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Hydrodynamic derivatives and motion response of a submersible surface ship in unbounded water
ARTICLE IN PRESS Ocean Engineering 37 (2010) 879–890
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Hydrodynamic derivatives and motion response of a submersible surface ship in unbounded water Michio Ueno National Maritime Research Institute, Marine Dynamics Research Group, 6-38-1 Shinkawa, Mitaka, Tokyo 181-0004, Japan
a r t i c l e in fo
abstract
Article history: Received 12 August 2009 Accepted 1 March 2010 Available online 15 March 2010
A submersible surface ship (SSS) is based on a novel concept that the SSS goes on surface like conventional ships in moderate seas but goes underwater in rough seas to the depth sufficient to avoid wave effects. The SSS has a wing system that produces downward lift to go underwater with preserving the residual buoyancy for its safety. The SSS is expected to be able to keep both safety and punctuality even if it encounters unexpected bad weather. The motion of the SSS is studied. The equations of motion are formulated and the procedures for estimating hydrodynamic derivatives are presented. The hydrodynamic derivatives are estimated for a SSS having a configuration, a hull with a pair of main wings and a pair of horizontal tail wings. Using these estimated hydrodynamic derivatives, calculation of the SSS motion is carried out. The calculation results show some specific aspects of the SSS especially for effects of the elevator of main wings and horizontal tail wings, aileron of main wings, rudder and propeller revolution. It is confirmed that the existence of static roll restoring moment and having large hull comparing with wing area play important roles in the motion of the SSS. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Submersible Underwater vehicle Motion equation Hydrodynamic derivative Motion response
1. Introduction The submersible surface ship (SSS) is a ship based on a novel concept, a ship that can avoid rough seas by going underwater, proposed by Hirayama et al. (2005a). The concept of SSS is new in the point that goes on surface like usual ships in calm and moderate seas but goes underwater in rough seas into the depth sufficient to avoid wave effects. In order to submerge the SSS uses wings producing downward lift. Even in submerged condition the SSS keeps residual buoyancy for its safety. The SSS is expected to be able to keep both safety and punctuality even if it encounters unexpected bad weather. A SSS of which configuration is a hull with a pair of main wings and a pair of horizontal tail wings has been studied by a group of the Yokohama National University (YNU). Among researches necessary to realize a SSS are the dynamics, the structure and strength, the powering system for underwater cruising, the navigation and routing. Researches on the navigation and the wing performance of a SSS are in progress and are expected to be reported in the near future. From a viewpoint of the dynamics of a SSS related researches so far are as follows.
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Hirayama et al. (2005a) carried out a trial tank test for a selfpropelled SSS model made by modifying a conventional container ship model and obtained the basic information about the wing area necessary to submerge a conventional ship and its vertical motion control. Hirayama et al. (2005b) also clarified by a tank test the relation between the submerged depth, the downward lift and resistant acting on the SSS hull having modified bow shape. Koyama et al. (2006) simulated a submerging motion of the SSS model in waves and discussed its vertical stability. Hirakawa et al. (2007) studied the vertical motion control of the SSS model in the submerging and emerging motion by a tank test and simulation calculations. In order to clarify feasibility of the SSS, not only such studies concerning vertical motion done by YNU group but also studies concerning the lateral motions are needed. Mori et al. (1988) proposed a high-speed semi-submersible vehicle with wings (HSV). The HSV submerges by using downward lift of wings, which is the same mechanism as the SSS. However, the objective of HSV is reducing the wave-making resistance by submerging into shallow depth, which is different from that of the SSS. Mori et al. confirmed that the HSV is successful in reducing the wave-making resistance but the reduction of frictional resistance still needed. Mori et al. (1991) showed using numerical calculation that the downward lift is effective in small submerged depth and that the nature of vertical motion is unstable but can be stabilized by controlling the main and tail wing angles. Although they studied vertical stability
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during submerged cruising, lateral motion such as sway and yaw motion of the HSV has not been investigated. There are many reports concerning the dynamics of underwater vehicles. Ishidera et al. (1985) reported the equations of sixdegree-of-freedom (6-DOF) motion for a remotely operated vehicle and the automatic depth and heading control system. Ura and Otsubo (1987) proposed a concept of unmanned autonomous gliding submersibles and analyzed its vertical motion. Maeda et al. (1988) reported the hydrodynamic characteristics of two types of unmanned untethered submersibles and estimated its stability derivatives. Towed vehicles are among proposed underwater vehicles (Muddie and Ivers, 1975; Dessureault, 1976; Ohkusu et al., 1987). Ohkusu et al. (1987) measured the hydrodynamic forces acting on a model of a depth-controlled vehicle towed by a long cable and discussed its vertical stability. Most of all these vehicles are statically stable in its steady advancing condition. That means the buoyancy is equal to the gravity in water. On the other hand, a SSS using downward lift to balance with the residual buoyancy are considered to have common features with airplanes. The gravity of an airplane corresponds to the buoyancy of a SSS while the upward lift of an airplane corresponds to the gravity and the downward lift of a SSS. However, one difference is that a SSS generally feels static restoring moments for pitch and roll motion, because points of action for the gravity and the buoyancy are generally different from each other. Airplanes have nothing to do with that kind of static restoring moments. Another difference is that the volumelift ratio of a SSS is larger than that of airplanes. That means the hydrodynamic effect of hull part of a SSS is relatively larger than that of airplanes. As long as the configuration of SSS proposed by YNU group is concerned, one more difference is that it has no vertical tail wing. In this report, the equations of motion of a SSS in unbounded water are presented. The procedure to estimate hydrodynamic derivatives for the linear equations are also presented. This procedure is applied to the SSS configuration of YNU group and calculations of the motion are carried out for investigating the effects of the elevator of horizontal tail wings and main wings, aileron of main wings, rudder and propeller revolution. Results of these calculations show some aspects of the SSS dynamics, especially those due to the existence of static roll restoring moment and having large hull part.
2. Equations of motion 2.1. Configuration and coordinate systems A SSS consisting of a hull, main wings and horizontal tail wings shown in Fig. 1 is supposed. The tail and main wings have
δa
q
r
δew δr
δet
z,w
Matrix E is defined by Eq. (2) in which f, y and c are Euler angles. 2 cos y cos c 6 ½E ¼ 4 sin f sin y cos ccos f sin c cos f sin y cos c þsin f sin c
3
cos y sin c
sin y
sin f sin y sin c þ cos f cos c cos f sin y sin csin f cos c
sin f cos y 7 5 cos f cos y
ð2Þ Angular velocities p, q and r in the body-fixed coordinate system are related to Euler angles by Eq. (3). 8 _ _ > > < p ¼ f c sin y _ sin f cos y q ¼ y_ cos f þ c ð3Þ > > : r ¼ y_ sin f þ c _ cos f cos y Assuming the symmetry about xy-plane, the moment of inertias and products of inertias of the SSS have relations represented by Eq. (4). Ixy ¼ Iyx ¼ 0,
Iyz ¼ Izy ¼ 0
ð4Þ
2.2. Linear equations of motion Let r, g, m and , stand for the density of water, the gravitational acceleration, mass and displaced volume of the SSS, respectively. The center of gravity, (xg, 0, zg) and the center of buoyancy (xb, 0, zb) of the SSS are in the xz-plane in the body-fixed coordinate system. A stability axis is employed as a body fixed O-xyz coordinate system after the example of airplane dynamics (Kato et al., 1982). The horizontal steady advancing condition is chosen as the reference condition, in which the velocity components are defined by Eq. (5). u ¼ U0 ,
v¼w¼p¼q¼r¼0
ð5Þ
In Eq. (5), U0 stands for the steady advancing velocity. The general 6-DOF motion is described as deviation from the reference condition in which v, w, p, q and r stand for the deviations of velocity components. The longitudinal velocity component u is expressed by Eq. (6).
δew
δet
p
x,u
elevators. The main wings have also an aileron. The hull is equipped with a propeller and a rudder as conventional ships. A general body-fixed coordinate system is shown in Fig. 1. The origin locates on the center plane at midship. x, y and z points forward, right and downward, respectively. u, v and w are translational velocities to x, y and z directions and p, q and r are angular velocities around x, y and z axes. In Fig. 1, dew, det, da and dr show the positive direction of the main wing elevator angle, tail wing elevator angle, aileron angle and rudder angle. Note that the main wings work both as the elevator and the aileron. The elevator has symmetric angles and the aileron has anti-symmetric angles for the port side and the starboard side wings. In the earth-fixed coordinate system Oe-xeyeze, xeye-plane is horizontal and ze-axis points vertically downward. The earthfixed coordinate (xe, ye, ze) and their time derivatives are related to those in the body-fixed one (x, y, z) by Eq. (1). 3 2 dxe x e 6 2 3 dt 7 7 6 x u 6 dxe 7 7 6 6y v 7 ð1Þ 7 4 5 ¼ ½E6 ye 6 dt 7 7 6 z w 4 dxe 5 ze dt
δa
y,v
Fig. 1. General body-fixed coordinate system.
u ¼ U0 þ u~
ð6Þ
In Eq. (6), u~ stands for the deviation of longitudinal velocity component from U0. For convenience the following expression, Eq. (7), is employed hereafter. u~ u
ð7Þ
ARTICLE IN PRESS M. Ueno / Ocean Engineering 37 (2010) 879–890
The nonlinear equations of 6-DOF motion for an underwater vehicle which can be applicable to a general underwater moving body are derived by Ueno and Sawada (1993). The same can be reduced to the linear equations of motion by applying the stability axis, the reference condition defined by Eq. (5), the expressions of Eqs. (6) and (7), and neglecting the second order terms and over u, v, w, p, q and r. The resultant linear equations of motion are shown in Eq. (8).
881
2.3. Relations between the stability axis and the body axis The body axis OB-xByBzB is defined as xB-axis is parallel to the base line of the SSS while yB-axis coincides with y-axis. Fig. 2 shows the body axis OB-xByBzB and the stability axis O-xyz. The angular difference between the stability axis and the body axis is ai around y or yB-axis. The body axis is useful since it does not depend on the posture in the steady advancing condition. The
8 _ þ ðmzg þm15 Þq_ þ m13 U0 q ¼ ðmrrÞg sin y þX ðm þ mx Þu_ þm13 w > > > > > _ _ þðmxg þ m26 Þr_ þðm þ mx ÞU0 r ¼ ðmrrÞg sin f cos y þY Þ v þ ðmz ðm þ m > y g þ m24 Þp > > > > _ _ _ > Þ w þðmx þm35 Þqðm þmx ÞU0 q ¼ ðmrrÞg cos f cos y þ Z m u þ ðm þ m z g 13 > > > > > ðmzg þm24 Þv_ þ ðIxx þJxx Þp_ þ ðIxz þm46 Þr_ þ fm13 vðmzg þ m15 ÞrgU0 > < ¼ ðmzg rrzb Þg sin f cos y þ L > > > ðmzg þm15 Þu_ þ ðmxg þ m35 Þw _ þ ðIyy þJyy Þq_ þ f2m13 u þ ðmx mz Þwðmxg þ m35 ÞqgU0 m13 U02 > > > > > ¼ ðmzg rrzb Þg sin yðmxg rrxb Þg cos f cos y þ M > > > > > _ þ ðIxz þm46 Þp_ þ ðIzz þJzz Þr_ þ fðmy mx Þv þ ðm24 m15 Þp þðmxg þ m26 ÞrÞgU0 ðmx > g þm26 Þv > > > : ¼ ðmxg rrxb Þg sin f cos y þ N In Eq. (8), mij stands for added masses and added moment of inertias where 1, 2 and 3 for i and j represent x, y and z directions, respectively. m11, m22, m33, m44, m55 and m66 are replaced by mx, my, mz Jxx, Jyy and Jzz, respectively. Some components of mij are zero due to the symmetry of the SSS configuration. Dot above variables means time derivatives of these variables. Right-hand side of Eq. (8) contains hydrostatic forces and moments due to the gravity and the buoyancy, and hydrodynamic external forces and moments X, Y, Z, L, M and N. The first terms in the right hand side of Eq. (8), originated from the buoyancy and the gravity represent the hydrostatic forces and moments. These hydrostatic terms may play an important role for the dynamics of underwater vehicles. One example can be found for the descending motion of a launcher of a deep-sea robot (Ueno et al., 2008). Taking into consideration the symmetry of the SSS configuration, hydrodynamic external forces and moments are assumed to be expressed in linear forms as in Eq. (9). 8 X ¼ X0 þ Xu u þXw w þ Xq q þXdew dew þ Xdet det þ Xn Dn > > > > > Y ¼ Yv vþ Yp p þ Yr r þYda da þ Ydr dr > > > > < Z ¼ Z0 þ Zu u þ Zw w þ Zq q þ Zdew dew þZdet det þ Zn Dn ð9Þ L ¼ Lv v þLp p þ Lr r þ Lda da þLdr dr > > > > > M ¼ M0 þ Mu u þMw w þ Mq q þ Mdew dew þ Mdet det þ Mn Dn > > > > : N ¼ Nv v þ Np p þ Nr r þ N da þ N dr da
ð8Þ
forces and moments defined in these two coordinate systems are related to each other by Eq. (10). 2 3 2 3 XB LB X L 7 6 7 1 6 Y ð10Þ 4 Y M 5 ¼ ½Ea 4 B MB 5 Z B NB Z N In Eq. (10) and hereafter, subscript B stands for those in the body axis and no subscript for those in the stability axis. Matrix Ea 1 is defined by Eq. (11). 2 3 cos ai 0 sin ai 60 7 1 1 0 ½Ea ¼ 4 ð11Þ 5 sin ai 0 cos ai Moment of inertias, product of inertias, added masses and added moment of inertias defined in the body axis are converted to those in the stability axis using Eqs. (12) and (13). 8 > Ixx ¼ IxxB cos2 ai 2IxzB sin ai cos ai þ IzzB sin2 ai > > > > < Iyy ¼ IyyB ð12Þ 2 2 > > > Izz ¼ IxxB sin ai þ 2IxzB sin ai cos ai þ IzzB cos ai > > : Ixz ¼ ðIzzB IxxB Þsin a cos a þ IxzB ðcos2 a sin2 a Þ i i i i ½mij ¼ ½Ea ½mijB ½Ea 1
ð13Þ
dr
In Eq. (9), Xu, Xw, Xq, Yv, Yp, Yr, Zu, Zw, Zq, Lv, Lp, Lr, Mu, Mw, Mq, Nv, Np and Nr are hydrodynamic derivatives concerning hull and wing systems. dew, det, da, dr and Dn are main wing elevator angle, tail wing elevator angle, aileron angle, rudder angle and deviation of propeller revolution from the reference condition. Xdew, Xdet, Xn, Yda, Ydr, Zdew, Zdet, Zn, Lda, Ldr, Mdew, Mdet, Mn, Nda and Ndr are also hydrodynamic derivatives concerning control parameters. X0, Z0 and M0 stand for the hydrodynamic longitudinal and vertical forces and pitch moment in the horizontal steady advancing condition.
