Cobblestone effect on SES

(1)

P

I

SVERRE STEEN

COBBLESTONE EFFECT ON SES

TEC1HIasc1fE UNWERSITE1T

Scheepshydrcmechanica

Archief

Mekelweg 2, 2628

Deif t

Tel: 015-2786873/Fax:2781836

DOKTOR INGENIØRAVHANDLING 1993:50 INSTITUTT FOR MARIN HYDRODYNAMIKK TRONDHEIM

(2)

Cobblestone Effect on SES

Dr.ing thesis

Sverre Steen

Department of Marine Hydrodynamics

The Norwegian Institute of Technology

(3)

PREFACE

The research reported in this thesis was funded by the Royal Norwegian Council for Scientific and Industrial Research (NTNF), Ulstein Eikefjord A.S. and Ulstein International A.S. as a part of the high speed marine vehicles research program initiated by NTNF.

Professor Ph.D. Odd M. Faltinsen at the Department of Marine Hydrodynamics, the Norwegian Institute of Technology has been supervisor on this work.

Derivation of the mathematical model of the vertical plane dynamics of an SES with a rigid planing aft seal was done together with my colleague Asgeir Sørensen at the Department of Engineering Cybernetics, the Norwegian Institute of Technology.

Carrying out the full scale trials and analyzing measured data were done in cooperation with Asgeir Sørensen. with assistance from the Norwegian Marine Technology Research Institute - MAR[NTEK A.S. and with funding from Ulstein Marine Electronics A.S. and Ulstein International A.S.

(4)

ACKNOWLEDGEMENT

I am grateful to my supervisor Professor Ph.D. Odd M. Faltinsen for his support and encouragement.

I would also express my gratitude to my colleague and friend Asgeir Sørensen at the Department of Engineering Cybernetics, the Norwegian Institute of Technology with whom I have been cooperating on this project. I am particularly grateful for the good cooperation developing the new mathematical model of SES dynamics, carrying out and analyzing the full scale measurements and writing the articles together. Without his eagerness and confidence the progress and success of this work would have been suffering.

The cooperation from Ulstein Eikefjord A.S. and Ulstein International have been of great value performing the full scale measurements. Their enthusiasm and patience, and their ability to solve practical problems in an instant are appreciated.

(5)

ABSTRACT

This thesis contains a study of high frequency vertical accelerations on Surface Effect Ships (SES). High frequency vertical accelerations are dominating at high forward speeds in low sea states. Reduction of the acceleration level in low sea states is crucial for the success of an SES as a passenger carrier. Since the air cushion compressibility effects do not scale according to Froude scaling laws, it is especially important to develop a complete an reliable mathematical model describing these effects. In this thesis, a detailed mathematical model of SES dynamics in the vertical plane is derived. This model accounts for both uniform and spatial pressure variations in the air cushion, whereas previous workers only have been considering uniform pressure variations. It also accounts for the effect of a flexible bag stern seal on the air cushion dynamic pressure response, and the interaction between the air flow under the flexible bag and the bag response. The bag response is treated as linear and quasi-static. The model is formulated in the frequency domain, using describing functions to account for the effect of the non-linear leakage of air under the bow and stern seals. The spatially varying pressure is described by a one-dimensional velocity potential that satisfies the Helmholtz equation in the cushion region and appropriate boundary conditions on the surfaces enclosing the air cushion volume.

The results show dominating vertical accelerations in a frequency range important for passenger comfort and crew work ability. The most important factor for the determination of the level of the spatial pressure variations is the wave and motion induced leakage area variations under the bow and stern seals. The use of a flexible bag instead of a rigid planing seal at the stern increases the level of vertical accelerations in low sea states, due to larger wave induced leakage area variations in these conditions. The use of a flexible bag also decreases the spatial pressure resonance frequencies. It is found that the use of a ride control system in the bag has a favorable effect on the vertical accelerations. It is identified that a redesign of the bag that minimizes the wave induced leakage area variations in low sea states would improve the passenger comfort significantly.

A non-linear time-domain model of heave and dynamic uniform air cushion pressure is developed. It shows that the quasi-linear mathematical model overpredicts the uniform pressure variations in high sea states. mainly due to linearization of important non-linear relations in the description of air leakage.

(6)

A , AP 11O A FP 1-10 A AP A FP 1-0c A RCS 110 A33 A35 A53 A55 Ab A ACh AP AL AB ARCS b B33 B35 B53 B55

NOMENCLATURE

area of the bag facing the cushion [m2] mean leakage area under the stern seal [m2] mean leakage area under the bow seai [m2] mean leakage area at the stern [m2J mean leakage area at the bow [m21

equilibrium leakage area from the air cushion through the ride control system [m2] hydrodynamic added mass coefficient in heave

hydrodynamic added mass coefficient in heave due to pitch motion hydrodynamic added mass coefficient in pitch due to heave motion hydrodynamic added mass coefficient in pitch

cross-sectional area of the bag [m2] air cushion waterplane area [in2]

cross-sectional area of the cushion at the bag [rn2] outlet area of fan [m2]

total dynamic leakage area from the air cushion [m2] leakage area of the bag [in2]

frequency dependent modal amplitude function for odd mode j due to action n total dynamic leakage area from the air cushion through the ride control system [m2] side hull sectional area [m2]

cushion beam [ml sidehull beam [m]

hydrodynamic damping coefficient in heave

hydrodynamic damping coefficient in heave due to pitch motion hydrodynamic damping coefficient in pitch due to heave motion hydrodynamic damping coefficient in pitch

(7)

C hydrostatic restoring coefficient in motion degree i due to motion of degree j

c orifice coefficient d side hull draft [m]

fe wave frequency of encounter [Hz]

f

resonance frequency of coupled heave - uniform pressure [Hz]

F3aC hydrodynamic heave excitation force amplitude acting on the sidehulls [N] F50e _{hydrodynamic pitch excitation moment amplitude acting on the sidehulls [Nm]}

F3bSP _{heave force due to spatial pressure due to bag [N]}

F5bS? pitch moment due to spatial pressure due to bag [Nm]

F3 heave force due to modal spatial pressure [N]

FSSP pitch moment due to modal spatial pressure [Nm]

g gravitational acceleration [nils2] feedback gain of ride control system i

h distance between bag surface and water surface [m]

H Heaviside function h0 cushion height [m] h cushion height at bag [m]

h air gap under bag where the air flow separates from the bag [m]

hL bag height [m]

H significant wave height [ml

he

_{aft seal submergence [ml} hSFP bow seal submergence [m]

.h/dPh change in leakage gap under the bag due to change in bag pressure [m] h/dp change in leakage gap under the bag due to change in cushion pressure [mi i imaginary unit

155 mass moment of inertia around the Yg15 [kgm2]

k wave number for water waves

k acoustic wave number

Kb equivalent bag stiffness constant related to the effect of dynamic bag pressure

K equivalent bag stiffness constant related to the effect of spatially varying cushion pressure

(8)

pressure

K1 air cushion stiffness constant

K2 linearized equivalent air outflow velocity constant K3 _{air cushion airflow constant}

L cushion length [rn]

m craft mass [kg]

NBAP _{bias of quasi-linearized leakage area aft}

NBFP bias of quasi-linearized leakage area at the bow

NRAI _{gain-value of quasi-linearized variable leakage area aft} NRFP _{gain-value of quasi-linearized variable leakage area at the bow} Po mean cushion pressure [N/rn2]

p atmospheric pressure [N/rn2]

Ph dynamic bag pressure [N/rn2]

p mean bag pressure [N/rn2] p mean bag pressure [N/rn2]

total cushion pressure [N/rn2]

p spatial pressure from the modal solution [N/rn2]

Pbsp spatial pressure from the solution accounting for the effect of bag [N/rn2]

P,, frequency dependent modal amplitude function for mode j due to action n

p dynamic uniform cushion pressure [N/rn2]

