Heeling Arm Definition l4567

Heeling arm definition This section describes how to define heeling arms and is valid for both the parent heeling arms that can be cross referenced into the heeling arm criteria, and for the Old heeling arm criteria where the heeling arm is specified for each criterion separately. There are several heeling arms that are used for the criteria. They are defined below. · General heeling arm · General heeling arm with gust · General cos+sin heeling arm · enger crowding · Wind · Turning · Lifting heeling · Towing heeling · Areas and levers · Important note: heeling arm criteria dependent on displacement

Also see the next section: Heeling arms for specific criteria - Note on unit conversion section on page 131. Note: When you are working with the parent heeling arms, make sure you copy them into a custom heeling arms folder before editing them. Same as for the Parent criteria, the Parent heeling arms will be reset to their default values each time you start up Hydromax. General heeling arm

The general form of the heeling arm is given below: where: is the heel angle, is the magnitude of the heeling arm, describes the shape of the curve. Typically n=1 is used for enger crowding and vessel turning since the horizontal lever for the enger transverse location reduces with the cosine of the heel angle. For wind n=2 is often used for heeling because both the projected area as well as the lever decrease with the cosine of the heel angle. However, some criteria, such as IMO Severe wind and rolling (weather criterion) have a heeling arm of constant magnitude, in this case n=0 should be used. Make sure you read Important note: heeling arm criteria dependent on displacement on page 131. General heeling arm with gust

Some criteria require a Gust Ratio, this is the ratio of the magnitude of the wind heeling arm during a gust to the magnitude of the wind heeling arm under steady wind. Error! Objects cannot be created from editing field codes.

Both the steady and the gust heel arm have the same shape.

where: is the heel angle, is the magnitude of the heeling arm, describes the shape of the curve. It should be noted, that in this case, the definition of gust ratio is the ratio of the heeling arms. Some criteria specify the ratio of the wind speeds; if it is assumed that the wind pressure is proportional to the square of the wind seed, the ratio of the heel arms will be the square of the ratio of the wind speeds. General cos+sin heeling arm

Some criteria, notably lifting of weights, require a heeling arm with both a sine and cosine component:

It should be noted that provided the indices are both unity, the same heeling arm form may be used for computing towing heeling arms of the form: in this case a constant angle (in the case of towing, the angle of the tow above the horizontal) is included. It may be shown that this is equivalent to: where:

,

,

and

Make sure you read Important note: heeling arm criteria dependent on displacement on page 131. enger crowding heeling arm

The magnitude of the heel arm is given by:

where: is the number of engers is the average mass of a single enger is the average distance of engers from the vessel centreline is the vessel mass (same units as ) The heeling arm parameters are specified as follows: Option number of engers: n

Description Number of engers

Units none

enger mass: M distance from centreline: D cosine power: n

Average mass of one enger Average distance of the engers from the centreline Cosine power for curve - defines shape

mass length none

Wind heeling arm

In the case of the wind pressure based formulation, the wind heeling arm is given by:

where: is a constant, theoretically unity is the windage area at height is the vessel mass is the wind pressure is the vertical centre of hydrodynamic resistance to the wind force In the case of the wind velocity based formulation, the wind heeling arm is given by:

where: is now effectively an average drag coefficient for the windage area multiplied by the air density and has units of density is the wind speed. And the other parameters are described as above. constant: a

Constant which may be used to modify the magnitude of the heel arm, normally unity for pressure based formulation or 0.5 ρairCD for the velocity formulation; where ρairis the density of air and CD is an average drag coefficient for the windage area Pressure or Velocity (type “P” or “V”)

none for pressure based formulation;

wind pressure or velocity

Actual velocity of pressure - depends on wind model

area centroid height: h

Height of defined total or additional windage area may specify either a total windage area Or, an area to be added to the windage area computed by Hydromax based on the hull sections

mass/(time2length) or length/ time length

wind model

total area: A additional area: A

mass/length3for velocity based formulation

length2 length2

height of lateral resistance: H

length

H = mean draft / 2 H = vert. centre of projected lat. u'water area H = waterline cosine power: n

There are four options for specifying H (all options are calculated with the vessel upright at the loadcase displacement and LCG): specified H is taken as half the mean draft. H is taken as the vertical centre of underwater lateral projected area. H is taken as the waterline Cosine power for curve - defines shape

