NO339535B1 - Floating unit and method for reducing stomping and rolling movements of a floating unit - Google Patents

Description

The present invention relates to improving the movement characteristics of a single-cylinder floating unit.

It has been observed that this type of floating unit may be subject to excessive rolling/stomping movements in certain sea conditions. Similar floating device designs attempt to eliminate such movements by providing as much damping as possible to the floating device.

Floating single cylinder structures have previously been described in, for example, published PCT application WO 02/090177 A1 and published US application No. 2012/0298027 A1. In the US publication, a single cylinder hull is described which is designed to reduce ice loads and provide more ice breaking mechanisms than conventional vessel structures. The floating vessel is provided with a conical shaped part and a different sided polygonal shape of the hull to break the ice more effectively. Pounding/rolling, heaving and breaking motions are induced by moving ballast water in the ballast tanks, thereby more effectively breaking the ice.

The aim of the claimed invention is to reduce pitching/rolling movements for a floating single cylinder unit in the sea.

A further aim of the present invention has been to increase the damping of a single-cylinder floating unit and at the same time simplify the manufacturing process of such a floating unit.

These objectives are achieved with a floating device as defined in claim 1 and method as defined in claim 7. Further embodiments of the floating device are defined in the dependent claims 2-6.

The strategy during the development of the claimed invention was to find a way to reduce unwanted movement by changing the shape of the floating device rather than increasing the damping. The floating structure can of course be provided with means for damping as much as possible in addition.

The reason for movements that are experienced but cannot be explained by linear theory is probably parametric coupling between different movement components, so-called Mathieu movements. The conditions that cause this behavior can be described by a differential equation that has a solution in some regions, while in other regions it is unstable and the solution will diverge to infinity.

The Mathieu differential equation can be written:

where e is the relative size of this variation in the stiffness expression and tts is the ratio between the natural frequency and the stiffness frequency. The equation shows that the "stiffness expression" has a regular variation with a frequency that can be different from the natural frequency of the system. The relative size of this

the variation in the stiffness expression (e) and the ratio between the natural frequency and the stiffness frequency (tn) then govern the solution.

Figure 1 shows the regions for stable and unstable solutions of the equation. As can be seen in Figure 1, a ratio of tu=0.5 leads to the greatest instability, but tu=l, 1.5 and 2 can also lead to instability. The size of the stiffness variation (e) also plays a role in that increasing the stiffness variation will increase the region of instability.

The patent-pending invention is a liquid unit of the cylinder type, preferably

with an octagonal cross-section, but can also be given other shapes, for example circular. The floating unit can also be provided with an internal moonpool. The floating unit is characterized by the average diameter (D) and draft (T). If the patented invention is designed as an octagon with eight flat side panels, the diameter is defined as the inscribed circle in the octagon. If other polygonal shapes are chosen, the diameter will correspondingly be defined as the inscribed circle in the polygon. An octagonal shape leads to increased viscous damping achieved by sharp corners and simplified production because flat steel sheets are much easier to produce than curved panels.

For a floating device such as the claimed invention, the above differential equation can represent the pitch/roll motion, and the stiffness variation can be represented by the variation in the metacentric height (GM) caused by heave motion.

The frequency of the wave movements is controlled by the natural period of the wave, or simply the wave excitation frequency. With, for example, a hiv excitation period of 12 seconds and a natural rolling period of 24 seconds, the frequency ratio will be td= 0.5 and the possibility of entering the unstable region is very high as shown in figure 1. If the natural If the hiv period and the wave excitation coincide, this will of course intensify the problem.

Previously, as mentioned above, in order to avoid excessive pitching/rolling motions, i.e., entering the unstable regions shown in Figure 1, floating units have been provided with as much damping as possible.

Another possibility to avoid entering the unstable region would be to prevent the frequency ratio from entering the unstable regions. Considering the fact that the excitation comes from a stochastic process (the waves), it will still be very difficult to prevent the frequency ratio from entering the dangerous regions.

