FREE OSCILLATION IN SURGE AND SWAY OF A MOORED FLOATING DRY DOCK

Measurements are presented of the free period of oscillation In surge and sway of the AFDL-20 (floating dry dock with 2100 long ton displacement) and of the forces and movements Induced. The Dock is spread moored fore and aft, respective! by one I.-1/2 inch die lock chain about 260 feet long with rise of about 35 feet and scope of 8. These measurements are compared with those obtained from oscillating a I to 40 linear scale model and from analytics and the agreement is pronounced good.


INTRODUCTION
For many years the Bureau of Yards and Docks has been concerned with resear on the forces Induced on moored vessels by waves.In such a study -unlike a larg< number of those In the field of hydrodynamics which involve consideration of signifi cantly free or fixed objects -the concern is with objects which are forced to move against the restraint of elastic type moorings.Since, as in the hydrodynamics field in general, it is very rewarding to study at a reduced scale in the laboratory, It is necessary frequently to model the charac teristlcs of the ships moorings as well as of the ship itself, where although the technique for the latter seems well established, that for the former Is not.
To provide correlated data with which to evaluate the ability to model ships 1 mooring characteristics, a relatively small floating dry-dock, was spread moored in a simple manner, and a I to 40 linear scale model of it were caused to oscillate significantly in surge and sway In sensibly still water and the period of the free oscillations was measured.The results obtained from the model, as extrapolated to the Prototype by means of the Froude Model Law since Inertlal forces seem dominan were compared with those obtained from the Prototype.By means of a tug temporarily attached to it, the Dock was displaced particular amounts in surge or sway as the case might be and then permitted to oscillate freely.The output from the chatn dynamometers was recorded as a funtlon of time so that direct measurement of both the chain tension and period of oscillation could be made.Movement of the Dock was determined as a function of time by means of direct reading by surveyors of the positions of scales attached to the Dock.

Because an analytical approach is desired in general, considerable attention was given also to the application of basic mechanics to provide a comparlsion with results obtained from both the
The initial tension In the chains was varied during the experiments to provide a variation In the restoring force.This was done either by waiting for the tide to vary the still water level or by changing the length of the chain.(Table I) An attempt was made to conduct the experiments only when the wind, currents and waves were at a negligible level.In the case of the latter persistent surgesthose wtth about I, 3, and 12 minute period -had to be tolerated but these, like the locally generated wind waves and the other environmental disturbances/ were not considered to have affected adversely the results obtained.In no case was It possible to obtain pure surge or sway, so that coupling at what is considered a low level had to be tolerated.

MODEL
where c Is the vertical distance from the low point on the chain to the directrix (Figure 2-a).With this value the horizontal component T" can be obtained simply as: Because the chain tension (T) and not its horizontal component is measured It is necessary as a check to calculate this from'the relationship:

T = w(c + y 0 +b) (7)
where y 0 Is the vertical distance from the midpoint of the line connecting the anchor and Dock ends of the chain to the low point on the catenary system (Figure 2~a) such that: The horizontal distance x Q corresponding to y Q Is given by: For Case II, it Is necessary after computing the distance from the directrix to low point on chain (c) on the basis of equation (5) to calculate the run (2a) to determine whether or not this checks the assumed value.This Is done on the basis of the relationship: 2a * csinh" 1 2L/c (10) When the assumed and calculated runs agree then the chain tension at the Dock end and its horizontal component are calculated as in Case I.
In general, as Indicated by a consideration of the catenary equations, the relationship between restoring force and movement will be non-linear such that for surge: where k Is the spring factor, x the movement In surge and n an exponent defining the non-linearity of the moorings.
In the case of sway:

F (x) = T H bow (cos<?0 + T" stern (cos <=>C) (13)
where oC Is the horizontal angle between the chain and the direction of motion (sway).

If damping and added mass effects are neglected, graphical methods as outline for example by Timoshenko (1937) may be used to obtain the natural period (T ) whic will vary of course with amplitude of initial displacement. Thus from equation (I): mx -F(x) (14)
where graphical methods or the equivalent must be used when F(x) Is non-linear.However, when F(x) is linear then the natural period from equation ( 14) Is the very familtar: T n -2 TT (mA)'/ 2 (15)

ANALYTICAL RESULTS
The relationships between restoring force and movement was calculated by u of equations ( 12), ( 13) and (4) through (10) for three initial tensions with results of the type shown In Figures 3 and 4. The relationship is found to be non-linear althc depending upon the initial tension, there is in all cases a range over which this non-linearity is very weak.5 and 6) indicates that for any particular initial tension the natural period varies Inversely with the initial displacement In a nonlinear manner.7 and 8 where the <  FLOATING DRY DOCK used to define the curves Is taken from the nearly linear portion of the curves In Figure 5 (T n of 42,25.5 and 12.5 seconds for JT of respectively 4.5, II and 22 kips) and Figure 6 (T n of 315,225 and 175 seconds for IT of respectively 4.5, 10.5 and 21 kips).It Is Indicated that, within such a linear restoring force-displacement range, the natural period varies Inversely with Initial tension such that:

Kips
T ny = 530/IT 0 * 366 (17 where T n Is In seconds and IT Is In kips and x and y refer respectively to surge and sway.Equations ( 16) and ( 17) are not proper when initial displacements are Into the definitely non-linear range.
It Is Interesting to note that by linearizing the restoring force data over a reasonable range of initial displacements and then Introducing the slope of the restoring force-displacement line as the spring factor (k) In equation (15) It is possible (Table II) to obtain values for the natural period which agree well In many cases with those obtained by more elaborate means for the true condition which is non-linear.
Table II.Natural periods computed using the true and linearized curves of restoring force versus movement (Figures That reasonable agreement is obtained Is due to the fact that In this study the restoring force-displacement relationship is not strongly non-linear over a considerable range of movements.Linearizations permit natural periods to be computed relatively easily where the results obtained in many cases may be well within the accuracy desired in ordinary engineering applications. A more complicated mooring system for the Dock, consisting of eight chains (I each fore and aft and 3 each port and starboard) each with a large concentrated load as described by Wlegel et al (1956) was studied analytically using equations (4) through (15).The comparison between the natural periods of oscillation in both surge and sway as calculated, on the basis of the developed restoring force-displacement curves, and as measured on the Model Is indtcated to be very good.

