Calculation and measurement of forces on a high speed vehicle in forced pitch and heave

Calculation and measurement of forces on a high

speed vehicle in forced pitch and heave

G. Delhommeau

DHN (LMF. URA CNRS 1217). Ecole Centrale de Nantes. 1 rue de la Noë, 44072 Nantes. France

P. Ferrant

SIREHNA, Immeuble Atlanpole, 1 rue de la Noë, 44071 Nantes, France

&

M . Guilbaud

LEA (URA CNRS 191) CEAT. Université de Poitiers, 43 rue de Vaérodrome. 86036 Poitiers. France

(Received 6 June 1991; accepted 1 November 1991)

An experimental apparatus to measure added mass and damping coefficients in pure heave or pitch or the coupling coefficients in combined motion on a surface effect ship (SES) sidewall with a forward speed in the small est section of a hydrodynamic water tunnel has been built. To cheelc the results obtained, the whole experimental set has been modified to be used m a towing tank with a 3-3 larger model with same values of reduced parameters. Comparison with both kinds of measurements is satisfactory. The influence of Froude 'i;;"^ber;^^2n has of motion, frequency and phase lag between motions for combined motion has been studi;d; some tests have been also performed with two symmetrical idewalls (catamaran). Measured coefficients have been compared with coinputed ones obtained with the code A Q U A + from the ECN. In spite of the large moUon ampUtudes, agreement between tests and linear computations is quite acceptable.

1 I N T R O D U C T I O N

Calculation of vehicle seakeeping in waves is a difficult problem, particularly when forward speed is high, corre-sponding to a non-dimensional parameter T = ojUlg, based on circular frequency ca and forward speed U, greater than 0-25. For example, f o r a SES, values of x greater than 10 can be encountered. I n these conditions, for performance or seakeeping computations, exper-imental values of added mass and damping are needed; furthermore these values can be used to check com-putational methods. A n experimental set-up f o r the determination of added mass and damping f r o m unsteady force measurements on models oscillating har-monically on a free surface has been studied and built. I t was planned to be used in a recirculating water channel with a small test-section, and a model length less than 0-5 m . For this scale, high frequencies (up to 12 cycles per s) must be achieved at Froude numbers close to 1 to simulate sea conditions that such a SES can encounter

when moving i n head seas. So, high values of mertia forces, compared with hydrodynamic ones, can be found for such models; as the latter are obtained by subtracting measured forces when the model is oscillating i n air f r o m forces measured when oscillating in water, an important discrepancy can appear. Great care has to be taken to reduce the mass of the part of the experimental set-up which is located under the dynamometer and thus is responsible f o r inertia forces (struts, models). The dynamometer is very compact and close to the model. N o check o f t h e resuhs can be done at high Froude numbers because few tests are available, but interest is increasing for response o f high speed ships to a seaway (cf., f o r example, Keunig'). Furthermore, previous unsteady measurements in the same tunnel on models in waves have been successfully compared with measurements m a towing tank using an 8 times larger model.^ Nevertheless, to check the results at non-zero Froude numbers, the whole experimental set-up has been modified to be used in a towing tank with a 3-3 times larger scale.

Fig. 1. Experimental setup.

Tiie test results have also been compared with the results o f computations with the code A Q U A + f r o m the D H N - E C N . This code solves the radiation problem with forced motion and forward speed, i n perfect fluid. As the model is very thin, the linearized free surface conditions written on the undisturbed free surface is simplified by the thin ship approximation; the problem can thus be reduced to a radiation problem without forward speed. The solution is based on a panel method with Kelvin singularities, so the free surface boundary conditions is satisfied by each singularity through the Green function. Then l i f t and pitching moment in phase (added mass) and in quadrature (damping) with motion are computed.

The second part of this text is devoted to the descrip-tion o f t h e experimental set-up and of the test procedure; equations to determine added mass and damping are presented. The numerical computations are presented in the third paragraph. Test measurements i n water channel and in towing tank are compared to each other and also with the numerical results in the fourth paragraph.