3. Hydrodynamic derivatives A procedure to estimate the hydrodynamic derivatives for the SSS is presented in this chapter. Comparable procedures for airplanes (USAF, 1968; Kato et al., 1982) are consulted together
xe V U0+u x α xB αi
-w
(xct,zct)
^
O,OB
U0 i
^
x xB
^
zTB Dt
Fig. 2. Stability axis O-xyz and body axis OB-xByBzB.
^
T ze
zB z
Mt
^
Dw
^
Mw (xcw,zcw)
Lw
Lt zB z
Fig. 3. External forces acting on a SSS in the condition of which the angle of attack is a.
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with considerations about the features of SSS such as having large hull and no vertical tail wing. The interaction between hull and wings are neglected by assuming small motion deviations from the reference condition. Although the interaction effects between hull, propeller and rudder can be taken into account as those for conventional ships, it is simplified here to clarify the basic properties of the SSS motion. 3.1. Components in external forces Fig. 3 shows a condition of the SSS in which the velocity of SSS is V with the angle of attack a. The velocity V is defined by Eq. (14). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ ðU0 þ uÞ2 þ v2 þ w2 ð14Þ External forces X, Y, Z and moments L, M, N in Eq. (8) are represented by Eq. (9) in which linear hydrodynamic derivatives are used. In order to estimate these linear hydrodynamic derivatives, another perspective is introduced and alternative expression for the external forces in Eq. (8) is obtained as shown in Eq. (15). 8 > X ¼ T cos ai þ X o h 4 þ X o w 4 þX o t 4 > > > > > Y ¼ Y o h 4 þ Y o w 4 þY o t 4 > > > < Z ¼ T sin a þ Z o h 4 þ Z o w 4 þ Z o t 4 i > L ¼ L o h 4 þL o w 4 þ L o t 4 > > > > > M ¼ TzTB þ M o h 4 þM o w 4 þ M o t 4 > > > : N ¼ N o h 4 þN o w 4 þN o t 4
ð15Þ
8 oh4 X ¼ X0o h 4 þ DX o h 4 ðuÞ ¼ X0o h 4 þ Xuo h 4 u > > > > > > Y o h 4 ¼ DY o h 4 ðv,p,rÞ ¼ Yvo h 4 v þYpo h 4 p þ Yro h 4 r > > > > > < Z o h 4 ¼ Z0o h 4 þ DZ o h 4 ðu,w,qÞ ¼ Z0o h 4 þZuo h 4 u þZwo h 4 w þZqo h 4 q
For the tail wing, the downwash effect by the main wing is taken into consideration and the next expressions, Eq. (20) are employed for the tail wings. 8 > ¼ C 1 @e > C > Lt Lt < @a @e > > > : CDt ¼ CDt 1 @a
ð20Þ
Thrust T can be estimated by Eq. (21). T ¼ rD4P n2 KTðJÞ
ð21Þ
In above equation, KT, n and DP represent the thrust coefficient, propeller revolution and propeller diameter. The thrust coefficient KT is assumed to be represented by a second polynomial of the advance coefficient J as in Eq. (22). J¼
ð1wP ÞðU0 þuÞ nDP
ð22Þ
In Eq. (22), wP stands for the wake coefficient and a0, a1 and a2 are constants. For the horizontal steady advancing condition Eq. (8) is reduced to Eq. (23). 8 /hS ^ ^ > > > T0 cos ai þ X0 ðD w0 þ D t0 Þ ¼ 0 > < _ _ /hS T0 sin ai þ Z0 þ ð L þ L Þ þðmrrÞg ¼ 0 w0 t0 > > > > ^ w0 þ M ^ t0 ÞðL^ w0 xCw þ L^ t0 xCt Þðmxg rrxb Þg þ m13 U 2 ¼ 0 : T0 zTB þM /hS þ ðM 0
ð23Þ
ð16Þ
ð17Þ
If hull related derivatives in Eq. (16) are defined in the body axis they can be converted to those in the stability axis by using ^ D ^ and M ^ in Eq. (17) represent lift, drag and pitch Eq. (10). L, moment acting on the main or tail wings. The distinction can be made by subscripts w for the main wings and t for the tail wings. ^ t are defined around the hydrodynamic ^ w and M Note that M centers. These coordinates of hydrodynamic centers in the vertical plane are represented by (xCw, zCw) and (xCt, zCt) for the main and tail wings, respectively. Hereafter, zCw and zCt are assumed to be zero. aw and at stand for the angle of attack to the main and tail wings. These are represented by Eq. (18). wþ xCw,t q U0
ð19Þ
0
> L o h 4 ¼ DL o h 4 ðv,p,rÞ ¼ Lvo h 4 v þLpo h 4 p þLro h 4 r > > > > > > M o h 4 ¼ M0o h 4 þ DM o h 4 ðu,w,qÞ ¼ M0o h 4 þ Muo h 4 u þMwo h 4 w þMqo h 4 q > > > > oh4 : N ¼ DN o h 4 ðv,p,rÞ ¼ Nvo h 4 v þNpo h 4 p þNro h 4 r
aw,t
8 1 > > L^ ¼ rSV 2 CL > > 2 > > < 1 ^ D ¼ rSV 2 CD > 2 > > > 1 > > ^ : M ¼ rScV 2 CM 2
KTðJÞ ¼ a0 þ a1 J þ a2 J2 ,
In Eq. (15), superscripts /hS, /wS and /tS stand for terms originated from the hull, main wings and tail wings, respectively. T stands for the thrust which is assumed to direct parallel to the xB-axis at z is equal to zTB. The hull, main wings and tail wings terms are assumed to be represented as in Eqs. (16) and (17).
8 ^ w,t cos aw,t X /w,tS ¼ L^ w,t sin aw,t D > > > > /w,tS /w,tS > > Y ¼ DY ðv,p,rÞ > > > < Z /w,tS ¼ L^ cos a þ D ^ w,t sin aw,t w,t w,t /w,tS /w,tS > ¼ DL ðv,p,rÞ >L > > > /w,tS ^ w,t L^ w,t xCw,t D ^ w,t zCw,t > ¼M >M > > : N/w,tS ¼ DN/w,tS ðv,p,rÞ
Coefficients for the lift, drag and moment CL, CD and CM are defined by Eq. (19).
ð18Þ
In Eq. (23), T0 stands for the thrust in the reference condition. 3.2. Vertical derivatives Based on the preliminary considerations in the previous section, hydrodynamic derivatives for vertical motion are estimated by differentiating Eq. (15). Resultant formulae are in Eqs. (24)–(26) in which vertical bar followed by 0 represents the differentiation at the reference condition. 8 @KT > Þ > j cos ai þXuo h 4 rU0 ðSw CDw0 þSt CDt0 > Xu ¼ rD3P n0 ð1wP Þ > @J 0 > > > > < 1 @CDw @CDt þ St CLt0 Xw ¼ rU0 Sw CLw0 @a 0 @a 0 2 > > > > > 1 @CDw > @CDt > þ x ¼ r U x S C S C X > q w t 0 Cw Lw0 Ct Lt0 : @a @a 2 0
0
ð24Þ 8 @KT Þ > > Zu ¼ rD3P n0 ð1wP Þ sin ai þ Zu/hS þ rU0 ðSw CLw0 þ St CLt0 > > @J 0 > > > ( > ) > < @CLt 1 @CLw þC þC þS Zw ¼ Zwo h 4 rU0 Sw t Dw0 Dt0 @a 0 @a 0 2 > > > > ) > > > @CLt 1 @CLw > oh4 > þ rU0 xCw Sw þCDw0 þxCt St þCDt0 : Zq ¼ Zq 2 @a 0 @a 0
ð25Þ
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8 @KT > 3 > M ¼ r D n ð1w Þ zTB þ Mu/hS þ rU0 ðSw cw CMw0 þSt ct CMt0 Þ > u P 0 > > @J 0 > > > > > t xCt CLt0 Þ > < rU0 ðSw xCw CLw0 þS @C 1 @CLw /hS þSt xCt Lt > Mw ¼ Mw þ rU0 Sw xCw > > @a 0 @a 0 2 > > > > > 1 @C Lw /hS > 2 þ St xCt 2 @CLt > > Mq ¼ Mq rU0 Sw xCw : @a 0 @a 0 2 ð26Þ 3.3. Lateral derivatives 3.3.1. Y derivatives Yv, Yp and Yr are assumed to be estimated as components originated from the hull neglecting the effects of the main and tail wings. So Yv, Yp and Yr are obtained as in Eq. (27). Yv ¼ Yv/hS ,Yp ¼ Yp/hS ,Yr ¼ Yr/hS
ð27Þ
883
Fig. 5. Sweepback angle.
concerned. Z b=2 1 y ycðyÞdy S=2 0
ð31Þ
Taking the effect of main and tail wings into consideration and adding the hull component to them, Lv can be obtained by Eq. (32). @L½vS @L½vD 1 @CLw þ þ ¼ L/hS rU0 Sw y w Gw Lv ¼ L/hS v v @v 0 @v 0 @a 0 2 @C 1 tan L Þ þSt y t Lt Gt þ rU0 ðSw y w CLW0 tan Lw þ St y t CLt0 t @a 0 2 ð32Þ
3.3.2. L derivatives The derivative Lv is assumed to originate from the roll moment induced by lateral velocity, L[v], which consists of the dihedral angle part L[v]D and the sweepback angle part L[v]S. Dihedral angle G is defined as shown in Fig. 4. Note that the angle is defined opposite direction to that of airplanes. Taking the lateral velocity v and dihedral angle G into consideration, L[v]D can be expressed as in Eq. (28). Z b=2 Z 0 1 @C vG 1 @C vG L½vD ¼ rV 2 cy dy l a þ rV 2 cy dy l a þ @a V @a V b=2 2 0 2 rU0
Z @CL vG @a
b=2
cy dy
ð28Þ
0
In Eq. (28), b stands for the breadth of wings and Cl stands for the local lift coefficient at y. Chord length c is to be zero for y from 0 to the half breadth of the hull. Sweepback angle L is defined as shown in Fig. 5. Taking into consideration the lateral velocity v and sweepback angle L, L[v]S can be expressed as in Eq. (29). Z 0 @C 1 v rV 2 cy dy lðnsÞ a cos L þ L½vS ¼ @ a 2 V b=2 Z b=2 @C 1 v rV 2 cy dy lðnsÞ a cos L þ @ a 2 V 0 Z b=2 cy dy ð29Þ rU0 CL0 tan Lv 0
In Eq. (29), subscript (ns) stands for the lift coefficient in case that the sweepback angle is zero. The following relations (Kato et al., 1982), Eq. (30) are also employed in Eq. (29). @CLðnsÞ a cos L ¼ CL ð30Þ @a Common integral in Eq. (28) for L[v]D, and Eq. (29) for L[v]S can be replaced by y that is defined by Eq. (31). Subscript w or t is added to y depending on the main wings and the tail wings
The derivative Lp is assumed to originate from the roll moment induced by roll angular velocity, L[p], which is expressed as in Eq. (33). Z 0 Z b=2 1 @C py 1 @C py rV 2 cy dy l a þ rV 2 cy dy l a L½p ¼ þ @a @a V V b=2 2 0 2 rU0
@CL p @a
Z
b=2
cy2 dy
ð33Þ
0
Taking the effect of main and tail wings into consideration and adding the ship hull component, Lp can be obtained by Eq. (34). @L½p þ Lp ¼ L/hS p @p 0 ! Z bw=2 Z bt=2 @CLt @CLw 2 2 ¼ L/hS r U c y dy þ c y dy ð34Þ w t 0 p @a @a 0
0
0
0
The derivative Lr is assumed to originate from the roll moment induced by yaw angular velocity, L[r], which is expressed as in Eq. (35). Z b=2 Z b=2 1 L½r ¼ rðVryÞ2 cyCl dy 2rU0 CL r cy2 dy ð35Þ b=2 2 0 Taking the effect of main and tail wings into consideration and adding the ship hull component, Lr can be obtained by Eq. (36). Z bw=2 Z bt=2 @L½r /hS 2 Lr ¼ L/hS þ ¼ L 2 r U ðC c y dy þ C ct y2 dyÞ w 0 Lw0 r v Lt0 @r 0 0 0 ð36Þ
3.3.3. N derivatives Nv is approximated by the viscous force due to the hull and obtained by Eq. (37). Nv ¼ Nvo h 4
ð37Þ
As for Np, the yaw moment due to roll angular velocity is taken into consideration and the following relation, Eq. (38) is employed for differentiation by p.
y v z Fig. 4. Dihedral angle.
