum of a velocity-square dependent drag and an acceleration-dependent inertial
force. This is the basis for the so-called Morisons equation.
Data reduction of the measured forces in-line with the direction of oscil-lation is herein based on Morisons equation and two different analyses of the force records, namely Fourier analysis and the least-square method. 2.1 In-line torce coefficients
The problem studied is a uniform harmonic flow about a circular cylinder placed with its axis normal to the flow. In this situation, the cylinder will experience a force in-line with the fluid motion which may be consi-dered to be composed of two parts, an inertia force due to the accelera-tion of the flow, and a drag force due to shear forces and separaaccelera-tion. When the flow separates, the shear force is of minor importance compared with the pressure drag force.
The in-line force which Consists of the inertia force F and the drag force Fd can be written in terms of Morison's equation,
F = F. + Fd = Vol CM + l p Ar CD u u (7)
Here CM and C are the inertia and drag Coefficients, U the instantaneous velocity of the flow. Vol = 1/4 irD2L for circular cylinder where L and D are the length and diameter of the measuring cylinder. For the ship sec-tion Vol = B'DL, i.e. beam draft, length of the ship secsec-tion respectively. Ar = DL where D is the diameter of the circular cylinder or draft of the ship section.
For an oscillating flow represented by u = - Um cos O , with O = 2ir t/T the
Fourier averages of C0 and CM for a circular cylinder are given by Keulegan and Carpenter [o] as
2m F cosO C = - m de D
8'
12
O-pULO
2m
U T 2m F sine a m J de it D O 1pU2LO2m
Fiere F5 represents the measured force. C0 and CM are the first two terms irr a complete series expansion of the normalized force and represent the mean value over one period of oscillation. Additional coefficients in the series may be calculated in the manner similar to that done by Keulegan and Carpenter.
Another ay is to evaluate the difference between the measured and
calcu-lated forces. By minimising the err-or between the measured force and the calculated force from equation (7), according to the method of least-squares, the following coefficients for a circular cylinder are found:
2m F cose .1 cosO 4 m CD1 =
-pULD
dO U T 2m F sinO C a a de C M1s 3 ' 1 2 - M riD O-pULO
2m
The Fourier analysis and the method of least-square yield identical CM values and the C0 values differ only slightly.
In addition to those cited above, the following measured maximum force coefficients are used
maximum of the measured force in a cycle
(10)
CFmax
- 1pU2 LD
2m
and the root-mean-square (rms) value given by1 m CF
/ T
F 2 = 1 2 dt O -pU LO2m
(9) and (8)2.2 Transverse force (lift) coefficients
As a result of the formation and shedding of vortices, circular cylinders in an oscillatory flow can experience transverse forces. This particular component of the total force has been of practical interest because under certain circumstances it can be comparable with, or even greater than, the in-line force. The transverse force can give rise to fluid-elastic oscilla-tions and since the frequency of oscillation is much higher than for the in-line force, it may have a major influence on the fatique life of some structural members. It has also been shown that even small transverse os-cillations of the body regulate the wake motion, alter the spanwise corre-lation and change drastically the magnitude of both the in-line and trans-verse forces.
In the present study no attention is given to structural movements. The
test cylinders are rigid and are held in position with imperceptibly small motions as described. Thus, in the following, we will only study force coefficients for rigid cylinders in uniform harmonic motion.
The transverse force may be analyzed in various ways. The two force
coef-ficients chosen here are the maximum lift coefficient defined by
r maximum peak of the transverse force
"Lmax 1 2
iP LOU
and the rms value of the transverse force given by
Lm CL
/ T
F (t) 2 = - dt 1pU2 L D2m
The time history of the lift force has shown to be very "irregular" and it is thus necessary to use the mean value of at least SO oscillations in ca)-culating the coefficients according to Sarpkaya [s]
2.3 Governing parameters
From dimensional analysis, the time dependent forces acting on a smooth cylinder in an oscillating flow may be written in non-dimensional form:
UT
UD
F m m t 1pU2LD- g)-6
u T2m
= g(KC RUT
UD
where we recognize KC = as the Keulegan Carpenter number and R = as the Reynolds number.
