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Performing captive model tests with a hexapod
Ocean Engineering 171 (2019) 49–58
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Performing captive model tests with a hexapod a
a
a,b,c
Haiwen Tu , Lei Song , De Xie a b c
a
a
a,b,c,∗
, Zeng Liu , Zhengyi Zhang , Jianglong Sun
T
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), Wuhan, Hubei, 430073, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, Shanghai, 200240, China Hubei Key Laboratory of Naval Architecture & Ocean Engineering Hydrodynamics (HUST), Wuhan, Hubei, 430073, China
ARTICLE INFO
ABSTRACT
Keywords: Captive mode test Hydrodynamic coefficient Six-DOF motion platform
In this study, a hexapod, six-degree of freedom (DOF) platform has been implemented to perform the captive model tests. With this novel device, a physical ship model can be forced to produce independent or coordinated motions in six-DOF that cannot be accomplished by a traditional one. This paper starts with the demonstration of the principle of the novel captive model test device. The motion relations between the driving legs of the testing platform and the physical ship model have been derived, based on which the mathematical model of hydrodynamic force has been established. Next, the testing procedures have been given by performing hydrodynamic coefficients tests of a rectangular frame and two ship models using the novel test device. Finally, the test results are compared with the theoretical values to verify the reliability of the device. The test device proposed in this paper provides a novel alternative to the traditional captive model test devices. It can identify the complex hydrodynamic coefficients of the mathematical model simultaneously in a single device, which is of great significance for the efficient prediction of hydrodynamics performance of given ships and offshore structures.
1. Introduction Experimental test is a classical and effective method to study ship's hydrodynamic performance as well as to validate and improve the theoretical models (Chen et al., 2006; Feng et al., 2015; Sun et al., 2016; Tu et al., 2018). In order to predict the ship's hydrodynamic performance, it is necessary to perform captive model test where the ship model is forced to move with the motions of the testing devices (Lau et al., 2004; Shin and Choi, 2011; Yun and Kim, 2012; Zhang and Zou, 2013). Forces and moments on the hull are measured during these forced movements, and data is collected and then used to estimate hydrodynamic coefficients in mathematical modeling. Generally, the Rotating Arm (RA) and the Planar Motion Mechanism (PMM) are the two main mechanisms employed by the traditional captive model tests (Park et al., 2016). Through the RA, vertical PMM, and horizontal PMM, the physical model can be forced to produce rotary motion, independent or coordinated motion of heave and pitch, and independent or coordinated motion of sway and yaw, respectively. The RA is an early test device of captive model (Feldman, 1987). In 1944, the Davidson basin in the United States built the first RA to study the maneuverability of submarines, other countries subsequently built similar RA devices. The RA device played a significant role in the 1950s. However, it was very rare to build the RA device in recent years due to
the larger size, higher cost and the test limitation. Instead, the PMM was selected to carry out the captive model tests. In 1952, Goodman first developed the PMM to determine the stability derivative of the aircraft at Langley wind tunnel. In 1957, Goodman and Gertler installed the PMM to the tank carriage to determine the hydrodynamic coefficients of the vertical plane of submarines in Taylor tank. Since then, the PMM was developed from the vertical PMM which could only be used to study the hydrodynamic performance of submarines to the horizontal PMM which could be used to study the hydrodynamic performance of surface ship. In the 1960s, the Danish tank, the University of Delft, the University of California, the University of Tokyo and the British naval tank successively built the PMM devices. In 1972, Denmark and America built the large amplitude PMM, which could measure the nonlinear hydrodynamic coefficients of the model in the horizontal plane. In the early 1980s, the advent of computer-controlled tank carriage led to further development of the captive model device. The PMM was considered to be the most effective device to study the ship hydrodynamic performance. Rhee et al. (2000) adopted the PMM and the coning motion device in model test to obtain the hydrodynamic coefficients of a submerged body. Lee et al. (2011) evaluated the added mass of a spheroid-type unmanned underwater vehicle by vertical PMM test, and the vertical PMM test results agreed well with the theoretical calculations and the CFD results. Millan and Thorburn. (2010) and Burgess et al. (2011) described a new
∗ Corresponding author. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), Wuhan, Hubei, 430073, China. E-mail address: [email protected] (J. Sun).
