"RECENT ADVANCES IN THE ANALYSIS OF HIGH SPEED PLANING HYDRODYNAMICS AND DYNAMICS"

"RECENT ADVANCES IN THE ANALYSIS OF HIGH SPEED PLANING HYDRODYNAMICS AND DYNAMICS" Lab ratcdwii vor 1tt -z Dr. Armim Troesch - University of Michigan, Ann Arbor ABSTRACT High-speed planing craft are seeing increased use as recreational, commercial, and naval vehicles. The growth in popularity of this type of craft is demonstrated by the number of conferences on the subject, e.g., ASNE/HPVC '92, FAST '93, FAST '95, and FAST '97. However, operational difficulties associated with the powering and dynamics of planing hulls have been 'extensively documented. Unlike displacement vessels, the dynamics and hydrodynamics of planing craft generally do not lend themselves to a linear analysis. Their high speeds, small trim angles, and shallow drafts produce significant nonlinearities. This paper will review new methods in evaluating steady hydrodynamic performance and unsteady dynamic performance including dynamic stability and response in a seaway. Recently developed vortex lattice methods and slender body theories are used to predict steady, calm water performance. Next, by incorporating experiments and theory, a method for the examination of vertical plane stability of planing vessels in calm water and in waves is described. Much of the work presented here has appeared in scientific archival journals, not commonly available to the practicing naval architect. For that reason an extensive list of references is also given. 1. INTRODUCTION The analytical study of planing hydrodynamics began as early as 1930, when von Karman (1929) and Wagner (1931) examined the landing of seaplanes. Since that time much effort has been expended on studying the steady-state behavior of planing hulls. The rationale for current methods which follow this early work (e.g., Shuford, 1957), has a genesis in low-aspect-ratio-wing theory (or slender body theory), although not of a very consistent form. Perhaps the most popular and still most widely used methodology is that due to Savitsky (1964). Savitsky's method, based upon prismatic hull forms, is mostly empirical and is easily adapted to tabular-type calculations. Payne (1988) gives a review of many of the relevant articles and follows with a more detailed examination of published experimental results (Payne, 1995). To summarize Payne (1988). the current methods for the prediction of steady lift and drag forces combine theoretical and experimental results. Due to the empirical nature of the methods, their applicability is restricted to a limited class of geometrically similar hull forms. Compared to the study of the steady, calm water performance, planing dynamics have received even less attention, e.g. experiments by Fridsma (1969 and 1971), linear vertical plane stability and motions by Martin (1978(a) and (b)), nonlinear seakeeping simulators e.g. Zarnick (1978) or Payne (1990), or linear seakeeping statistics by White and Savitsky (1988). When discussing dynamic behavior, two primary concerns of the high speed craft designer and operator are the areas of operations and safety. Operations, or the operability of the planing craft, is related to rider discomfort and speed loss in a seaway. Severe shock loads cause a reduction in operating personnel effectiveness and provide strong motivation for shock mitigation strategies or devices. The safe dynamic performance, or the survivability of planing hulls generally refers to stability issues such as the occurrence of coupled yaw-roll instability, i.e. "chine walking", coupled heave-pitch

instability, i.e., "porpoising," or vertical motion instability in a seaway leading to "pitch-pole" capsizing. Both of these areas, operability and survit'ability, are linked to vessel dynamics, but they are different in that dynamic operability refers to small, frequently occurring motions that can potentially degrade the overall performance of the craft while instability refers to large, possibly catastrophic behavior. The following sections describe state-ofthe-art steady and dynamic planing hull performance prediction techniques. The eventual goal of these collected efforts is a rational technology for calm and rough water performance in seas of all headings allowing designers and builders of planing boats to design better performing, more economical, and safer craft. 1. The time dependent amplitudes of motion are given as flk(t) where (1h,1) are the vertical displacement of the center of gravity and the rotation of the body relative to the inertial axis respectively. Surge is generally small and therefore effectively decouples from the other two degrees of freedom. The vertical center of gravity, measured from the keel is icg and the longitudinal center of gravity, measured from the transom, is cg. The beam is given as B, the deadrise as, an the mean trim angle is defined as t, positive bow up. In keeping with traditional planing hull nomenclature, the definition of trim, positive bow up, conflicts with the definition of pitch, r, positive bow down. 1.1. Critique of the Various Methods It should be emphasized that the goal stated in the previous paragraph of developing accurate planing performance predictors for all conditions and hull shapes has not yet been realized. While the progress has been substantial, the methods are still based upon simplified hydrodynamic models, using assumptions such as zero gravity, empirical sectional adde.d mass and damping coefficients, two dimensional flows, and "a speed dependent Archimedes' force." Realizing the models' limitations and including a dose of healthy skepticism, one can successfully use these current methods (which incorporate much of the relevant physics) in conjunction with model experimen&, prototype testing, and practical experience to ore effectively design high speed craft. 