Z @N y @N y2 ¼ ¼ @p 0 U0 @a 0 U0
1 _ _ @dL @ðdDÞÞ A @a 0 b=2 b=2
0
Z @CD rU0 CL0 @a 0
b=2 0
cy2 dy
ð38Þ
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Applying Eq. (38) to the main wings and tail wings and adding the hull component, Np can be obtained by Eq. (39). Np ¼ Np/hS þ rU0
(
Z @CDw CLw0 @a 0
bw=2 0
Z @CDt cw y2 dy þ CLt0 @a 0
bt=2
) ct y2 dy
0
ð39Þ The derivative Nr is assumed to originate from the yaw moment induced by yaw angular velocity, N[r], which is expressed as in Eq. (40). Z b=2 Z b=2 1 ^ ¼ ðdL^ adDÞy rc dyðV 8yrÞ2 ðCl aCd Þy ð40Þ N½r ¼ b=2 b=2 2 Upper and lower signs in Eq. (40) correspond to the left and right wing, respectively. Differentiation of Eq. (40) about r is shown in Eq. (41). Z b=2 Z b=2 @N½r 2 ¼ r U cy C dyj 2 r U C cy2 dy ð41Þ 0 0 D0 d 0 @r 0 b=2 0 Applying Eq. (41) to the main wings and tail wings, and adding the hull component to them, Nr can be obtained by Eq. (42). Z bw=2 Z bt=2 cw y2 dy þCDt0 ct y2 dyÞ ð42Þ Nr ¼ Nr/hS þ2rU0 ðCDw0 0
0
In Eq. (45), ya1 and ya2 stand for y-coordinates of edges of the aileron as shown in Fig. 6. Then the Lda can be obtained by Eq. (46). qL½da Lda ¼ qda 0 Z ya2 qCLw qa ¼ rU02 cw y dy ð46Þ qa 0 qda 0 ya1
3.4.3. Rudder The mathematical models for conventional ships’ manoeuvrability including that for the rudder force are reviewed by Kose et al. (1981). Using the estimation procedure for the conventional ships, the normal force on the rudder FN, shown in Fig. 7, is approximated by Eq. (47). FN
3.4.1. Elevator The elevators are equipped with the main wings and the tail wings. The positive elevator angle is defined as that induces positive Z-force. Derivatives concerning the elevator are estimated by differentiating Eq. (15) about de and Eq. (43) is obtained. 8 ^ @a > @X @D > ¼ 1 rU 2 S@CD @a > ¼ Xde ¼ > 0 > > @a 0 @de 0 @de 0 @a 0 @de 0 2 > > > _ > < @Z @L @a 1 2 @CL @a ð43Þ Zde ¼ ¼ ¼ r U S 0 @d @d > d @ a @ a @ 2 e e e > 0 0 0 0 0 > > > > > @M @L^ @a 1 @CL @a > > ¼ xC ¼ rU02 SxC > Mde ¼ : @a 0 @de 0 @de 0 @a 0 @de 0 2 Eq. (43) takes subscript w or t depending on the elevator of the main wings or the tail wings, respectively. 3.4.2. Aileron The aileron is assumed to be fitted in the main wing. The positive aileron angle is defined as to induce positive roll moment. The derivatives Yda and Nda are assumed to be zero as in Eq. (44). @Y @N 0,Nda ¼ 0 ð44Þ Yda ¼ @da 0 @da 0 The derivative Lda is assumed to originate from the roll moment induced by the aileron angle da, L[da], which is estimated by Eq. (45). Z ya2 Z ya2 1 @C @a @C @a L½da ¼ 2 rV 2 lw da cw y dy rU02 Lw da cw y dy @a @da @a @da ya1 2 ya1 ð45Þ
ð47Þ
Sr stands for the rudder projected area. fa stands for the rudder lift slope defined by Eq. (48) in which Ar. represents the rudder aspect ratio fa ¼
3.4. Control derivatives
1 1 rSr fa Ur2 sin dr rSr fa Ur2 dr 2 2
6:13Ar 2:25 þ Ar
ð48Þ
Ur in Eq. (47) stands for the effective rudder inflow velocity. Ur is estimated by Eq. (49) using the procedure for conventional ship (Kose et al., 1981). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !) ( 8KT 1þ 1 ð49Þ Ur ð1wP ÞU 1 þ 0:6 p=J2 Force and moments induced by the rudder angle dr, Y[dr], N[dr] and L[dr], can be obtained as in Eq. (50) in which it is assumed that the rudder angle dr is small and the relation between the body axis and the stability axis, Eq. (10), is used. 8 Y FN > < ½dr N½dr FN zrB sin ai FN xrB cos ai ð50Þ > : L F z cos a F x sin a N rB N rB i i ½dr xrB and zrB in Eq. (50) stand for the coordinates in the body axis indicating the point of action of the rudder normal force FN as shown in Figs. 7 and 8. Then the control derivatives for rudder are obtained by Eq. (51). 8 1 2 > > Ydr ¼ rSr fa Ur0 > > 2 > > < 1 2 Ndr ¼ rSr fa Ur0 ðxrB cos ai þ zrB sin ai Þ ð51Þ > 2 > > > 1 > 2 > : Ldr ¼ rSr fa Ur0 ðxrB sin ai þzrB cos ai Þ 2
-xrB FN δ
xB
r yB
Fig. 7. Rudder angle and rudder normal force.
x ya1 ya2 Fig. 6. Aileron coordinates.
zrB zB
xB
Fig. 8. zrB coordinate of rudder.
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Table 1 Principal dimensions, estimated hydrodynamic derivatives and assumed coefficients. Hull dimensions Length PP, L(m) Breadth, B(m) Displaced volume /L3 Residual buoyancy (N) xg, xb /L zg/L zb/L m/(0.5rL3) Ixx/(0.5rL5) Iyy, Izz/(0.5rL5) Ixz/(0.5rL5) Wing dimensions xCw/L xCt/L Sw/L2 St/L2 cw/L ct/L sw/L st/L
G L Rudder and propeller dimensions Sr/L2 Ar xrB/L zrB/L DP/L zTB/L Horizontal steady advancing condition U0(m/s) n0(rps) Hull angle (1) Main wing angle (1) Tail wing angle (1) Propeller thrust (N)
Added mass coefficients 2.000 0.290 0.009209 135.02 0.018 0.0 0.025 0.014976 0.0000176 0.000456 0.0 0.01875 0.385 0.015 0.0096 0.075 0.060 0.100 0.080 0.0 0.0 0.0010788 1.8219 0.5 0.033 0.057 0.060 1.5 26.7 5.0 10.0 20.0 41.20
3.4.4. Propeller revolution Derivatives Xn, Zn and Mn are estimated by differentiating Eq. (15) about n and they are obtained by Eq. (52). 8 > > Xn ¼ @T cos ai > > @n 0 > > > > < @T Zn ¼ sin ai ð52Þ @n 0 > > > > > @T > > > : Mn ¼ @n zTB 0 In Eq. (52), the derivative of T is given by Eq. (53) using Eqs. (21) and (22). @T @ ¼ rD4P ðn2 KT Þ ¼ rD4P n0 ð2a0 þa1 J0 Þ ð53Þ @n 0 @n 0
4. Calculation of motion 4.1. Estimate of hydrodynamic derivatives Trial calculations are carried out for confirming fundamental properties of the equations of motion, Eq. (8) and estimating procedures for hydrodynamic derivatives. The relation between velocity components in body-fixed coordinate system and earthfixed coordinate system, Eq. (1) and the relation between angular
mx/(0.5rL3) mx, my/(0.5rL3) m13, m15, m35 m24, m26, m46 Jxx/(0.5rL5) Jyy/(0.5rL5) Jzz/(0.5rL5) Propeller coefficients wp a0(KT 0th coefficient) a1(KT 1st coefficient) a2(KT 2nd coefficient) Wing coefficients CDw CDt* dCDw/da dCDt*/da CLw CLt* dCLw/da dCLt*/da CMw CMt da/ddew da/ddet Hull derivatives Yv/hS/(0.5rL2U) Yp/hS/(0.5rL3U) Yr/hS/(0.5rL3U) Lv/hS/(0.5rL3U) Lp/hS/(0.5rL4U) Lr/hS/(0.5rL4U) Nv/hS/(0.5rL3U) Np/hS/(0.5rL4U) Nr/hS/(0.5rL4U) Zw/hS/(0.5rL2U) Zq/hS/(0.5rL3U) Mw/hS/(0.5rL3U) Mq/hS/(0.5rL4U)
0.000238 0.013316 0.0 0.0 0.0000414 0.000564 0.000419 0.184 0.527 0.455 0.0 0.0457 0.075 0.288 0.246 0.549 0.732 2.098 1.678 0.0 0.0 1.0 1.0 0.0232 0.0 0.00484 0.0 0.000975 0.0 0.00770 0.0 0.00444 0.0139 0.00290 0.00462 0.00266
velocity components in body-fixed coordinate system and Euler angles, Eq. (3) are also used. Note that these relations are nonlinear. Left part of Table 1 shows the principal dimensions of the SSS used for feasibility studies at YNU (Hirayama et al., 2005a, 2005b; Koyama et al., 2006; Hirakawa et al., 2007) and its horizontal steady advancing condition measured in their experiments. The vertical distance between the center of gravity and the center of buoyancy is assumed 0.025L where L stands for the model length between perpendiculars. The steady horizontal advancing velocity is 1.5 m/s and the hull angle is 51 in the direction of bow down. The main wings and tail wings angles are also in the nose down direction for making the total downward lift equal to the residual buoyancy. Whole of the main wing works as the elevator and aileron and whole of the tail wing works as the elevator. The dihedral angle and the sweepback angle for both main and tail wings are zero. Coefficients in Eq. (22) representing the propeller characteristics are estimated using the same size propeller data and shown in the right part of Table 1. Right part of Table 1 shows the estimated hydrodynamic derivatives for the SSS based on the procedure presented here. Son measured the hydrodynamic forces and moments acting on the same hull shape model and derived the hydrodynamic derivatives (Son, 1983). The hull components of hydrodynamic derivatives for the SSS are estimated using these basic data. The hull components in lateral hydrodynamic derivatives of the SSS are assumed to be twice as large as the corresponding Son’s data with taking into consideration the increase of mass of the SSS in submerged condition. Attention is paid to that Son’s data include
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10.0
ew (deg)
et (deg)
886
0.0 -10.0 0
20
40
10.0 0.0 -10.0
60
0
20
40
U0+u
1.5
v
1.0
w
U0+u, v, w (m/s)
U0 + u, v, w (m/s)
2.0
0.5 0.0 -0.5 0
60
t (s)
t (s)
20
40
2.0
U0+u
1.5
v
1.0
w
0.5 0.0 -0.5
60
0
t (s)
20
40
60
t (s)
p
4.0
0.0 -2.0
p, q, r (deg/s)
p, q, r (deg/s)
r
-4.0 0
20
40
p
1.0
q 2.0
60
q
0.5
r 0.0 -0.5 -1.0 0
t (s)
20
40
60
1.0
5.0
0.5
(deg)
10.0
0.0
0.0
,
, (deg)
t (s)
-5.0
-0.5
-10.0 0
20
40
-1.0
60
0
t (s)
20
40
Xe (m) 0
20
40
60
80
Xe (m)
100
0
-1 0 1
20
40
60
80
100
-1 Ze (m)
Ze (m)
60
t (s)
0 1
Fig. 9. Tail wing elevator response (elevator amplitude, 51; period, 10 s). Fig. 10. Main wing elevator response (elevator amplitude, 51; period 10 s).
mass and added mass terms represented in left hand side of Eq. (8). These are the same kind of terms as the hydrodynamic derivatives represented in the right hand side of Eq. (9). The experimental data such as total lift and drag forces in the horizontal steady advancing condition shown in the references (Hirayama et al., 2005a, 2005b; Hirakawa et al., 2007) are adopted as basic quantities. The coefficients related to lift and drag of main and tail wings are estimated using the semi-empirical formula (Blevins, 1992). The effective aspect ratios for the main and tail wings are obtained as those for a pair of wings. The downwash effect qe/qa is assumed to be 0.2. In order to satisfy the equilibrium in horizontal steady advancing condition represented by Eq. (23), a correction parameter 0.644 multiplied to the estimated lift and drag related coefficients. Total lift generated by the hull and the main and tail wings in the steady advancing
condition is equal to the residual buoyancy, 135.02N as shown in Table 1. Total resistance or propeller thrust in the steady advancing condition calculated using coefficients listed in Table 1 is 41.20N. Time constants and maximum change rates for the elevators, aileron and rudder are assumed to be 0.1 s and 20.01/s. Those for propeller revolution are assumed to be 0.5 s and 4.0 rps/s.