https://doi.org/10.1016/j.oceaneng.2018.10.037 Received 10 March 2018; Received in revised form 21 September 2018; Accepted 23 October 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
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PMM, capable of both hydrodynamic maneuvering studies and ice maneuvering studies. It paved the way for an expanded NRC-IOT research program related to in-ice maneuvering. Avila and Adamowski (2011) undertook forced oscillation tests using a PMM to evaluate the hydrodynamic coefficients of full-scale ROV, and the experimental results were compared with the ones estimated by system identification method. Park et al. (2016) conducted an experimental study on hydrodynamic coefficients for high-incidence-angle maneuver of a submarine. The horizontal PMM was used to carry out pure sway, pure yaw, and combined sway/yaw tests, while pure heave, pure pitch, and combined heave/pitch tests was completed by the vertical PMM device. Although the classic RA method and the PMM have been widely implemented to carry out the captive model tests in the past decades, they still face some limitations and challenges, especially for physical models with complex motions. Both RA and the PMM have motion limitations. In particular, the RA can only generate rotary motion, the vertical PMM can only generate pure heave, pure pitch, and combined heave/pitch motions, the horizontal PMM can only generate pure sway, pure yaw, and combined sway/yaw motions. When measuring the hydrodynamic coefficients of model with complex motions, the RA, vertical PMM, and horizontal PMM need to be implemented in conjunction with each other to complete the captive model tests. It makes the experiment complex, inefficient and costly. Fortunately, the six-degree of freedom (DOF) mechanism has been emerged as a promising solution to the aforementioned issue. By using a six-DOF mechanism, it is possible to produce independent or coordinated movement in six-DOF in a single device, which is of great importance for the effective measurement of hydrodynamic coefficients for physical models with complex motions. The six-DOF motion mechanism was first proposed by Gough (1957), and was formally presented by Stewart (1965). Recently, the six-DOF motion mechanism had gained widely attention and application. It was applied to flight simulator, ship motion simulator, submarine simulator and vehicle driving simulation system. The multipurpose carriage in MARINTEK's towing tank includes a 6 degrees-offreedom hexapod motion platform, allowing forced motion tests, including large-amplitude PMM tests for maneuvering studies. The hexapod is mounted on transverse rails and a pivot axis, making it capable of large sway and yaw motions. A series of PMM tests were carried out on the scaled model of Gunnerus by the 6 DOF hexapod motion platform (Ross et al., 2015; Hassani et al., 2015). Kramer et al. (2016) used the 6 degrees-of-freedom hexapod motion platform to study the effect of drift angle on a ship-like foil with varying aspect ratio and bottom edge shape at the MARINTEK towing tank facilities. At present, to the authors' best knowledge there is no relevant literature on the usage of the 6 DOF motion mechanism in captive model testing. The objective of this study is to propose a novel captive model testing device, which employed a hexapod, six-DOF platform to provide independent or coordinated movement in six-DOF in a single device. Through this device, the efficiency is improved for the determination of the hydrodynamic coefficients for given physical models of ships and offshore structures. The rest of the paper is organized as follows. First, the principle of the novel captive model test device is shown. The motion relations between the driving legs of the testing platform and the physical ship model have been derived, based on which the mathematical model of hydrodynamic force has been established. Next, the testing procedures have been given by performing hydrodynamic coefficient tests of a rectangular frame and two ship models using the proposed testing device. Finally, the test results are compared with the theoretical values to verify the reliability of the device.