All the force coefficients mentioned above can then be written as a func-tion of the same parameters:
C = f (KC,R
0 1
nT
t
CM = f2(KC,R ,
and so on.
There is no simple way to deal with those time dependent coefficients even for the most manageable time-dependent flow. The evaluation of the instan-taneous values of the in-line forces is not always valid because of the random nature of the vortex shedding and the resulting asymmetry in the in-line force.
One method go is to eliminate time as an independent variable in equ-ation (15) and consider time-invariant averages of the force coefficients.
C0 = f(KC,R) CM = f2(KC,R)
and so on.
It has been found that the maximum amount of experimental control over the dimensionless variables can be obtained if the original variable which can be regulated, each occur in only one dimensionless product. In a U-tube water tank, the amplitude of oscillation and thus U5 is easily varied experimentally. Because of that, Sarpkaya [a] has defined a new parameter, called the frequency parameter as
2
Rn D
KC
-For a given U-tube the oscillation period T is constant. For a series of experiments conducted with a cylinder of a given diameter D in water of constant temperature, is held constant. Then the variation of a force coefficient with KC may be plotted for constant values of 5. Nevertheless the Reynolds number is not forgotten and can easily be founi from
= KC 5
The force coefficients herein is then analyzed according to the relationship:
00 = f1(KC,ß) CM = f2(KC,ß)
and so on.
3.0 DATA LOGGING
The computer programs for running registrations on the U-tube oscillating water tank and analysing the recorded time series are written in FORTRAN 77 and run on a VAX 730 computer. All registrations are run by a program which only reads the output from the A/U-converter and stores it on a disc-file as integer data in the form they are converted to by the A/D-converter. This representation takes half the disc space of data converted to Voit or physical units. It is then possible to run very long registrations, because
it is only the disc-capacity that limits the registration length. All kinds of analysis of the time series of the force and displacement
signals are then based on the integer data files stored on disc. Different kinds of analysis can be run on the same time series, and there is no problem in changing all the calibration factors if needed.
All analyses are run interactively and could be done just after each run. The user then has good control on how the experiment is running, and can take any action immediately in the calculations or running of the experi-ment if needed. All results are written both on the screen and to different files which can be printed out afterwards. If the integer data files are stored, parameter studies can be run later on the same time
series.
3.1 Calibration of Force.
The force measuring system in either the lift or drag direction is shown sche-matically in figure 3.1.
L.
L
FB = k3(VA-VAO) + k4(V8 -VBO)
The four calibration factors k1, k2, k3 and k4 for each force direction (in-line and transverse) can be calculated by using two different positions of the load P.
A calibration sequence, starting with a zero reading of the voltage from the two force transducers, and reading the voltage for two different positions of the load P, is run by the computer. From every sequence the calibration factors k1, k2, k3 and k4 are calculated, and all dats are then stored on a disc file. This sequence can be repeated a number of times for different loads. The weighted mean calibration factors are finally cal-culated from a series of calibration sequences.
F,gure 3.1
Forces on the measuring section
A load P on the measuring section in a distance 2 from side A can be divi-ded into two forces. FA and F8 at the ends of the measuring section. This
load P will give a voltage output from the force transducers VA and VB which is different from the zero reading VAO and V80. We can then set up two force equations:
FA = kl(VA-VAO) + k2(VB-VBO)
measuring
section
Force
transducer
(17)The weighted mean calibration factors are found by giving an offset of I gram the same thfluence on the calibration factors for ail loads. This is
À
F8
A
B
VB
VA
because the measuring error is assumed to be constant. If a simple Sean value is used, a 1% offset of the load will give the same influence on the caliDration factors for all loads. This means that the calibration factors found by using small loads have a larger influence on the mean value than for larger loads for a constant measuring error.