2 E X P E R I I V I E N T A L S E T - U P A N D P R O C E D U R E The oscillator is made o f two horizontal parts (Fig. 1); the highest one is motionless and the lower one can have a heaving motion guided by four adjusted shafts. The horizontal axis f o r pitching motion is tied rigidly under the moving part. The heaving and pitching motions are created by an electric motor and two excentric-rod systems; the first one is on the motionless plate and the second one on the moving plate (Fig. 1). Movements (ranging f r o m 1 to 12 cycles per second) are transmitted through cranked belts. Mean positions of model and motion amplitudes can be changed {a = 0-30 mm for heave, ijj = 0-6° for pitch). The 3 component

dynamo-Fig. 2. Test model.

meter and the model are hung to the pitch axis. Pure heave or pitch motions or combined heaving and pitching motion with variable phase lag can be achieved.

The dynamometer is made of three strain gauge units measuring bending moments. These modules can be easily changed when model scale is changed. They are equipped with semi-conductor strain gauges wired in Wheatstone bridges.

Models are sidewalls f r o m the surface elfect ship M O L E N E S (Fig. 2) f r o m the D C N (Bassin des Carènes, France) tested either in the free-surface test-section (0-23m wide, OTSm high and 0-8m long) of a hydro-dynamic channel i n the C E A T Poitiers or in the towing tank (5 m wide, 3 m deep and 70 m long) o f t h e ECN; the model scales were respectively 1/35th or 1 / l l t h with model length L = 0-355 m or M 5 m . The immersions are D = 8-3 mm or 27-3 m m and the masses are G'095 and I T kg. Measuring units are different and models have been built as light as possible to reduce inertia forces and moments.

Signals f r o m the dynamometer and f r o m the linear displacement transducer controlling either heave motion or pitch motion are connected to a data acquisition system after low pass filtering. This system, connected to a microcomputer, consists o f a 16-input scanner, a 12-bit analog to digital converter and a 4000-word memory; this system enables data acquisition up to 20 kilocycles per s. Measurements are synchronized by the motion; 20 measurements are done each period on each signal during 15 periods except for low frequency acquisition in basin where the duration of test is limited; 8 periods only can be used there. A numerical bandpass filtering is achieved on each signal to compute the amplitude and the phase lag refered to motion signal. Models are first oscillated only in air so that inertia forces and moments can be obtained. Then oscillations are done with models partially in water and total forces and moments are recorded. By subtracting inertia forces and moments f r o m total ones, hydrodynamic lift F., moment M are computed and also l i f t phase lag (j>^ and pitching moment are (f)^. Phase lags are refered to the motion {(p = 0

corresponds to model at higher position f o r heave or combined motion, leading edge at high position f o r pitch).

Experimental forces and moments are made non-dimensional by:

and Cv M ( 1 )

with U = forward speed if U ^ 0 and U = .fgL i f U = 0; we have a = a/L f o r heaving or combined motion and a = ^, f o r rotation, \j/ amplitude o f the rotation in degrees.

Once these coefficients are known, added mass M^-, damping B^j (where index / is for effort and index j for motion) can be computed f r o m movement equations i n pure motion, assuming corresponding hydrostatic restor-ing coefficients Cj, are known. For heave motion, we have:

{C^a - F.cos (pf) F, sin

= ^ " =

and for pitch motion:

(C551// — M cos 4. A f .

am

M s i n (3) Once the previous coefficients for pure motions are obtained, combined motion can be used to compute cross coupling coefficients. From l i f t measurements, we have:

M35 = [-F, cos {4>, - 4>,f) + C.iip — B^^coa sin

+ (C33fl - M,,aoj') cos </),p]/[a)^^] (4)

B35 = [-F, sin ( 0 , - (pip) - B^^ma cos (p,^

+ {M,,aof - C,,a) sin 0,p]/[cuiA] (5) Similarly f r o m pitching moment measurements, we have:

M j 3 = — [M cos (j)^ + M55 \]/a)- cos (/),p

- Bssonl/ sin (pi^ - C551// cos 0,p - Cs^a]!

X [coV] (6) ^53 = [ - M sin - MSJCDV sin 0,p

+ Bi^ij/oj cos (/),p - Csji// sin (j),p]l[anl/] (7) where C35 = C33 are the cross-coupfing restoring coef-ficients and 0,p the phase lag f r o m pitch motion on heave motion.