Fig. 1. The structure of the testing device of captive model.
mechanism and six-component balance. Among them, the six-DOF motion mechanism is composed of fixed platform, universal joint, six driving legs and moving platform. The installing frame is rigidly fixed on the carriage beam along the longitudinal direction of the towing tank. The fixed platform of the six-DOF motion mechanism is rigidly connected with the installing frame. The moving platform of the sixDOF motion mechanism is rigidly connected with the ship model by the six-component balance that installed at the center of gravity of the model. Between the moving platform and the six-component balance, there is a cylinder used for connection. Different lengths of cylinders are used to accommodate different loading conditions or different ship models. The fixed platform is connected with the moving platform by six telescopic driving legs and universal joints. The position and posture of the moving platform is controlled by telescopic movement of the driving legs. The testing device is controlled by computer. The servo motors drive the electric cylinders to produce telescopic movement once the parameters are input by the control system. Then, the moving platform and the ship model will generate displacement and rotation motions in different directions. The device can drive the ship model to produce independent or coordinated motions in six-DOF. The forces and moments are measured by the six-component balance during these motions, and the data are used to estimate hydrodynamic coefficients for mathematical modeling. The performance indicators of the testing device are shown in Table 1, including the maximum amplitude of displacement, velocity and acceleration of the six-DOF motions. 2.2. Motion analysis The testing device forces the ship model to generate the specified motion. It is driven by the driving legs of the six-DOF motion mechanism. The motion analysis is to establish the motion relation between the six driving legs and the ship model, including displacement, velocity and acceleration. To describe the motions of the device and the ship model, we introduce two right-handed Cartesian coordinate systems. One is the fixed coordinate system O-XYZ, its origin is the geometry center of the fixed Table 1 Performance indicators of the testing device of captive model.
2. Description of mechanism 2.1. Description of structure The structure of the test device is shown in Fig. 1. The test equipment consists of control system, installing frame, six-DOF motion 50
Motions
Displacement
Velocity
Acceleration
Frequency (rad/s)
Sway Surge Heave Roll Pitch Yaw
150.0 mm 150.0 mm 100.0 mm 20.0° 15.0° 15.0°
250.0 mm/s 250.0 mm/s 250.0 mm/s ± 15.0°/s ± 15.0°/s ± 15.0°/s
0.3g 0.3g 0.3g ± 20.0°/s2 ± 20.0°/s2 ± 20.0°/s2
0.628–6.28 0.628–6.28 0.628–6.28 0.628–6.28 0.628–6.28 0.628–6.28
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The length of the driving leg can be calculated by the follow formula:
Lix2 + Liy2 + Liz2
Li = | L i | =
(5)
The median length of each driving leg is denoted as |Li |m . Then the flexible quantity of each driving leg is expressed as follows:
Si = |Li |
(6)
|Li |m
By taking a derivative with respect to time for formula (1), the velocity vector VPi of each hinged point Pi can be written as: (7)
VPi = TPi + R
By taking a derivative with respect to time for T = TZ TY TX , we can get the following formula: (8)
T = Sk TZ TY TX + TZ Sj TY TX + TZ TY Si TX , Fig. 2. Coordinate systems and the relations of space vectors.