1.2. Problem Definition The discussion in is paper will concentrate on vertical plane performance, e.g. steady forward speed, heave and pitch motions, and porpoising. This is not to infer that the transverse plane motions, which include chine walking and maneuvering are unimportant, but rather that we are solving the more tractable problems first. Consider a planing hull with the right handed coordinate system as defined in Figure Lii) Figure 1. Coordinate system and wetted length definitions. The wetted surface is comprised of the pressure area and the spray area. Following the conventions of Savitsky (1964), Savitsky & Brown (1976), and Latorre (1983), the keel and chine wetted lengths are labeled as Lk(t) and L(t) respectively. When the hull is traveling forward with constant speed and no vertical oscillation, the wetted lengths are essentially constant. When the hull is undergoing vertical motion, the wetted lengths become functions of time. The wetted length and equivalently the wetted surface of an oscillating planing hull is strongly time dependent. For low to moderate amplitudes of motion on a hull with moderate B I 2

deadrise, e.g. 20 degrees, the keel wetted length can be treated kinematically as the intersection between the keel and a stationary free surface. As the vertical velocity of the bow increases with increasing heave and pitch amplitudes, surface disturbances are pushed ahead of the bow and the keel wetted length must include free surface dynamics. The chine wetted length is influenced by the time-dependent spray-jet dynamics at all speeds and the kinematic intersection between the keel and the mean free surface provides only a rough approximation (Troesch, 1992). There are two distinctly differeñt flow regimens in the planing hull flow physics: chine unwetted (i.e. the region between Lk(t) and L(t)) and chìne wetted (i.e. the region aft of L(t)). For the purposes of this paper, the chine-unwetted condition is depicted in Figure 2a and the chine wetted in Figure 2b. The chine is always a point of zero dynamic pressure. Chine unwetted or chine wetted flow depends, by the criteria adopted here, on whether or n the dynamic pressure gradient, in the directio tangent to the hull surface, is also zero at the chine. Of particular significance to lift, drag, and trimming moment calculations, the chine unwetted area typically corresponds to a high lift area contributing to a high lift to drag ratio while the chine wetted area corresponds to low lift but large wetted surface drag. pressure (a) Figure 2. Schematic of chine unwetted condition with pressure distribution, forward station. Schematic of chine wetted condition with pressure distribution, after station. 2. CALM WATER PERFORMANCE Prediction of planing hull calm water lift and drag involves one of the more challenging problems in free surface hydrodynamics (Lai & Troesch, 1995). The dynamically supported planing hull generates complex surface flows including spray jets and reentrant breaking waves. Due to the extreme difficulty in accurately solving the fully nonlinear boundary value problem, various approximate models have been put forth. Generally, the theories may be considered to be classed as either two-dimensional slender body theories (or less rigorous two dimensional strip theories) or fully three-dimensional theories. Gravitational effects on the free surface become higher order as the speed increases, and consequently many theories are derived for zero gravity. 2.1. Two Dimensional Strip and Slender Body Theories Wagner (1933) initiated planing slender body theory when in 1932 he modeled the planing hydrodynamics as a water impact problem. More recently, this slender body, two-dimensional water entry model has been extended by Vorus (1992, 1996) and Zhao & Faltinsen (1992). In the period between the above three referenced works, 1932-1996, there have appeared numerous articles on the "added mass" (i.e., impact or water entry) strip theory of planing. Payne (1988, 1993, and 1995) discusses many of these methods and references them in a comprehensive bibliography. Due to their simplicity, the added massstrip theories have gained wide acceptance. However, this approach misses some of the relevant physics of planing and has to be calibrated using experimental results. For hull forms that follow the family of hulls on which the added mass method is based, e.g. prismatic hulls with no keel camber, the method works well indeed. The recent slender body theories, e.g. Tulin (1956) or Vorus (1996), are derived from well defined mathematical models. This suggests, at least in principle, that the methods can be applied to more general hull shapes, 3