Added mass and damping coefficients are presented in non-dimensional form: CM.,. = ^ pL" pmL" (8) with TRIM ( d e g r e e s ) e - 2 0 2 T r •10 0 10 S I N K A G E ( m m) A

Fig. 3. Evolution of C33 versus sinkage and trim (for L = l T 5 m model). n = A f o r / « = 4 f o r / 7! = 5 f o r 4,5,6, a n d ; = 1,2,3; 1,2,3, a n d j = 4,5,6; 4,5,6, a n d j = 4,5,6;

L being a reference length chosen equal to the length between perpendiculars. The reduced frequency is given by:

/ I

CO (9)

Calculations are made with the Unearized hydrostatic restoring coefiicients. The shape of the hull varies quickly around the waterline, so the assumption of linearization (constant hydrostatic coefficients) is very approximative. Curves 3 and 4 show the variation of these restoring coefficients in heave and pitch with the trim and sinkage of the hull. For great displacements (a = 20 m m and ip = 3°), the relative variation of the coefficients are 30% for heave and 50% for pitch. The influence o f the two pure restoring coefficients on added-mass is very import-ant at low frequency. Variations of 10% on C33 or C55 gives variation of about 30% on C M 3 3 and 80% on C M 5 5

TRIM ( d e g r e e s ) • - 4 - 2 0 2 4 0 I 1 1 1 1 r 1 1 — ^ 3 0 E 2 0 S 1 0 Ü 4^ A * A H = 3 f o r i = 1,2,3, a n d j = 1,2,3; - 2 0 - 1 0 0 10 2 0 S I N K A G E ( m m ) *

Fig. 4. Evolution of C 5 5 versus sinkage and trim (for L = M 5 m model).

10'^CM33 60

40

20

O 2ÏÏ 4TT

A simplified approach avoiding the computation of the complete forward-speed Green function is adopted here by modifying the 3D zero forward-speed seakeeping code A Q U A D Y N , developed by D H N - E C N , and used i n international tests since 1978.'"'

Forward speed effects are accounted for under the thin ship hypothesis {dtpldx and ~ 0) in the same manner as in the strip theory. However, owing to the 3D modelization, a better accuracy is obtained for the surge motion.

Applications of the resulting computer code, A Q U A - f , were presented at the Workshop on com-parative study of computer programs, held in Bergen, Norway in 1982.'°

3.2. Mathematical formulation Fig. 5. Influence of variation of hydrostatic restoring

coef-ficients C33 on CM33.

for (5 = 1-5 ( c f , by example, Fig. 5 for CM33). This influence decreases rapidly when 8 increases. A t (5 = 5, percentages are reduced to 2 and 17% respectively. Effects of variation of C33 = C53, C33 or C55 on coupled added-mass and damping in combined motions are quite negligible (less than 1% for 10% restoring coefficients variation). For tests with forward speed, the dynamic steady l i f t has been computed using a wave resistance computing code.^''* The results show that dynamic l i f t increases the heave restoring coefficient of 5-7% and the pitch one o f 8-6%.

Series of tests have been done in channel and in towing tank for equal values of reduced parameters: Froude number {F = Uj^JgL, U free-stream velocity) F = 0 and

1, heave amplitude ajD = 0-5 and 0-75, pitch amplitude \j/ = 3°, and for combined motion ajD = 0-5, 1// = 3°,

(f>ip = 0 or 180°. Further tests have been done only i n

channel: influence of Froude number (F = 0-75 and 1-15), of pitch ampHtude (ij/ = 2 and 4° at F = 0 and F = \) and for combined motion at F = 1, influence of phase lag between motions ((/),p = 0, 90, 180, 270° at ajD = 0-75 and ij/ = 3°). Finally, tests with two symme-trical models (catamaran) have been led in a towing tank; results are similar to those obtained with one sidewall.

We consider a right-handed Cartesian coordinate system moving in steady translation with the mean forward velocity U of the ship, the origin being in the plane of the undisturbed free surface, with the x-axis i n the direction of ship's forward speed and z-axis pointing upwards.

Under the assumption of irrotational flow, the linear-ized problem for the velocity potential in infinite depth is defined by:

Ad" 8^

S>\ lim

0 in the fluid domain = VE • Holco ° n

dt'

dx dx' = 0 free surface condition 1

z = 0

dt dx wave elevation (10)

with: V = grad<b, being the outward normal to CQ (mean body surface).