Q0 mean volume flux from fan into the cushion [rn3/s]

mean volume flux from the cushion to the bag [rn3/s] QA' _{volume flux of air from the cushion at the stern [rn3/s]}

Qf volume flux through the lift fan as a general function of air cushion pressure [m3/s]

Q'" volume flux of air from the cushion at the bow [rn3/s]

QRCS

volume flux of air from the cushion through the side control system [rn3/s] Q total volume flux of air into the air cushion [m3/s]

(9)

r1 mode shape function for mode j

R8 length of upper internal web in bag [m]

R length of lower internal web in bag [m] s arc length coordinate

S00 arc length from the forward fastening point of the bag to the lowermost point on the bag surface [m]

SAR arc length of the upper lobe of the bag material [rn] SBC arc length of the mid lobe of the bag material [m]

SCDO arc length of the lower lobe of the bag material [m] S sea wave spectrum

r time [s]

T tension in the bag material [N]

T1 tension in the upper lobe of the bag material [N]

T2 tension in the mid lobe of the bag material [N] T tension in the lower lobe of the bag material [N] T peak period [s]

U craft forward speed [mis]

u1 contribution to equivalent piston velocity from changes in bag shape [mis]

u, contribution to equivalent piston velocity from air flow from cushion to bag [rn/si

¿13 contribution to equivalent piston velocity from air flow under the bag [mis] Ue exit velocity for air flow under the bag [rn/si

Vb bag volume [m3]

V mean bag volume [m3]

V cushion volume [m3]

V wave volume addition to cushion volume [ni3]

V0 equilibrium air cushion volume [m3]

V&/dpb change of bag volume caused by a change of bag pressure [m3] Vb/pC change of bag volume caused by a change of cushion pressure [ms] aVJPb change of cushion volume caused by a change of bag pressure [m3]

V/PC change of cushion volume caused by a change of cushion pressure [m3] x,y,z coordinates in system with the origin in the air cushion center [m]

(10)

XgYgZg coordinates in system with the origin in the center of gravity [m]

XA,ZA coordinates of the point A of the bag (in local coordinate frame of the bag) [m] XB,ZB coordinates of the point B of the bag (in local coordinate frame of the bag) [m]

x0z coordinates of the point C of the bag (in local coordinate frame of the bag) [m]

x0, coordinate from center of gravity to air cushion center in x-direction [m]

Xe the point where the airflow separates from the bag surface (in local coordinate frame

of the bag) [in]

XF coordinate from air cushion center to fan in x-direction [m]

XRCS coordinate from air cushion center to ride control system in x-direction [m]

x4 coordinate from the forward fastening point of the bag to the lowermost point on the bag surface in x-direction [m]

Z,eÇ' relative motion between the lower edge of the stern seal and the water surface [in]

Zre! relative motion between the lower edge of the bow seal and the water surface [m]

Greek Symbols

ratio of specific heat for air

variable leakage area under the stern seal Em2] variable leakage area under the bow seal [m2]

AARCS commanded variable leakage area through the ride control system [m2]

A1 air gap under bag [m]

¿p pressure difference [N/rn2] Kronecker delta

E phase angle Ç wave elevation [in]

wave amplitude [m]

fl heave [ml

(11)

acoustic wavelength [in] dim.less dynamic bag pressure

ltba dim.less dynamic bag pressure amplitude

i-tu dim.less dynamic uniform pressure

dim.less dynamic uniform pressure amplitude

Rm dim.less dynamic uniform pressure mean value

J-Isp dim.less spatial pressure from the modal solution

lisps dim.less spatial pressure amplitude from the modal solution

Pbs dim. less spatial pressure from the solution accounting for the effect of bag

libspa dim. less spatial pressure amplitude from the solution accounting for the effect of bag

ç

relative damping ratio for mode j

relative damping ratio for coupled heave - uniform pressure

P density of air [kg/rn3]

Pa density of air at atmospheric pressure [kg/rn3]

Pb density of air at bag pressure [kg/rn3] Pb0 density of air at mean bag pressure [kg/rn3]

PC density of air at cushion pressure [kg/rn3] PC0 density of air at mean cushion pressure [kg/rn3]

Ps density of sea water [kg/rn3]

rms-value of the time dependent part of the variable leakage area

one-dimensional spatial pressure velocity potential due to the effect of a flexible bag spatial pressure velocity potential due to action n

spatial pressure velocity potential

one-dimensional spatial pressure velocity potential due to action n

14J one-dimensional spatial pressure velocity potential

(1)0 water wave circular frequency [radis]

water wave circular frequency of encounter [radis]

(12)

i

INTRODUCTION

1.1 BACKGROUND AND MOTIVATION

During the last half part of the 1980'ies, there was an expanding market for high speed vessels worldwide. Many new fast ferry services were opened. and the use of larger high speed vessels for cargo and combined car and passenger transportation was considered. Many new advanced concepts had been suggested. However, major technological challenges had to be overcome, and the expected increase in speed and size represented a challenge to the expertise available from many diverse disciplines. In Norway there were several yards involved in development and building of high speed vessels. In 1989 the Royal Norwegian Council for Scientific and Industrial Research (NTNF) initiated the Norwegian High Speed

(15)

OUTBOARD PROFILE

INBOARD PROFILE

Figure 1.2 Surface Effect Ship - general characteristics. Butler (1985).

Eikefjord Marine and NTNF in cooperation with the Norwegian Institute of Technology, and the Norwegian Marine Technology Research Institute - MARINTEK initiated my scholarship to study the seakeeping of SES.

The SES concept consists of two slender hulls in a catamaran configuration, with a pressurized air chamber between the sidehulls (see Figure 1.2). The pressurized air cushion is sealed in each end of the tunnel by flexible seals, allowing the waves to pass through the air cushion. The aft seal is usually a flexible bag, consisting of a ioop of flexible material open against the side hulls with one or two internal webs restraining the aft face of the loop into a two or three-loop configuration. The bow seal is usually a finger seal, consisting of a row of vertical loops of flexible material. An SES air cushion with a bow finger seal and a 3-loop flexible bag aft is sketched in Figure 1.3. The air cushion carries about 80% of the vessel weight, the craft is lifted and the wetted surface is reduced. This gives significantly

lower drag in the high speed range, compared to a conventional catamaran of the same size. In addition, SES seems to have more favorable motion characteristics in moderate and high sea states with less motion sickness among the passengers than on catamarans of the same size. However, in low sea states and almost calm water, significant high frequency vertical

(16)

Figure 1.3 Sketch of an SES air cushion with a bow finger seal and a 3-ioop flexible bag seal aft. Toyama, Ono and Nishihara (1992).

accelerations have been experienced onboard the SES. These high frequency vertical accelerations are commonly called the cobblestone effect, and are viewed upon as a serious drawback of the SES as a passenger carrier.

