A NOVEL MANOEUVERING MODEL BASED ON LOW-ASPECT-RATIO LIFT THEORY AND LAGRANGIAN MECHANICS

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1 A NOVEL MANOEUVERING MODEL BASED ON LOW-ASPECT-RATIO LIFT THEORY AND LAGRANGIAN MECHANICS Andrew Ross, Tristan Perez Thor I. Fossen, Dept. of Engineering Cybernetics, Norwegian University of Science and Technology NTNU, Norway. Centre for Ships and Ocean Structures (CeSOS), Norwegian University of Science and Technology NTNU, Norway. Abstract: This paper proposes a physically motivated reappraisal of manoeuvring models for ships and presents a new model developed from first principles by application of low aspect-ratio aerodynamic theory and Lagrangian mechanics. The coefficients of the model are shown to be related to physical processes, and validation is presented using the results from a planar motion mechanism dataset. Keywords: Manoeuvring models, Lagrangian mechanics, hydrodynamics, aerodynamics. 1. INTRODUCTION The development of effective manoeuvring models is a vitally important task in the marine control community. The development of manoeuvring models for surface ships has a long history. Due to the complex interaction of different effects, the hydrodynamic forces of manoeuvering vessels are generally modelled by a series expansion, and the coefficients of the model are obtained by regression analysis of data from scale model tests (e.g., planar motion mechanism and rotating arm). There two types of model commonly used. The first type represents the complex interactions through a Taylor series expansion, with even terms to second order, and odd terms generally to third order. This model was proposed by Abkowitz (1964), taking into consideration port-starboard symmetry. These models have the mathematical elegance of the Taylor expansion, but they have no inherent physical meaning or interpretation. The second model commonly used is a expansion in terms of second order modulus terms ( x x). This approach was introduced by Fedayevsky and Sobolev (1963); and, as discussed by Clarke (23), the model effectively represents the crossflow drag at large angles of incidence. It should be mentioned that the differences between these two types of are quite fundamental, and the different coefficients are in general irreconcilable. In this paper, we take a first-principle approach and consider the different theories that explain the phenomena involved to a great extent. In particular, we consider a Lagrangian approach to derive the added mass and Coriolis due to added mass and use Kirchhoff s Equation to derive the corresponding equations of motion. We treat the manoeuvering vessel as a low aspect-ratio wing. Furthermore, we model the effect of roll on drag and lift by analogy to the effects that an angle of dihedral has on an aircraft wing. This is combined with cross-flow drag theory, which describes much of the damping components at large angles of incidence.

2 With this approach, the physical properties of the system, such as symmetry in the mass matrix or skew-symmetry of the Coriolis matrix are exploitable. Such properties are fundamental in energy-based control methods, such as Lyapunov theory; furthermore, the model is presented in a vectorial, not component, form. This makes the model useful for the control community, as it constitutes a physical and detailed framework. To check the fidelity of the new model, its parameters are fitted to PMM dataset and validated against other PMM runs. The results demonstrate its ability to capture the behaviour of the ship model accurately and effectively. 2. RIGID-BODY EQUATIONS OF MOTION For a manoeuvring ship, four degrees of freedom (DOF) are typically sufficient: surge, sway, roll and yaw. The Newton-Euler equations of motion in body-fixed coordinates can be expressed (Fossen 1994, Fossen 22): M RB ν + C RB (ν)ν=τ RB (1) η = J(η)ν, (2) where ν = [u, v, p, r] is a vector of body-fixed velocities, M RB is a rigid body inertia tensor about the origin of the body-fixed frame: m my g M RB = m mz g mx g mz g I x I xz, my g mx g I zx I z The rigid body Coriolis-centripetal terms are: C RB (ν) = mz gr m(x gr + v) my gp m(y gr u) mz gr my gp I yzr + I, xyp m(x gr + v) m(y gr u) I yzr I xyp with {x g, y g, z g } being the coordinates of the centre of gravity in the body-fixed frame. The vector τ = [X, Y, K, N] is made of forces and moments acting on the body, η = [x, y, φ, ψ] is the vector of generalised positions relative to the inertial frame, and J(η) is a kinematic transformation: cosψ sinψ J(η) = sin ψ cosψ cosφ 1 cosφ 3. HYDRODYNAMIC FORCES. The vector of forces τ RB can be decomposed into the control forces produced by the propulsion (3) system and control surfaces (rudders, fins, flaps, and interceptors) and the hydrodynamic forces that result from the motion of the hull through the water. In this paper, we are concerned with the latter, which can be expressed in terms of the generalised velocities, accelerations and positions: τ H = M A ν C A (ν)ν D(ν) ν g (η), 3.1 Added Mass The added mass terms are part of the radiation forces and the coefficients of the added-mass matrix can be obtained from potential theory: X u M A = M A = Y v Yṗ Yṙ, 3.2 Restoring K v Kṗ Kṙ N v Nṗ Nṙ The term g (η) describes the forces due to the mass distribution of the ship and its buoyancy. For relatively small angles of roll this can be approximated by: g (η) = ρg GM T sin φ, where GM T is the transverse metacentric height, and is the displacement volume of the vessel. For larger roll angles, this term can be modelled by fitting a polynomial to the intact stability curve of the vessel. 3.3 Added Mass Coriolis centripetal Forces The kinetic energy of the fluid due to the motion of the ship can be expressed as T A = 1 2 ν M A ν. (4) The different force components can be obtained by applying Kirchhoff s equations: X A = r T A v d T A dt u Y A = r T A u d dt T A p K A = d dt T A v N A = v T A u u T A v d dt T A p. The solution to these equations is: [ XA Y A K A N A ] = MA ν C A (ν)ν

3 For a constant and symmetric added mass matrix, C A (ν)ν is: C A (ν) = Y v v + Yṗp + Yṙr X u u. Y v v Yṗp Yṙr X u u (5) 3.4 Damping Terms The damping terms are among the most complex ones to model. These forces come from various phenomena through which the kinetic energy of the vessel is transfer to the fluid Lift and Drag A ship can be modelled as a low aspect ratio wing (Hooft 1994, Leite et al. 1998). The lift and drag forces act in the stability axes as indicated in Figure 1. These can be written as: L = 1 2 ρu2 SC L (β, Re) (6) D = 1 2 ρu2 SC D (β, Re), (7) where L is lift, D is drag, ρ is the water density, U is the total velocity, S is a characteristic area such as L 2 pp, C L is the nondimensional lift coefficient, and C D is the non-dimensional drag coefficient. The sideslip angle β can be expressed as β = arctanv/u = arccosu/u = arcsinv/u. D o b y b x b L tan β = v u U u v Fig. 1. Lift and Drag during manoeuvring. The lift coefficient is modelled as proportional to the sine of the sideslip angle (Lewis 1989b): C L = C Lβ sinβ, (8) where C Lβ is a constant of proportionality. Treating this force as a function of longitudinal position, i.e., applying the local velocity = u 2 + (v + xr) 2, gives us: C L (x) = C Lβ sin β (x) = C Lβ v + xr, which leads to X L (x) = L (x) sinβ (x) v = L (x) = 1 2 ρsc Lβ (v + xr) 2. If this is integrated this over the length of the ship, the total longitudinal force is: X L = X vv v 2 + X rv rv + X rr r 2 Proceeding in a similar fashion we can also obtain Y L (x) = L (x)cosβ (x) u = L (x) = 1 2 ρsc Lβ (uv + uxr) Y L = Y uv uv + Y ur ur The drag coefficient is conventionally modelled as a quadratic function of sideslip, with a zero angle drag coefficient (Hoerner and Borst 1975) as: C D = C D + C Dββ sin 2 β, (9) where C D is a dimensionless drag coefficient a at angle of sideslip, and C Dββ describes the induced drag proportional to sin 2 β. If low speed operations are to be considered, then we propose that the drag force should be augmented by a linear component that dominates near u = m/s. Without a linear component, exponential convergence is not present, and the system is not physically correct. This linear drag decays with velocity to reflect that, at higher speeds, the drag is dominated by higher order effects. This is somewhat similar to the situation in aerodynamics, in which induced drag dominates at low airspeed, decaying as the velocity grows, until the contribution from profile drag becomes prevalent at higher speeds (Lewis 1989b). Such a term can be added as: D l = D U exp ( au)u, where a is set to give a smooth transition between linear and nonlinear regimes: a 1 2 achieves this, although this is nothing more than an empirical value. At forward speed, the drag coefficient is that of equation (9), with a term that we propose acts linearly with the total velocity, U, which agrees with the results of Skjetne et al. (24) and Ayaz et al. (26). This is a slight abuse of notation, as the equation is seemingly no longer dimensionally correct. The drag coefficient C DU actually varies linearly with the Reynold s number, but we do not include the constant length and viscosity for brevity: these are hidden within C DU. Sectionally, the drag coefficient is then given by: C D (x) = C D + C DU + C Dββ sin 2 β (x) ( ) 2 v + xr = C D + C DU + C Dββ.