4.2. Elevator response Figs. 9 and 10 show the response to the tail wing elevator and the main wing elevator, respectively. The elevator starts to work after the horizontal steady advancing condition when time t is equal to zero. The designated elevator angle is sinusoidal of which
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M. Ueno / Ocean Engineering 37 (2010) 879–890
10.0 5.0 0.0 0
20
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60
U0+u, v, w (m/s)
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U0+u v w
0
20
40
60
p, q, r (deg/s)
t (s) 20.0
p
15.0
q
10.0
r
5.0 0.0 -5.0 0
20
40
60
, (deg)
t (s) 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0
4.3. Rudder response
20
40
60
t (s) 12 10 8 6 Xe (m)
becomes periodic one. The pitch angle amplitude is about 51 for the tail wing elevator response and about 0.91 for the main wing elevator response. The reason why the response to the main wing elevator is smaller than that to the tail wing elevator is that the point of action of control force from the center of gravity is farther for the tail wing elevator than for the main wing elevator. The trajectory of SSS in vertical plane is shown in the bottom graph of Figs. 9 and 10. Initial movement direction is downward for the tail wing elevator while that is upward for the main wing elevator. That is because the longitudinal coordinate of tail wing locates aft of the center of gravity while that of main wing locates fore. Note that the vertical scale is different from horizontal one. Although the motion in vertical plane must be under influence of the static pitch restoring moment due to the buoyancy and the gravity, its effect cannot be distinguished in this calculation. Slight upward trajectory is due to the nonlinear calculation in Eq. (1). An example of the self-propelled SSS model test done by the YNU group shows that the periodic motion with fore wings angle amplitude of about 101 at 4.6 s period induces the motion of which the amplitude of pitch angle, speed and heave are about 101, 0.2 m/s and 0.1 m, respectively (Hirayama et al., 2005a). Note that their first SSS model has the fore wings. The calculation shown in Fig. 9 for the response to the tail wing elevator cannot be directly compared to the model test results obtained by the group of YNU, because many different features such as the wing configuration and bow shape assumed are different. However, the orders of response amplitude ratios in their tank test results are comparable to those in Fig. 9.
0
4 2 0 -2 -4 0
2
4
6
8
10
12
6
8
14
16
Ye (m) Xe (m) -4
-2
0
2
4
10
12
-1 Ze (m)
887
0 1 Fig. 11. Rudder response (rudder angle, 51).
amplitude is 51 and period is 10 s for the both cases. The response of SSS is in the vertical plane and surge, heave and pitch motions are induced. The transient response decay soon and the motion
Fig. 11 shows the response to the rudder. The rudder angle order is 51 stepwise to the starboard side after the horizontal steady advancing condition when time t is equal to zero. The right turning with negative lateral velocity v is induced. The negative lateral velocity for right turning means that the bow directs inward. That is same as the conventional surface ship’s right turning with starboard rudder angle. So the basic mechanism of turning by rudder of SSS is considered to be same as that of conventional ships. The mechanism of turning of conventional ships is as follows. Not only the rudder force produces turning moment directly but also the rudder force induces lateral motion of the ship. In this lateral motion, the ship hull plays a role as a huge lifting body in the horizontal plane. This huge lifting body in oblique motion produces large lateral force and turning moment. This hull originated force and moment are dominant for turning. The roll angle f shows short transient starboard down direction just after the rudder starting to work. This is the direct consequence of roll moment by rudder. Soon after that, the roll angle evolves to the opposite direction and reaches to the steady port side down heel angle. This is the effect of right turning which makes the inflow speed into left wing larger than that into right wing. That makes left wing’s downward lift larger than right wing’s one. That results in negative roll moment which overcomes the rudder originated roll moment. The resultant heel angle reaches to the point where the roll moment by rudder, the roll moment by wings and the static roll restoring moment due to the buoyancy and the gravity come to equilibrium. The heel angle makes the vertical downward lift inclined and the horizontal component of that works as centrifugal force during turning. This is a specific feature in the mechanism of turning by rudder of the SSS, which is not seen in conventional ship’s turning. The heel angle of conventional ships during turning depends on the hydrodynamic forces acting on rudder and hull, and the centrifugal force originated from ship mass.
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δa (deg)
888
5.0 0.0 0
20
40
60
t (s)
U0+u, v, w (m/s)
2.0 1.5
U0+u v
1.0
w
0.5 0.0
during steady turning. This induces the increase of inflow angle of attack and then the increase of downward lift. The forward velocity U0 +u shows almost same value as the horizontal steady advancing condition. But it slightly increases from 1.500 to 1.509 m/s. This induces the increase of downward lift, because the increase of forward speed increases the base dynamical pressure. All these changes result in that the vertical component of downward lift barely becomes larger than the residual buoyancy, since the vertical velocity w shows positive or downward direction. The resultant trajectory of this turning motion by rudder is slight downward spiral as shown in the bottom graph in Fig. 11. Note that the vertical scale is different from horizontal one.
-0.5 0
20
40
60
p, q, r (deg/s)
t (s) 10.0 8.0 6.0 4.0 2.0 0.0 -2.0
p q r
0
20
40
60
t (s)
, (deg)
2.0 1.5 1.0 0.5 0.0 -0.5 0
20
40
60
t (s) 20
Xe (m)
15
10
5
0
-5 -5
0
5
10
15
20
25
Ye (m) Xe (m) -10
-5
0
5
10
15
20
Ze (m)
-1 0 1 Fig. 12. . Aileron response (aileron angle, 21).
The heel angle makes the vertical component of downward lift decrease because it inclines. The trim angle y which should be affected by the static pitch moment shows bow up direction
4.4. Aileron response Fig. 12 shows the response to the aileron. The aileron angle order is 21 stepwise to produce positive roll moment or starboarddown moment. The aileron starts to work after the horizontal steady advancing condition when time t is equal to zero. Note that the aileron holds the constant angle. In case of an airplane, holding a constant aileron angle makes an airplane spin continuously. However, the response of SSS to the constant aileron angle does not become spin motion. The reason is that the SSS is under influence of the static roll restoring moment. Airplanes do not feel such static roll restoring moment. The Transient roll motion shows large overshoot before reaching steady condition. The steady heel angle f shows starboard down direction corresponding to the induced roll moment direction. This steady heel angle represents equilibrium position of the roll moment by aileron under the influence of inflow velocity difference between right and left wings and the static roll restoring moment. The heel angle makes the downward lift direction inclined and the horizontal component of the downward lift is produced. Airplanes use this horizontal component of lift as centrifugal force to turn. The positive or starboard down roll angle as shown in Fig. 12 makes the downward lift incline port side. That means that the horizontal component of the inclined downward lift directs port side. If this horizontal component of downward lift worked as the centrifugal force to turn, the SSS turns left. On the contrary, Fig. 12 shows that the SSS turns right. This implies that the mechanism of SSS turning by aileron is thoroughly different from that of airplanes. The mechanism of SSS turning by aileron is considered as follows. The horizontal component of the inclined downward lift directing port side induces negative sway or port side lateral velocity v. As a matter of fact, Fig. 12 shows negative lateral velocity. The lateral velocity results in inflow angle of attack into the hull of SSS to play a role as a huge lifting body. As mentioned in the previous section for the rudder response, this huge lifting body in oblique motion produces large lateral force and turning moment. And this hull originated force overcomes the horizontal component of the inclined downward lift in opposite direction. So the SSS turns right by positive aileron angle. This difference between the SSS and airplanes comes from the difference of the ratio of hull volume to lift force by wings. In other words, hull effect is dominant for the SSS but wing effect is dominant for airplanes in turning motion. Note that the above mentioned turning mechanism by aileron is for the SSS having the configuration shown as Fig. 1. The point is that it has no vertical tail wing. The turning response to aileron may be different one if the SSS has a vertical tail wing of sufficient area. The evolved steady heel angle is starboard side down. The trim angle y which should be affected by the static pitch restoring
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n (rps)
40.0 20.0 0.0 0
20
40
60
U0+u, v, w (m/s)
t (s) 3.0
U0+u
2.0
v w
1.0
0
20
40
60
t (s)
p, q, r (deg/s)
2.0
p q
1.0
r 0.0 -1.0 -2.0 0
20
40
60
t (s)
, (deg)
horizontal steady advancing condition plus a sinusoidal variation. This sinusoidal variation’s amplitude is 5 rps and period is 10 s. The sinusoidal variation of propeller revolution starts to work after the horizontal steady advancing condition when time t is equal to zero. Variation of the propeller revolution affects the propeller thrust and then forward speed. Since the dynamical pressure changes depending on forward speed, the variation of propeller revolution induces the variation of downward lift and drag. That means response to the propeller revolution is qualitatively similar to that to the elevator. Comparing Fig. 13 with Figs. 9 and 10 confirms this fact.
5. Conclusions
0.0 -1.0
4.0
2.0
0.0 -2.0 -4.0 0
20
40
60
t (s) Xe (m) 0
20
40
60
80
100
-2 Ze (m)
889
The linear equations of motion of a SSS in unbounded water are presented. Procedures for estimating hydrodynamic derivatives of the linear equations are also presented. Based on theoretical and semi-empirical formulae together with the experimental data for the SSS reported by YNU group, hydrodynamic derivatives for the SSS are estimated. Using these estimated hydrodynamic derivatives, the responses to the main and tail wings’ elevators, rudder, aileron and propeller revolution are calculated. The elevator of tail wing is more effective than that of main wing in vertical motion. Turning motion by rudder for the SSS is similar to that of conventional ships. However, the difference is that the SSS uses horizontal component of inclined downward lift as a part of centrifugal force. The mechanism of turning by aileron for the SSS is quite different from that of airplanes. The direction of turning by aileron for the SSS is opposite to that of airplanes. The reason is that the SSS is under influence of static roll restoring moment and has large hull that works as a huge lifting body. The response to the propeller revolution change is similar to that of the elevator since it affects thrust, velocity and dynamical pressure. Since the hydrodynamic derivatives are estimated with some assumptions and correction factors to satisfy the equilibrium condition, calculated results cannot be discussed quantitatively in detail. However, the basic characteristics of the SSS motion, especially the important roles of the static roll restoring moment and the large hull, are clarified through considerations of these calculation results. These facts seem to indicate the validity of the equations of motion and the estimation procedure of hydrodynamic derivatives presented here. Experimental research is needed for quantitative validation.
0 Acknowledgements
2 Fig. 13. Propeller revolution response (propeller revolution amplitude, 5 rps; period, 10 s).
moment shows bow down direction. The forward velocity U0 + u shows almost unchanged but slightly decreased value from 1.500 to 1.495 m/s. These heel and trim angles, and forward speed tendencies are all opposite to those in the response to rudder mentioned in the previous section. The vertical velocity w also directs opposite direction and the resultant trajectory by aileron is slight upward spiral as shown in the bottom graph in Fig. 12. Note that the vertical scale is different from horizontal one. 4.5. Propeller revolution response Fig. 13 shows the response to the propeller revolution. The designated propeller revolution is the constant revolution for the
The author would like to express his deep appreciation to the support and discussion of Prof. Hirayama and members of Yokohama National University group. A part of this work was supported by the KAKENHI (2036392). References Blevins, R.D., 1992. Applied Fluid Dynamics Handbook. Krieger Publishing Company, pp. 353–361. Dessureault, J.G., 1976. ‘‘Batfish’’ A depth controllable towed body for collecting oceanographic data. Ocean Engineering 3, 91–111. Hirakawa, Y., Kondo, S., Takayama, T., Hirayama, T., 2007. Trial experiment on the submersible surface ship utilizing downward lift (part 4). Conference Proceedings, vol. 4. The Japan Society of Naval Architects and Ocean Engineers, pp. 143–144. Hirayama, T., Takayama, T., Hirakawa, Y., Koyama, H., Nishimura, K., Kondo, S., 2005a. Trial experiment on the submersible surface ship utilizing downward lift. Conference Proceedings, vol. 5. The Japan Society of Naval Architects of Japan, pp. 141–142.
