Optimal Yacht Rig Design using Mathematical Programming Jarrad Wallace Department of Engineering Science University of Auckland New Zealand j wal [email protected] Abstract Traditional rig design methodologies are often based on empirical “rules-of-thumb" that frequently result in a rig that is over built and excessively heavy, which is due primarily to the coarse approximations made in calculating member loads and subsequent large factors of safety required to avoid mechanical failure. This paper investigates the feasibility of using mathematical models in improving the rig design process, and applying optimization techniques to minimise the weight of the structure. It presents a new method of quickly and efficiently evaluating rig loads and deflections under multiple loading conditions, incorporating complementarity constraints to handle tension-only members. This is then extended to investigate the non-linear optimization of member properties and rig geometry in attempting to achieve minimum rig weight, as a direct method of improving yacht performance. 1 Introduction The traditional practice of designing masts for yachts has been around for many years and in some cases consists of following some empirically derived “rules-of- thumb” to create a rig that roughly matches the performance characteristics of the hull it will operate in. More often than not though, these rules create a rig that is big, heavy and over-built, having large safety factors to account for the rig loading approximations and ensure that it will not fail. Inevitably, the resultant structure is less than ideal and ultimately hinders the performance of the yacht. At the other end of the spectrum, the process of exactly calculating the required properties that the mast must have in order to perform perfectly across all predictable loading conditions is a difficult and involved series of calculations. It requires a large commitment of computational and human resources. There is a need in the industry for an efficient method of evaluating the mechanical response of a rig under known loading conditions, so as to be able to determine its suitability and seaworthiness. This paper investigates the feasibility of a Mathematical Programming (MP) approach to evaluating a rig, and possible optimization for minimum weight as a direct method of improving yacht performance. The MP was developed in the GAMS software package for both ease of use and the adaptability it affords for future development, GAMS (2003). 2 Physics of rig design As the focus of this paper is to discuss the MP, not the physics of sailing, the distribution and application of sail forces on to the rig will only be briefly described, as a means to communicate a general overview of yacht design and hopefully add to the reader’s understanding of this study. Yachts, in their simplest state, are wind-powered vessels that rely on the efficiency of their sails and underwater form to generate forwards drive force. Since sails are purposely made from soft deformable material to best catch the wind, they require the support of the mast to hold them to the wind, Henderson (1985). In this manner, the mast acts primarily as a column and requires lateral support itself, provided by the rigging (or stays) and spreaders. Working together, these parts form a complete structure commonly termed the rig. For this study, the term rig refers to the most common style in use today, the Bermudian sloop rig. as shown in Figure 1. Figure 1. Example of a Bermudian sloop rig. Courtesy of Alan Andrews Yacht Design Inc. 2.1 Force and moment equilibrium It is the job of the rig to transfer the forces from the sails to the hull of the yacht, coupling the aerodynamic and hydrodynamic systems and effectively creating the drive force, as mentioned above. When the lift forces produced by the sails are applied to the hull, they create a moment that is trying to turn the yacht over on its side, which we will call the Overturning Moment (OM). To oppose this moment and stop the yacht from rolling completely over on its side is the Righting Moment (RM). The RM is created by the ballast of the yacht, primarily in the keel and hull, and is related to the heel angle such that the RM increases as the yacht heels more. As you can imagine, the self-weight of the rig adds to the OM when the yacht is heeled, and as such is reducing the available proportion of OM contribution from the sail's lift for a given RM. Hence, if the lift force in the sails is reduced, then so is the resultant drive force component, and the performance of the yacht is directly affected. 2.2 Sail Forces Balancing OM against RM also allows a simple method of approximating the magnitude of the sail forces that are applied to the rig in each of a series of real-life sailing loadcases. These forces are predominantly located at the head (uppermost corner) of the sail, and have components in both the lateral and vertical directions (refer Figure 2). By determining the lever arm of each sail and its relative percentage of the total OM based on sail area, the resultant overturning force (lateral component) can be easily calculated. The compressive force (vertical component) is more heuristic, but can be evaluated based on an approximation of the apparent wind speed (as seen by the sails). These forces are then applied in the MP as the sail forces that must be carried by the rig in the various loadcases (refer Table 1). 88 Figure 2. Sail forces acting on rig sailing upwind under mainsail and genoa. Courtesy of Cookson Boats. Once we have the forces that the sails apply to the rig, we are able to attempt the next step of calculating the resultant loads in all members of the rig, under each loading condition. Description Load Position X Force Y Force % OM based Component (N) (N) on sail area Upwind Main & Genoa Main Halyard Top of Mainsail 2,190 -18,350 60.4% (18 knots AWS) Genoa Halyard Top of Headsail 1,621 -16,044 39.6% Upwind Main & Jib Main Halyard Top of Mainsail 2,345 -22,654 64.7% (20 knots AWS) Jib Halyard Top of Headsail 1,446 -16,506 35.3% Spinnaker Reaching Spin Halyard Top of Spinnaker 3,649 -16,472 100% Spinnaker Knockdown Spin Halyard Top of Spinnaker 5,068 -19,310 100% Table 1. Sail loads by loadcase. 3 Slack stays and the complementarity problem The primary role of stays in a rig is to support the mast, working in conjunction with the spreaders. Spreaders act as small compression struts whose role is to improve the angle of the stays to the mast. In this way, stays are designed specifically as tension-only members, as they have the added advantage of improved material properties over compression members that have to deal with buckling and other non­ linear failure modes. Typically, stays are made of wire or thin rod and have very low material “stiffness”, such that they cannot support their own weight. Complexities involved in modelling the behaviour of tension-only members become apparent when they go slack and effectively remove themselves from the structure. As a slack member, the load in the stay is zero, while by Hooke's Law the constraint that relates member change in length d i with load s, through spring constant k, would have the stay in compression, Przemieniecki (1968). ^ - k.dl This linear relation violates the physics of the structure, giving a need for a non­ linear constraint where the effective length of the member is allowed to decrease once the stay goes slack, without the outcome producing a compressive load in the stay. This is taken care of by the complementarity constraint, s > 0 _L k.dl - s > 0 . 89 Complementarity enforces the Hooke's Law member extension constraint when the member is in tension, but allows the effective length of the stay, defined by the distance between the end points, to reduce below the actual stay length without introducing compression. The application of complementarity conditions to evaluate tension-only member loads in rig design is understood by the author to be a new model for rig design, and provides the basis for an MP that can solve rig design problems in a single solve statement. 4 Embedded structural analysis With the ability to efficiently evaluate slack stays, a MP can be constructed to calculate the loads in all members and the resultant deflection of the structure under the given sail forces. This is achieved through the use of Finite Difference approximations and Structural Equilibrium Equations (SEE). The SEE are derived from mechanics of trusses theory, as the structure performs like a truss and all members in the rig except the mast are pin-jointed at the ends. Mechanics of beams theory is included to evaluate the mast members, and incorporated into the truss theory equations to maintain a level of simplicity and elegance. 4.1 Model formulation The MP is formulated as a Mixed Complementarity Problem (MCP), as defined by Ferris & Munson (2000), having linear and non-linear equality constraints as well as complementarity constraints. The MCP formulation can be compared to a linear matrix-based structural analysis, such as Matrix Stiffness Method (MSM), with linear elastic material properties, except it also incorporates non-linear geometry. This is an important addition, as it takes into account the actual deformed member positions when balancing the member forces, rather than assuming the members remain in their initial positions, as with linear MSM, Przemieniecki (1968). This assumption works satisfactorily for small strain, but yacht rigs experience large strain and require the non­ linear geometry to maintain a reasonable level of accuracy.