At the end, the difference between the loads and the calculated forces using the weighted mean calibration factors is printed out. The standard devi-ation of the difference between input and calculated loads gives a good
impression of the repeatability of the force measuring system. By plotting input and calculated force for the two force transducers, the linearity and hysteresis can be checked.
To check the properties of each of the force transducers, and to find the cross-torque a special calibration program was made. The program runs a calibration on only one single force transducer without the measuring
section. The force transducer was placed in a Special support where it
could be rotated and the angle of rotation could be measured. The
trans-ducer could be loaded in both directions simultaneously during the calibration. From this test both linearity and hysteresis could be found. By loading the transducer in one direction, the output from the amplifier in the other direction was watched when the transducer was rotated. The force trace from this test was found to have a sinuous shape when going through zero, with a very small offset of about 5 mVolt (cross-torque) for a maximum output in the opposite direction. A small difference in the angular direction would then decrease the output considerably. This
decrease is not cross-torque, but the real component of the actual force in this direction.
From this study it was shown that the force transducers acted perfectly when loaded in both directions simultaneously. The calibration factors k1, k2, k3 and k4, from equation (17) for the force transducers mounted together with the measuring section, were calculated from the calibration factors found by this last method, and the result was almost the same.
From this point of view, the calibration procedure when calibrating both force transducers with the measuring section mounted seem to be sufficient when changing only the measuring sections.
3.2 Calibration of the Wave Probe.
The wave probe is calibrated by reading the voltage output from the wave-probe amplifier for different water levels. This is done by the wave wave-probe calibration subroutine. A straight line through the points calculated by least-squares method gives the calibration factor. At the end of the cali-bration sequence the measured- and the calculated points according to the calibration factor are printed out. This gives a good estimate of the accu-racy of the calibration factor and the linearity of the system.
3.3 Location of oscillation cycles.
All calculations of force coefficients are done for each oscillation cycle of the water. When calculating the drag coefficient, the phase of the flow velocity is needed and for the inertia coefficient. the phase of the flow acceleration. Both of those phases are found from the displacement signal. A sampling frequency of loo Hz gives about 286 points per cycle. To locate the start point for each cycle, the displacement signal is scanned for zero crossings. As the signal is digitally represented, we have to look for the first sample after the zero crossing. The location of a cycle in this man-ner will therefore suffer from a random phase shift, the amount of which will depend on the number of points used to represent a cycle. With 266 points per cycle this corresponds to a change of 1.260 between adjacent data points.
Even though the displacement signal is very smooth, there will always be some small ripples in the data points, giving the possibility of locating wrong zero crossing points. An offset in the position signal will also give wrong zero crossings. The fan blowing in only one of the two upright arms will introduce such an offset in relation to the position of the water when it is at rest.
It will be shown that a phase shift of 10 can be of Importance when inte-grating the drag and inertia coefficients. For this reason, special care has been taken when locating the zero crossings to avoid the problems men-tioned above.
Before starting the fan making the water oscillate, zero voltage out-puts for all channels are read by the computer. Such a zero reading can be
done before each run, but it is not necessary if there is no drift in the amplifiers. The zero voltages read are then stored on a disc-file together with the date and time for this registration. This zero reading is called static zero.
For location of the zero crossings a routine called start/stop routine is run. tt looks through all the data points for the position signal
locating every positive-going or negative-going crossings (i.e. positive or negative gradients). When approaching a zero crossing, a numerical filter with a cut-off frequency of 10 Hz is activated, smoothing the data points. To find the true zero point, a linear interpolation between adjacent points on either side of the zero is performed.