e.g. twin hulls, catamarans, hydro-planes., etc. The analysis required to understand the theory, though, is quite advanced and generally not familiar to small craft designers. In addition the computations require workstation level computing power, something that is currently not readily available to the general small craft industry. 2.1.1. Practical application of slender body theory To illustrate the potential of the method, an example of a successful application of the slender body theory described by Vorus (1996) to actual planing hulls is described next. Students from the University of Michigan (UM) have participated in an international, intercollegiate solar boat regatta. This yearly event, sponsored by the American Society of Mechanical Engineers, gives students an opportunity to design and build boats powered by battery and solar power. The contest itself involves a sprint race for top speed and a two hour endurance event. University of Michigan students and faculty advisors selected a hull form based upon the 1920's Hickman "Sea Sled" concept. The original sea sled "looked like someone had taken a perfectly normal V- bottomed boat and cut it down the centerline, then reassembled it so the original sides were in the center and the centerlines were on the sides." (Seidman, 1991) The UM team applied the slender body theory and computer programs developed by Vows (1996), taking advantage of the increased pressure associated with chine unwetted or chine dry flow as shown in Figure 2a. The UM boat is described as the Inverse "V" or "Vn V" for short. As can be seen from Figure 2, the characteristics of the flow, which include the hull pressure distribution, jet velocity, and free surface deformation, change dramatically as the jet edge passes over severe hull geometric variations. When the jet head reaches a location on the hull's surface where the surface curvature exceeds that which would normally occur in an unrestrained jet, such as at a chine, the pressure drops significantly. If the hull deadrise can be varied in such a fashion that the flow remains chine unwetted over most of the boat length, then the hull will' have a significantly higher lift to drag ratio than that of a comparable hull which has large- chine wetted sections. In addition, by chaiging the deadrise to be interior rather exterior, extra lift is generated by the jet reversal under the hull. This was the philosophy followed in the design of the VnV. See Figure 3 for a photograph of the VnV hull mold during construction. Based upon slender body planing analysis (Vows, 1996), the longitudinal variation in deadrise was optimized such that a maximum lift to drag ratio was achieved. Figure 3. Construction of the male plug for the UM solar boat, VnV. Figure 4. The University of Michigan's VnV in the sprint race. 4

powered by two surface-piercing propellers and three 36 volt marine-grade batteries, the UM boat won the sprint race by achieving a top speed in excess of 30mph (48kmph). Figure 4 sho the boat at speed. 2.2. Three Dimensional Theories Three dimensional planing hull models have also been developed. Most of these are based upon some computational fluid dynamics code and are numerically intensive. Generally, the two classes of models are those that include or those that do not include gravity in the free surface boundary condition. The two dimensional approaches discussed in the previous sub-section represent che limit of infinite Froude number, where the influence of gravity is neglected. Representative articles of three dimensional modeling combined with gravity are Wang & Rispin (1971), Wellicome & Jahangeer (1978) and Doctors (1974). Wang & Rispin (1971) complete an asymptotic expansion in terms of the inverse Froude number, while Wellicome & Jahangeer and Doctors distribute pressure panels on the mean horizontal plane, i.e., z=o, approximation to the hull's surface. Their analysis, based upon the linearized free surface condition, provides for downstream wave propagation and yields reasonable estimates for the hydrodynamic force when the speed is low. When the Froude number is high, however, the results begin to diverge from experimental results. Of significance to the more recent work described by Lai & hoesch (1995) and Savander (1996), none of the aforementioned three dimensional methods satisfy a continuity of velocity (i.e., a Kutta condition) on the chine where the flow leaves the hull surface in the transverse direction. These papers, however, do satisfy a Kutta condition at the trailing edge or transom. If a Kutta condition is to be applied on all of the hull's surfaces from which the flow separates, then methods described by Lai & Troesch (1995 and 1996) or Savander (1996) should be applied. Lai and Troesch employ a vortex lattice method to solve the nonlinear free surface boundary value problem. The wetted surface is defined prior to the calculation, using empirical results from Brown (1971) and Martin (1978a), or analytical results from Vorus (1992). Savander uses a three-dimensional boundary