Option

Description

Units

length length length none

Turning heeling arm

The magnitude of the heel arm is derived from the moment created by the centripetal force acting on the vessel during a high-speed turn and the vertical separation of the centres of gravity and hydrodynamic lateral resistance to the turn. The heeling arm is obtained by dividing the heeling moment by the vessel weight. The heeling arm is thus given by:

where (in consistent units): is a constant, theoretically unity is the vessel velocity is the radius of the turn is the vertical separation of the centres of gravity and lateral resistance The heeling arm parameters are specified as follows: Option constant: a vessel speed: v turn radius: R turn radius, R, as percentage of LWL Vertical lever: h

h = KG h = KG - mean draft / 2 h = KG - vert. centre of projected lat. u'water area cosine power: n

Description Constant which may be used to modify the magnitude of the heel arm, normally unity Vessel speed in turn Turn radius may be specified directly Or, as some criteria require, as percentage of LWL

Units none

There are four options for specifying h (all options are calculated with the vessel upright at the loadcase displacement and LCG): specified h is taken as KG - position of G above baseline in upright condition h is taken as KG less half the mean draft. h is taken as the vertical separation of the centres of gravity and underwater lateral projected area.

length

Cosine power for curve - defines shape

none

length/time length %

length length length

Lifting heeling arm

This is used to simulate the effect of lifting a weight from its stowage position. The magnitude of the heel arm is given by:

where: is the mass of the weight being lifted is horizontal separation of the centre of gravity of the weight in its stowage position and the suspension position is vertical separation of the centre of gravity of the weight in its stowage position and the suspension position is the vessel mass (same units as ) The heeling arm parameters are specified as follows: Option Mass being lifted: M vertical separation of suspension from stowage position: v

horizontal separation of suspension from stowage position: h

Description Mass of weight being lifted Vertical separation of suspension point from weight’s original stowage position on the vessel. This value is positive if the suspension position is above the original stowage position. Horizontal separation of suspension point from weight’s original stowage position on the vessel This value is positive if the horizontal shift of the weight should produce a positive heeling moment.

Units mass length

length

Towing heeling arm

The magnitude of the heel arm is given by:

where: is the tension in the towline or vessel thrust, expressed as a force. is horizontal offset of the tow attachment position from the vessel centreline is vertical separation tow attachment position from the vessel’s vertical centre of thrust is the vessel mass is the power index for the cosine term which may be used to change the shape of the heeling arm curve is the (constant) angle of the towline above the horizontal. It is assumed that the towline is sufficiently long that this angle remains constant and does not vary as the vessel is heeled. The heeling arm parameters are specified as follows: Option tension or thrust: T vertical separation of propeller centre and tow

Description Tension in towline or vessel thrust Vertical separation tow attachment position from the vessel’s vertical centre of thrust.

Units force length

attachment: v horizontal offset of tow attachment: h

angle of tow above horizontal: tau cosine power: n

This value is positive if the towline is above the thrust centre. Horizontal offset of the tow attachment position from the vessel centreline. This value is positive if the offset is in the direction of the tow. Angle of tow above the horizontal Cosine power for curve - defines shape

length

angle none

Areas and levers

Some criteria require the evaluation of above and below water lateral projected areas and their vertical centroids. The may also specify additional areas and vertical centroids or the total areas and vertical centroids. In all cases the vertical centroids are given in the Maxsurf/Hydromax co-ordinate system; i.e.: from the model’s vertical datum, positive upwards. Centroids of area are calculated for the upright vessel (zero trim and heel) at the mean draft. The areas are calculated from the hydrostatic sections used by Hydromax; thus, increasing the number of sections will increase the accuracy of the area calculation; further, only “Hull” surfaces are included in the calculation - “Structure” surfaces are ignored. The vertical position of the keel, K, is assumed to be at the baseline (as set up in the Frame of Reference dialog), even if the baseline does not correspond to the physical bottom of the vessel. Important note: heeling arm criteria dependent on displacement

Some heeling arm criteria are dependent on the displacement of the vessel for the calculation of the Heeling Arm. For example, the value “A” in:

,is manually calculated from: , where M = heeling moment Δ = displacement. This means that the heeling arm will vary with the displacement. Hydromax will not take the change in displacement into . When evaluating these criteria that are dependent on displacement, care has to be taken to make sure any change in displacement is taken into . For large angle stability this means that every loadcase will have its own set of criteria. For Limiting KG and Batch analysis, there are two options: 1. Calculate the worst-case lever based on the displacement and VCG that result in the worst lever and see if the criterion is actually a limiting one for KG. 2. Calculate limiting KG at single displacements and change the heeling arm for each displacement.