where Si, Sj and Sk are anti-symmetric matrices which can be written as:
platform. The other is the moving coordinate system O1-X1Y1Z1, its origin is the center of the ship model gravity. The positive direction is according to the right-handed system with positive X1 in the ship bow, positive Y1 in the port, and positive Z1 upward. In the initial state, the OZ axis and the O1Z1 axis overlap, the OX axis is parallel to the O1X1 axis, and the OY axis is parallel to the O1Y1 axis, as shown in Fig. 2. The hinged point of the driving leg and the fixed platform is Bi (i = 1,2, …,6), the hinged point of the driving leg and the moving platform is Pi (i = 1,2, …,6). The position vectors of O1, Bi and Pi in the fixed coordinate system are R, Bi and Pi, respectively, the position vector of Pi in the moving coordinate system is Pi , the length vector of driving leg from Bi to Pi in the fixed coordinate system is Li. The geometrical relation of the vectors is shown in Fig. 2. Pi in the moving coordinate system can be transformed into Pi in the fixed coordinate system by coordinate transformation, the formula is as follows:
0 0 Si = 0 0 0 1
cos sin 0
sin cos 0
1 0 TX = 0 cos 0 sin
0 0 , TY = 1
cos 0 sin
0 sin 1 0 0 cos
cos cos sin
Bi = TPi + R
Bi
(9)
(11)
where I3 is 3 × 3 identity matrix, X = ( , , , x , y , is the velocity vector of ship model. The velocity vector of the six hinged points Pi can be written as VP = (VP1, VP2, VP3, VP4, VP5, VP6)T . The relation of velocity vectors between the six hinged points Pi and the ship model can be expressed as follows:
z )T
(12)
VP = J1 X ,
Where J1 =
TZ TY Si TX P1 TZ Sj TY TX P1 Sk TZ TY TX P1
I3
TZ TY Si TX P2 TZ Sj TY TX P2 Sk TZ TY TX P2
I3
TZ TY Si TX P3 TZ Sj TY TX P3 Sk TZ TY TX P3 I3
.
TZ TY Si TX P4 TZ Sj TY TX P4 Sk TZ TY TX P4 I3 TZ TY Si TX P5 TZ Sj TY TX P5 Sk TZ TY TX P5 I3 TZ TY Si TX P6 TZ Sj TY TX P6 Sk TZ TY TX P6 I3
cos cos
18 × 6
The relation of the velocity vectors between each driving leg and each hinged point Pi can be expressed as follows:
Li =
cos sin cos + sin sin sin cos cos cos sin
Li VP = eiT VPi, |Li | i
(13)
sin sin (3)
where eiT is the unit direction vector of the driving leg. The velocity vector of the six driving legs can be written as L = (L1 , L 2 , L3 , L4 , L5 , L6 )T . The relation of the velocity vectors between the six driving legs and the six hinged points Pi can be expressed as follows:
The position vector R, Bi, Pi and the rotation matrix T can be obtained once the geometric parameters of the six-DOF motion mechanism and the posture of the ship model are given. Using the relation between the space vectors in Fig. 2, the length vector Li (i = 1,2, …,6) of the driving leg can be expressed as:
Li = Pi
1 0 0 0
VPi = (TZ TY Si TX Pi , TZ Sj TY TX Pi , Sk TZ TY TX Pi , I3)3 × 6 X ,
(2)
cos sin sin sin sin sin sin + cos cos sin
0 0
Then the relation of the velocity vectors between each hinged point Pi and ship model can be expressed as follows:
,
0 sin cos
cos sin
0 1 0 0 0 , Sk = 1 1 0 0 0
(10)
Then, the rotation matrix T can be obtained by the following formula.
T = TZ TY TX =
0 0
VPi = (TZ TY Si TX Pi , TZ Sj TY TX Pi , Sk TZ TY TX Pi )( , , )T + (x , y , z )T
where T is the rotation matrix from the moving frame to the fixed frame. The posture change from the moving coordinate system to the fixed coordinate system requires rotation of γ angle around the Z axis first, and then rotation of β angle around the Y axis, and then rotation of α angle around the X-axis. According to the rotation formula of the coordinate system, the rotation matrix around each axis can be expressed as:
TZ =
0
1 , Sj =
By substituting formula (8) into (7), the velocity vector can be further derived as:
(1)
Pi = TPi + R,
0
(4)
L = J2 VP, 51
(14)
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e1T 01 × 3
where J2 =
01 × 3 01 × 3 e2T
01 × 3 01 × 3 0
0
0
0
01 × 3 e3T
0
0
0
0
e4T 01 × 3
3×9
01 × 3 01 × 3 e5T
.
Fig. 4. The coordinate systems.