integral method to determine the lifting surface corrections to the slender body two-dimensional solutions of Vorus (1996). Typical results are shown in Figures 5-7. In Figure.s 5 and 6, the pressure distributions are shown for a prismatic planing hull. Figure 5 displays the complete distribution for a twenty degree deadrise hull with a running trim angle of 5 degrees. The mean wetted length to beam ratio; X. is 2.5. The high pressure area in the chine unwetted area is apparent. Figure 6 shows a comparison between experiment and vortex lattice calculations. Pressures at two longitudinal cuts along the hull are given: one cut at the centerline and one cut at a quarter beam outboard from the center line. The experimental results are from Kapryan & Boyd (1955). Figure 5. Pressure distribution calculated from a vortex lattice method for a prismatic planing hull. = 20 degrees, t = 5 degrees, X = 2.5 (Lai & Troesch, 1996). Figure 7 from Savander (1996) shows the pressure distribution on the half plane for a typical water ski boat traveling at 25mph (40 kmph). As can be seen from the figure, the longitudinal variation in deadrise produces a C. 5

reduction in the pressure, potentially leading to negative gauge pressures and subsequent losses in lift in the bow region. Full scale measurements confirmed the existence of these Suction regions. This effect could not have been predicted by the two dimensional strip theories based upon added mass coefficients as described in section 2.1. o 0.5 15 2 23 3 3.6 Figure 6. Center and quarter beam pressure distributions for a prismatic planing hull. = 20 degrees, =6 degrees,. = 2.91 (Lai & Troesch, 1995). C,xV Redmond, Washington). USAEROIFSP is a source-doublet panel code that satisfies the complete nonlinear body and free surface boundary conditions, including the effects of gravity. Two quantities used to judge the accuracy of USAERO/FSP for this type of application were the determination of the wetted surface and the vertical lifting force for prismatic hulls. Both the lift coefficient and the apex angle of the wetted surface in the chines dry area can be determined from well established empirical relationships if the mean wetted length and trim angle are given (Savitsky, 1964 and Savitsky & Brown, 1976). The intersection of the hull and undisturbed water datum is given by the apex angle between the waterline and hull centerline. The more significant the rise in the spray sheet, the more the dynamic apex angle will increase over the reference apex angle. Since the highest pressures in chine unwetted flow are encountered near the apex, errors in the modeling of the spray sheet dynamics will significantly influence the lift force predictions. Calculations using the current version of USAERO/FSP (Wang, 1995) have shown a lower apex angle and corresponding lower lift. This suggests that fundamental work is still needed to accurately describe three dimensional spray sheet flows associated with planing before standard, nonlinear CFD codes can achieve the same level of accuracy as the more specialized planing hydrodynamics codes, e.g. Payne (1990), Lai & Troesch (1995), and Savander (1996). 3. PLANING HULL DYNAMICS IN THE VERTICAL PLANE Figure 7. Pressure distribution on the half plane for a non prismatic ski boat. Note reduction in forward pressure due to keel camber. (Savander, 1996). The calculation of planing hull hydrodynamics has also been completed by Wang (1995) using the commercial CFD code USAERO1FSP (Analytical Methods, Inc. As discussed above, the generation of lift for high speed craft is fundamentally different than that of displacement craft. While displacement vessels rely almost exclusively on hydrostatic buoyancy forces to keep them afloat, planing hulls generate much of their lift dynamically. In this area, planing hull technology shares many close parallels with the science of aircraft lift and drag. Similarly, the dynamic behavior of planing craft and the dynamics of displacement hulls differ in a fundamental way. Hydrostatics can be used to achieve a reasonable approximation of the 6

vó and pitch responses for a displacement ji ewman, 1970). Disregading1he higher 0der effects of system inertia and damping, the only significant hydrodynamic forces cting on this type of vessel are the hydrostatic restoring force (i.e. the vertical force per unit vertical displacement and the moment to trim a unit rotation) and the incident wave force (i.e. the Froude-KrYl0' force). The system inertia and damping are of higher order and at low speeds the system does not exhibit any resonant behavior. Planing hull dynamics are jntrinsically more difficult. The system stiffness is no longer related to a static spring but rather to a dynamic spring, one that is a nonlinear function of the craft's speed and its rapidly changing wetted surface. In addition, high planing speeds leading to high wave frequency of encounters make resonant motions common. This increased complexity of planing hull dynamics compared to that of. displacement craft has resulted in different design methodologies for studying dynamic behavior. A significant, though not isolated, example of planing hull dynamic instability is