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Hydrodynamic derivatives and motion response of a submersible surface ship in unbounded water Michio Ueno National Maritime Research Institute, Marine Dynamics Research Group, 6-38-1 Shinkawa, Mitaka, Tokyo 181-0004, Japan
a r t i c l e in fo
abstract
Article history: Received 12 August 2009 Accepted 1 March 2010 Available online 15 March 2010
A submersible surface ship (SSS) is based on a novel concept that the SSS goes on surface like conventional ships in moderate seas but goes underwater in rough seas to the depth sufficient to avoid wave effects. The SSS has a wing system that produces downward lift to go underwater with preserving the residual buoyancy for its safety. The SSS is expected to be able to keep both safety and punctuality even if it encounters unexpected bad weather. The motion of the SSS is studied. The equations of motion are formulated and the procedures for estimating hydrodynamic derivatives are presented. The hydrodynamic derivatives are estimated for a SSS having a configuration, a hull with a pair of main wings and a pair of horizontal tail wings. Using these estimated hydrodynamic derivatives, calculation of the SSS motion is carried out. The calculation results show some specific aspects of the SSS especially for effects of the elevator of main wings and horizontal tail wings, aileron of main wings, rudder and propeller revolution. It is confirmed that the existence of static roll restoring moment and having large hull comparing with wing area play important roles in the motion of the SSS. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Submersible Underwater vehicle Motion equation Hydrodynamic derivative Motion response
1. Introduction The submersible surface ship (SSS) is a ship based on a novel concept, a ship that can avoid rough seas by going underwater, proposed by Hirayama et al. (2005a). The concept of SSS is new in the point that goes on surface like usual ships in calm and moderate seas but goes underwater in rough seas into the depth sufficient to avoid wave effects. In order to submerge the SSS uses wings producing downward lift. Even in submerged condition the SSS keeps residual buoyancy for its safety. The SSS is expected to be able to keep both safety and punctuality even if it encounters unexpected bad weather. A SSS of which configuration is a hull with a pair of main wings and a pair of horizontal tail wings has been studied by a group of the Yokohama National University (YNU). Among researches necessary to realize a SSS are the dynamics, the structure and strength, the powering system for underwater cruising, the navigation and routing. Researches on the navigation and the wing performance of a SSS are in progress and are expected to be reported in the near future. From a viewpoint of the dynamics of a SSS related researches so far are as follows.
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Hirayama et al. (2005a) carried out a trial tank test for a selfpropelled SSS model made by modifying a conventional container ship model and obtained the basic information about the wing area necessary to submerge a conventional ship and its vertical motion control. Hirayama et al. (2005b) also clarified by a tank test the relation between the submerged depth, the downward lift and resistant acting on the SSS hull having modified bow shape. Koyama et al. (2006) simulated a submerging motion of the SSS model in waves and discussed its vertical stability. Hirakawa et al. (2007) studied the vertical motion control of the SSS model in the submerging and emerging motion by a tank test and simulation calculations. In order to clarify feasibility of the SSS, not only such studies concerning vertical motion done by YNU group but also studies concerning the lateral motions are needed. Mori et al. (1988) proposed a high-speed semi-submersible vehicle with wings (HSV). The HSV submerges by using downward lift of wings, which is the same mechanism as the SSS. However, the objective of HSV is reducing the wave-making resistance by submerging into shallow depth, which is different from that of the SSS. Mori et al. confirmed that the HSV is successful in reducing the wave-making resistance but the reduction of frictional resistance still needed. Mori et al. (1991) showed using numerical calculation that the downward lift is effective in small submerged depth and that the nature of vertical motion is unstable but can be stabilized by controlling the main and tail wing angles. Although they studied vertical stability
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during submerged cruising, lateral motion such as sway and yaw motion of the HSV has not been investigated. There are many reports concerning the dynamics of underwater vehicles. Ishidera et al. (1985) reported the equations of sixdegree-of-freedom (6-DOF) motion for a remotely operated vehicle and the automatic depth and heading control system. Ura and Otsubo (1987) proposed a concept of unmanned autonomous gliding submersibles and analyzed its vertical motion. Maeda et al. (1988) reported the hydrodynamic characteristics of two types of unmanned untethered submersibles and estimated its stability derivatives. Towed vehicles are among proposed underwater vehicles (Muddie and Ivers, 1975; Dessureault, 1976; Ohkusu et al., 1987). Ohkusu et al. (1987) measured the hydrodynamic forces acting on a model of a depth-controlled vehicle towed by a long cable and discussed its vertical stability. Most of all these vehicles are statically stable in its steady advancing condition. That means the buoyancy is equal to the gravity in water. On the other hand, a SSS using downward lift to balance with the residual buoyancy are considered to have common features with airplanes. The gravity of an airplane corresponds to the buoyancy of a SSS while the upward lift of an airplane corresponds to the gravity and the downward lift of a SSS. However, one difference is that a SSS generally feels static restoring moments for pitch and roll motion, because points of action for the gravity and the buoyancy are generally different from each other. Airplanes have nothing to do with that kind of static restoring moments. Another difference is that the volumelift ratio of a SSS is larger than that of airplanes. That means the hydrodynamic effect of hull part of a SSS is relatively larger than that of airplanes. As long as the configuration of SSS proposed by YNU group is concerned, one more difference is that it has no vertical tail wing. In this report, the equations of motion of a SSS in unbounded water are presented. The procedure to estimate hydrodynamic derivatives for the linear equations are also presented. This procedure is applied to the SSS configuration of YNU group and calculations of the motion are carried out for investigating the effects of the elevator of horizontal tail wings and main wings, aileron of main wings, rudder and propeller revolution. Results of these calculations show some aspects of the SSS dynamics, especially those due to the existence of static roll restoring moment and having large hull part.
2. Equations of motion 2.1. Configuration and coordinate systems A SSS consisting of a hull, main wings and horizontal tail wings shown in Fig. 1 is supposed. The tail and main wings have
δa
q
r
δew δr
δet
z,w
Matrix E is defined by Eq. (2) in which f, y and c are Euler angles. 2 cos y cos c 6 ½E ¼ 4 sin f sin y cos ccos f sin c cos f sin y cos c þsin f sin c
3
cos y sin c
sin y
sin f sin y sin c þ cos f cos c cos f sin y sin csin f cos c
sin f cos y 7 5 cos f cos y
ð2Þ Angular velocities p, q and r in the body-fixed coordinate system are related to Euler angles by Eq. (3). 8 _ _ > > < p ¼ f c sin y _ sin f cos y q ¼ y_ cos f þ c ð3Þ > > : r ¼ y_ sin f þ c _ cos f cos y Assuming the symmetry about xy-plane, the moment of inertias and products of inertias of the SSS have relations represented by Eq. (4). Ixy ¼ Iyx ¼ 0,
Iyz ¼ Izy ¼ 0
ð4Þ
2.2. Linear equations of motion Let r, g, m and , stand for the density of water, the gravitational acceleration, mass and displaced volume of the SSS, respectively. The center of gravity, (xg, 0, zg) and the center of buoyancy (xb, 0, zb) of the SSS are in the xz-plane in the body-fixed coordinate system. A stability axis is employed as a body fixed O-xyz coordinate system after the example of airplane dynamics (Kato et al., 1982). The horizontal steady advancing condition is chosen as the reference condition, in which the velocity components are defined by Eq. (5). u ¼ U0 ,
v¼w¼p¼q¼r¼0
ð5Þ
In Eq. (5), U0 stands for the steady advancing velocity. The general 6-DOF motion is described as deviation from the reference condition in which v, w, p, q and r stand for the deviations of velocity components. The longitudinal velocity component u is expressed by Eq. (6).
δew
δet
p
x,u
elevators. The main wings have also an aileron. The hull is equipped with a propeller and a rudder as conventional ships. A general body-fixed coordinate system is shown in Fig. 1. The origin locates on the center plane at midship. x, y and z points forward, right and downward, respectively. u, v and w are translational velocities to x, y and z directions and p, q and r are angular velocities around x, y and z axes. In Fig. 1, dew, det, da and dr show the positive direction of the main wing elevator angle, tail wing elevator angle, aileron angle and rudder angle. Note that the main wings work both as the elevator and the aileron. The elevator has symmetric angles and the aileron has anti-symmetric angles for the port side and the starboard side wings. In the earth-fixed coordinate system Oe-xeyeze, xeye-plane is horizontal and ze-axis points vertically downward. The earthfixed coordinate (xe, ye, ze) and their time derivatives are related to those in the body-fixed one (x, y, z) by Eq. (1). 3 2 dxe x e 6 2 3 dt 7 7 6 x u 6 dxe 7 7 6 6y v 7 ð1Þ 7 4 5 ¼ ½E6 ye 6 dt 7 7 6 z w 4 dxe 5 ze dt
δa
y,v
Fig. 1. General body-fixed coordinate system.
u ¼ U0 þ u~
ð6Þ
In Eq. (6), u~ stands for the deviation of longitudinal velocity component from U0. For convenience the following expression, Eq. (7), is employed hereafter. u~ u
ð7Þ
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The nonlinear equations of 6-DOF motion for an underwater vehicle which can be applicable to a general underwater moving body are derived by Ueno and Sawada (1993). The same can be reduced to the linear equations of motion by applying the stability axis, the reference condition defined by Eq. (5), the expressions of Eqs. (6) and (7), and neglecting the second order terms and over u, v, w, p, q and r. The resultant linear equations of motion are shown in Eq. (8).
881
2.3. Relations between the stability axis and the body axis The body axis OB-xByBzB is defined as xB-axis is parallel to the base line of the SSS while yB-axis coincides with y-axis. Fig. 2 shows the body axis OB-xByBzB and the stability axis O-xyz. The angular difference between the stability axis and the body axis is ai around y or yB-axis. The body axis is useful since it does not depend on the posture in the steady advancing condition. The
8 _ þ ðmzg þm15 Þq_ þ m13 U0 q ¼ ðmrrÞg sin y þX ðm þ mx Þu_ þm13 w > > > > > _ _ þðmxg þ m26 Þr_ þðm þ mx ÞU0 r ¼ ðmrrÞg sin f cos y þY Þ v þ ðmz ðm þ m > y g þ m24 Þp > > > > _ _ _ > Þ w þðmx þm35 Þqðm þmx ÞU0 q ¼ ðmrrÞg cos f cos y þ Z m u þ ðm þ m z g 13 > > > > > ðmzg þm24 Þv_ þ ðIxx þJxx Þp_ þ ðIxz þm46 Þr_ þ fm13 vðmzg þ m15 ÞrgU0 > < ¼ ðmzg rrzb Þg sin f cos y þ L > > > ðmzg þm15 Þu_ þ ðmxg þ m35 Þw _ þ ðIyy þJyy Þq_ þ f2m13 u þ ðmx mz Þwðmxg þ m35 ÞqgU0 m13 U02 > > > > > ¼ ðmzg rrzb Þg sin yðmxg rrxb Þg cos f cos y þ M > > > > > _ þ ðIxz þm46 Þp_ þ ðIzz þJzz Þr_ þ fðmy mx Þv þ ðm24 m15 Þp þðmxg þ m26 ÞrÞgU0 ðmx > g þm26 Þv > > > : ¼ ðmxg rrxb Þg sin f cos y þ N In Eq. (8), mij stands for added masses and added moment of inertias where 1, 2 and 3 for i and j represent x, y and z directions, respectively. m11, m22, m33, m44, m55 and m66 are replaced by mx, my, mz Jxx, Jyy and Jzz, respectively. Some components of mij are zero due to the symmetry of the SSS configuration. Dot above variables means time derivatives of these variables. Right-hand side of Eq. (8) contains hydrostatic forces and moments due to the gravity and the buoyancy, and hydrodynamic external forces and moments X, Y, Z, L, M and N. The first terms in the right hand side of Eq. (8), originated from the buoyancy and the gravity represent the hydrostatic forces and moments. These hydrostatic terms may play an important role for the dynamics of underwater vehicles. One example can be found for the descending motion of a launcher of a deep-sea robot (Ueno et al., 2008). Taking into consideration the symmetry of the SSS configuration, hydrodynamic external forces and moments are assumed to be expressed in linear forms as in Eq. (9). 8 X ¼ X0 þ Xu u þXw w þ Xq q þXdew dew þ Xdet det þ Xn Dn > > > > > Y ¼ Yv vþ Yp p þ Yr r þYda da þ Ydr dr > > > > < Z ¼ Z0 þ Zu u þ Zw w þ Zq q þ Zdew dew þZdet det þ Zn Dn ð9Þ L ¼ Lv v þLp p þ Lr r þ Lda da þLdr dr > > > > > M ¼ M0 þ Mu u þMw w þ Mq q þ Mdew dew þ Mdet det þ Mn Dn > > > > : N ¼ Nv v þ Np p þ Nr r þ N da þ N dr da
ð8Þ
forces and moments defined in these two coordinate systems are related to each other by Eq. (10). 2 3 2 3 XB LB X L 7 6 7 1 6 Y ð10Þ 4 Y M 5 ¼ ½Ea 4 B MB 5 Z B NB Z N In Eq. (10) and hereafter, subscript B stands for those in the body axis and no subscript for those in the stability axis. Matrix Ea 1 is defined by Eq. (11). 2 3 cos ai 0 sin ai 60 7 1 1 0 ½Ea ¼ 4 ð11Þ 5 sin ai 0 cos ai Moment of inertias, product of inertias, added masses and added moment of inertias defined in the body axis are converted to those in the stability axis using Eqs. (12) and (13). 8 > Ixx ¼ IxxB cos2 ai 2IxzB sin ai cos ai þ IzzB sin2 ai > > > > < Iyy ¼ IyyB ð12Þ 2 2 > > > Izz ¼ IxxB sin ai þ 2IxzB sin ai cos ai þ IzzB cos ai > > : Ixz ¼ ðIzzB IxxB Þsin a cos a þ IxzB ðcos2 a sin2 a Þ i i i i ½mij ¼ ½Ea ½mijB ½Ea 1
ð13Þ
dr
In Eq. (9), Xu, Xw, Xq, Yv, Yp, Yr, Zu, Zw, Zq, Lv, Lp, Lr, Mu, Mw, Mq, Nv, Np and Nr are hydrodynamic derivatives concerning hull and wing systems. dew, det, da, dr and Dn are main wing elevator angle, tail wing elevator angle, aileron angle, rudder angle and deviation of propeller revolution from the reference condition. Xdew, Xdet, Xn, Yda, Ydr, Zdew, Zdet, Zn, Lda, Ldr, Mdew, Mdet, Mn, Nda and Ndr are also hydrodynamic derivatives concerning control parameters. X0, Z0 and M0 stand for the hydrodynamic longitudinal and vertical forces and pitch moment in the horizontal steady advancing condition.