1.2 OVERVIEW OF THE PRESENT WORK

To reduce or eliminate the high frequency vertical accelerations on SES two main remedial actions ame considered, altering the design of the craft or it's subsystems, and active control of the air flow into or out of the air cushion. The necessary basis for both actions is a thorough understanding of the physical characteristics of the dynamic air cushion pressure and the related high frequency vertical plane accelerations. To investigate the cause of the high frequency vertical accelerations, a series of full scale measurements was made on the 35 m SES produced by Brødrene Aa and later also the Ulstein Group. These measurements showed that the high frequency vertical accelerations in low sea states where concentrated on

(17)

acceleration. Consequently a new prototype aft seal system, consisting of rigid panels made of composite material hinged up in the wet deck and planing on the water surface, was tested on the 35 m SES. Measurements on the SES equipped with this rigid planing seal showed that the two peaks of highest frequency appeared at higher frequencies than for a similar craft equipped with a bag seal. It also showed lower response around the two highest response peaks than the craft with flexible bag. By studying power spectra, and cross-power spectra between points of different longitudinal position in the air cushion, it was found that the two response peaks of highest frequency were due to resonance of acoustic pressure waves in the longitudinal direction of the air cushion. Selected results from the full scale measurements are given in Chapter 2. A mathematical model for the vertical plane dynamics of an SES with a rigid planing aft seal is developed in Chapter 3, by means of a boundary value problem solved using a modal approach for the spatial varying pressure in the air cushion. Numerical simulation results using the mathematical model are found to agree qualitatively well with the fullscale measurements on the craft with rigid planing aft seal. The mathematical model is used for development of an active acceleration reduction system, called Ride Control (RCS). The mathematical model is also useful for studies of different cushion designs. Parts of this work is published in Sørensen, Steen and Faltinsen (1992, 1993). In Chapter 4 the mathematical model of the vertical plane dynamics of SES is extended to take the effect of the flexible bag aft seal into account. The bag dynamic response is described by a linear quasi-static model, taking the airflow under the bag into account. Comparison between numerical simulation results and measured response indicate that the mathematical model describes the influence of the flexible bag on the craft dynamics in a satisfactory manner. It is found that the reduction of the acoustic resonance frequencies in the air cushion is due to the dynamic response of the bag to dynamic cushion pressure variations. For both aft seal types, it is found that the magnitude of the response is sensitive to variable leakage of air from the cushion, and therefore sensitive to changes influencing the leakage. The ability of the seals to minimize the motion induced leakage in low sea states is found to be important. The flexible bag has a leakage gap in calm water, resulting in wave induced leakage area changes also in low sea states. The rigid planing seal is planing on the water in low sea states and calm water, resulting in small wave and motion induced leakage area changes for low sea states. This difference in leakage gap between the two seal types is found to be the main reason why the craft with flexible bag has a larger level of vertical accelerations in low sea

(18)

states than the craft with rigid planing aft seal. The mathematical model provides the possibility to study the complex coupled system of bag and cushion dynamics in a fast and efficient way. It is useful for design of motion control systems, and for design studies of different bag and cushion designs. The complete mathematical model, valid for SES with flexible bag and taking dynamic leakage area variations into account, is presented in Steen and Faltinsen (1993). The mathematical models mentioned so far are both linear models. To investigate the importance of a variety of nonlinear effects in the air cushion dynamics, a nonlinear model of the air cushion was developed. It shows that the linear mathematical model tends to overpredict the response for large wave heights, and that the most important non-linearities regarding the magnitude of the dynamic pressure response are related to the

air flow Out of the air cushion.

1.3 REVIEW OF PREVIOUS WORK

The SES concept evolved during the sixties as an improvement of the air cushion vehicle. The surface piercing sidehulls reduced the air leakage from the air cushion and made water propulsion possible. Most of the development was sponsored by the US Navy, and was viewed upon as a step to reach their goal of a "100 knot Navy". In addition to new challenges in machinery and construction materials, the research work on SES can be divided in three main topics:

- transverse stability resistance and propulsion - seakeeping capabilities

Previous work on the two first topics will only get a brief presentation, while the last topic will be treated in some more detail.

(19)

introduction to SES stability, and a survey of the results from the research project is given by Blyth (1986. 1987, 1988). For later SES designs with wider sidehulls, the transverse stability is good when measured in terms of static stability. The research on dynamic stability has not yet reached far.

The calm water resistance of an SES differs from that of a catamaran or monohull with respect to wave making drag from the air cushion and drag on the air cushion seals. The wave making drag from the air cushion is calculated analytically by a number of authors; the most well-known is perhaps L.J. Doctors. His review article on this subject (Doctors, 1992) is comprehensive, and contains many valuable references. Particularly important references from a practical point of view are Doctors and Sharma (1972, 1973), and Tatinclaux (1975).

The ultimate goal of modelling of SES dynamic behaviour, as well as dynamic behaviour of other kinds of craft, is to create an accurate and complete mathematical model of the total system response due to any actual excitation, especially that arising from operation over real sea surface waves. The model should be valid as a complete design tool once it is validated. It should be generally valid and not restricted to specific design configurations. This is an ambitious goal, even for conventional ships, so it should not be surprising that this goal is not completely reached yet. However, a variety of models exist, and a review of their basic characteristics and the theory behind is given.

Maybe the most distinct difference between the simulation models is whether they are linear (frequency-domain) or non-linear (time-domain) models. SES has many non-linear dynamic characteristics, but experiments have shown that the motions and accelerations show quite linear response characteristics under realistic sea conditions (Ochi and Moran (1979), McHenry et. al. (1991))

For conventional ships the use of Froude-scaled model test for determination of dynamic capabilities is widely used. For SES this method is not directly applicable because of the lack of scaling of the atmospheric pressure, which in turn lead to erroneous scaling of the stiffness and damping of the air cushion, see Moran (1976). Kaplan, Bentson and Davis (1981) identifies the usefulness of model testing for SES development through the validation of a predictive computer program by comparison with model test data. This might not be a very accurate method, because the effects of air compressibility are hardly measurable at

(20)

model scale. Hence, the validation of the way the program handles the important air cushion compressibility effects will be poor. This leads us to another distinction between the simulation models; whether they are basically analytic or basically empirical.

All simulation models contain differential equations of rigid body motions coupled with the dynamic cushion pressure and other SES specific forces. Specific areas of interest are:

- free surface response to cushion pressure

- local air flow between the seals and water surface - seal dynamics

- side hull hydrodynamic forces, including the effect of the presence of the air cushion - fan dynamics

The deflection of the free surface in the air cushion will interact with the dynamic air cushion pressure and influence the total craft response. Doctors (1974) and Kim and Tsakonas (1981) deal with this problem from an analytical point of view. Kaplan (1989) analyses this problem, and concludes that 'The overall influence of such generated waves is found to have only a small effect on the motion responses, from both frequency response and spectral rms-value considerations.' Nesteg.rd (1990) shows how this effect can be included in a computational model. A related problem is the diffraction of the wave system passing

through the air cushion. Moran (1975) concludes on an experimental basis that the wave system passing through the air cushion may differ radically from the undisturbed wave profile commonly assumed. The wave system generated by the sidehulls were found to have a significant influence on the incoming wave field in the air cushion. On the opposite, Bentson and Kaplan (1979) reported good agreement between computations and experiments, without taking the diffraction effect into account.

The local effects of the air flow between the seals and the free surface are important for the determination of the total air leakage from the cushion, and thereby for the dynamic air cushion pressure. The local leakage area under the seals depends on the craft vertical

(21)

takes the position of the separation point or the mixing of spray into account. The non-linearity of the leakage aiea variations is recognized to be important. All the non-linear models take it into account. Some linear models use statistical linearization methods to linearize this hard non-linearity. The linear model presented in this thesis takes the interaction between the bag shape and the air flow under the bag into account, and also uses describing functions to linearize the non-linearity of the leakage area variations.

Seal dynamics is a complex subject, both due to the diversity of the seal designs and due to the complicated dynamics involved in most seal designs. The main research effort has been directed against resistance and wear of the seals. Malakhoff and Davis (1982) give a survey of different seal types and materials. Inch et al (1989) is treating seal design from a practical point of view. Malakhoff and Davis (1981) reports on experimental investigation of drag and wear of bow finger skirts. Doctors (1977) presents a theoretical model for flexible planing surfaces applicable to many planing seal designs Fukasawa (1991) treats the dynamic response of a floating membrane structure in waves. This treatment might be extended to cover a flexible bag in contact with the free water surface. This thesis uses a quasi-static model of the bag dynamics to investigate the effect of the flexible bag on the craft response.