described in Codega & Lewis (Codega & Lewis, 1987). The United States Coast Guard purchased 20 high-speed surf rescue boats for search and rescue operations. While able to perform most of their required missions, the boats would become unstable when operating at high speed in waves, especially if turning maneuvers were attempted. Each hull exhibited the unstable behavior to some degree, but each was unique in its individual response. "Some were very easy to force into the unstable mode but very controllable once there. In others, the instability was difficult to induce, but the result was very severe." (Codega & Lewis, 1987) To the limits of manufacturing tolerances, all the boats were the same indicating that the cause of the instabilities was beyond the factors normally considered in planing boat design. Though the cause of :his coupled roll-yaw instability was eventually identified and corrected, it serves to illustrate the difficulty of finding and fixing undesirable dynamic behavior at the design stage. Planing hull designers, lacking the extensive resources available to the displacement vessel community and saddled with a significantly more difficult problem, have to rely more upon their previous experience than actual calculations. If an evaluation of a planing craft's dynamic performance is to be made, the current options appear to be a limited linear analysis (e.g. Martin, 1978b), previous model tests (e.g. Fridsrna, 1969 and 1971), empirical formulae based upon model tests (e.g., Savitsky & Brown, 1976, and Blount & Fox, 1976), simulation (e.g., Zarnick, 1978 and Payne, 1990), initiating new model. tests, or constructing a prototype for full scale testing. Due to the significant nonlinearities associated with planing dynamics, simulation appears, to be gaining acceptance as a low cost alternative for designers. "The role that simulation should play in design, particularly in preliminary design, however, is not clear. While the desk-top computer trade magazines extol the virtues of the newer physics and mathematics simulators, they also acknowledge that there are sorne edges to this simulated world, and if you step over, the simulation breaks down badly (Swaine, 1992). Since it is one of the goals of design to define those very "edges, that is, define the design wave, the design response, the design bending moment, etc. that the system should successfully withstand, accurate knowledge of ail of the system's wedges" is essential. The attraction of a planing hull simulator is that many complicated and nonlinear aspects of the planing dynamics problem can be accurately included. The primary disadvantage of simulation, a disadvantage also inherent in experimental model test programs, is that the parameter range under consideration is usually limited and therefore the determination of all the system's wedges" is generally not possible. \Vhile the availability of faster and larger computers has made it possible to include more sophisticated dynamics modeling in planing simulators, a finite simulation (or a finite experimental model test program) of a highly nonlinear system can still only give a partial view of the total system characteristics. If simulation is to play an important part in planing hull design, the designer should be able to identify beforehand the initial conditions and parameter ranges related to 7

critical performance areas." (Troesch & Hicks. 1994). n order (o more effectively use a simulator, an evaluation of the dynamic planing hull system should include modern geometric methods of nonlinear analysis (e.g., Troesch & Falzarano, 1992, Troesch & Hicks, 1994, Hicks, et ai, 1995). A brief description of the methodology is given below. Considering only vertical plane dynamics (i.e. heave and pitch), the equations of motion about the center of gravity are Z=mfl3(t) M = I55r5(t) (I) where m is the planing hull's mass and 155 is the pitch mass moment of inertia about the center of gravity. The vertical force, Z, and moment, M, include the sum of all hydrodynamic and propulsive contributions including trim tabs and propellers. In the absence of any excitation, the accelerations are zero and the hull assumes an equilibrium position which is a balance of the various force and moment components. The iterative method to find this mean attitude is an essential part of the calm water, steady forward speed problem described earlier. Equation (1) appears deceptively simple. In reality, the hydrodynamic forces and moments are functions of the unknown rigid body accelerations, velocities, and displacements These functional relationships involve the solution of nonlinear integrodifferential equations. In order to practically apply the nonlinear analysis methods, the equations of motion must be written as ordinary differential equations. Using the insight gained from forced-oscillation model test results (e.g. de Zwaan, 1973 and Troesch, 1992), the physics of planing dynamics/hydrodynamics can be modeled (i.e. approximated) as forces in phase with the motion acceleration, the motion velocity, and a functional representation dependent upon the motion displacement (Hicks, et al, 1995). Applying these assumptions, the