3. Hydrodynamic derivatives A procedure to estimate the hydrodynamic derivatives for the SSS is presented in this chapter. Comparable procedures for airplanes (USAF, 1968; Kato et al., 1982) are consulted together
xe V U0+u x α xB αi
-w
(xct,zct)
^
O,OB
U0 i
^
x xB
^
zTB Dt
Fig. 2. Stability axis O-xyz and body axis OB-xByBzB.
^
T ze
zB z
Mt
^
Dw
^
Mw (xcw,zcw)
Lw
Lt zB z
Fig. 3. External forces acting on a SSS in the condition of which the angle of attack is a.
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with considerations about the features of SSS such as having large hull and no vertical tail wing. The interaction between hull and wings are neglected by assuming small motion deviations from the reference condition. Although the interaction effects between hull, propeller and rudder can be taken into account as those for conventional ships, it is simplified here to clarify the basic properties of the SSS motion. 3.1. Components in external forces Fig. 3 shows a condition of the SSS in which the velocity of SSS is V with the angle of attack a. The velocity V is defined by Eq. (14). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ ðU0 þ uÞ2 þ v2 þ w2 ð14Þ External forces X, Y, Z and moments L, M, N in Eq. (8) are represented by Eq. (9) in which linear hydrodynamic derivatives are used. In order to estimate these linear hydrodynamic derivatives, another perspective is introduced and alternative expression for the external forces in Eq. (8) is obtained as shown in Eq. (15). 8 > X ¼ T cos ai þ X o h 4 þ X o w 4 þX o t 4 > > > > > Y ¼ Y o h 4 þ Y o w 4 þY o t 4 > > > < Z ¼ T sin a þ Z o h 4 þ Z o w 4 þ Z o t 4 i > L ¼ L o h 4 þL o w 4 þ L o t 4 > > > > > M ¼ TzTB þ M o h 4 þM o w 4 þ M o t 4 > > > : N ¼ N o h 4 þN o w 4 þN o t 4
ð15Þ
8 oh4 X ¼ X0o h 4 þ DX o h 4 ðuÞ ¼ X0o h 4 þ Xuo h 4 u > > > > > > Y o h 4 ¼ DY o h 4 ðv,p,rÞ ¼ Yvo h 4 v þYpo h 4 p þ Yro h 4 r > > > > > < Z o h 4 ¼ Z0o h 4 þ DZ o h 4 ðu,w,qÞ ¼ Z0o h 4 þZuo h 4 u þZwo h 4 w þZqo h 4 q
For the tail wing, the downwash effect by the main wing is taken into consideration and the next expressions, Eq. (20) are employed for the tail wings. 8 > ¼ C 1 @e > C > Lt Lt < @a @e > > > : CDt ¼ CDt 1 @a
ð20Þ
Thrust T can be estimated by Eq. (21). T ¼ rD4P n2 KTðJÞ
ð21Þ
In above equation, KT, n and DP represent the thrust coefficient, propeller revolution and propeller diameter. The thrust coefficient KT is assumed to be represented by a second polynomial of the advance coefficient J as in Eq. (22). J¼
ð1wP ÞðU0 þuÞ nDP
ð22Þ
In Eq. (22), wP stands for the wake coefficient and a0, a1 and a2 are constants. For the horizontal steady advancing condition Eq. (8) is reduced to Eq. (23). 8 /hS ^ ^ > > > T0 cos ai þ X0 ðD w0 þ D t0 Þ ¼ 0 > < _ _ /hS T0 sin ai þ Z0 þ ð L þ L Þ þðmrrÞg ¼ 0 w0 t0 > > > > ^ w0 þ M ^ t0 ÞðL^ w0 xCw þ L^ t0 xCt Þðmxg rrxb Þg þ m13 U 2 ¼ 0 : T0 zTB þM /hS þ ðM 0
ð23Þ
ð16Þ
ð17Þ
If hull related derivatives in Eq. (16) are defined in the body axis they can be converted to those in the stability axis by using ^ D ^ and M ^ in Eq. (17) represent lift, drag and pitch Eq. (10). L, moment acting on the main or tail wings. The distinction can be made by subscripts w for the main wings and t for the tail wings. ^ t are defined around the hydrodynamic ^ w and M Note that M centers. These coordinates of hydrodynamic centers in the vertical plane are represented by (xCw, zCw) and (xCt, zCt) for the main and tail wings, respectively. Hereafter, zCw and zCt are assumed to be zero. aw and at stand for the angle of attack to the main and tail wings. These are represented by Eq. (18). wþ xCw,t q U0
ð19Þ
0
> L o h 4 ¼ DL o h 4 ðv,p,rÞ ¼ Lvo h 4 v þLpo h 4 p þLro h 4 r > > > > > > M o h 4 ¼ M0o h 4 þ DM o h 4 ðu,w,qÞ ¼ M0o h 4 þ Muo h 4 u þMwo h 4 w þMqo h 4 q > > > > oh4 : N ¼ DN o h 4 ðv,p,rÞ ¼ Nvo h 4 v þNpo h 4 p þNro h 4 r
aw,t
8 1 > > L^ ¼ rSV 2 CL > > 2 > > < 1 ^ D ¼ rSV 2 CD > 2 > > > 1 > > ^ : M ¼ rScV 2 CM 2
KTðJÞ ¼ a0 þ a1 J þ a2 J2 ,
In Eq. (15), superscripts /hS, /wS and /tS stand for terms originated from the hull, main wings and tail wings, respectively. T stands for the thrust which is assumed to direct parallel to the xB-axis at z is equal to zTB. The hull, main wings and tail wings terms are assumed to be represented as in Eqs. (16) and (17).
8 ^ w,t cos aw,t X /w,tS ¼ L^ w,t sin aw,t D > > > > /w,tS /w,tS > > Y ¼ DY ðv,p,rÞ > > > < Z /w,tS ¼ L^ cos a þ D ^ w,t sin aw,t w,t w,t /w,tS /w,tS > ¼ DL ðv,p,rÞ >L > > > /w,tS ^ w,t L^ w,t xCw,t D ^ w,t zCw,t > ¼M >M > > : N/w,tS ¼ DN/w,tS ðv,p,rÞ
Coefficients for the lift, drag and moment CL, CD and CM are defined by Eq. (19).
ð18Þ
In Eq. (23), T0 stands for the thrust in the reference condition. 3.2. Vertical derivatives Based on the preliminary considerations in the previous section, hydrodynamic derivatives for vertical motion are estimated by differentiating Eq. (15). Resultant formulae are in Eqs. (24)–(26) in which vertical bar followed by 0 represents the differentiation at the reference condition. 8 @KT > Þ > j cos ai þXuo h 4 rU0 ðSw CDw0 þSt CDt0 > Xu ¼ rD3P n0 ð1wP Þ > @J 0 > > > > < 1 @CDw @CDt þ St CLt0 Xw ¼ rU0 Sw CLw0 @a 0 @a 0 2 > > > > > 1 @CDw > @CDt > þ x ¼ r U x S C S C X > q w t 0 Cw Lw0 Ct Lt0 : @a @a 2 0
0
ð24Þ 8 @KT Þ > > Zu ¼ rD3P n0 ð1wP Þ sin ai þ Zu/hS þ rU0 ðSw CLw0 þ St CLt0 > > @J 0 > > > ( > ) > < @CLt 1 @CLw þC þC þS Zw ¼ Zwo h 4 rU0 Sw t Dw0 Dt0 @a 0 @a 0 2 > > > > ) > > > @CLt 1 @CLw > oh4 > þ rU0 xCw Sw þCDw0 þxCt St þCDt0 : Zq ¼ Zq 2 @a 0 @a 0
ð25Þ
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8 @KT > 3 > M ¼ r D n ð1w Þ zTB þ Mu/hS þ rU0 ðSw cw CMw0 þSt ct CMt0 Þ > u P 0 > > @J 0 > > > > > t xCt CLt0 Þ > < rU0 ðSw xCw CLw0 þS @C 1 @CLw /hS þSt xCt Lt > Mw ¼ Mw þ rU0 Sw xCw > > @a 0 @a 0 2 > > > > > 1 @C Lw /hS > 2 þ St xCt 2 @CLt > > Mq ¼ Mq rU0 Sw xCw : @a 0 @a 0 2 ð26Þ 3.3. Lateral derivatives 3.3.1. Y derivatives Yv, Yp and Yr are assumed to be estimated as components originated from the hull neglecting the effects of the main and tail wings. So Yv, Yp and Yr are obtained as in Eq. (27). Yv ¼ Yv/hS ,Yp ¼ Yp/hS ,Yr ¼ Yr/hS
ð27Þ
883
Fig. 5. Sweepback angle.
concerned. Z b=2 1 y ycðyÞdy S=2 0
ð31Þ
Taking the effect of main and tail wings into consideration and adding the hull component to them, Lv can be obtained by Eq. (32). @L½vS @L½vD 1 @CLw þ þ ¼ L/hS rU0 Sw y w Gw Lv ¼ L/hS v v @v 0 @v 0 @a 0 2 @C 1 tan L Þ þSt y t Lt Gt þ rU0 ðSw y w CLW0 tan Lw þ St y t CLt0 t @a 0 2 ð32Þ
3.3.2. L derivatives The derivative Lv is assumed to originate from the roll moment induced by lateral velocity, L[v], which consists of the dihedral angle part L[v]D and the sweepback angle part L[v]S. Dihedral angle G is defined as shown in Fig. 4. Note that the angle is defined opposite direction to that of airplanes. Taking the lateral velocity v and dihedral angle G into consideration, L[v]D can be expressed as in Eq. (28). Z b=2 Z 0 1 @C vG 1 @C vG L½vD ¼ rV 2 cy dy l a þ rV 2 cy dy l a þ @a V @a V b=2 2 0 2 rU0
Z @CL vG @a
b=2
cy dy
ð28Þ
0
In Eq. (28), b stands for the breadth of wings and Cl stands for the local lift coefficient at y. Chord length c is to be zero for y from 0 to the half breadth of the hull. Sweepback angle L is defined as shown in Fig. 5. Taking into consideration the lateral velocity v and sweepback angle L, L[v]S can be expressed as in Eq. (29). Z 0 @C 1 v rV 2 cy dy lðnsÞ a cos L þ L½vS ¼ @ a 2 V b=2 Z b=2 @C 1 v rV 2 cy dy lðnsÞ a cos L þ @ a 2 V 0 Z b=2 cy dy ð29Þ rU0 CL0 tan Lv 0
In Eq. (29), subscript (ns) stands for the lift coefficient in case that the sweepback angle is zero. The following relations (Kato et al., 1982), Eq. (30) are also employed in Eq. (29). @CLðnsÞ a cos L ¼ CL ð30Þ @a Common integral in Eq. (28) for L[v]D, and Eq. (29) for L[v]S can be replaced by y that is defined by Eq. (31). Subscript w or t is added to y depending on the main wings and the tail wings
The derivative Lp is assumed to originate from the roll moment induced by roll angular velocity, L[p], which is expressed as in Eq. (33). Z 0 Z b=2 1 @C py 1 @C py rV 2 cy dy l a þ rV 2 cy dy l a L½p ¼ þ @a @a V V b=2 2 0 2 rU0
@CL p @a
Z
b=2
cy2 dy
ð33Þ
0
Taking the effect of main and tail wings into consideration and adding the ship hull component, Lp can be obtained by Eq. (34). @L½p þ Lp ¼ L/hS p @p 0 ! Z bw=2 Z bt=2 @CLt @CLw 2 2 ¼ L/hS r U c y dy þ c y dy ð34Þ w t 0 p @a @a 0
0
0
0
The derivative Lr is assumed to originate from the roll moment induced by yaw angular velocity, L[r], which is expressed as in Eq. (35). Z b=2 Z b=2 1 L½r ¼ rðVryÞ2 cyCl dy 2rU0 CL r cy2 dy ð35Þ b=2 2 0 Taking the effect of main and tail wings into consideration and adding the ship hull component, Lr can be obtained by Eq. (36). Z bw=2 Z bt=2 @L½r /hS 2 Lr ¼ L/hS þ ¼ L 2 r U ðC c y dy þ C ct y2 dyÞ w 0 Lw0 r v Lt0 @r 0 0 0 ð36Þ
3.3.3. N derivatives Nv is approximated by the viscous force due to the hull and obtained by Eq. (37). Nv ¼ Nvo h 4
ð37Þ
As for Np, the yaw moment due to roll angular velocity is taken into consideration and the following relation, Eq. (38) is employed for differentiation by p.
y v z Fig. 4. Dihedral angle.