01 × 3
displacement history of the ship in surge, sway, heave, roll, pitch and yaw motions are shown in Fig. 3. The largest error of linear displacement is 1.24 mm. The largest error of angular displacement is 1.56 e−03 rad.
3×9 01 × 3 01 × 3 e6T By substituting formula (12) into (14), the relation of the velocity vectors between the six driving legs and the ship model can be given as:
2.3. Mathematical model of hydrodynamic force
(15)
L = J2 J1 X
By taking a derivative with respect to time for formula (15), the relation of the accelerations between the six driving legs and the ship model can be given as:
The ship is subjected to various hydrodynamic forces while it is moving in the water. To predict the navigation performance of the ship, the mathematical model of hydrodynamic force should be established. Three right-handed Cartesian coordinate systems are introduced, including the fixed coordinate system, the moving coordinate system and the semi-fixed coordinate system (shown in Fig. 4). The fixed coordinate system and the moving coordinate system are defined in section 2.2. The semi-fixed coordinate system O2-xyz moves with the ship speed. It coincides with the moving coordinate system in equilibrium. As a note, the Y, Y1 is positive to starboard and Z, Z1 downwards in
(16)
L¨ = J2 J1 X¨ + J2 J1 X + J2 J1 X
Then, we have built the relations of displacement, velocity and acceleration between the six driving legs and the ship model. Once the theoretical motion of the ship model is given, the six driving legs can drive the ship model to achieve the specified movement. The comparisons between the actual value and the theoretical value of the
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. The displacement history of surge, sway, heave, roll, pitch and yaw motions. 52
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maneuvering. In maneuvering, the ship is subjected to unsteady hydrodynamic forces. Surge, sway, and yaw motions in the horizontal plane are considered in the moving coordinate system O1-X1Y1Z1. Under the ‘slow motion’ assumption, we describe the total hydrodynamic surge X and sway Y forces and yaw moment N acting on the ship as general functions of the state variables at a certain instant such as surge, sway, and yaw velocities (u, v, r, respectively) and accelerations (u , v , r , respectively) and rudder deflection angle δ such that (X , Y , N ) = f (u , v, r , u, v , r , ) . The general functions can be expanded by using the Taylor series expansion. Then, according to the momentum theorem of the rigid body, the linear equations of ship maneuvering motion can be expressed as:
mass coefficients, jk are the generalized damping coefficients. The subscript in turn represents the acting force and the direction of motion. For example, µ42 x¨2 represents the added inertial force caused by the ship model's swaying (k = 2) in the rolling (j = 4) direction, namely the corresponding component of the coupling force. Due to hull shape characteristics, the generalized added mass coefficients and damping coefficients can be expressed as:
{µjk } =
µ11 0 0 0 0 0 0 µ 22 0 µ 24 0 µ 26 0 µ33 0 µ35 0 0 , 0 µ42 0 µ44 0 µ46 µ µ 0 0 0 0 53 55 0 µ62 0 µ64 0 µ66
mu = Xu u + Xu u
11
(17)
m (v + u1 r ) = Yv v + Yr r + Yv v + Yr r + Y
{
jk }
0 0 0 0 0
=
Izz r = Nv v + Nr r + Nv v + Nr r + N where m is the ship's mass, Izz is the mass moment of inertia. u = u u1, u1 is the ship advance speed at the reference state. The simplified derivatives expressions, e.g., Xu = X / u , Xu = X / u and so on, are known as the hydrodynamic derivatives. In seakeeping, the ship is subjected to various hydrodynamic forces. It is assumed that the six-DOF motions of the ship are independent, the wave amplitude is small, and the ship motion amplitude is also small. The equation of ship oscillation motion can be expressed as:
µ 22 x¨2 + µ42 x¨2 + µ62 x¨2 +
jk
+ Fjk = 0
jk
+ Mjk = 0
0
44
0
53
0
55
62
0
64
0
26
0 46
0
(21)
66
22 x2
+ F22 = 0 x + M42 = 0 42 2 62 x2 + M62 = 0
(22)
(23)
2sin (
F2, µ sin ( t )
F2, cos ( t )
µ42 ( A
2sin (
t )) +
42 A cos ( t ) =
M4, µ sin ( t )
M4, cos ( t )
µ62 ( A
2sin (
t )) +
62
A cos ( t ) =
M6, µ sin ( t )
M6, cos ( t )
t )) +
22
A cos ( t ) =
(24) The added mass and damping of the sway motion can be written as:
µ 22 = 62
,
35
0
42
µ 22 ( A
F2, µ 2
, µ42 =
A M6, = A
M4, µ A
2
, µ62 =
M6, µ A
2
,
22
=
F2, , A
42
=
M4, , A (25)
Therefore, when the ship model is forced to make pure sway motion, the added mass and damping of the sway motion can be derived. The added mass and damping of the other motions can also be obtained by the same method.