The radiation condition at infinity is taken into account i n the free surface condition by the term o f vanishing viscosity e', issued f r o m the limiting absorption principle.

The potential O can be spht into two parts:

3 N U M E R I C A L C O M P U T A T I O N 3.1. Principle

When solving the Hnearized seakeeping problem with forward speed using a Kelvin source method, the involved numerical computations become more and more difficult as the parameter i increases, especially for T > 1/4, as was mentioned by Guével and Bougis^ and Wu and Eatock-Taylor.''

where is a steady wave resistance potential, and «ï^ an harmonic time-dependent potential.

Under the thin ship hypothesis, these two potentials are uncoupled. Using the complex notation:

A{t) = A* cos ojt + A** sin mt = R{Ac""')

solution o f (cf " ) : A$d = 0 3 ^

5=1 J + UAscy,)e-'"]

lim o f f ^ i - 2ie.'m^i + g - ^ + 2iUm 5 $ ,

-2Ue' 8^ dx + U' 3 ^ dx' -'=0 dz 0 3 ^ dx (11) wliere:

(a) g is the index of the forced motions: 1,2,3 for translations following x, y, and z and 4,5,6 for rotations around Ox, Oy, and Oz.

(b)

e,-Ko for<7 = 1,2,3 ^^^^ (e,_3 A OPo) • Ho for q = 4,5,6

(c) Po is a current point o f CQ and e^{q = 1,2,3) are the unit vectors of the axis Ox, Oy and Oz.

Using again the thin ship hypothesis, we neglect the terms d^^jdx and d'^jdx'. The potential Oj is then solution o f a radiation problem without forward speed and can be obtained after solving six elementary radi-ation problems ^j^^, q = 1 , . . . 6: A $ ^ , = 0 dno Ico " lim - a;^$„ E ' ^ 0 + 2ie'a><!? Rq + g SO dz 1-0 Oj is obtained by: = 0 9=1

The radiation pressures are given by:

P

UA,<^,

(13)

(14)

pm' A^^R, - ipmUA(,^„2 + ipcoAs^m

9=1

I f we note M the complex added mass:

with: 3«

=

{p = 4,5,6) are given by:

F^p = m'Y. A^M^^ - imUA.M^, + imUA.M^, 9=1 (19) 9=1 with: U M;, = M,5 / - M,3 and ML Ü for ^ / 5,6 (20) (21) W i t h the previous assumptions, the difference with zero-forward speed radiation forces is only due to the coupling of forward speed with the rotation of the normal at PQ on Q . The complementary terms are called

mj terms i n the formulation of Newman.'^

When the body moving with forward speed is sub-mitted to forced harmonic motions, the total force on the body F / i s obtained by adding to the radiation forces , the hydrostatic restoring forces F".

rpT ^ pR • pH with: < 6

]FP"=1 C,,A, 9=1

(22)

being the matrix o f linearized hydrostatic restoring coefhcients. This matrix depends only on the geometry of the water-plane area of the body at rest.

3.3 Numerical solution of the boundary value problem The computer code is based on a boundary element method for solving the diffraction radiation problem of bodies oscillating i n waves with forward speed under the thin ship approximation, in infinite or finite uniform depth.

Singularities are sources of constant strength distributed over triangular or quadrilateral panels.

The originality of the method lies in the computation o f t h e Green function. The derivatives of terms in l/i?, which are independent of the frequency, are only cal-culated for the first frequency by classical Hess and Smith formulas with exact analytical quadrature for Httle dis-tances (J?/£2 < 7, Q being the greatest diagonal of the panel) and by single-point quadrature for great distances (F/Q > 7).

The remaining part of the Green function which depends only on two parameters, horizontal and vertical distances X and Z , is interpolated into a file of four elementary functions by direct formulas avoiding system-atic research o f the values to be interpolated.''

The size of the file is lower than 256 K bytes for four bytes real numbers, with 46 values in Z and 328 values i n X, logarithmically spaced.

. BZ 1 m • O a d - O "È O CM33 / . 1 . 1 . 1 . 1 1 •^ T . T H N K - Z .75 a T U N N E L - 0 . 7 5 .BBIB D *^-a ° — a + ° ? ° a S 1 . 1 . 1 . 1 1 1 2 4 e s Fig. 6. Pure heave (F =

1 0

0),

1 2 1 4

With this method, the computing time for the Green function is half the one needed to compute terms in XjR i n infinite depth and four times in finite uniform depth. Most o f the computing time is devoted to the solution o f t h e linear system of the diff"raction-radiation problem f o r more than 100 panels on the half body.