Z @N y @N y2 ¼ ¼ @p 0 U0 @a 0 U0
1 _ _ @dL @ðdDÞÞ A @a 0 b=2 b=2
0
Z @CD rU0 CL0 @a 0
b=2 0
cy2 dy
ð38Þ
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Applying Eq. (38) to the main wings and tail wings and adding the hull component, Np can be obtained by Eq. (39). Np ¼ Np/hS þ rU0
(
Z @CDw CLw0 @a 0
bw=2 0
Z @CDt cw y2 dy þ CLt0 @a 0
bt=2
) ct y2 dy
0
ð39Þ The derivative Nr is assumed to originate from the yaw moment induced by yaw angular velocity, N[r], which is expressed as in Eq. (40). Z b=2 Z b=2 1 ^ ¼ ðdL^ adDÞy rc dyðV 8yrÞ2 ðCl aCd Þy ð40Þ N½r ¼ b=2 b=2 2 Upper and lower signs in Eq. (40) correspond to the left and right wing, respectively. Differentiation of Eq. (40) about r is shown in Eq. (41). Z b=2 Z b=2 @N½r 2 ¼ r U cy C dyj 2 r U C cy2 dy ð41Þ 0 0 D0 d 0 @r 0 b=2 0 Applying Eq. (41) to the main wings and tail wings, and adding the hull component to them, Nr can be obtained by Eq. (42). Z bw=2 Z bt=2 cw y2 dy þCDt0 ct y2 dyÞ ð42Þ Nr ¼ Nr/hS þ2rU0 ðCDw0 0
0
In Eq. (45), ya1 and ya2 stand for y-coordinates of edges of the aileron as shown in Fig. 6. Then the Lda can be obtained by Eq. (46). qL½da Lda ¼ qda 0 Z ya2 qCLw qa ¼ rU02 cw y dy ð46Þ qa 0 qda 0 ya1
3.4.3. Rudder The mathematical models for conventional ships’ manoeuvrability including that for the rudder force are reviewed by Kose et al. (1981). Using the estimation procedure for the conventional ships, the normal force on the rudder FN, shown in Fig. 7, is approximated by Eq. (47). FN
3.4.1. Elevator The elevators are equipped with the main wings and the tail wings. The positive elevator angle is defined as that induces positive Z-force. Derivatives concerning the elevator are estimated by differentiating Eq. (15) about de and Eq. (43) is obtained. 8 ^ @a > @X @D > ¼ 1 rU 2 S@CD @a > ¼ Xde ¼ > 0 > > @a 0 @de 0 @de 0 @a 0 @de 0 2 > > > _ > < @Z @L @a 1 2 @CL @a ð43Þ Zde ¼ ¼ ¼ r U S 0 @d @d > d @ a @ a @ 2 e e e > 0 0 0 0 0 > > > > > @M @L^ @a 1 @CL @a > > ¼ xC ¼ rU02 SxC > Mde ¼ : @a 0 @de 0 @de 0 @a 0 @de 0 2 Eq. (43) takes subscript w or t depending on the elevator of the main wings or the tail wings, respectively. 3.4.2. Aileron The aileron is assumed to be fitted in the main wing. The positive aileron angle is defined as to induce positive roll moment. The derivatives Yda and Nda are assumed to be zero as in Eq. (44). @Y @N 0,Nda ¼ 0 ð44Þ Yda ¼ @da 0 @da 0 The derivative Lda is assumed to originate from the roll moment induced by the aileron angle da, L[da], which is estimated by Eq. (45). Z ya2 Z ya2 1 @C @a @C @a L½da ¼ 2 rV 2 lw da cw y dy rU02 Lw da cw y dy @a @da @a @da ya1 2 ya1 ð45Þ
ð47Þ
Sr stands for the rudder projected area. fa stands for the rudder lift slope defined by Eq. (48) in which Ar. represents the rudder aspect ratio fa ¼
3.4. Control derivatives
1 1 rSr fa Ur2 sin dr rSr fa Ur2 dr 2 2
6:13Ar 2:25 þ Ar
ð48Þ
Ur in Eq. (47) stands for the effective rudder inflow velocity. Ur is estimated by Eq. (49) using the procedure for conventional ship (Kose et al., 1981). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !) ( 8KT 1þ 1 ð49Þ Ur ð1wP ÞU 1 þ 0:6 p=J2 Force and moments induced by the rudder angle dr, Y[dr], N[dr] and L[dr], can be obtained as in Eq. (50) in which it is assumed that the rudder angle dr is small and the relation between the body axis and the stability axis, Eq. (10), is used. 8 Y FN > < ½dr N½dr FN zrB sin ai FN xrB cos ai ð50Þ > : L F z cos a F x sin a N rB N rB i i ½dr xrB and zrB in Eq. (50) stand for the coordinates in the body axis indicating the point of action of the rudder normal force FN as shown in Figs. 7 and 8. Then the control derivatives for rudder are obtained by Eq. (51). 8 1 2 > > Ydr ¼ rSr fa Ur0 > > 2 > > < 1 2 Ndr ¼ rSr fa Ur0 ðxrB cos ai þ zrB sin ai Þ ð51Þ > 2 > > > 1 > 2 > : Ldr ¼ rSr fa Ur0 ðxrB sin ai þzrB cos ai Þ 2
-xrB FN δ
xB
r yB
Fig. 7. Rudder angle and rudder normal force.
x ya1 ya2 Fig. 6. Aileron coordinates.
zrB zB
xB
Fig. 8. zrB coordinate of rudder.
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Table 1 Principal dimensions, estimated hydrodynamic derivatives and assumed coefficients. Hull dimensions Length PP, L(m) Breadth, B(m) Displaced volume /L3 Residual buoyancy (N) xg, xb /L zg/L zb/L m/(0.5rL3) Ixx/(0.5rL5) Iyy, Izz/(0.5rL5) Ixz/(0.5rL5) Wing dimensions xCw/L xCt/L Sw/L2 St/L2 cw/L ct/L sw/L st/L
G L Rudder and propeller dimensions Sr/L2 Ar xrB/L zrB/L DP/L zTB/L Horizontal steady advancing condition U0(m/s) n0(rps) Hull angle (1) Main wing angle (1) Tail wing angle (1) Propeller thrust (N)
Added mass coefficients 2.000 0.290 0.009209 135.02 0.018 0.0 0.025 0.014976 0.0000176 0.000456 0.0 0.01875 0.385 0.015 0.0096 0.075 0.060 0.100 0.080 0.0 0.0 0.0010788 1.8219 0.5 0.033 0.057 0.060 1.5 26.7 5.0 10.0 20.0 41.20
3.4.4. Propeller revolution Derivatives Xn, Zn and Mn are estimated by differentiating Eq. (15) about n and they are obtained by Eq. (52). 8 > > Xn ¼ @T cos ai > > @n 0 > > > > < @T Zn ¼ sin ai ð52Þ @n 0 > > > > > @T > > > : Mn ¼ @n zTB 0 In Eq. (52), the derivative of T is given by Eq. (53) using Eqs. (21) and (22). @T @ ¼ rD4P ðn2 KT Þ ¼ rD4P n0 ð2a0 þa1 J0 Þ ð53Þ @n 0 @n 0
4. Calculation of motion 4.1. Estimate of hydrodynamic derivatives Trial calculations are carried out for confirming fundamental properties of the equations of motion, Eq. (8) and estimating procedures for hydrodynamic derivatives. The relation between velocity components in body-fixed coordinate system and earthfixed coordinate system, Eq. (1) and the relation between angular
mx/(0.5rL3) mx, my/(0.5rL3) m13, m15, m35 m24, m26, m46 Jxx/(0.5rL5) Jyy/(0.5rL5) Jzz/(0.5rL5) Propeller coefficients wp a0(KT 0th coefficient) a1(KT 1st coefficient) a2(KT 2nd coefficient) Wing coefficients CDw CDt* dCDw/da dCDt*/da CLw CLt* dCLw/da dCLt*/da CMw CMt da/ddew da/ddet Hull derivatives Yv/hS/(0.5rL2U) Yp/hS/(0.5rL3U) Yr/hS/(0.5rL3U) Lv/hS/(0.5rL3U) Lp/hS/(0.5rL4U) Lr/hS/(0.5rL4U) Nv/hS/(0.5rL3U) Np/hS/(0.5rL4U) Nr/hS/(0.5rL4U) Zw/hS/(0.5rL2U) Zq/hS/(0.5rL3U) Mw/hS/(0.5rL3U) Mq/hS/(0.5rL4U)
0.000238 0.013316 0.0 0.0 0.0000414 0.000564 0.000419 0.184 0.527 0.455 0.0 0.0457 0.075 0.288 0.246 0.549 0.732 2.098 1.678 0.0 0.0 1.0 1.0 0.0232 0.0 0.00484 0.0 0.000975 0.0 0.00770 0.0 0.00444 0.0139 0.00290 0.00462 0.00266
velocity components in body-fixed coordinate system and Euler angles, Eq. (3) are also used. Note that these relations are nonlinear. Left part of Table 1 shows the principal dimensions of the SSS used for feasibility studies at YNU (Hirayama et al., 2005a, 2005b; Koyama et al., 2006; Hirakawa et al., 2007) and its horizontal steady advancing condition measured in their experiments. The vertical distance between the center of gravity and the center of buoyancy is assumed 0.025L where L stands for the model length between perpendiculars. The steady horizontal advancing velocity is 1.5 m/s and the hull angle is 51 in the direction of bow down. The main wings and tail wings angles are also in the nose down direction for making the total downward lift equal to the residual buoyancy. Whole of the main wing works as the elevator and aileron and whole of the tail wing works as the elevator. The dihedral angle and the sweepback angle for both main and tail wings are zero. Coefficients in Eq. (22) representing the propeller characteristics are estimated using the same size propeller data and shown in the right part of Table 1. Right part of Table 1 shows the estimated hydrodynamic derivatives for the SSS based on the procedure presented here. Son measured the hydrodynamic forces and moments acting on the same hull shape model and derived the hydrodynamic derivatives (Son, 1983). The hull components of hydrodynamic derivatives for the SSS are estimated using these basic data. The hull components in lateral hydrodynamic derivatives of the SSS are assumed to be twice as large as the corresponding Son’s data with taking into consideration the increase of mass of the SSS in submerged condition. Attention is paid to that Son’s data include
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10.0
ew (deg)
et (deg)
886
0.0 -10.0 0
20
40
10.0 0.0 -10.0
60
0
20
40
U0+u
1.5
v
1.0
w
U0+u, v, w (m/s)
U0 + u, v, w (m/s)
2.0
0.5 0.0 -0.5 0
60
t (s)
t (s)
20
40
2.0
U0+u
1.5
v
1.0
w
0.5 0.0 -0.5
60
0
t (s)
20
40
60
t (s)
p
4.0
0.0 -2.0
p, q, r (deg/s)
p, q, r (deg/s)
r
-4.0 0
20
40
p
1.0
q 2.0
60
q
0.5
r 0.0 -0.5 -1.0 0
t (s)
20
40
60
1.0
5.0
0.5
(deg)
10.0
0.0
0.0
,
, (deg)
t (s)
-5.0
-0.5
-10.0 0
20
40
-1.0
60
0
t (s)
20
40
Xe (m) 0
20
40
60
80
Xe (m)
100
0
-1 0 1
20
40
60
80
100
-1 Ze (m)
Ze (m)
60
t (s)
0 1
Fig. 9. Tail wing elevator response (elevator amplitude, 51; period, 10 s). Fig. 10. Main wing elevator response (elevator amplitude, 51; period 10 s).
mass and added mass terms represented in left hand side of Eq. (8). These are the same kind of terms as the hydrodynamic derivatives represented in the right hand side of Eq. (9). The experimental data such as total lift and drag forces in the horizontal steady advancing condition shown in the references (Hirayama et al., 2005a, 2005b; Hirakawa et al., 2007) are adopted as basic quantities. The coefficients related to lift and drag of main and tail wings are estimated using the semi-empirical formula (Blevins, 1992). The effective aspect ratios for the main and tail wings are obtained as those for a pair of wings. The downwash effect qe/qa is assumed to be 0.2. In order to satisfy the equilibrium in horizontal steady advancing condition represented by Eq. (23), a correction parameter 0.644 multiplied to the estimated lift and drag related coefficients. Total lift generated by the hull and the main and tail wings in the steady advancing
condition is equal to the residual buoyancy, 135.02N as shown in Table 1. Total resistance or propeller thrust in the steady advancing condition calculated using coefficients listed in Table 1 is 41.20N. Time constants and maximum change rates for the elevators, aileron and rudder are assumed to be 0.1 s and 20.01/s. Those for propeller revolution are assumed to be 0.5 s and 4.0 rps/s.