(19)
where = 2 f , f is the frequency, A is the amplitude. xk , xk and x¨k are the displacement, velocity and acceleration of the ship model. For the ship model that performs forced oscillation in calm water, the hydrodynamic force is composed of two components, one is the added mass force that proportion to the acceleration, and the other is the damping force that proportion to the velocity. The forced motion equations can be expressed:
x¨k µjk + xk
0
0
By substituting formula (19) and (23) into (22), we can get the following formula:
The ship model is forced to make harmonic motion by using the captive model device. Its motion of the k-th mode can be expressed as:
x¨k µjk + xk
0 0
M62 = M6, µ sin ( t ) + M6, cos ( t )
3.1. Test principle
t)
33
24
F22 = F2, µ sin ( t ) + F2, cos ( t )
To study the test process and reliability of the hydrodynamic testing device, we carried out experiments to measure hydrodynamic coefficients of a rectangular frame and two ship models. The forced sway motion is performed for the rectangular frame. The test results are compared with the numerical results to verify the reliability of the testing device. The forced surge and sway motions are performed for the ship-I model, and a coupled forced surge and sway motion is performed for the ship-II model. The hydrodynamic coefficients of these two ship models are collected and analyzed.
2sin (
0
0
M42 = M4, µ sin ( t ) + M4, cos ( t )
3. Measurement tests of hydrodynamic coefficient
A
0 0
By using the least square method to fit the hydrodynamic force F22 and moment M42 , M62 , we can get the following formula:
where j = 1, 2, 3, 4, 5, 6 represent surge, sway, heave, roll, pitch and yaw motions, respectively. Mj is the inertia force coefficient, 2Nj is the damping force coefficient, Cj is the restoring force coefficient, and Fj is the wave exciting force.
xk = Asin ( t ), xk = A cos ( t ), x¨k =
22
Take the sway motion as an example, we will derive the expression of the added mass and damping. The linear equations of the sway motion can be written as follows:
(18)
Mj x¨j + 2Nj xj + Cj x j = Fj
0
3.2. Test model and equipment installation The three models used in the tests are shown in Fig. 5. The principal dimensions of the test models are summarized in Table 2. The equipment used in the experiment includes carriage system, six-DOF motion mechanism, six-component balance and ultrasonic range finder, as shown in Fig. 6. The performance parameters of the test equipment are shown in Table 3.
(20)
3.2.1. Installation of six-DOF motion mechanism First, the upper and lower parts of the installing frame are rigidly fixed on the carriage beam by bolts. Then, the inverted trapezoid key on the fixed platform of the six-DOF motion mechanism is aligned to the
where k, j = 1, 2, 3, 4, 5, 6 represent surge, sway, heave, roll, pitch and yaw motions, respectively Fjk and Mjk are the hydrodynamic force and the moment and are obtained by tests. For the heave, roll and pitch motions, the restoring force is considered. µjk are the generalized added 53
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(a)
(b)
(c) Fig. 5. Test models: (a) rectangular frame model, (b) ship-I model, (c) ship-II model. Table 2 Principal dimensions of the models. Main parameters
Length (m) Breadth molded (m) Depth molded (m) Design draft (m)
Models Rectangular frame
Ship-I
Ship-II
4.000 0.200 0.200 0.100
3.810 0.682 0.374 0.158
1.350 0.300 0.230 0.040
(a)
(b)
Fig. 7. The installation of the six-DOF motion mechanism. (c)
mechanism is rigidly fixed on the carriage system by the installing frame to guarantee the platform will not shake in the process of the test. The installation is shown in Fig. 7.