As a further alternative, the inventor has therefore contemplated a third possibility to avoid the possibility of entering the unstable regions, which is to reduce the variation of stiffness GM (e in figure 1) as much as possible. This may appear to be the most effective way to reduce exposure to Mathieu instability.

A floating unit is provided which comprises a single center cylinder having a longitudinal center axis. The central cylinder comprises a lower main section and an upper section extending up from the main section and having at least one outward facing side, i.e., facing away from the interior of the floating unit towards the surroundings, forming an angle of inclination (cp) with a vertical axis. The upper section penetrates a still water surface when the floating unit is in use. The main section of the floating structure has an equivalent diameter (D), a draft (T) in still water such that there is a distance h from a waterline (25) to the cylindrical part (14) and a metacentric height (KM) which is given as a function of the diameter (D), the draft (T) in still water, the angle of inclination (cp) and a variable part (x) of the draft resulting from the motions of the floating unit such that KM = KM(D,T ,h,(p,x).The angle of inclination (cp) in the upper section lies within the range of ±20% of the optimal angle of inclination (cp) which satisfies

the equation

Alternatively, this can be expressed so that the optimal angle of inclination (cp) in the upper section (12) is the angle that satisfies the equation

and where the angle of inclination in the upper section lies within the range of ±20% of the optimum angle of inclination.

When the upper section is provided with such an angle of inclination, the parametric link between heave and ram/roll will be disconnected.

The upper section, seen from the side, can be conically designed or alternatively, the upper section, seen from the side, can have a curved shape.

A horizontal cross-section of the cylinder of the floating structure, perpendicular to the longitudinal axis of the floating structure, may have a regular polygonal shape, for example a regular octagonal shape. Alternatively, the horizontal cross-section of the cylinder of the floating structure, perpendicular to the longitudinal axis of the floating structure, may be circular.

A method for reducing pitch/roll motions for a single center cylinder floating unit is also disclosed wherein the floating structure comprises a single center cylinder having a longitudinal center axis. The central cylinder comprises a lower main section and an upper section extending upwards from the main section and having at least one outward facing side, i.e., facing away from the interior of the floating unit and towards the surroundings, forming an angle of inclination ((p) with a vertical axis. The upper section penetrates a still water surface when the floating unit is in use. The main section of the floating structure has an equivalent diameter (D), a draft (T) in still water such that there is a distance h from a waterline (25) to the cylindrical part (14) and a metacentric height (KM) which is given as a function of the diameter (D), the draft (T) in still water, the angle of inclination ((p) and a variable part (x) of the draft resulting from the movements of the floating unit (10) such that KM = KM(D,T,h,cp,x).The angle of inclination (cp) of the upper section (12) is chosen to be within the range of ±20% of the optimum inclination angle ((p) which satisfies the equation

The equivalent diameter (D) mentioned several times above is equal to the actual diameter if the main section has a circular cross-section. If the cross-section of the main section is polygonal, the equivalent diameter (D) is taken as the diameter of the circle which gives the same area as the area of the actual polygonal cross-section.

A non-limiting embodiment of the present invention will now be further described with reference to the figures where Figure 1 is discussed above and shows the stability diagram for Mathieu's equation.

Figure 2 shows KM as a function of draft for different diameters (D).

Figure 3 shows the derivative of the KM function as a function of the draft (T) for different diameters. Figure 4 shows a graph showing D-T combinations where the derivative of KM is equal to 0. Figure 5 shows a plot where 8KM(D,T,cp,x) is plotted for several angles as a function of the draft (T). Figure 6 shows a plot of the inclination angle which will not give any variation for the metacentric height. Figure 7 shows a side view of a floating unit according to the present invention.

Figure 8 shows a cross-section of the floating unit shown in Figure 7.

Figure 9 shows an embodiment of the present invention with a slope.

Figure 10 shows a plot where the optimal inclination angle as a function of draft is plotted for diameters from 50 meters to 130 meters in steps of 20 meters.

In the following, we will discuss how a design according to the patent-pending invention can enter the Mathieu instability region and how the patent-pending invention solves this problem.