For recent SES designs, with relatively wide sidehulls, the hycirodynarnic forces on the sidehulls are important, especially in the low frequency range. A common assumption is that the two sidehulls are hydrodynamically independent of each other. Faltinsen and Zhao (1991 a, 1991 b) developed a 2½-dimensional strip-theory for high Froude numbers,

applicable to SES sidehulls. The ½-dimension accounts for the 3-dimensional effect of the divergent wave system. Nesteg.rd (1990) developed a theory that takes into account the interaction between the air cushion and the hydrodynamic forces on the sidehulls. The work of Faltinsen and Zhao (1991 a, 1991 b) and Nestegârd (1990) is used in the development of the linear motion prediction part of the program FASTSEA. The effect of the air cushion on the hydrodynamics of the sidehulls is taken into account. The interaction between the unsteady cushion pressure variations and the free surface deflection is included. Spatial pressure variations are not taken into account. The code is robust and computationally efficient. FASTSEA is capable of motion prediction for catamarans, monohulls, SES, and hydrofoil catamarans. FASTSEA also considers added resistance and speed loss as functions of the computed motion response.

(22)

important by many workers in the field of SES dynamics. The dynamic effects might be related to surge and stall of the fan, changes in fan rotational speed, compressibility and inertial effects of the air in the fan and ducting system. It is known from experiments that the dynamic effects are highly dependent on fan geometry, rotational speed and pressure. Durkin and Luehr (1978) concludes that the dynamic effects are mainly related to the air inertia in the fan ducting system. This conclusion is supported by Sullivan, Gosselin and Hinchey (1992) through their careful experimental study. Sullivan, Gosselin and Hinchey find that in their case the dynamic fan response influences the overall cushion dynamics They conclude that an experimental investigation is necessary in each case to obtain detailed knowledge of the dynamic effects, given the present state of knowledge. All practical numerical models for the dynamic of SES use quasi-static fan pressure-density relations.

The importance of including spatial cushion pressure variations in the calculations of the vertical plane dynamics of SES is first addressed by Nakos et. al. (1991). They analyze the spatial pressure variations that arise as a result of the wave train passing through the air cushion, and shows how the frequency of the first and second resonances can be found. As a part of the work for this thesis, SØrensen, Steen and Faltinsen (1992, 1993) analyzes the vertical plane motions of an SES including the effect of spatial cushion pressure variations by use of a modal approach. The resulting model is used for the development of a ride control system, see also SØrensen (1993). In this thesis and in Steen and Faltinsen (1993), this model is extended to take variable leakage area and the effect of a flexible bag aft seal into account. The purpose of a mathematical model of SES dynamics will usually be to make a computational model for use in the design process. To show the present "state of the art in

SES dynamics. a table of existing seakeeping or motion prediction programs is given in Table 1.1. The computer program Simpacc resulting from the present work is included in the table for comparison, even if it hardly can be viewed as a complete seakeeping program, since it only includes vertical plane dynamics. While the previous workers in the field of SES dynamics have concentrated on response to excitation in a frequency range up to the dynamic

(23)

e 1.1 Comparison of SES motion preddctlon programs.

1.4 ORGANiSATION OF THIS THESIS

This thesis is divided into 6 chapters. Chapter 2 contains full scale measurements on the 35 m SES with flexible bag aft seal and with rigid panel aft seal. The measurements serve both as a motivation for this study, and for comparison with the results from the mathematical simulation model. Chapter 3 contains the mathematical model of the SES with rigid planing aft seal, and results and discussion of this model. Chapter 4 describes the bag dynamics and its effect on the overall craft dynamics. The mathematical model of the vertical plane

Program Oceanics DTRC/ORJ Textron Mantime

Dynamics

Fastsea Simpacc

Type Nonlinear Nonlinear Nonlinear Linear Linear Linear

Ship type SES ACV SES, ACV SES SES, Cat.,

Monohull Foilcat. SES Rigidbody deg. of freedom 6 5 (No yaw) 6 5 (No surge) 6 2 Heave, Pitch Free surface deflection No Yes (relaxation equation) Yes (steady-state emp.) No Yes No Seal flow shut-off

Yes Yes Yes Yes

(describing functions) No Yes (describing functions) Seal dynamics modelled Yes (seal forces) No No No No Yes (effect on cush. dyn.) Spatial cushion pressure var. No No No No No Yes Ride control simulation

(24)

dynamics for a craft with flexible bag is formulated. Simulation results are given, and the model and the assumptions behind it are discussed. Chapter 5 investigates the importance of non-linear effects on the air cushion dynamics by use of a non-linear time simulation model. Chapter 6 contains conclusions and recommendations for further work.

(25)

2

FULL SCALE MEASUREMENTS

Full scale measurements on a 35 m SES equipped with a conventional flexible bag have been carried out on a few different occasions. Full scale measurements on a craft with

a rigid planing aft seal was carried out on one occasion. During all

the presented measurements, the craft was running at about 45 knots in head sea waves in different sea states Selected results from the full scale measurements will be presented and discussed. The full scale measurement results serve mainly as a motivation for the analytical studies. Because of the lack of reliable wave measurements during the full scale measurements they can not be used for a complete verification of the mathematical model and computer simulation program, but can only be used for a qualitative comparison of the simulations with the full scale results.

All the measurements presented here were made together with Asgeir Sørensen. and a presentation similar to this one is therefore appearing in his dr.ing-thesis (Sørensen 1993).

Figure 2.1 Measurement Arrangement

2.1 MEASUREMENT ARRANGEMENT

Even if the measurements were made at different occasions, the measurement arrangement and equipment were mainly the same at all occasions. The basic measurement arrangement is presented here. Individual differences for each measurement is presented with the results. The measurements were made with field equipment from Marintek. The sensor signals were prefiltered by a 4th order analog Butterworth filter with crossover frequency at

Accelerometers

+

Low pass AID - Personal

Filter Transformer Computer

(26)

40 11z. The measurements were subsequently sampled and stored on a Personal Computer at a rate of 100 Hz. The duration of each measurement series was 120 s. The measurement arrangement is illustrated in Figure 2.1. The following variables were sampled and stored simultaneously:

Vertical accelerations at the bow, near the centre of gravity and at the stern, see Figure 2.2.

- Pressure variations at the fore, near the middle, and at the aft of the cushion,

see Figure 2.2.

Panel angle or bag pressure on some measurements

Besides the sampled variables on the PC, the following parameters were observed visually and registered manually for each measurement:

Craft speed (observed from the speed log, the GPS navigator, or by measuring the time used to travel a distance carefully measured in advance)

- Significant sea wave height H and main wave direction

iiutu t iuuuuu u.I.i. t. 1.1_i_lu

._ -

ili.

lu

tI.I. l

t.---I

iiiu .itiii

-'i

_{IlBIliu i. i. 1*1.1.}

I Iiiu

u

isiti.

1Ii IuiIUtu

ui

i u

itigi,uu

u

1*i I

,.mi,.i i i

i.

i 1u1*1 i

1.Iu IUIU utu.'l.iu i

1i1

I. iIU

_.i.i.

UI.,.

T

iuiu I

i,.i.iu i

i

I.i.i. lu i.t.i.i i I I

_{luuIul ui.i.}

i u lu iw,.,

,.j.ui

lu

(27)

2.2 METHODS OF ANALYSIS

The time series obtained during the full scale measurements are analyzed with respect to qualities like resonance phenomenons, energy content at different frequencies and mutual phase differences between the different signals. To carry out such an analysis, it is convenient to use spectral analysis techniques. A short presentation of the spectral analysis techniques used is given below. For a complete treatment, see any standard textbook on this subject, for example Newland (1984).