Output from this routine is the point number of the first sampling point after each zero crossing, the distance from the true zero point to this sampling point and the period time in seconds of each cycle. The standard deviation of the period time is usually 0.001 second giving a standard deviation in calculation of the zero point of about 0.0005 seconds or 0.060. When the Start and stop point of each cycle is known, the mean value in voltage from the beginning of the first cycle to the end of the last one is calculated for each channel. This mean value is called dynamic zero. This
is done by a mean value routine which also finds the maximum and minimum voltage in every cycle for every channel. Those values are checked against the limits of the A/il-converter (± 10.24 Volt), giving a warning if they have been exceeded.
The first time the start/stop routine is run for a measuring sequence, the static zero is used. It is usually re-run using the dynamic zero to correct the offset error introduced by the fan. When doing this it is not necessary to update the static zero too often because if any D.C. offset is intro-duced, it will be taken care of. The only problem is a significant drift in
the zero voltage during the run. As described later, any drift will be detected in the maximum velocity routine, and can be taken care of then.
3.4 Maximum Fluid Velocity.
The maximum fluid velocity in every cycle is calculated from the maximum amplitude of the displacement signal. This amplitude is found from a finite
discrete Fourier transform of the displacement signal performed in the maxi-mum velocity routine. The 5 lowest terms, written in terms of sines and cosines, are printed out for every cycle, and give a very good control of the water oscillation in the tank.
The constant term gives the different offset using static or dynamic zero, and any drift in those values. By use of dynamic zero this term is found to be less than 0.1 mm, and no drift has been observed.
The water in the tank is believed to be a harmonic oscillator. Thus the first order term should give zero for the cosine term and gives the amplitude of oscillation in the sine term, and no higher order components should occur. This has been shown to be not quite true and is mainly because of two reasons. First of all, the displacement signal is digitised starting with the first point after zero-crossing. The phase of the Fourier transform is in relation to this first point in the sequence, giving a phase angle in relation to the zero crossing point. The routine used for the discrete Fourier transform also assumes that the period T is equal to the number of points in the cycle N, multiplied by the sampling period AT. This is not true and gives a mean phase angle through the cycle which can be written
T-NET
[deg] (18)
36
T
For a period of 2.86 seconds and a sampling frequency of 100 Hz this phase angle will be _0.630 < < 0.630. Both the phase angles mentioned give a contribution to the first order cosine term of the Fourier transform. Secondly, the wave probe and amplifier will introduce measuring errors and noise resulting in high-order terms in Fourier transform. The amplitude of those high harmonics are on the other hand in the order of the fundamental and the phase angles introduced are then believed to be small and are not accounted for.
The amplitude of oscillation and phase angle according to the first point in every cycle is then calculated from the first order terms of the Fourier transform. This phase angle a has been corrected according to equation (18) and is in the order of 10, and differs from cycle to cycle. This phase angle is not exactly the same as found in the start/stop routine. The nain reason for this is that in the start/stop routine the displacement signal
is only smoothed just around the zero-crossing point but in the Fourier analysis the best fit of the Fourier components to the data point of the displacement signal for the whole period is found. This procedure on the other hand only uses points behind the zero crossing and will usually not fit to the end point of the previous cycle.
By observing the Fourier components, the displacement signal is found to be very close to a sineous wave with on1y negligible higher harmonics. Then
the velocity is believed to be harmonic and the force trace can be analyzed according to the procedure described earlier. The standard deviation on the velocity from period to period seems to be less than 0.5 of the velocity.
3.5 Calculation of the Force Coefficients.
When calculating the force coefficients it is possible to choose between static- and dynamic zero for the forces. The difference between those two
zeros is calculated for every test series and has proved to be in the order of 0.02 Newton. The difference differs from test series to test series, especially for the lift force, and is usually less for long test series. The difference between those two zeros only affects the maximum- and the
rms coefficients.
All the ftrce coefficients are usually calculated from a numerical
integration of the in-line and transverse forces. Those two forces are the sum of the same two forces from each of the two force transducers, Upon this signal the discrete Fourier transform of the in-line force is calcu-lated in this study.