4 C O M P A R I S O N O F C O M P U T E D AND M E A S U R E D C O E F F I C I E N T S

Comparison is shown in Figs. 6-11 where coefficients are plotted versus reduced frequency 5. Symbols are f o r tests and curves for calculations. The lower part of each graph presents the added mass and the upper one, the damping. Figures 6 and 7 deal with pure heaving motion and Figs 8 and 9 with pure pitching motion; the combined motion

CA33 . 0 0 1 0 CÏVi33 sa 1 0 . o s s s * o -fe o + T . T H N K m.y'U Q T U N N E L m ^ D 0 . 7 5 0 . 7 5 a >a o - I . 1 . 1 , L 0 2 4 S S 1 0 1 2 1 .

Fig. 7. Pure heave (F = 1).

CA5S . 0 0 0 1 0! 0 0 0 0 5 CM35 , 00010 , 0 0 0 0 3 1 - Q 0 ° ^° % ° % ° * ^ 1 • 1 . -1 1 ' • ' • — - J + T . T R N K ^ - 3 " 0 T U N N E L i j j 3 -D 1 1 -"a ^ r a — 0 + • ^ ° ^ — H " 0 . r . 1 , 1 . 1 . 1 0 I S ₁ 4 Fig. 8. Pure pitch {F = 0).

is illustrated by Figs 10 (no phase lag between motions ^tp = 0°) and 11 (motions in opposition of phase 4 > i p = 180°). As indicated in the second paragraph, the

discrepancy is important at low frequency (Figs 6-9). Furthermore, at F = 0 (Figs 6 and 8), some problems occur because waves produced by the body are reflected by the waUs, particularly in tunnel where they are close to the models; as soon as there is motion the waves flow away and reflection doesn't affect the model (Figs 7 and 9). Good agreement can be seen for both types of tests, at F = 1, particularly for high frequencies in spite o f the quite different water depths. Results coincide within few per cents f o r sub and supercritical flows for our model (which has length L to beam B and length to draft D ratios L/B = 28 and L/D = 40); this similarity has been already mentioned f o r critical and supercritical flows (cf.' with L/B = 8 and L/D = 32). Computed and measured

CA5.5 , 0 0 0 1 0 CM55 0001 0 . 0 0 0 0 s o -45 + T . T H N K V = 3 ' o T U N N E L \ | / - 3" ' -!-o crt- o + O V t . f . r . I . I . I 0 2 4 S 3 1 0 1 2

2 5 10*CA 10*CM 3 0 2 0 1 0 0 - 1 0 - 2 0 - 3 0 - 4 0 C F 1 5 3 C R 3 5 • - ^ - i 5- - - » - R - - S* « # 0 » -C M 5 3 C M 3 5 TJ*-3—ar-g—*—S'l C M o r C H S a - P I T C H / H E R V E T . T R N K C M a r C R S S - H E H V E . ' P I T C H T . T H N K C M o r C H S 3 - P I T C H / H E R V E T U N N E L C M a r C R 3 5 - H E H V E - ' P I T C H T U N N E L I . I I 1 1 1 1 1 • — 1 2 • Added mass a I 0 _{1 2} 1 4 (2) • Damping ( 3 ) Hydrostatic 0 1+3 Ci) Total lift

Fig. 10. Combined motions F = 1 = 3°).

(ij!>TP = 0° a / D = 0-5

values of added mass are i n close agreement for both motions and both Froude numbers. For damping coef-ficients, measured values are nearly located on a parallel to the computed coefficients, but with slightly lower values. Particularly when frequency increases, the measured damping doesn't tend to zero as computed values do. This is due to the viscous damping, which is not predicted by perfect fluid computations. So viscous damping seems to be nearly independent of the frequency and doesn't tend to zero at high frequencies.