Auto Power Spectrum

The auto power spectrum called power spectrum shows how the energy of the signal is distributed along the positive frequency axis. The power spectrum is always real and positive. If the signal contains pure periodic components at given frequencies, e.g. a sinusoidal, the power spectrum will contain lines (infinite peak values) at the same frequencies. Therefore, dominating resonances in the dynamic system will appear as peak values in the power spectrum around the corresponding resonance frequencies. The power spectrum of an output signal of a dynamic system is influenced by both the properties of the corresponding input signal and the inherent physical properties of the dynamic system. This means that the power spectrum of different time series of a particular sensor signal will vary depending on the actual input signai. For instance in the case the SES is advancing with speed U in head sea waves, the frequency of encounter power spectrum of the excess pressure at the aft of the air cushion can be represented by the relation

= I(f)i2Sç(0)o)

2ir

4,p (f»

Ça _{i +_____U}2w

g

where fe is the wave frequency of encounter in Hz, PAPOL)1Ç0 is the transfer function of the pressure signal at the aft of the cushion relative to the incident wave elevation amplitude a

and S(w0) is the sea wave spectrum. Hence, assuming that the inherent physical properties of the SES dynamics do not vary too much, the power spectrum of the output signal is quite sensitive to variations in sea states represented by the sea wave spectrum.

(28)

Cross Power Spectrum

The cross power spectrum is a complex function containing amplitude and phase information. It shows how two signals are correlated in the frequency domain. The cross power spectrum amplitude shows how the correlated part of the energy of the two signals are distributed along the positive frequency axis. The cross power phase indicates the phase difference between the two correlated signals as a function of the frequency. For instance the cross power spectrum of the excess pressure at the aft relative the excess pressure at the fore is given by

ppfe

(f)

(fe) (2.2)

Coherence

The coherence function expresses the degree of correlation between two signals. It get values between zero and one. If the coherence function is equal to one at a certain frequency, there exists a linear relation between the two signals at this frequency. If the coherence function is equal to zero at a certain frequency, there does not exist a linear relation between the two signals at this frequency, i.e. the two signals are not correlated. The coherence function of the excess pressure signals at the aft end relative to the fore end of the cushion is given by the following relation

2 4p,..p,)feH 2

-

(f)

2.3 RESULTS FROM FULL SCALE MEASUREMENTS

(29)

2.3.1 SES with Flexible Bag Aft Seal

Figures 2.3 to 2.13 are based on time series taken when the craft was advancing with the speed U-45 1tots in head sea waves with significant wave height estimated to be H=0.3-0.4 m. The equilibrium excess air cushion pressure was Po=47° mm Wc. The ride control system was turned off. Figures 2.14 to 2.17 are based on time series taken at zero forward speed in calm water with the fans on maximum speed. The equilibrium excess air cushion pressure was Po=500 mm Wc. The ride control system was turned off.

Figures 2.3 to 2.5 show the full scale power spectra of vertical accelerations at the bow, near the centre of gravity and at the stem. All three acceleration signals show concentration of energy around 2 Hz and 5 Hz. The pronounced peak values around 2 Hz and 5 Hz indicate the occurrence of two resonances of the vertical accelerations in the frequency range below 8 Hz.

Figures 2.6 to 2.9 show the power spectra of the pressure variations at the fore, the middle and the aft of the air cushion, and the air pressure in the bag. The pressure signals at the fore and aft of the air cushion show concentration of energy around 2 Hz and 5 Hz, while the pressure signal in the middle of the air cushion only shows concentration of energy around 2 Hz. The pronounced peak values around 2 Hz and 5 Hz indicate the occurrence of two resonances of the air cushion pressure in the frequency range below 8 Hz. It is important to notice that the peak at 5 Hz is pronounced at the ends and not in the middle of the air cushion. The power spectrum of the pressure in the bag has the same shape as the power spectrum of the pressure aft, with peaks at 2 Hz and 5 Hz, but it is seen that the pressure

variations in the bag are somewhat amplified compared with the pressure aft in the air cushion. The standard deviation of the pressure variations in the bag is about 1.3 times higher than the standard deviation of the pressure variations aft in the air cushion.

Figure 2.10 shows the cross power spectrum between the pressure at the aft and at the fore in the air cushion. The plot shows absolute value and phase angle. The absolute value shows peaks at 2 Hz and 5 Hz, similar to those seen in the power spectra of pressure fore and aft in the air cushion. The phase angle is around 0° up to 3 Hz, then there is a phase shift, and from 4 Hz to above 8 Hz the phase is about 180°. Beyond 9 Hz the phase is about 360°.

(30)

the air cushion. It is seen that the coherence is almost one in the frequency range around 2 Hz and around 5 Hz, indicating a linear relation between the two signals. Also at 10 Hz the coherence is large. This means that there is a very good correlation between the pressure signals from the fore and the aft of the air cushion around 2 Hz, around 5 Hz, and at 10 Hz. Figure 2.12 shows the cross power spectrum of pressure in the aft of the air cushion and the pressure in the bag. The plot of the absolute value shows peaks at 2 Hz and 5 Hz, similar to what is seen from the power spectra in figures 2.8 and 2.9. The plot of the phase between the pressure aft and the pressure in the bag shows that they are basically in phase up to 7 Hz. They are also in phase around 10 Hz. The phase difference from 7 Hz to 9 Hz has little significance, because there is very little correlated energy in this frequency range, as is seen from the absolute value of the cross power spectrum.

Figure 2.13 shows the coherence function between the pressure in the aft of the air cushion and the pressure in the bag. It is seen that the coherence is almost one up to 7 Hz. There is a drop in coherence from 7 Hz to 9 Hz. The drop in coherence from 7 Hz to 9 Hz could be expected, due to very little energy in this frequency range. From 9 Hz to 12 Hz, the coherence is again almost one. This means that the bag pressure is mainly dependent only on the cushion pressure aft.

Figure 2.14 shows the cross power spectrum between the pressure in the aft and at the fore part of the air cushion when the craft has zero forward speed in calm water. The fans were running at maximum speed, resulting in a mean air cushion pressure of 500 mm Wc. Air was leaking out under the bow and stern seals. The absolute value shows peaks at 2 Hz and 5 Hz, quite similar to those seen in the cross power spectrum in Figure 2.10. The phase angle is around 0° up to 3 Hz, then there is a phase shift, and from 4 Hz to above 8 Hz the phase is about 180°. Beyond 9 Hz the phase is about 360°.

Figure 2.15 shows the coherence function of the pressure at the fore and aft ends of the air cushion. It is seen that the coherence is nearly one in the frequency ranges around 2 Hz and around 5 Hz. This means that there is good correlation between the pressure signals

(31)

From 4 Hz to 6 Hz and around 10 Hz they are about 300 out of phase. The phase differences from 3 Hz to 4 Hz and from 7 Hz to 9 Hz have little significance, since there are little correlated energy in these frequency ranges.

Figure 2.17 shows the coherence function of the pressure in the aft of the air cushion and the pressure in the bag. It is seen that the coherence is almost one in the frequency range around 2 Hz arid around 5 Hz, indicating a linear relation between the two signals. Also at

10 Hz there is some coherence. This means that there is a very good correlation between the pressure signals from the fore and the aft of the air cushion around 2 Hz, around 5 Hz, and some correlation at 10 Hz.

Figure 2.3 Full scale measured power spectrum of vertical accelerations at the bow on a 35 m SES with flexible bag aft seal, running at 45 knots in head sea waves with significant wave height estimated to be H=0.3-0.4 m.

o 2 4 6 8 10

(32)

E

0.8

0.6

0.0

Figure 2.4 Full scale measured power spectrum of vertical accelerations at the centre of gravity on a 35 m SES with flexible bag aft seal, running at 45 Iazots in head sea waves with significant wave height estimated to be H=O.3-O.4 m.

0.8 0.6 0.4 0.2 0.0 0 2 4 6 Freq. (Hz) 8 10 C4 0.4 E 0.2

(33)

o J 4000 3000 L) ₂₀₀₀ E E 1000 o 2 4 6 8 10 Freq. (Hz)

Figure 2.6 Full scale measured power spectrum of air cushion pressure variations at the bow on a 35 m SES with flexible bag aft seal, running at 45 knots in head sea waves with significant wave height estimated to be Hz0.30.4 m.