From the Fourier transform of the in-line force signal in every cycle, the drag and inertia coefficients are calculated from the first order sine and cosine term. The phase angle between those two terms is corrected according to the phase angle a found from the Fourier transform of the displacement signal. The maximum- and the rms coefficients can also be calculated by use of for instance the 13 lowest terms of the Fourier transform. This implies that components with a frequency up to 4.5 Hz are taken into account. It has been found that there is no contribution to the force signal above this frequency. The in-line force as a function of time during the cycle is then written
a 13
F(t) + E acos(n8+a+ ) + b sin(nO+s+$
n n n
n= i
where an and bn are the cosine and sine terms of the Fourier transform respectively, and O = 2,rt/T, and e is the phase angle found from the Fourier transform of the displacement signal. is a phase shift intro-duced to the high frequency components of the analog force signal by the Butterworth filter. These phase angles are given by figure 1.10. The correc-tion due to the filter is very small for a cut-off frequency of 20 iz. This is due to no phase shift on the first order term of the Fourier transform, which is most significant, and small corrections for the terms which are of some magnitude.
The force signals could also be filtered numerically to take away high frequency noise before running the numerical integration. A numerical filter introduces no phase angle and could have a very steep response function. This has, on the other hand, proved to be unnecessary because the numerical integration itself acts like a filter smoothing away high frequency noise. The phase angles used are relative to the distance between the first sampling point and the zero crossing found in the start/stop
routine.
The transverse force coefficients are herein only calculated by a numerical integration. This is mainly to limit the computer time letting the analysis be run interactively between each run. In this study the drag and inertia force coefficients have been of most interest. On the other hand, Fourier analysis could be done for the forces from each of the four force trans-ducers gr any combination of them.
Calculation of force coefficients by the means of a Fourier analysis or numerical integration has shown to give almost the same results. On the other hand the more computer time-consuming Fourier analysis of the forces could be said to be unnecessary. Nevertheless the Fourier analysis gives more information and is therefore included for the most interesting forces or force-combinations in the actual study.
3.6 Output from the calculations
When analyzing the recorded time series the program is usually run interactively and then the output is written onto the screen and Onto dif-ferent data files simultaneously. The results can then be observed during the calculation and action taken if anything is going wrong. Afterwards the results can be printed out from the data files. An example of an out-put from the analyzing program is found in the appendix.
The results are also tabulated into different files and those files can be printed or plotted out after a set of runs, giving the force-coefficients as a fntion of KC or R0 value.
Any combination of drawings of the different forces and the displacement signal can be made. The calculated drag and inertia coefficients can be sub-stituted into Morison's equation to give the predicted force or the remain-der term for drawing. Drawing of the in-line and transverse force from the two force transducers gives information about the correlation of the forces from the two ends of the cylinder.
3.7 Testïnq of the computer program
Both the data registration and the data analysis part of the computer program have been tested thoroughly. The main concern in the data registration part of the program is the properties of the A/D-converter. Running only a static calibration of the A/D-converter has not proved to
be sufficient. A test of the A/D-converter should at least include a sta-tic calibration, check of cross-torque and check of dynamic response. Only
the first two points have been assessed because of lack of equipment and knowledge. The A/D-converter was a little off when calibrated, but this is not so important to correct, since ali calibrations of the different trans-ducers are run by the 4/0-converter, and then the calibrations factors include this error. A small cross-torque between the different channels was also found, and a subroutine has been sede to correct for this error because it was not easy to get it repaired.
The data analysis part of the program was tested by inputting a computed force trace, instead of one registered by the 4/0-converter, and then a complete calculation was run. By using different phase angles between the
force and the displacement signal, and comparing the output from the com-puter program with the analytical results, the properties of the program could be found. All the mean values of the force coefficient were in exact agreement with the analytical one, but for each period there was some scatter on the inertia and drag coefficients. The coefficients found from Fourier analysis were shown to have larger scatter than those coming from numerical integration. This scatter is, on the other hand, one order of magnitude less than the scatter introduced when using a measured signal.
