Recent advances in the analysis of high speed planing hydrodynamics and dynamics

"RECENT ADVANCES IN THE ANALYSIS OF HIGH SPEED

PLANING HYDRODYNAMICS AND DYNAMICS"

Dr. Armim Troesch -University of Michigan, Ann Arbor

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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

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

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-of-the-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.

1.1. 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

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.

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

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.

(a)

pressure

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

mass-strip 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

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.

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 DimensionalTheories

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).

C.

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

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.

Figure 6. Center and quarter beam pressure

distributions for a prismatic planing hull. = 20 degrees,

=6 degrees, . = 2.91 (Lai & Troesch, 1995).

C,xV

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.

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

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

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 typeof 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

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)

(I)

M = I55r5(t)

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 isan 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}

integro-differential 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 steady-state 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

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.

J

Icg11 lcg cruical (a) (b)

Figure 8. Schematic of typical Hopf

bifurcation curves Single branch. Multiple branches. (Troesch & Hicks, 1994)

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

kg/B

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.

0.8 113(t) _0.4 B o -0.4 (a) 1.2

'l",".'

lU!i!t I!!!!

'u"

0 5

10 15 20 25 30 35 40 45

t/T (b) -0.4 0 5

_{10 15 20 25 30 35 40 45}

tri;

-0.4

uuiiiimim

iiiiiiii

0

5 1015202530354045

tri;

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)

(c) 1.2 kg/B 018

i3(t)

0.4 B O

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 Figure8.

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

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

of

multi-Yregion

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)

8 u ç) 4 time (sec) 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. _As 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:

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.

-

____

Iv1rnMM!.uhII

IfIOßhIlIVlVII)1II1III1III

40 45 50 55 60 65 7i

IIIIL

average

LiiI!I1!Il!Ii!1i

runnin: trim

Ii

40 45 50 55 60 65 7

ACKNOWLEDGMENTS

The author would like to acknowledge the

Office of Naval Research under

Contracts DOG-G-N00014-94-l-0652 and

DOD-G-N00l4-95-l-1124 with program managers Dr.

Edwin Rood and Dr. Thomas Swean,

respectively, for support in the preparation of

the paper and

CNPq of

the Brazilian Government for travel support to SOBENA

'96.

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