As mentioned, the claimed invention is a simple cylindrical design where the cross-section is preferably in the form of an octagon, but may also be circular or generally in the form of a regular polygon.

The octagonal or polygonal shape is not relevant for the assessments related to the present invention, so that in the mathematical expressions below we will assume that the patent-applied invention has a circular cross-section with diameter D and depth T. The diameter can be chosen so that n/ 4* D2 will give the same cross-sectional area as the octagon of the actual floating unit concerned.

A circular cylinder comprising a main section with a constant cross-section as a function of height will therefore be considered.

The metacentric height, which is the distance from the keel to the metacenter KM, can be written:

where KB is the distance from the keel to the center of buoyancy and BM is the metacentric height which is the distance from the center of buoyancy to the metacentre.

BM is given by the following expression:

where I is the moment of inertia of the water plane area, i.e., for a circular water line which means that I =7i/64<*>D<4>, and V is displacement, i.e., V =7i/2<*>D<2* >T.

By simplifying the expression, BM can be written as:

This expression can be derived with respect to T. The following is then achieved:

The depth T which does not provide any variation for BM is:

Solving this equation for D/T:

A plot of the metacentric parameter KM as a function of D is shown in Figure 2. Note that variations in KM will be equal to variations in GM because GM = KM - KG and KG remains constant for a given load condition.

A plot of the derivative of KM with respect to the draft T, i.e.

d(KM)/dT = '/2-D2/(l 6*T2), is shown in Figure 3. In Figure 3, the variation of KM is plotted for several diameters as a function of the draft T.

As an example, a floating unit with a diameter of 90 m is chosen here, but a floating unit having a different diameter D will obviously have different results.

For a draft of 31 meters, the value is equal to zero (ie, 8KM(T,90) = 0 in Figure 3). For smaller drafts, the variation becomes greater and larger numerical values. It is clear that for drafts below 30 m the condition to obtain Mathieu instability is present because the parameter e becomes a significant number.

Figure 3 also shows that for each diameter there is a draft (T) which gives a zero value for the variation KM. This relationship is shown in figure 4 where the line is based on both a numerical solution of 5KM = 0 and the equation D/T = (8), where both give the same result.

However, it is not always possible to design a floating unit with D-T combinations as shown above. A diameter of, for example, 90 m requires a draft of approximately 31 m. In many situations this is not possible, but a draft of let's say 21 m will lead to a situation where the KM value will have a variation due to dynamic variation of the draft . How to solve this problem will be explained in detail.

Figures 7-9 show an embodiment of the patent-pending floating unit 10 with a single center cylinder 11 supporting a cover structure 16. The center cylinder 11 comprises a cylindrical main section 14 with an octagonal cross section and a conical upper section 12 extending from the top of the main section and through the water surface 30 of still water. The upper section 12 comprises outward facing sides 13 which face the surroundings. On top of the upper section 12 the floating unit is provided with the deck structure 16. Normally the outward facing sides 13 of the conically shaped upper section 12 will be inclined outwards from the longitudinal axis 22 in the upward direction, but as explained below this is not always the case . The floating structure may be provided with a moonpool 18 extending through the main section 14 and the upper section 12.

In Figure 8, the main section 14 and the upper section 12 are shown to have a cross-section normal to the floating unit 10's longitudinal axis 22 which has a regular octagonal shape with side surfaces 20. The shape of the cross-section can nevertheless also be circular or of a regular polygonal shape different from octagonal. The advantage of using a polygonal shape is that flat plates rather than curved plates can be used in the production of the center cylinder 11 of the floating unit 10 which is simpler and cheaper than using curved plates. In addition, the corners 21 which are formed where two side surfaces 20 are joined together contribute to the damping of the floating unit.

By providing the floating unit with the upper conical section 12 penetrating the still water surface 30, i.e., the conically shaped upper section 12 extending above and below the water surface, the pitch/roll and heave movements of the floating single cylinder unit can be decoupled . The inclination angle cp that the outward facing sides 13 of the conical upper section 12 make with the longitudinal axis (ie, generally with a vertical line) is shown in Figures 7 and 9. The magnitude of the inclination angle cp depends on the main parts of the floating unit 10 as this will be explained below.