EXPERIMENTAL RESULTS FROM THE PROTOTYPE
The oscillations of the Dock occurred tn the range where restoring force varied In a nearly linear manner with displacement.This range varied with Inttlal line tension being up to 0.6 and 2 feet in surge for initial tension of respectively 22 ant 4.5 kips and up to 15 feet in sway for the same initial tensions.
A very marked increase in restoring force occurred when the Dock moved into the definitely non-linear region where, although the tug used to produce initial displacement had the capability to move the Dock into the lower part of the definitely non-linear range, the free oscillations were never sustained in this region due likely to the high rate of damping involved.
The results obtained are in the form of oscillograms of oscillations in force (those from the chain dynamometers) as a function of time.A facsimile of two sucl oscillograms is presented In Figure 9.The oscillations appear to be approximately sinusoidal and to exhibit a dying down typical of a system with greater than critlc< damping.In the case of surge the amplitudes appear to decrease in nearly equal decrements as an arithmetical series (.7, .5, .3,.1,etc. feet for maximums in th case of run '90, Figure 10).This seems to be characteristic of a Coulomb type damping -an apparent anomaly In a hydrodynamic system -where the nonlinearity is located in the damping system.The oscillations tend to maintain themselves at a nearly constant period even though the amplitude of the motion tends to die down in the manner discussed previously.This linearity of period with movement as well as time permits consideration of the measured periods as a function only of the initial tension.An inverse relationship is indicated (Figures 7 and 8) as predicted by the analytics.The comparison between the experimental and analytically derived periods on this basis is considered good with the greatest differences occurring at the lowest initial tensions.

EXPERIMENTAL RESULTS FROM THE MODEL
Prototype and the Model" TEST FACILITIES AND PROCEDURES PROTOTYPE The vessel used Is a floating drydock £ AFDL-20 of 2100-long tons displacemi (Figure was moored In the harbor of Port Hueneme, California In about 35 feet of water by one chain , 1-1/2 inch die lock of scope 8,respectively fore and aft (Figure 2).Strain gage type dynamometers as described by O'Brien and Jones (1955) were used to measure chain tension at the Dock end.

A
I to 40 linear scale Model was constructed by the David Taylor Model Basin and balanced dynamically and moored by the University of California under a contract with the Bureau of Yards and Docks as described by Wiegel et al (1956).The Dock ends of the lines were fitted with dynamometers whose design and Installation entailed considerable effort.As In the Prototype the Model was displaced In either surge or sway and then permitted to oscillate freely with force data recorded as a function of time so that the period could be deduced.Movements were not measured rigorously; in some cases estimates of initial displacement were made

Fig. 3 .Fig. 4 .
Fig. 3. Restoring force versus surge for a high initial tension.15 30)inS Nl NOI1V11DSO 30 aOliBd 3383 Dock end of the chain, when resolved Into their horizontal components by use of measured chain slopes, give the restorii forces as expressed by equations (12) and (13).The comparison between these values and those obtained analytically is considered In the main to be good -th< results presented In Ftgures 3 and 4 are examples -although there is considerabli scatter in the data in places.This is considered to reflect both the non-linear an coupled nature of the Dock movement which was not purely In either surge or swo and to certain vagaries In the measurement of this movement by surveyors readtn; from some distance on scales attached to the moving Dock.The magnitude of the added mass effect was evaluated at the extremes of the oscillations where the acceleration (x) was maximum and the velocity and thus th damping was zero.For surge equations (Igraphically (Figure10).These values along with values for the measured restoring force and mass (m) when substituted in equation (18) gave values for C^ close enough to unity to permit the added mass effect to be declared negligible in this study.
The principal effort was devoted to obtaining the natural period of oscillation in surge as a function of Initial tension and displacement.The data is contained on oscillograms of force -that is the Dock end of the lines -versus time.From these data the periods were measured.The Model values for the period were extrapolated to the Prototype by means c the Froude Law -a kinematical relationship which states that for identical conditi< of gravity the length ratio Is equal to the square of the time ratio -which in this study means that the natural periods obtained in the Model were multiplied by 6.3 -the square root of the length ratio of 40 prototype to I model -to obtain th corresponding value In the Prototype.Movement of the Dock was not measured directly.It Is recognized that this movement can be obtained by indirect means from the force oscillograms -this wa; done in three cases of particular interest by use of calculated curves of restoring force versus movement -but such an effort on a general basis was considered beyond the scope of this paper.However, from the limited indirect studies of movement which were made, it appears that in general the free oscillations occurrc In a range of amplitudes within which the variation of restoring force with movetm was close enough to linear to permit the Model results for period to be reviewed as functions of initial tension only* Such a comparison is made in Figure7where, although the scatter is consider) with the mid Initial tension, the comparison with the Prototype and analytically derived values Is thought to be good with the usual Inverse relationship between period and initial tension indicated.The scatter, besides being due to the inabil of the experimenters to cause the Dock to oscillate purely in surge, is attributed to the fact that the restoring force-displacement relationship is not truly linear K»OJ VtlHNI <UI») DKM ONnOlSH

Table I
Data on the characteristics of the mooring chains and free oscillations in surge and sway