Fig. 6. Test equipment: (a) six-DOF motion mechanism, (b) six-component balance, (c) ultrasonic range finder.
3.2.2. Installation of six-component balance The position of the center of the ship model gravity should be found and marked. Then two solid blocks are installed in this position, and the steel base is fixed on the solid blocks by four bolts. The bottom of the six-component balance is connected to the steel base by the steel flange
keyway on the lower part of the installing frame. The fixed platform and the installing frame are further fixed by bolts. At last, the electromechanical control box is fixed on the installing frame and the electric lines are connected to the control system. The six-DOF motion Table 3 Performance parameters of the test equipment. No
Equipment
Type
Performance parameters
1 2 3 4
Carriage system Six-DOF motion mechanism Six-component balance Ultrasonic range finder
SIEMENS 6R24 Huster-SDMM FC-K6D68 UC2000-30 GM-IUR2-V15
Speed: 0.1～8.0 m/s; Accuracy: 0.1% See Table 1 Range: 5kN/50Nm; Accuracy: 0.1% Sensing distance: 80–2000 mm
54
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Fig. 10. The towing tank laboratory.
Fig. 8. The installation of the six-component balance.
to ensure rigid connection to the ship model. The vertical positions of the center of the ship model and the six-component balance gravity should be overlapped. The installation of the six-component balance is shown in Fig. 8. 3.2.3. Installation of ultrasonic range finder The displacement history of the ship model motion is measured by the ultrasonic range finder which is fixed on the carriage through an extended rod. A baffle is installed on the ship model along its width direction to insure that the sound waves are sent vertically upon it. The measured displacement is the absolute value of the ship model motion. The installation of the ultrasonic range finder is shown in Fig. 9. It can realize the displacement measurement of the surge motions. If we need to measure the displacement of sway and heave motions, only the positions of the ultrasonic range finder and the baffle need to be changed.
Fig. 11. Forced sway motion of the rectangular frame.
3.3. Test procedure
8
The experiment was performed at the towing tank laboratory of the School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), as shown in Fig. 10. The principle dimensions of the towing tank are 175 × 6 × 4 m. The motion parameters of the six-DOF motion mechanism are set before the test. In the process of the captive model tests, the force/displacement sensors are connected with the data acquisition system. The collection
6 4 F(N)
2 0 -2 -4 -6 -8
Ultrasonic range finder
0
5
10
15
20
25
t(s)
Baffle
Fig. 12. The horizontal force-time curve of sway motion with period T = 5s.
frequency of the system is set to 100 Hz. The data acquisition system continues to collect data until the model runs smoothly for 10 to 12 cycles. The second time data collection is carried out after the water surface is calm. 4. Experimental results Fig. 11 shows a forced sway motion of the rectangular frame. The amplitude of the motion is 0.04 m. The period of the motion is from 1s to 5s. Fig. 12 and Fig. 13 show the horizontal force-time curves of sway motion with period T = 5s and T = 1.25s, each curve has been performed a Fourier fit. The forces here represent the hydrodynamic part, and do not include the inertial part. The added mass coefficients are
Fig. 9. The installation of the ultrasonic range finder. 55
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160 120 80 F(N)
40 0 -40 -80 -120 -160
0
1
2
3
4
5
6
7
8
t(s) Fig. 16. The added mass coefficients of surge motion of the ship-I model.