4 The sectional drag force is then: D (x) = 1 2 ρs2 C D (Re, β) [ = 1 ( ) ] v + xr 2 2 ρs2 C D + C DU + C Dββ = 1 2 ρs ( 2 C D + 3 C DU + C Dββ (v + xr) 2) This force acts in the longitudinal direction as: X D (x) = D (x)cosβ (x) (1) After integration we find that the force can be expressed as X D = X uu u 2 + X uuu u 3 + X vv v 2 + X rr r 2 + X vr vr + X uvv uv 2 + X rvu rvu + X urr ur 2. (11) The sway force due to drag can be found in a similar fashion, and finally the yaw and roll moments depend on the total lift and drag and the location of the centre of pressure. The total lift and drag forces and moments are found to be: X LD = X uu u 2 +X uuu u 3 +X vv v 2 +X rr r 2 +X vr vr + X uvv uv 2 + X rvu rvu + X urr ur 2, (12) Y LD = Y uv uv + Y ur ur + Y uur u 2 r + Y uuv u 2 v + Y vvv v 3 + Y rrr r 3 + Y rrv r 2 v + Y vvr v 2 r. (13) K LD = Y LD z cp = K uv uv + K ur ur + K uur u 2 r +K uuv u 2 v+k vvv v 3 +K rrr r 3 +K rrv r 2 v+k vvr v 2 r (14) N LD = Y LD x cp = N uv uv + N ur ur + N uur u 2 r +N uuv u 2 v+n vvv v 3 +N rrr r 3 +N rrv r 2 v+n vvr v 2 r. (15) Non-linear Lift (Cross-flow drag) The lift and drag forces described in the previous section arise from circulatory effects. However, since the ship hull is being treated as a low aspect-ratio wing, it is necessary to include an additional nonlinear lift component, with an associated induced drag term. This component of lift arises from the deflection of water and the imparting of momentum onto this water. Hoerner and Borst (1975) stated that this comes from a pair vortex sheets that curl around the lateral edges of an aircraft s wings. In this paper s model, this corresponds to the presence of a single vortex sheet that curls vertically around the bottom of the hull (not horizontally around the bow and stern). The form that this nonlinear component takes is C L = k C nl sin 2 β cosβ. (16) where k is a form coefficient. The induced drag that arises from this is C D = k C nl sin 3 β. (17) The cross-flow drag principle assumes that the viscous drag is a function solely of the fluid velocity athwartships. Given this, the sectional crossflow drag coefficients can be used to calculate the sway, yaw and roll forces acting on the ship. Explanations of this principle can be found in the work by Faltinsen (199), Beukelman and Journee (21) or Golding et al. (26). After finding the sectional cross-flow drag forces and integrating over the length of the hull, it can be shown (Norrbin 1971) that the non-linear lift forces can be approximated by Y cf = K cf = N cf = Y v v v v + Y r v r v + Y v r v r +Y r r r r (18) K v v v v + K r v r v + K v r v r +K r r r r (19) N v v v v + N r v r + N v r v r +N r r r r. (2) Influence of Roll The roll angle φ influences the lift and drag characteristics of the hull. Especially for ships with a low metacentric height, roll has a significant effect on the manoeuvring characteristics (Blanke and Jensen 1997). This emphasises the need for detailed modelling of the the roll-sway-yaw interactions. In an aircraft wing, the axial tilt (or v-shape) of a wing is called dihedral if tilted upwards, and anhedral if tilted downwards. Dihedral is shown by Γ in Figure 2. Γ Wings Fig. 2. Dihedral interpretation of roll angle. This effect can be characterised by equating the roll angle with an angle of dihedral or anhedral: anhedral when the ship rolls into the sway direction, and dihedral when the ship away from the sway direction. The term dihedral will be used to mean both. The dihedral angle of a rolling hull is shown in Figure 2, where φ and Γ are conceptually the same. φ