matrix form of the equations of motion becomes [A]{q(t)) + [B]{î(t)} = (2) _{F )} +{Fe(t)} where A and B are [2x21 constant matrices representing the mass plus the added mass and damping coefficients, respectively, F"(r1(()) is a vector functiok representing the total restoring force and moment for a given hull attitude, 1(t), F:0) is a constant vector representing the mean lift and trimming moment, and F(t) is a vector function with sinusoidal time dependence representing the incident wave exciting force and mcment for regular waves. Following the techniques described by Troesch & Falzarano (1992) and Troesch & 1-licks (1994), Eq. (2) can be examined to determine critical performance areas and to investigate the many different options at the design stage. The first step would be to restrict the parameter range to a manageable size. Decisions about hull loading condition and geometry (e.g., speed, displacement, trim angle, center of gravity, length, beam, deadrise, number and location of chines, spray rails, etc.) will presumably have been made earlier during the calm water powering analysis. While these parameters may represent an optimum steadystate propulsion condition, they can and should be adjusted to achieve a safer or more comfortable ride. The choice of operating environmental conditions increases the size of the parameter matrix. Critical wave lengths, headings and heights should be identified and non-critical conditions eliminated. With the guidance provided by the analysis of Eq (2), a more accurate and efficient simulation study can be conducted. As an example (Troesch & Hicks, 1994), the critical cg value (i.e., bifurcation point) at which porpoising occurs was estimated using continuation methods (Seydel, 1988) in conjunction with Eq. (2). Porpoising is defined as periodic heave and pitch oscillations in the absence of incident waves. Figure 8 is a schematic of the types of motion possible, where Hopf bifurcation curves with single and multiple branches are shown. Figure 8a 8

sketches a typical bifurcation where one stable branch of periodic solutions (non zero heave and pitch for 1cg<1cg) connects to an equilibrium line (zero heave and pitch for cg > cg,). This curve is representative of the behavior of simulated motions for lower speeds where there is a single solution for a given parameter (i.e., cg) value. Figure 8b sketches a bifurcation curve with two stable periodic branches connected by a possibly unstable branch. This curve is representative of the behavior of the simulated motions shown in Figures 9 where a single parameter value may have more than one possible steady state solution. (a) output in an unforced state, that is, no incident waves. The motions are plotted versus time normalized by the linear natural period. The condition of the hull in the figure is LIB = 7 with a beam Froude number of C, = 5.0. The trim and mean wetted length ratio are initially 5.8 degrees and 3.1 respectively. These three time series correspond to increasingly aftward shifts of the leg which exhibit increasingly unstable, porpoising-like behavior. 'u" (a) 0.8 lu!i!t I!!!! 113(t) 0.4 B o -0.4 0 5 10 15 20 25 30 35 40 45 t/t 1.2'l",".' Icg11 kg/b 1.2 018 (b) J lcg cruical (b) Figure 8. Schematic of typical Hopf bifurcation curves Single branch. Multiple branches. (Troesch & Hicks, 1994) kg/b Figure 9 is a simulation of Eq. (1) following Zarnick (1978). The simulator was run in the unforced condition. This verified the critical cg value at which the hull became unstable and demonstrated the boundedness of the heave and pitch motions once the hull began to porpoise. Figure 9 shows a series of heave time histories produced by the simulator which aie typical of the graphical simulation i3(t) 0.4 B O -0.4-0.4 0 5 10 15 20 25 30 35 40 45 tri; uuiiiimim iiiiiiii 0 5 1015202530354045 tri; (c) Figure 9. Simulated unforcd heave time histories (porpoising): C. = 5.0 and UB= 7.0. (a) leg/b = 2.09. (b) leg/b = 2.03. (e) leg/b = 1.98. (Troesch & Hicks, 1994) 9

The simulation in Figurè 9 clearly shows that the heave motion oscillates between two bounded magnitudes. Initially the response follows the smaller of the two amplitudes of motion, with only a few instances of large oscillation (a). As the cg progresses aft, the attraction of the larger amplitude is more clearly defined by more occurrences of 113(t) oscillating at the higher amplitude (b), until a point is reached where 113(t) moves periodically between the two amplitudes (c). This type of behavior, i.e., attraction to two different oscillation amplitudes, is explained by curves such as those shown in Figure 8. The above results have practical significance for the planing hull designer or builder. While porpoising may appear to be an undesirable operating condition, it is not uncommon to observe recreational boat owners running their planing hulls at high speeds while experiencing small vertical oscillations. In reality these boats are porpoising but an experienced operator may feel that the motions are acceptable. Blount and Codega (1991) state that when a boat porpoises in a certain operating condition, it will continue to do so whenever that condition is repeated, allowing