With reference to figure 9, the distance from the waterline 25 to the cylindrical part is "h". The conical upper section 12 will also extend a distance above the waterline, preferably at least the distance "h".

It is possible to develop an expression for a metacentric parameter KM by taking into account the inclination of the conical upper section 12. The main contribution of the inclination is that it increases the waterline moment of inertia, but it will also affect the displacement and the vertical position of the center of buoyancy.

By creating the slope as a section of a cone, both the KB and BM values can be calculated. In general, one can define KM as a function of the following parameters:

where D is the equivalent diameter of the unit as explained above, i.e., if the cross-section of the floating unit is polygonally shaped (for example as an octagon), the diameter is chosen so that 7t/4*D2 gives the same cross-sectional area as the polygon of the actual the floating unit in question.

T is the static (still water) draft,

<D is the inclination angle that must be found,

x represents the vertical movement from the original draft as a result of the movements of the floating unit.

The derivative of the expression for KM can now be found:

The resulting function can be plotted as shown in Figure 5 where the variation in the KM value 5KM(D,T,cp,x) is plotted for four angles of inclination and as a function of the draft T. Again, as an example, D = 90 m in all four cases. As can be seen from Figure 5, it will normally be possible to find a value for the inclination angle (p which will lead to 8KM(D,T,cp,x) = 0 for any combination of diameter D and draft T.

The optimum angle of inclination can be found for any desired draft, but in the example below a value x = 0 is chosen, i.e. the draft in still water, as this is considered the most correct draft. It will be possible to solve the equation 5KM(D,T,(p,x) = 0 for many different values of the draft and a corresponding number of varying inclination angles will be found. The result will be that the upper part 12 of the floating unit no longer will be conically shaped, but rather have a curved shape similar to an upside down champagne bottle. This approach would theoretically be the most correct shape of the upper section 12, but because the upper section 12 in this case would have a double-curved shape it would the upper section can be both difficult and expensive to manufacture. Therefore, for practical purposes, a constant value for the draft can be chosen, for example x = 0, whereby a conically shaped upper section 12 results having an angle of inclination corresponding to the selected value for the draft.

Solving the equation 5KM(D,T,<q>),x) = 0 with respect to the inclination angle cp, i.e., the value x is set to zero, gives the inclination angle which will ensure that KM does not vary (and thus also GM). Thus, the parametric link between heaving movements and stomping/rolling movements is disconnected.

Figure 6 shows the result of solving the equation 5KM(D,T,cp,x) = 0 for a diameter of 90 m. As can be seen from Figure 6, for a draft T = 20 m, an inclination angle cp~23° should be chosen to ensure decoupling of heaving and pitching/rolling movements. It is clear that values slightly above or below the angle of inclination for a given diameter and draft will also work, if not as well as the optimum value. In the example above, the most optimal value for the tilt angle would be about 23°, but tilt angles in the range of about ± 20% of the most optimal tilt angle can be considered usable, i.e., in this example, the tilt angle should at least fall within the range of about 18° - 28°.

It should also be noted that in the above example, for drafts greater than about 32-33 m (when the diameter is 90 m), the inclination angle is negative. This means that the conical upper section 12 should be inclined towards the longitudinal axis, i.e., with a gradually decreasing diameter in the upward direction of the longitudinal axis, to ensure decoupling of heaving and pitching/rolling movements.

As explained, the tendency to get Mathieu motions depends on the D/T ratio, but it is possible to define a pitch angle in the waterline region that will lead to no variation in GM stiffness and thus no driving forces for parametric coupling between heave and pitch -/scroll movements for any combination of D and T.

The angle of inclination required to achieve this is defined by the mathematical equation given above which is also given in more detail in the appendix below.

In what follows, it will be described in more detail how the angle of inclination (p can be calculated with reference to figures 5 and 9-10. The calculation is made with the calculation tool MathCad® and the formulas are taken from this program. In the example, D = 90m is used as a starting point (diameter), T = 30 (operational draft), cp = 20° (inclination angle), h = 5 (distance between parallel part and working water line).