3000 2500 2000 c-J 1500 E E 1000 500 O 2 4 6 8 10 Freq. (Hz)

Figure 2.7 Full scale measured power spectrum of air cushion pressure variations at the air cushion centre on a 35 m SES with flexible bag aft seal, running at 45 knots in head sea

(34)

5000 4000 3000 c-4 L) E

--

2000 1000

L

0 2 4 6 8 10 Freq. (Hz)

Figure 2.8 Full scale measured power spectrum of air cushion pressure variations at the stern on a 35 m SES with flexible bag aft seal, running at 45 knots in head sea waves with significant wave height estimated to be HOE3-O.4 m.

10000 8000 .r 6000 C-,' L) E g, ₄₀₀₀ 2000 O

(35)

Figure 2.10 Full scale measured cross power spectrum between cushion pressure at the stern and at the bow of a 35 m SES with flexible bag aft seal, running at 45 knots in head sea waves with significant wave height estimated to be H=O.3O.4 m.

0.8 q 0.6 C.) q) C) L: o L) 0.4-02

i

4 6 8 Frequency of Encounter [Hz]

- Abs. Va'ue Phase Angle 10

2 4 6 8 10 12

Frequency of Encounter [Hz]

Figure 2.11 Full scale measured coherence function between cushion pressure at the stern and at the bow of a 35 m SES with flexible bag aft seal, running at 45 knots in head sea waves with significant wave height estimated to be Hr=O.3O.4 m.

(36)

o

L)

Figure 2.12 Full scale measured cross power spectrum between cushion pressure at the stern and air pressure in the flexible bag aft seal of a 35 m SES, running at 45 knots in head sea waves with significant wave height estimated to be H=O.3-O.4 m.

1.2 0.6-

0.4-

02-o o

Abs. Value PhaseAngle

200 150 100 -150 -200 4 6 8 10 12 0.8- vi 10 12 4 6 8 Frequency of Encounter [Hz]

(37)

Figure 2.14 Full scale measured cross power spectrum between cushion pressure at the stern and at the bow of a 35 m SES with flexible bag aft seal, with zero forward speed in calm water. 0.8 C) 0.6 C.) C C) C) o 0.4 0.2

Abs. Value PhaseAngle

200 100 O -o -300 o0 3 4 5 6 7 10 Frequency of Encounter [Hz]

Figure 2.15 Full scale measured coherence function between cushion pressure at the stern and at the bow of a 35 m SES with flexible bag aft seal, with zero forward speed in calm water.

3 4 5 6 7 10

(38)

Figure 2.16 Full scale measured cross power spectrum between cushion pressure at the stem and air pressure in the flexible bag aft seal of a 35 ,n SES, with zero forward speed in calm water. 0.8 0.6 o o o 0.4 0.2 3 4 5 6 7 Frequency of Encounter [Hz]

(39)

2.3.2 SES with Rigid Panel Aft Seal

Figures 2.18 to 2.25 are based on time series taken when the craft was advancing with the speed U=45 knots in head sea waves with significant wave height estimated to be H=0. 15-0.3 m. The equilibrium air cushion pressure was Po=50° mm Wc. The ride control system was turned off. Since the sea state was lower, the measured response in these time series must be expected to be lower compared to the results presented in the previous section. Figures 2.26 to 2.27 are based on time series taken at zero forward speed in calm water with the fans on maximum speed. There was a large leakage gap under the bow seal, where the excess air was leaking out. No leakage took place under the stern seal. The equilibrium excess

air cushion pressure was Po=50° mm Wc. The ride control system was turned off.

Figures 2.18 to 2.20 show the power spectra of the vertical accelerations at the bow, near the centre of gravity and at the stern. None ofthe three acceleration signals show response below 4 Hz. The resonance around 2 Hz seen in the previous Section may not be excited in this sea state. The power spectra of the bow and stern acceleration signals have concentration of energy around 6 Hz, while the power spectrum of the acceleration signal from near the centre of gravity shows a smaller but broader response in the frequency range between 4.5 Hz to 8.5 Hz.

Figures 2.21 to 2.23 show the full scale power spectra of the pressure variations at the fore, the middle and the aft of the air cushion. All three pressure signals show concentration of energy around 6 Hz. One should notice that the response due to the pressure signal at the aft is greater than those at the bow and in the middle of the air cushion.

In Figure 2.24 the amplitude of the cross power spectrum of the pressure signal at the fore versus the aft shows concentration of energy around 6 Hz. The phase of the cross power spectrum shows small phase difference between the pressure at the fore and the aft inthe frequency range below 4 Hz. Between 4 Hz and 5.5 Hz the phase difference increases from 00 to 1800. Between 5.5 Hz and 10 Hz the phase difference between the two correlated

pressure signals is almost constant 180°.

In Figure 2.25 the coherence function shows that the two pressure signals at the fore and the aft are almost coherent in the frequency range l-4 Hz and 5.5-10 Hz. In these frequency ranges there is a high degree of correlation between the pressure at the bow and

(40)

the pressure aft. Below 1 Hz, in the frequency range 4.5-5.5 Hz, and beyond 10 Hz, the correlation between the two pressure signals is not so good. This could very well be due to the low energy content in these frequency ranges.

In Figure 2.26 the power spectra of vertical acceleration at the bow and stern are presented together with the phase between the acceleration signals. The measurement was made on a 35 m SES at zero forward speed in calm water. The fans were running at maximum speed. The absolute value shows concentration of energy in a narrow peak at 2 Hz. The phase of the cross power spectrum shows zero phase difference in this frequency range. Outside the frequency range near 2 Hz there is almost no energy.

In Figure 2.27 the cross power spectrum of the pressure signal at the fore versus the aft for a 35 m SES at zero forward speed in calm water is presented in terms of amplitude and phase angle. The fans were running at maximum speed. The absolute value shows concentration of energy in a narrow peak at 2 Hz. The phase of the cross power spectrum shows zero phase difference in this frequency range. Outside the frequency range near 2 Hz there is almost no energy.

In Figure 2.28 the coherence function of the pressure signal at the fore versus the aft for a 35 m SES at zero forward speed in calm water is presented. The coherence function is equal to one around 2 Hz, showing a linear relation between the two signals in this frequency range. Outside this frequency range the coherence is less, probably due to the low energy content. However, there is also a peak in the coherence around 6 Hz.

(41)

0 2 4 6 8 lO

Freq. (Hz)

Figure 2.18 Full scale measured power spectrum of vertical accelerations at the bow on a 35 in SES with rigid panel aft seal, running at 45 knots in head sea waves with significant wave height estimated to be H3O.l5-O.3 m.

.prt

0 2 4 6 8 10

Freq. (Hz)

Figure 2.19 Full scale measured power spectrum of vertical accelerations at the centre of gravity on a 35 in SES with rigid panel aft seal, running at 45 knots in head sea waves with

significant wave height estimated to be H=O.l5-O.3 m. 0.25 0.20 C" _0.15 E 0.10 0.05 0.00 0.20 0.15 0.10 0.05 0.00

(42)

0.6 0.5 0.4 0.3 0.2 0.1 O 0.0 0 2 4 6 10 Freq. (Hz)

Figure 2.20 Full scale measured power spectrum of vertical accelerations at the stem on a 35 m SES with rigid panel aft seal, running at 45 knots in head sea waves with significant wave height estimated to be H=O.l5-O.3 m.

500 400 300 E E 200 100

(43)

E E 1600 1400 1200 1000 600 400 800 200

Figure 2.23 Full scale measured power spectrum of air cushion pressure variations at the stern on a 35 m SES with rigid panel aft seal, running at 45 knots in head sea waves with significant wave height estimated to be H=0.15-0.3 m.

o 2 4 8 10

Freq. (Hz)

Figure 2.22 Full scale measured power spectrum of air cushion pressure variations at the air cushion centre on a 35 m SES with rigid panel aft seal, running at 45 knots in head sea waves with significant wave height estimated to be 15-0.3 n.