5 Hoerner (1965) detailed the effect that an angle of dihedral has on the induced drag function using the following: CL 2 C Di = πa R cos 2 Γ, where Γ is the angle of dihedral, C L is the lift coefficient, and A R is the aspect ratio of the wing. In this manoeuvring model, the same structure is used to describe the dihedral effect using the roll angle, i.e. Γ = φ. This formula is used to augment the drag equation (1) as follows: C D (φ) = C cos 2 φ β2 = C sin β 2 ( 1 + tan 2 φ ) C D (φ) = C v2 U 2 tan2 φ = C v2 U 2 φ2, (21) where C is a generic coefficient that is replaced later. This enters into the drag equation as: D (φ) = 1 v2 ρu2 2 U 2 Cφ2 = 1 2 ρv2 Cφ 2. (22) Using tan 2 φ φ 2 introduces errors of approximately 5% at φ = 15, but the errors escalate quickly above this. For large roll angles, such as manoeuvring in extreme seas, it becomes important to keep the trigonometric relationship, or to introduce an additional coefficient from the series expansion of the tan function: tan x = x + x x x < π 2 tan φ 2 φ φ4 D (φ) = 1 2 ρv2 ( φ 2 C Dφφ + φ 4 C Dφ4 ). In this paper, the drag function given in (22) is assumed to be good enough. The effect that dihedral has on lift is described in Hoerner and Borst (1975) work on fluid dynamics, in which the generic equation describing the lift coefficient in relation to the angle of attack is given by: dα 1 = dc L 2π cos 2 Γ + 1 πa P cos 2 Γ. We apply this equation and subtract the lift coefficient at zero angle of dihedral (using cos 2 φ = 1 sin 2 φ) to give: C L (φ) = C cos 2 φβ C L (φ) = C sin 2 φβ (23) = C Lφφ φ 2 v u. (24) These terms enter into the original lift and drag equations to give: L = 1 2 ρ ( u 2 + v 2) v u C Lφφφ 2 (25) = 1 2 ρsuvc Lφφφ 2 (26) D = 1 2 ρsc Dφφv 2 φ 2. (27) In the body fixed frame, these changes in lift and drag are given by: X LD = X L vvφφ v2 φ 2 Y LD = Y L uvφφ uvφ2. The drag forces experienced by the ship can increase in the region of 1-15 % during a moderately aggressive manoeuvre, while the lift forces drop by a similar percentage, and so it is important to take this into account in the modelling process Total Forces Collecting all the effects and adding linear and cubic roll damping, we can obtain a expression for the forces as: N (ν)ν (C (ν)+d(ν)) ν with X N, Y N, K N, N N being: = [ X N Y N K N N N ], (28) X N = X L uuu 2 X L uuuu 3 X L vvv 2 X L rrr 2 X L rvrv X L uvvuv 2 X L rvurvu X L urr ur2 X L uvφφ uvφ2 + Y v vr + Yṗpr + Yṙr 2 (29) Y N = Y L uv uv Y L ur ur Y L uur u2 r Y L uuv u2 v Y L vvvv 3 Y L rrrr 3 Y L rrvr 2 v Y L vvrv 2 r Y L uvφφuvφ 2 Y v v v v Y r v r v Y v r v r Y r r r r X u ur (3) K N = K p p K ppp p 3 K L uv uv KL ur ur K L uur u2 r K L uuv u2 v K L vvv v3 K L rrr r3 K L rrvr 2 v K L vvrv 2 r K L uvφφuvφ 2 K v v v v K r v r v K v r v r K r r r r (31) N N = N L uv uv NL ur ur NL uur u2 r N L uuv u2 v N L vvv v3 N L rrr r3 N L rrv r2 v N L vvr v2 r N L uuφφuvφ 2 N v v v v N r v r v N v r v r N r r r r Yṗpu Yṙru + (X u Y v )uv. (32) 4. MODEL VALIDATION Figure 3 shows a comparison between different models fits to a PMM dataset. The data is separate from the set used to estimate the coefficients.