The good agreement between measurements in towing tank and in tunnel can be observed whatever the coupling coefficients are (Figs 10 and 11). The added-mass cross-coupling coefficients tend to zero when the frequency increases. A t low frequency, discrepancy appears

2 5 1 0* C A 1 5 1 0 5 0 - 5 - I B 3 0 2 0 1 0 0 - 1 0 - 2 0 - 3 0 - 4 0 . _ C R 5 3 _ C R 3 5 « t l t f i O • » t r -C M 5 3 C M 3 5 » C M o r C R 5 3 - T . T R N K + C M o r C R S S - T . T R N K o C M o r C R 5 3 - T U N N E L X C M o r C R S S - T U N N E L _1 I 1 1 I 1 I I e 1 0 1 2 1 4

Fig. 11. Combined motions F = 1 {(f>rp = 180° a/D *P = 3°).

0-5

Fig. 12. Components o f unsteady l i f t amplitude (heave).

between measurements and computations. Computed values of CM53 are nearly constant but measured values seem to increase rapidly for = 0° and to decrease also rapidly for = 180°. For CM35 computations predict a decrease but measurements show some slow increase. For damping coefficients, agreement is better at low frequency. The coefficients are nearly constant f o r 5 > 2 as f o r measurements than for calculations, but measured values of C^53 are non-zero at high frequencies f o r 4 > f p = 180°. A t low values o f S , curves of computed,

values of both damping coefficients are nearly parallel to measured values.

It must be noticed, f o r combined motions, particularly at = 180°, amplitudes of part of models can be very important. For instance, both motions tend to h f t the forepart of the model above the water level, so the instan-taneous immersed part of the body is not constant and can be quite different f r o m the mean immersed part assumed in the computations. The useful range f o r SES in head seas correspond to the medium range o f studied frequencies. I n spite of relatively low wave frequency, the frequency of encounter is high due to the high speed of advance. So, for these conditions, added-mass and damping coefficients can be considered as constant, as i t can be seen on the last figures. Furthermore, simplified computations using asymptotic formulas give good results in 5 ^ 6, that is in the useful range that encounter the SES.

Figure 12 shows the variation of the lift coefficient amplitude in heaving motion versus reduced frequency 5. The coefficient can be written as:

C, = I c o ' C M , , - C 3 3 - f - icoCAjil (23) The different curves correspond to various components

of lift: the curve 1 is for added-mass, curve 2 is for damping and curve 3 for hydrostatic part; curve 4 is for the sommation of added-mass and hydrostatic com-ponents and curve 5 f o r total lift. It can be observed there that hydrostatic component is predominant at low fre-quencies and than at high frefre-quencies, it is added-mass which is predominant. The damping coefficient is weak, and its influence is only large close io 5 = M 5 , where added-mass and hydrostatic components cancel each other. So, in spite of the underestimation of this last term, lift can be correctly predicted by numerical computations as it can be seen in Ref. 14. A t low frequency, the dis-crepancy between tests and computations may perhaps be attributed to the bad estimation of the predominant hydrostatic term as explained in a previous paragraph. 5 C O N C L U S I O N

We have presented the experimental results obtained by use of an experimental set-up enabling to oscillate models in pure heave and pitch or in combined motion. From measurements of unsteady forces and moments, added-mass and damping coefiicients are obtained i n pure motions, and in combined motion, the cross-coupling coefiicients. This set-up has been designed to be used in a smafl test-section of an hydrodynamic tunnel (width 230 mm). As a check of these first results obtained, the experimental set-up has been modified to be used in a towing tank with a model 3-3 times larger. The com-parison of the measurements of the two series of tests is very satisfactory, except in some cases where a bottom effect i n the tunnel has been identified.

So this study has emphasized the interest of smafl, cheap facilities where reliable measurements can be achieved i f tests are carefully performed. Nevertheless, the depth of immersion is 8 mm, so that high relative errors on the position of the model can be made.

The comparison of test results with values obtained by linear computations is quite acceptable i n spite of the underestimation of damping. For cross-couphng coef-ficients, discrepancy appears at low frequency; i t can be probably attributed to diflSculties to determine hydro-static coefiiicients when assuming constant the immersed part of the model.

For a fast ship as a SES, comparison of test and computed values shows that asymptotic formula gives good results in the major part of frequency range encountered.

A C K N O W L E D G E M E N l S

This job has been supported by contracts D R E T N o . 87/40RF and N o . 88/470. This support is thankfully acknowledged by the authors.

R E F E R E N C E S

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