O 2 4 6 8 10 Freq. (Hz) 500 400 300 E E 200 100

(44)

Figure 2.24 Full scale measured cross power spectrum between cushion pressure at the bow and at the stern of a 35 m SES with rigid panel aft seal, running at 45 knots in head sea waves with significant wave height estimated to be H=O.l5-O.3 m.

G) 0.6-ç) U) C) o

0.8-0

0.2 o o

- Abs. Value Phase Angle

234567

300 200 100 t, Q) O) o D Q) (I) (t lr\(.( --200 -300 10 10 3 4 5 6 7 Frequency of Encounter [Hz]

(45)

Abs. Value, Bow Abs. Value. Stem - - -. Phase Angle

Figure 2.26 Full scale measured power spectra of vertical acceleration at the bow and sterm absolute values and phase difference. For a 35 m SES with rigid panel aft seal, at zero forward speed in calm water.

Frequency of Encounter [Hz]

Abs. Value Phase Angle

Figure 2.27 Full scale measured cross power spectrum between cushion pressure at the bow and stem of a 35 m SES with rigid panel aft seal, at zero forward speed in calm water.

2000 1800- 1600- 1400- 1200-E . 1000- 800- 400- 200-0 300 200 t, i00 -100 0 1 2 ₃ 4 ₅ ₆ ₇ ₈ ₉ ₁₀-200 0.6 0.5-c'J 0.4- 0.1-'-'V.) I." 'i '.4 'II 'I: 300

200g

i00<

--100 o -o 2 3 4 5 6 7 8 9 10 -200

(46)

0.8 o 0.6 L) o cl) o

o

0.4-0.2 o o 1 2 3 4 5 6 7 ₈ 9 10 Frequency of Encounter [Hz]

Figure 2.28 Full scale measured coherence function between cushion pressure at the bow and stern of a 35 m SES with rigid panel aft seal, at zero forward speed in calm water.

2.4 DISCUSSIONS AND CONCLUSIONS OF THE FULL SCALE TEST RESULTS

The response spectra shown in the previous sections show that there are two dominating resonances in the coupled system of vertical motions and cushion pressure in the frequency range below 8 Hz. The power spectra of the measurements on the craft with bag show response both around 2 Hz and around 5 Hz. Study of the power spectra of vertical accelerations show that the acceleration level around 2 Hz is almost the same at the bow, near the centre of gravity and aft. This means that the accelerations must be dominated by heave acceleration. Study of the cross power spectrum shows that the pressure fore and aft in the

(47)

to excite the 2 Hz vibrations. Other measurements with the craft with panel show a resonance peak at 2 Hz.

Study of the power spectra of vertical accelerations of the craft with rigid panel aft seal show that the peak at 6 Hz is much lower near the centre of gravity than at the bow and stem. This indicates that the accelerations at 6 Hz are dominated by pitch accelerations. Study of the cross power spectrum between the pressure at the fore and at the aft of the air cushion show that the pressure signals is in good correlation, but 180° out of phase with each other. This indicates the existence of a standing acoustic wave with wave length twice the length of the air cushion. The frequency of such an acoustic wave is given by

co 340

-

-The existence of such a standing acoustic wave is also in good agreement with the dominating pitch accelerations. This means that in the frequency range beyond 4 Hz, the assumption of uniform pressure variations is no longer valid for a 35 m SES. Spatial pressure variations have to be considered. The measurements of the craft with bag show the saine signs of a standing acoustic wave, but with the peak appearing at 5 Hz instead of at 6 Hz. One of the main ambitions of this work is to find the reason for this difference in resonance frequency. Even if the lack of reliable wave measurements prevents a rigorous comparison, it is the impression that the craft with bag experiences higher acceleration levels at 5 Hz than the craft with bag does at 6 Hz, when the sea state is estimated to be the same. The builder of these craft has got the same impression. The construction of the rigid planing seal system was done partly to reduce the high frequency vertical accelerations. Another purpose of this work is to find the reason for the increased vertical accelerations experienced on a craft with bag, and to find measures to reduce the amplifying effects of the bag. Since the pressure in the bag is in phase with the pressure at the aft part of the air cushion, excitation due to interaction between the bag and the water is probably not important in this case.

The measurements of the craft at zero forward speed in calm water show that the concentration of energy at 2 Hz and 5-6 Hz is not caused by a concentration of energy in the incoming wave system at these frequencies. The response at zero speed andcalm water must be caused by some kind of self-excited oscillation, since there was no external disturbance present. Self-excited oscillations are usually found at the resonance frequencies of the

(48)

dynamic system, so the occurrence of self-excited oscillations at 2 Hz and 5-6 Hz does not indicate that the response in waves at these frequencies are caused by self-excited oscillations. The occurrence of self-excited oscillations indicate that non-linear effects are dominating the response in the conditions where the self-excited oscillations occur. For the craft equipped with a rigid panel, the self-excited oscillations at 2 Hz are dominated by uniform pressure variations, causing pitch accelerations around a point near the stern. This is seen from the phase plots of Figures 2.26 and 2.27, and from the absolute values of Figure 2.26, where it is seen that the acceleration level is much larger at the bow than at the stern.

The significant sea wave height during the measurements was estimated visually. Accurate measurement of the very low waves considered here (below 0.5 m) require special equipment. Wave measurements will also increase the time needed for set-up and calibration of the test equipment. Together this means that the extra costs for wave measurements are large, and regretfully it was not possible to get funding for it. Visual estimation of sea wave height is difficult, and partly based on experience. The estimates given in this chapter were verified by the experienced crew operating the craft. The sea states considered was purely wind generated and almost unidirectional. This made the estimation easier and probably more accurate. On the contrary the very low wave height probably made the relative error larger. Because of the large uncertainty of the wave measurements, and because the response is sensitive to the sea state,

one should be

careful comparing numerical simulations quantitatively with the presented full scale measurements.

(49)

3

CRAFT WITH RIGID PANEL AFT SEAL

Analyzing the SES equipped with a rigid panel aft seal is a simplification of the analysis of the SES with flexible bag aft seal. To analyze the SES with flexible bag, the mathematical model for the SES with rigid panel is extended to take account for the flexibility of the bag. This is done by adding a new spatial pressure degree of freedom that takes the flexibility of the bag into account. The extension to take account for the flexibility of the bag will be presented in the next chapter. The mathematical model of the SES with rigid planing aft seal to be derived in this chapter is valid for SES with all kinds of aft seal systems as long as their flexibility is negligible when exposed to normal dynamic cushion pressure variations.

Previous workers in the field of SES vertical plane dynamics have assumed that the pressure in the air cushion was spatially uniform. Full scale measurements with a 35 m SES presented in Chapter 2 show that the acoustic resonances in the air cushion excited by incident sea waves can result in significant pitch accelerations in the high frequency range. In this chapter the work by Kaplan and Davis (1974, 1978) and Kaplan, Bentson and Davis. (1981) is extended by including the effect of spatial pressure variations in the longitudinal direction of the air cushion. The effect of variable leakage area under the bow and stern seals is taken into account by use of describing functions. It is shown how the effect of a ride control system (RCS) can be included in the mathematical model. Parts of this chapter are also presented in Sørensen, Steen and Faltinsen. (1992, 1993).

