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.

 View a video that shows how to for example create a custom heeling arm criterion.

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

§   User Defined Heeling Arm

§   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 233.

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.

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 passenger crowding and vessel turning since the horizontal lever for the passenger 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 183.

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.

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.

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 183.

A user-defined heeling arm may be used in the criteria. With the heeling arm, the user can specify the number of points and the shape of the heeling arm curve.

This heeling arm can then be cross-referenced into any of the heeling arm criteria. First, the number of points is specified and then for each point the angle and magnitude of the curve can be specified. These should be comma delimited for example <45 , 1.2> for a heeling arm magnitude of 1.2 meters at 45 degrees angle of heel. (To aid input of the data, if only one value is supplied it is taken as the heel angle – and the magnitude is left unchanged, and if a value preceded by a comma is given, this is taken as the magnitude – and the heel angle is left unchanged.) A single coefficient may be adjusted and this is used as a multiplication factor (whist the shape of the curve remains unchanged).

The magnitude of the heel arm is given by:

where:

 is the number of passengers

 is the average mass of a single passenger

 is the average distance of passengers from the vessel centreline

 is the vessel mass (same units as )

The heeling arm parameters are specified as follows:

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.

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:

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:

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:

Some criteria require the evaluation of above and below water lateral projected areas and their vertical centroids. The user 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.

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 account.

When evaluating these criteria that are dependent on displacement, care has to be taken to make sure any change in displacement is taken into account. 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.