4.2. Elevator response Figs. 9 and 10 show the response to the tail wing elevator and the main wing elevator, respectively. The elevator starts to work after the horizontal steady advancing condition when time t is equal to zero. The designated elevator angle is sinusoidal of which
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r (deg)
M. Ueno / Ocean Engineering 37 (2010) 879–890
10.0 5.0 0.0 0
20
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60
U0+u, v, w (m/s)
t (s) 2.0 1.5 1.0 0.5 0.0 -0.5
U0+u v w
0
20
40
60
p, q, r (deg/s)
t (s) 20.0
p
15.0
q
10.0
r
5.0 0.0 -5.0 0
20
40
60
, (deg)
t (s) 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0
4.3. Rudder response
20
40
60
t (s) 12 10 8 6 Xe (m)
becomes periodic one. The pitch angle amplitude is about 51 for the tail wing elevator response and about 0.91 for the main wing elevator response. The reason why the response to the main wing elevator is smaller than that to the tail wing elevator is that the point of action of control force from the center of gravity is farther for the tail wing elevator than for the main wing elevator. The trajectory of SSS in vertical plane is shown in the bottom graph of Figs. 9 and 10. Initial movement direction is downward for the tail wing elevator while that is upward for the main wing elevator. That is because the longitudinal coordinate of tail wing locates aft of the center of gravity while that of main wing locates fore. Note that the vertical scale is different from horizontal one. Although the motion in vertical plane must be under influence of the static pitch restoring moment due to the buoyancy and the gravity, its effect cannot be distinguished in this calculation. Slight upward trajectory is due to the nonlinear calculation in Eq. (1). An example of the self-propelled SSS model test done by the YNU group shows that the periodic motion with fore wings angle amplitude of about 101 at 4.6 s period induces the motion of which the amplitude of pitch angle, speed and heave are about 101, 0.2 m/s and 0.1 m, respectively (Hirayama et al., 2005a). Note that their first SSS model has the fore wings. The calculation shown in Fig. 9 for the response to the tail wing elevator cannot be directly compared to the model test results obtained by the group of YNU, because many different features such as the wing configuration and bow shape assumed are different. However, the orders of response amplitude ratios in their tank test results are comparable to those in Fig. 9.
0
4 2 0 -2 -4 0
2
4
6
8
10
12
6
8
14
16
Ye (m) Xe (m) -4
-2
0
2
4
10
12
-1 Ze (m)
887
0 1 Fig. 11. Rudder response (rudder angle, 51).
amplitude is 51 and period is 10 s for the both cases. The response of SSS is in the vertical plane and surge, heave and pitch motions are induced. The transient response decay soon and the motion
Fig. 11 shows the response to the rudder. The rudder angle order is 51 stepwise to the starboard side after the horizontal steady advancing condition when time t is equal to zero. The right turning with negative lateral velocity v is induced. The negative lateral velocity for right turning means that the bow directs inward. That is same as the conventional surface ship’s right turning with starboard rudder angle. So the basic mechanism of turning by rudder of SSS is considered to be same as that of conventional ships. The mechanism of turning of conventional ships is as follows. Not only the rudder force produces turning moment directly but also the rudder force induces lateral motion of the ship. In this lateral motion, the ship hull plays a role as a huge lifting body in the horizontal plane. This huge lifting body in oblique motion produces large lateral force and turning moment. This hull originated force and moment are dominant for turning. The roll angle f shows short transient starboard down direction just after the rudder starting to work. This is the direct consequence of roll moment by rudder. Soon after that, the roll angle evolves to the opposite direction and reaches to the steady port side down heel angle. This is the effect of right turning which makes the inflow speed into left wing larger than that into right wing. That makes left wing’s downward lift larger than right wing’s one. That results in negative roll moment which overcomes the rudder originated roll moment. The resultant heel angle reaches to the point where the roll moment by rudder, the roll moment by wings and the static roll restoring moment due to the buoyancy and the gravity come to equilibrium. The heel angle makes the vertical downward lift inclined and the horizontal component of that works as centrifugal force during turning. This is a specific feature in the mechanism of turning by rudder of the SSS, which is not seen in conventional ship’s turning. The heel angle of conventional ships during turning depends on the hydrodynamic forces acting on rudder and hull, and the centrifugal force originated from ship mass.
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δa (deg)
888
5.0 0.0 0
20
40
60
t (s)
U0+u, v, w (m/s)
2.0 1.5
U0+u v
1.0
w
0.5 0.0
during steady turning. This induces the increase of inflow angle of attack and then the increase of downward lift. The forward velocity U0 +u shows almost same value as the horizontal steady advancing condition. But it slightly increases from 1.500 to 1.509 m/s. This induces the increase of downward lift, because the increase of forward speed increases the base dynamical pressure. All these changes result in that the vertical component of downward lift barely becomes larger than the residual buoyancy, since the vertical velocity w shows positive or downward direction. The resultant trajectory of this turning motion by rudder is slight downward spiral as shown in the bottom graph in Fig. 11. Note that the vertical scale is different from horizontal one.
-0.5 0
20
40
60
p, q, r (deg/s)
t (s) 10.0 8.0 6.0 4.0 2.0 0.0 -2.0
p q r
0
20
40
60
t (s)
, (deg)
2.0 1.5 1.0 0.5 0.0 -0.5 0
20
40
60
t (s) 20
Xe (m)
15
10
5
0
-5 -5
0
5
10
15
20
25
Ye (m) Xe (m) -10
-5
0
5
10
15
20
Ze (m)
-1 0 1 Fig. 12. . Aileron response (aileron angle, 21).
The heel angle makes the vertical component of downward lift decrease because it inclines. The trim angle y which should be affected by the static pitch moment shows bow up direction
4.4. Aileron response Fig. 12 shows the response to the aileron. The aileron angle order is 21 stepwise to produce positive roll moment or starboarddown moment. The aileron starts to work after the horizontal steady advancing condition when time t is equal to zero. Note that the aileron holds the constant angle. In case of an airplane, holding a constant aileron angle makes an airplane spin continuously. However, the response of SSS to the constant aileron angle does not become spin motion. The reason is that the SSS is under influence of the static roll restoring moment. Airplanes do not feel such static roll restoring moment. The Transient roll motion shows large overshoot before reaching steady condition. The steady heel angle f shows starboard down direction corresponding to the induced roll moment direction. This steady heel angle represents equilibrium position of the roll moment by aileron under the influence of inflow velocity difference between right and left wings and the static roll restoring moment. The heel angle makes the downward lift direction inclined and the horizontal component of the downward lift is produced. Airplanes use this horizontal component of lift as centrifugal force to turn. The positive or starboard down roll angle as shown in Fig. 12 makes the downward lift incline port side. That means that the horizontal component of the inclined downward lift directs port side. If this horizontal component of downward lift worked as the centrifugal force to turn, the SSS turns left. On the contrary, Fig. 12 shows that the SSS turns right. This implies that the mechanism of SSS turning by aileron is thoroughly different from that of airplanes. The mechanism of SSS turning by aileron is considered as follows. The horizontal component of the inclined downward lift directing port side induces negative sway or port side lateral velocity v. As a matter of fact, Fig. 12 shows negative lateral velocity. The lateral velocity results in inflow angle of attack into the hull of SSS to play a role as a huge lifting body. As mentioned in the previous section for the rudder response, this huge lifting body in oblique motion produces large lateral force and turning moment. And this hull originated force overcomes the horizontal component of the inclined downward lift in opposite direction. So the SSS turns right by positive aileron angle. This difference between the SSS and airplanes comes from the difference of the ratio of hull volume to lift force by wings. In other words, hull effect is dominant for the SSS but wing effect is dominant for airplanes in turning motion. Note that the above mentioned turning mechanism by aileron is for the SSS having the configuration shown as Fig. 1. The point is that it has no vertical tail wing. The turning response to aileron may be different one if the SSS has a vertical tail wing of sufficient area. The evolved steady heel angle is starboard side down. The trim angle y which should be affected by the static pitch restoring
ARTICLE IN PRESS M. Ueno / Ocean Engineering 37 (2010) 879–890
n (rps)
40.0 20.0 0.0 0
20
40
60
U0+u, v, w (m/s)
t (s) 3.0
U0+u
2.0
v w
1.0
0
20
40
60
t (s)
p, q, r (deg/s)
2.0
p q
1.0
r 0.0 -1.0 -2.0 0
20
40
60
t (s)
, (deg)
horizontal steady advancing condition plus a sinusoidal variation. This sinusoidal variation’s amplitude is 5 rps and period is 10 s. The sinusoidal variation of propeller revolution starts to work after the horizontal steady advancing condition when time t is equal to zero. Variation of the propeller revolution affects the propeller thrust and then forward speed. Since the dynamical pressure changes depending on forward speed, the variation of propeller revolution induces the variation of downward lift and drag. That means response to the propeller revolution is qualitatively similar to that to the elevator. Comparing Fig. 13 with Figs. 9 and 10 confirms this fact.
5. Conclusions
0.0 -1.0
4.0
2.0
0.0 -2.0 -4.0 0
20
40
60
t (s) Xe (m) 0
20
40
60
80
100
-2 Ze (m)
889
The linear equations of motion of a SSS in unbounded water are presented. Procedures for estimating hydrodynamic derivatives of the linear equations are also presented. Based on theoretical and semi-empirical formulae together with the experimental data for the SSS reported by YNU group, hydrodynamic derivatives for the SSS are estimated. Using these estimated hydrodynamic derivatives, the responses to the main and tail wings’ elevators, rudder, aileron and propeller revolution are calculated. The elevator of tail wing is more effective than that of main wing in vertical motion. Turning motion by rudder for the SSS is similar to that of conventional ships. However, the difference is that the SSS uses horizontal component of inclined downward lift as a part of centrifugal force. The mechanism of turning by aileron for the SSS is quite different from that of airplanes. The direction of turning by aileron for the SSS is opposite to that of airplanes. The reason is that the SSS is under influence of static roll restoring moment and has large hull that works as a huge lifting body. The response to the propeller revolution change is similar to that of the elevator since it affects thrust, velocity and dynamical pressure. Since the hydrodynamic derivatives are estimated with some assumptions and correction factors to satisfy the equilibrium condition, calculated results cannot be discussed quantitatively in detail. However, the basic characteristics of the SSS motion, especially the important roles of the static roll restoring moment and the large hull, are clarified through considerations of these calculation results. These facts seem to indicate the validity of the equations of motion and the estimation procedure of hydrodynamic derivatives presented here. Experimental research is needed for quantitative validation.
0 Acknowledgements
2 Fig. 13. Propeller revolution response (propeller revolution amplitude, 5 rps; period, 10 s).
moment shows bow down direction. The forward velocity U0 + u shows almost unchanged but slightly decreased value from 1.500 to 1.495 m/s. These heel and trim angles, and forward speed tendencies are all opposite to those in the response to rudder mentioned in the previous section. The vertical velocity w also directs opposite direction and the resultant trajectory by aileron is slight upward spiral as shown in the bottom graph in Fig. 12. Note that the vertical scale is different from horizontal one. 4.5. Propeller revolution response Fig. 13 shows the response to the propeller revolution. The designated propeller revolution is the constant revolution for the
The author would like to express his deep appreciation to the support and discussion of Prof. Hirayama and members of Yokohama National University group. A part of this work was supported by the KAKENHI (2036392). References Blevins, R.D., 1992. Applied Fluid Dynamics Handbook. Krieger Publishing Company, pp. 353–361. Dessureault, J.G., 1976. ‘‘Batfish’’ A depth controllable towed body for collecting oceanographic data. Ocean Engineering 3, 91–111. Hirakawa, Y., Kondo, S., Takayama, T., Hirayama, T., 2007. Trial experiment on the submersible surface ship utilizing downward lift (part 4). Conference Proceedings, vol. 4. The Japan Society of Naval Architects and Ocean Engineers, pp. 143–144. Hirayama, T., Takayama, T., Hirakawa, Y., Koyama, H., Nishimura, K., Kondo, S., 2005a. Trial experiment on the submersible surface ship utilizing downward lift. Conference Proceedings, vol. 5. The Japan Society of Naval Architects of Japan, pp. 141–142.
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Mori, K., Doi, Y., Takahashi, N., Tanemura, H., 1991. Numerical simulation of flows and motions of semi-submergible high speed ship with wings. Transactions of the West-Japan Society of Naval Architects (82), 41–52. Muddie, J.D., Ivers, W.D., 1975. Simulation studies of the response of a deeply towed vehicle to various towing ship maneuvers. Ocean Engineering 3, 205–212. Ohkusu, M., Kashiwagi, M., Koterayama, W., 1987. Hydrodynamics of a depth controlled towed vehicle. Journal of the Society of Naval Architects of Japan 162, 99–109. Son, Kyoung-Ho, 1983. A study on unstable behavior due to coupled motion of steering and rolling of a ship, Doctoral thesis, Osaka University. Ueno, M., Sawada, H., 1993. A study on hydrodynamic forces acting on an underwater vehicle, Papers of Ship Research Institute, 30-1, Ship Research Institute, Ministry of Transport, Japan, pp. 1–41. Ueno, M., Nimura, T., Ando, H., Maeda, K., Tamura, K., 2008. On the Descending Motion of a Deep-Sea Robot, vol. 16. Control Engineering Practice, Elsevier Ltd., pp. 446–456. Ura, T., Otsubo, S., 1987. Design of unmanned untethered submersible for quick swimming (1st report: study on gliding performance). Journal of the Society of Naval Architects of Japan 162, 110–117. USAF, 1968. Stability and Control, DATCOM. Flight Control Division, Air Force Flight Dynamics Laboratory, Wright-Paterson Air Force Base, Ohio.