3.1 THE RIGID PANEL AFT SEAL

One of the main reasons for the development of the rigid planing aft seal was to reduce the high frequency vertical accelerations experienced on the SES equipped with a flexible bag aft seal. Instabilities of the bag seal occurring due to interaction between the water surface and the air flow under the bag were viewed by the builder to be one of the main reasons for the high level of vertical accelerations experienced in the high frequency range. The stiffness of the planing seal, and the damping obtained through the fitted pneumatic suspension system should prevent the occurrence of instabilities. A sketch of the

(50)

Pneumatic suspension bas Suspension stabilizer .anel Downsto cable

Pa

GRP planar panel

Figure 3.1 Rigid planing aft seal, side view

rigid planing seal is shown in Figure 3.1. There are five or six sandwich panels mounted side by side. On each sandwich panel is mounted one or two GRP panels of relatively low stiffness. The GR.P panels are planing on the water surface. The low stiffness allows them to follow the incoming waves as long as the wave height is below a certain limit. The limiting wave height is given by the length and angle of the GRP panels. For larger waves the sandwich panels will move to take up the wave motion. The sandwich panels are rigid, and are heavily built to provide the necessary strength to resist wave impacts in high sea states. The large mass of the panels lead to poor sealing ability in high sea states, which result in larger involuntary speed loss than for a similar craft with a flexible bag aft seal. The large mass of the sandwich panels creates large inertial forces when exposed to large wave-induced motions in high sea states. Together with wave impact loads this might lead to structural failure of the seal. Full scale measurements with a 35 m SES fitted with this rigid planing aft seal system indicated that the system was successful in removal of the instabilities experienced on craft with flexible bag aft seal. Results from the full scale measurements are shown in Chapter 2. The rigid planing seal system was fitted to several SES craft in commercial operation. It was removed after problems with the strength and durability of the sandwich panels and the main panel hinge when the SES was operating in heavy seas.

PC

V

Viet deck

Sandwich nanel

(51)

V

Figure 3.2 Global coordinate frame

3.2 AXIS SYSTEM AND CONVENTIONS

For use in the equations of motions, a moving coordinate frame is defined so that the origin is located in the mean water plane below the centre of gravity. The positive direction of the xg-, Yg' and z1-axes is forward, to the port, and upwards respectively. The axis system is sketched in Figure 3.2. The equations of motion are formulated in this moving frame. Translation along the zg-axis is called heave and is denoted

î.

The rotation angle around the y-axis is called pitch and is denotedTl. Heave is defined positive upwards, and pitch is

defined positive with the bow down.

In analyzing the dynamic pressure it is convenient to define a temporary coordinate frame (x,y,z) with the origin located at the geometrical centre of the air cushion water plane

area. The axes are defined as before, except for the longitudinal transformation of the origin to the centre of pressure. This means that x=x+x Y=Yg' ZZg, where is the longitudinal coordinate between the centre of gravity and the air cushion water plane area centre. The air cushion waterplane area is assumed to be rectangular, with the waterplane area given by

CG

(52)

A=Lb. In this coordinate frame the cushion length is reaching from x=-L/2 at the stem to x=L12 at the bow.

The situation when the craft is advancing in regular head sea is studied. The sea waves are assumed to have a small wave slope and a circular frequency o. The circular frequency of encounter e is

w=o+kU

(3.1)

where k = 2it/X is the wave number, ? is the wave length and U is the craft forward speed. The incident surface wave elevation Ç for regular head sea is defined as

Ç(x,t) = Çsin(ot + kx) (3.2)

where a is the wave amplitude. The water waves are assumed to pass undisturbed through

the air cushion.

The airflow in the cushion and bag is assumed to be ideal and compressible, with pressure changes occurring rapidly enough to justify the assumption of adiabatic pressure changes. The cushion height and beam are assumed to be small relative to the cushion length. By keeping the maximum frequency below a certain limit, the spatial pressure can be represented by acoustical plane waves in the longitudinal direction only. The maximum wave frequency of encounter is specified by:

2icc

(i)_emax

<_-__

b and

Where c is the speed of sound in air. h0 is the equilibrium cushion height, and b is the cushion width. Under the above assumptions, local geometric shapes like locally non-vertical or non-parallel cushion walls, and the shape of the aft seal can be accounted for by the effect on the total air cushion volume.

2tc

co_{e maz} <

h3

(53)

3.3 THE GLOBAL CONTINUITY EQUATION

The rate of change of the mass of air inside the air cushion is equal to the net mass flux of air into the cushion. The linearized global continuity equation for air mass flow into and out of the cushion follows from

p0Q(r) - PaQout(t) = -__

I p(x,t)dx + p0V(r)

AhU2

(3.4)

-L/2

where Q, is the volumetric air flow into the air cushion from the lift fan system. Q0 is the

volumetric air flow out of the air cushion. V is the cushion volume, p is the air density at the atmospheric pressure pa' p is the density of the air at the cushion pressure p. p is the density of the air at the equilibrium cushion pressure Pa+PO. The cushion pressure PC iS given by

p(x,t) =

Pa +

p(t)

p5(x,t) (3.5) where p is the dynamic uniform pressure and PSP is the spatially varying pressure. From the

adiabatic pressure-density relation we get the expression for the air density in the cushion p

p(x,t)

= p50

Pa +

p(t) + p(x,t)

_(3.6)

Pa ' Po

where y is the ratio of specific heat for the air.

It is convenient to express the pressures in terms of dimensionless quantities. The dynamic uniform pressure is represented by the dimensionless quantity defined as

p(t) - p0

M5(t) = (3.7)

po

where Po is the mean cushion pressure. p can be expressed on the following dimensionless form

= PSP (x, t) (3.8)

Po

(54)

differentiate with respect to the time t, we find an expression for the time derivative of p as

cO

(t(t)

+ iÌ_sp(x,r))

l+

Po)

The rate of change of the air cushion volume can be written as

1;yt)

= A(fl3(t)-xfl5(t)) - 1?555.(t)

(3.10)

The last term in equation (3.10) represents the change of air cushion volume due to the change of sea wave elevation, denoted wave volume pumping. It can be expressed as

=

y

Q(t) =

(t)

=

bJ(x,t)dx

_AÇ5

where i is the imaginary unit.

The volumetric air flow into the cushion is found by assuming quasistatic fan response and by linearization of the fan characteristic curve about the craft equilibrium operating point, see Figure 3.3. It is assumed that q fans with constant rate of revolutions are feeding the cushion. Fan i is located at the longitudinal position XF. The fan characteristic curve can then be represented by

(

-

u sp F )

"aQ _{p0 (.i}

_{(t) +}

_t _{(x .,t)}

where Q0

is the equilibrium air flow rate of fan

i when P5=Po and (JQ'8p)0 is the corresponding linear fan slope about the craft equilibrium operating point Q0 and Po The total equilibrium air flow rate into the air cushion is then

kL

sin

-'2 kL '2 (3.9) 'sot e (3.11) (3.12)

Cobblestone effect on SES

P

SVERRE STEEN

COBBLESTONE EFFECT ON SES

TEC1HIasc1fE UNWERSITE1T

Archief

Mekelweg 2, 2628

Tel: 015-2786873/Fax:2781836

Cobblestone Effect on SES

Dr.ing thesis

Sverre Steen

Department of Marine Hydrodynamics

The Norwegian Institute of Technology

PREFACE

ACKNOWLEDGEMENT

ABSTRACT

NOMENCLATURE

f

he

ç

CONTENTS

i

INTRODUCTION

FULL SCALE MEASUREMENTS

a rigid planing aft seal was carried out on one occasion. During all

+

._ -

ili.

tI.I. l

t.---I

iiiu .itiii

-'i

I Iiiu

isiti.

1Ii IuiIUtu

ui

itigi,uu

,.mi,.i i i

i.

I. iIU

.i.i.

i,.i.iu i

I.i.i. lu i.t.i.i i I I

luuIul ui.i.

,.j.ui

= I(f)i2Sç(0)o)

ppfe

(f)

-

(f)

--

L

i

0.4-

0.8-0

234567

200g

i00<

o

-

one should be

CRAFT WITH RIGID PANEL AFT SEAL

V

î.

w=o+kU

<_-__

2tc

p0Q(r) - PaQout(t) = -__

I p(x,t)dx + p0V(r)

AhU2

p(x,t) =

p(t)

p(x,t)

p(t) + p(x,t)

p(t) - p0

(t(t)

l+

= A(fl3(t)-xfl5(t)) - 1?555.(t)

Q(t) =

(t)

_.i.i.

_{luuIul ui.i.}

_{(t) +}