CHAPTER 3
BALANCE
In this chapter it is described how the forces applied on the catamaran have been calculated in order to reach the translational and rotational balance of the structure. This has been necessary since the catamaran is an unconstrained structure.
However to run the f.e.m analysis it has been mandatory to define a constraint for the model. So the “Inertia Relief” solution of Nastran has been used. This function will be detailed later in the chapter.
3.1 Translational balance
Having the value of the sail aerodynamic force and the first estimate of weights, the value of the forces applied on the hull has been calculated as follow along the y and z axis (Fig. 3.1.1) in order to define the translational balance of the structure.
y) Fay – Fw - Fd - Fcnt= 0 z) 2Wh+Ws+Wm+Wcb + Faz – Fi = 0 with: Wh = hull weight Wcb = crossbeam weight Wm = mast-sail weight Ws = skipper weight
Faz = aerodynamic force z-component
Fay = aerodynamic force y-component
Fd = hull drag force
The value of the aerodynamic force components derives from the roll angle which value has been defined through the rotational balance.
Fig. 3.1.1 Forces applied
Fcnt
3.2 Roll moment
To determine the rotational balance has been necessary to find either the center of buoyancy either the metacentric height of the hull (Fig. 3.2.1) to calculate the metacenter around which the entire structure rotates (Fig. 3.2.2)
Fig. 3.2.1 Hull section - Metacenter
with: a = metacentric height C = buoyancy center G = center of gravity M = metacenter WL = water line
Fig. 3.2.2 Hull roll configuration
with:
d = weight force s = buoyancy force
K = orthogonal projection of G, on the line where s is applied C’ = new buoyancy center
alpha = roll angle
These points have been fixed on the hull on the base of experimental data deriving from similar a-class catamarans.
Basing on geometrical considerations (Fig. 3.2.3) the moment of the fluid dynamics and weight forces has been calculated as functions of the roll angle Φ .
Fig. 3.2.3 Roll balance
Fluid dynamic force momentum:
Mfa = Fa a + (Fd+Fw)b + Fcnt e Weights momentum: Mfw = (Wh+Ws) Lcb cosΦ + Wcb cosΦ + Wm Lb cos(γ+Φ) + Wh c Fcnt e
with:
M = metacenter
Lcb = crossbeam length
a= hull metacenter – sail center of pressure distance
b = metacenter- point of application of the drag and waves force distance c = metacenter- point of application of the hull weight force distance Lb = hull metacenter – sail and mast barycenter distance
The moment of the forces applied on the immersed hull does not affect much the balance due to the fact the metacenter is extremely close to the point of application of these forces.
Mfa and Mfw are opposite in direction. The former is quite constant once fixed the wind
speed and the suitable angle of attack of the sail profile (α) , the latter instead decreases as the roll angle increases (Fig. 3.2.4).
Their values coincide as Φ reaches 10 degrees. This is the optimal configuration the skipper wants to reach, relocating his weight on the hull, during the close to the wind sail.
Thus the roll angle considered in this analysis is Φ= 10 degrees as previously stated.
3.3 Yawing and pitching moment.
The determination of the yawing and pitching moments has been difficult because they strictly depend from several parameters such as the sail thrust, the shape of the hull, the rudder action, the hull-water interaction which, at the stage of the project, was difficult to calculate.
Balancing the entire structure also along the yawing and pitching axes has been necessary anyway. For this purpose it has been found an “optimum” point along the hull length where applying the skipper weight force. In addition the Nastran “Inertia Relief” tool contributed to correct the minimum error stabilizing the structure.
This allowed to run the analysis reaching a convergent solution.
However the constraint entity imposed by the software has been checked. It has been assured its value was below the limit which guarantees a valuable result in output. Following the description of the method used.
3.4 Nastran “Inertia Relief”
Inertia relief is an advanced option in Nastran that allows to simulate unconstrained structures in a static analysis.
The inertia of the structure is used to resist the applied loadings. So the assumption is that the structure is in a state of static equilibrium even though it is not constrained. Once a “Suport” bulk data entry with a list of six non redundant degrees of freedom, all the translations and rotations in this work particularly, has been provided, Nastran calculates the forces that result from a rigid body acceleration of the “Suport” degrees of freedom in the specified directions. Then it calculates the summation of all applied loadings in the same directions.
Accelerations are applied to the structure in the “Suport” directions to balance the applied loads. Since the problem is not constrained, rigid body displacement is still possible.
The next step performed by the software is to constrain the “Suport” degrees of freedom to a displacement of 0.0 and provide the relative motion of all other grid points with respect to the reference point.
Whenever a “Suport” entry is used in a static analysis, the epsilon and strain energy printed in the output should all be approximately zero. The values printed for the strain energy indicate the ability of the model to move as a rigid body.
