A review of the analytical methods used for seaplanes’ performance prediction <s> Heave and Pitch Equations of Seaplane Motion: <s> Modeling of Marine Vehicles. Environmental Disturbances. Stability and Control of Underwater Vehicles. Dynamics and Stability of Ships. Automatic Control of Ships. Control of High--Speed Craft. Appendices. Bibliography. Index. <s> BIB001 </s> A review of the analytical methods used for seaplanes’ performance prediction <s> Heave and Pitch Equations of Seaplane Motion: <s> In order to study the dynamic behavior of ships navigating in severe environmental conditions it is imperative to develop their governing equations of motion taking into account the inherent nonlinearity of large-amplitude ship motion. The purpose of this paper is to present the coupled nonlinear equations of motion in heave, roll, and pitch based on physical grounds. The ingredients of the formulation are comprised of three main components. These are the inertia forces and moments, restoring forces and moments, and damping forces and moments with an emphasis to the roll damping moment. In the formulation of the restoring forces and moments, the influence of large-amplitude ship motions will be considered together with ocean wave loads. The special cases of coupled roll-pitch and purely roll equations of motion are obtained from the general formulation. The paper includes an assessment of roll stochastic stability and probabilistic approaches used to estimate the probability of capsizing and parameter identification. <s> BIB002 </s> A review of the analytical methods used for seaplanes’ performance prediction <s> Heave and Pitch Equations of Seaplane Motion: <s> The towing tank experiments conducted at Yokohama National University from November 30 to December 9 in 2005 suggested a new way of suppressing a dangerous coupled motion between heave and pitch called porpoising. The research in this paper was developed on the observations made in the experiments and conducted numerical simulations to further investigate the parametric design space. Two linear-time-invariant models were developed: rigid-body planing craft (conventional float planes or flying boats), and flexibly supported planing craft. The latter could simulate the new method found in the experiments for suppressing porpoising. In this study, the stability of the oscillatory motions was analyzed to see the effect of design variables on the inception of porpoising. The parametric study of flexibly supported float planes in the context of porpoising was a new contribution in the conceptual design of seaplanes. <s> BIB003
A seaplane advancing at a steady forward speed with a train of regular sea waves will move in all six degrees of freedom (DOF). This means that the plane's motion is considered to have three translational components (surge, sway and heave) and three rotational components (roll, pitch and yaw). Figure 5 shows the six DOF system of a seaplane BIB002 . Eventually, in order to analytically predict the performance of a seaplane, six non-linear equations of motion must be set up and solved simultaneously. However, for a seaplane with lateral symmetry, the six equations are reduced to two sets of equations, connecting respectively, the surge, pitch and heave, and the roll, sway and yaw. In addition, as the seaplane is considered a slender body, the hydrodynamic forces associated with the translational forward motion are much smaller than the forces associated with the other motions. Not only that, but also the porpoising stability limit is studied from the equations of heave and pitch only. Consequently, the motion of seaplanes can be described by two coupled equations of heave and pitch  . A heaving and pitching motions of a seaplane can be seen as a two degree of freedom springmass system. According to Ogilvie  , a seaplane, if given heave or pitch displacements from stationary, will rapidly oscillate several times before it comes to rest. As a result, assuming that the motion is linear and harmonic, the heave and pitch equations of seaplane motion can be presented as follows:  Where: is the added-mass coefficient in the ℎ direction due to ℎ motion. is the damping coefficient in the ℎ direction due to ℎ motion. is the hydrostatic restoring force coefficient in the ℎ direction due to ℎ motion. are the complex amplitudes of the exciting forces and moments in the ℎ direction. are forces and moments given by the real part of the solution. The determination of the coefficients and exciting forces and moments is a major problem in the analytical motion prediction. According to   , the problem can be simplified by dividing the seaplane into transverse segments and then applying a strip theory to calculate the coefficients. The added mass and damping coefficients are calculated using two-dimensional hydrodynamic theory from references       BIB001 BIB003 . The general solution for each of the two equations of heave and pitch has two components. The homogeneous solution and the particular integral. The homogeneous solution is obtained when the system is considered under no external excitation forces or moments. On the other hand, the particular integral is obtained when the external excitation forces and moments are considered. First of all, the homogeneous solution of the heave and pitch equations of seaplane motion can be obtained by eliminating the right hand side of equations (1) and (2)  If a steady-state solution is assumed, then the heave and pitch can have the following forms: If equations (5) and (6) and substituted in equations (3) and (4)  For non-trivial solutions of 0 and 0 , the determinant of equation (9) is set to be zero. The characteristic equation can then be written in the following form:  Equation (10) is the fourth order characteristic equation of the system. This equation is solved to obtain four roots. The characteristics of motion of the seaplane will depend on the nature of the roots of this equation. In order to solve equation (10)  In addition to that, the particular integrals of the two equations have the following form: Then, the solution of the two equations of heave and pitch can be obtained by adding the homogeneous solution and the particular integral together as follows: In the case where the external waves are assumed to have the same frequency as one of the components of the homogeneous solution, say = 1 = 1.836, then the general solution reads as: Where 1 , 2 , 3 and 4 are determined by the external forces. It should be noted that since the objective is to obtain the solution of simple harmonic motion, only the real part of equation (21) is considered. The imaginary part would not have any effect on the values of heave and pitch as orthogonal vector components do not influence one another. In addition to that, since the initial conditions are position and velocity, there would be no imaginary part in the final solution. This means that the imaginary part is an invalid physical solution because position cannot be imaginary.