the operator tó anticipate and perhaps accept small oscillatory instabilities. This acceptance of usmallu instabilities suggests that a significant number of hulls are operated in a parameter region of linear instability, that is near the Hopf bifurcation point. However, severe or possibly csastrophic motions may occur as the craft is suddenly attracted to the other periodic solution and experiences a sudden increase in its heave and pitch motions. This unanticipated extreme behavior may represent the edge' which a sale design should not step over. After the critical parameter ranges for steady, calm waler operation have been identified, the forced motions, i.e., the response in a seaway, can also be determined. By using Equation 2 and the continuation methods described by Seydel (1988) and Troesch & Falzarano (1993), approximate magnification curves of the heave and pitch responses can be constructed. These curves provide guidance when selecting the range of values for the incident wave amplinídes and wave lengths used in the more accurate simulation or experimental model tests. An example of the results of simulation in incident waves is shown below for a typical high speed offshore racing hull. The various parameters for this case are as follows: LIB = 4.9, C,. = 4.5, and Ç = 1.5 ft (0.46m). Here Ç, Is the incident wave amplitude. The pitch magnification curve is plotted in Figure 10. The pitch rrns values in degrees are plotted as functions of wavelength. The magnification curve is multiple valued for wave lengths to boat length (A/BL) in the range of 6.8 to 7.4. Unstable and multiple solutions are found near the peak responses. The motions are characterized by sudden jumps to larger or smaller solutions, similar to the time history shown in Figure 9c. 0.7 O.6 0.4 o of multi- Yregion 01234567891 wave length/boat length Figure 10. Simulated pitch magnification curve as a function of incident wave length. Wave amplitudes = 1.5 ft (0.46m). Two pitch time histories for ).JBL = 7.0 are shown in Figure 11. The only difference between the two runs are the values used for initial conditions in the simulator. (This should not be surprising, since the nature of periodic solutions of highly nonlinear systems frequently exhibits a strong dependence upon initial conditions.) In addition to the different 'o

RJAS values, the time histories also exhibit a significant dynamic bias or shift in the average running trim. This is a result of the implicit asymmetric (quadratic) nonlinearities in the diagonal and coupled stiffness matrix, F(1(t)). (Troesch & Falzarano, 1993) u ç) 8 - Iv1rnMM!.uhII IfIOßhIlIVlVII)1II1III1III 4 40 45 50 55 60 65 7i time (sec) IIIIL average runnin: trim LiiI!I1!Il!Ii!1iIi 40 45 50 55 60 65 7 time (sec) Figure 11. Simulated pitch time histories for two different sets of initial conditions. Wave amplitude Ç., = 1.5 ft (0.46m); A/BL = 7.0. As with the case for porpoising analysis, the above magnification curves have practical significance. Clearly the simulated motions are not linearly related to the incident wave amplitude thus agreeing with the eperimental observations of Fridsma (1969) and Savitsky & Brown (1976). Doubling the incident wave will not necessarily produce a factor of two in the response, particularly near resonance. For the hull in the above example, operation in a seaway near resonance could have dangerous consequences. In waves of increasing amplitude, the above results and results shown by Troesch & Falzarano (1992), Troesch & Hicks (1994), and Hicks, et al (1995) suggest that the first effect on the motions of the vessel will be slight, with perhaps only a small change in the mean trim and mean wetted length. the amplitude increases, the craft can be suddenly attracted to other periodic solutions and experience a sudden increase in its heave and pitch motions. This unanticipated extreme behavior may present serious consequences for the boat's operator and crew. 4. CONCLUSIONS This paper has reviewed the state-of-the-art of the technology of planing boat hydrodynamics. General findings are as follows: As A developing, physics based technology relating to.alm and rough water performance in seas of all headings is becoming available to the designers and builders of planing boats. Many of the previous efforts dealing with planing dynamics have limited ranges of applicability; e.g., the restrictive range of the experimental parameters in series experiments or in computer simulations. As a result of these restrictions, critical performance areas can be easily missed or overlooked. Using elements of previously derived hydrodynamic and dynamic theories, it is becoming possible to develop a comprehensive planing hull model valid for a wide range of speeds, hull forms, and incident sea states. In particular, the new technologies include three dimensional CDF codes that will allow designers to determine the required thrust and speed for planing boats in calm water, and modern methods of dynamical system analysis which allow naval architects to predict areas of dangerous wave-induced motions including displacements, velocities, and accelerations. Il

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