6 The model derived within this paper fits more closely than a conventional model, especially in the roll mode. Surge Force (N) Sway Force (N) Yaw Moment (N.m Pure Yaw Motion at 18 kt Lagrangian model PMM Test Conventional model Roll Moment (N.m time (s) Fig. 3. Pure yaw PMM test comparison of new model and traditional series expansion model for hydrodynamic forces. 5. CONCLUSION In this paper, we have taken a first-principle approach and consider the different theories that explain the phenomena involved to a great extent: Lagrangian mechanics, low-aspect-ratio aerodynamics, and cross-flow drag. The fidelity of the new model has been demonstrated by its ability to capture accurately and effectively the behaviour of a manoeuvring vessel during a PMM test. REFERENCES Abkowitz, M. A. (1964). Lectures on Ship Hydrodynamics - Steering and Maneuverability. Technical Report Hy-5. Hydro- and Aerodynamic s Laboratory. Lyngby, Denmark. Ayaz, Z, D. Vassalos, J. Kostas and J. Spyrou (26). Maneouvring behaviour of ships in extreme astern seas. Journal of Ocean Engineering 33, Beukelman, W. and J.M.J. Journee (21). Hydrodynamic transverse loads on ships in deep and shallow waters. In: 22nd International Conference on Hydrodynamics and Aerodynamics in Marine Engineering. Blanke, M. and A. G. Jensen (1997). Dynamic Properties of Container Vessel with Low Metacentric Height. Transactions of the Institute of Measurement and Control TIMC- 19(2), Clarke, D. (23). The foundations of steering and maneuvering. In: Proceedings of the IFAC Conference on Control Applications. Plenary talk. Faltinsen, O. M. (199). Sea Loads on Ships and Offshore Structures. Cambridge University Press. Fedayevsky, K.K. and G.V. Sobolev (1963). Control and Stability in Ship Design. State Union Shipbuilding Publishing House. Leningrad, USSR. Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles. John Wiley and Sons Ltd. ISBN Fossen, T. I. (22). Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics AS. Trondheim, Norway. ISBN Golding, B., A. Ross and T.I. Fossen (26). Identification of nonlinear viscous damping terms for marine vessels. In: IFAC SYSID 6. Hoerner, S.F. (1965). Fluid Dynamic Drag: Theoretical, Experimental and Statistical Information. Hoerner Fluid Dynamics. Hoerner, Sighard F. and H.V. Borst (1975). Fluid-Dynamic Lift: Information on Lift and its Derivatives in Air and in Water. Hoerner Fluid Dynamics. Bakersfield, CA 9339, USA. Hooft, J.P. (1994). The cross-flow drag on a manoeuvring ship. Ocean Engineering 21(3), Leite, A.J.P., J.A.P Aranha, C. Umeda and M.B. de Conti (1998). Current forces in tankers and bifurcation of equilibrium of turret systems: hydrodynamic model and experiments. Applied Ocean Research 2, Lewis, E. V., Ed. (1989a). Principles of Naval Architecture. 2nd ed.. Society of Naval Architects and Marine Engineers (SNAME). Lewis, E.V., Ed. (1989b). Principles of Naval Architecture. 2nd ed.. Society of Naval Architects and Marine Engineers (SNAME). Norrbin, N. (1971). Theory and observations on the use of a mathematical model for ship manoeuvring in deep and confined water. Technical Report 63. Swedish State Shipbuilding Experimental Tank. Gothenburg. Skjetne, R.,. Smogeli and T.I. Fossen (24). A nonlinear ship maneuvering model: Identification and adaptive control with experiments for a model ship. Modeling, Identification and Control (MIC) 25, 3 27.