Further from the figures:

New diameter when the unit is moved x meters from operational draft:

The height of the two cones that define the cone segment:

Volume as a function of geometry and motion amp(x) from static waterline:

Vertical center of buoyancy (KB or VCB):

Moment of inertia of displaced waterline:

Metacentric height (BM):

Total metacentric height:

Variation of KM as a function of draft for different angles of inclination. Derive the KM function to find the variation as a function of x:

Starts a loop for depth from 20 to 40 meters in steps of 1 meter:

Plot the function for several angles of inclination to show that it is possible to find an angle of inclination that passes through 0. The plots are made for a diameter of D = 90 and cp:=20°,92:=30°, tp3:=10° and cp4:= 1 °. The plots are shown in Figure 10 where the derivative of KM as a function of depth for these inclination angles is shown.

Then starts a loop for diameters from 50 meters to 130 meters in steps of 20 meters: j:=0..4 Solves the equation

to find the angle that gives the derivative of KM = 0. A plot of the optimal inclination angle as a function of draft, with diameters from 50m to 130m in steps of 20m, is shown in figure 11.

Claims (7)

1. Floating unit (10) comprising a single center cylinder (11) having a longitudinal center axis (22), the center cylinder comprising a lower main section (14) and an upper section (12) extending upwards from the main section (14) and has at least one outward facing side (13) forming an angle of inclination (cp) with a vertical axis (22) and where the upper section (12) penetrates a still water surface (30) when the floating unit is in use, where the main section (14) of the floating structure (10) further has an equivalent diameter (D), a draft (T) for still water such that there is a distance h from a waterline (25) to the cylindrical part (14) and a metacentric height (KM ) which is given as a function of the diameter (D), the draft (T) in still water, the angle of inclination (cp) and a variable part (x) of the draft resulting from the movements of the floating unit (10), so that KM = KM(D,T,h,cp,x), characterized in that the angle of inclination (cp) of the upper section (12) lies within a range of ±20% of the optimal angle of inclination (cp), where the optimal angle of inclination satisfies the equation 2. Floating unit according to claim 1, characterized in that the upper section (12) has an outward-facing side (13) which, seen from the side, is conically shaped. 3. Floating unit according to claim 1, characterized in that the upper section (12) has an outward facing side (13) which, seen from the side, has a curved shape. 4. Floating device according to one of claims 1-3, characterized in that a horizontal cross-section of the cylinder (11) on the floating structure (10), perpendicular to the longitudinal axis (22) of the floating structure (10), has a regular, polygonal shape. 5. Floating device according to one of claims 1-4, characterized in that a horizontal cross-section of the cylinder (11) on the floating structure (10), perpendicular to the longitudinal axis (22) of the floating structure (10), has a regular octagonal shape. 6. Floating unit according to one of claims 1-3, characterized in that a horizontal cross-section of the cylinder (11) on the floating structure (10), perpendicular to the longitudinal axis (22) of the floating structure (10), is circular. 7. Method for reducing pitch/roll movements of a single center cylinder floating unit (10) comprising a single center cylinder (11) having a longitudinal center axis (22), wherein the central cylinder comprises a lower main section (14) and an upper section ( 12) which extends up from the main section (14) and has at least one outward facing side (20) which forms an angle of inclination (cp) with the longitudinal axis (22) and where the upper section (12) penetrates a still water surface (30) when the floating unit is in use, where the main section (14) of the floating structure (10) further has an equivalent diameter (D), a draft (T) for still water such that there is a distance h from a waterline (25) to the cylindrical part (14) and a metacentric height (KM) which is given as a function of the diameter (D), where the draft (T) for still water, the angle of inclination (cp) and a variable part (x) of the draft which is a result of the movements of the floating unit (10), so that KM = KM(D,T,h,( p,x), characterized in that the angle of inclination (cp) of the upper section (12) is chosen so that it lies within a range of ±20% of the optimal angle of inclination (cp), where the optimal angle of inclination satisfies the equation