Fig. 13. The horizontal force-time curve of sway motion with period T = 1.25s.
2.0 Numerical result
1.8
Experimental result
LDB
1.6 1.4 1.2 1.0 0.8 0.6
0.0
0.1
0.2
0.3
0.4
0.5
[
0.5
0.6
0.7
0.8 Fig. 17. Forced sway motion of the ship-I model.
Fig. 14. Comparison of sway added mass coefficient between the experimental result and the numerical result for the rectangular frame.
6s. The experimental results of the added mass coefficients are shown in Fig. 16. The horizontal axis represents the dimensionless frequency [b/(2g )]0.5 . Here is frequency and b is the width of the ship model. The vertical axis represents the added mass coefficient µ/ m . Here µ is the added mass and m is displacement of the ship model. According to the experimental results, the added mass coefficient varies in the range of 0.047–0.148. The added mass coefficient reaches a peak value near the dimensionless frequency of 0.4. Fig. 17 shows a forced sway motion of the ship-I model. The amplitude of the motion is 0.06 m. The period of the motion is from 2s to 6s. The experimental results of the added mass coefficients are shown in Fig. 18. According to the experimental results, the added mass coefficient presents strong nonlinear characteristic and varies in the range of 0.85–1.10. Fig. 19 shows a coupled forced motion of swaying and rolling of the ship-II model. The amplitude of the sway motion is 0.02 m. The
calculated by the method in section 3.1. The experimental results of the added mass coefficients are shown in Fig. 14. The horizontal axis represents the dimensionless frequency [B/(2g )]0.5 . Here is frequency and B is the width of the rectangular frame. The vertical axis represents the dimensionless added mass µ/ LBD . Here µ is the added mass, L is the length of the rectangular frame, D is the draft of the rectangular frame and is the water density. The added mass first increases and then decreases. The experimental results agreed well with the numerical results (Luo et al., 2011) except for a few points. It shows that the captive model device can measure the added mass coefficients accurately. Fig. 15 shows a forced surge motion of the ship-I model. The amplitude of the motion is 0.06 m. The period of the motion is from 2s to
Fig. 15. Forced surge motion of the ship-I model.
Fig. 18. The added mass coefficients of sway motion of the ship-I model. 56
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Fig. 19. The coupled forced motion of swaying and rolling of the ship-II model. Fig. 22. The experimental results of the coupled added mass coefficients μ44 + μ42.
Fig. 20. The experimental results of the coupled added mass coefficients μ22 + μ24. Fig. 23. The experimental results of the coupled damping coefficients λ44 + λ42.
It can be seen from the test results that the added mass coefficients of surge motion are within the range of 0.05–0.15, the added mass coefficients of sway motion are within the range of 0.9–1.2, and the added moment of inertia coefficients of roll motion are within the range of 0.05–0.35. These are in agreement with the descriptions in the Principles of Ship (Sheng and Liu, 2004). The reliability of the experiment is proved. 5. Conclusions This paper studies a hydrodynamic testing device of captive model. With this novel device, a physical ship model can be forced to produce independent or coordinated motions in six DOF that cannot be accomplished by a traditional one. First, the principle of the novel captive model test device is given, the test results show that the device can force the ship model to achieve the specified motion well. Next, the hydrodynamic coefficient tests of the rectangular frame and two ship models are carried out by the test device, and the test results agree well with the theoretical values, the reliability of the device is verified. The novel captive model test device can identify the complex hydrodynamic coefficients of the mathematical model simultaneously in a single device, which is of great significance for the efficient prediction of hydrodynamics performance of given ships and offshore structures. In future work, the captive model device will be used to carry out more tests to further verify and improve its accuracy.
Fig. 21. The experimental results of the coupled damping coefficients λ22 + λ24.
amplitude of the roll motion is 3 deg. The period of the motion is from 1.25s to 10s. Fig. 20 shows the experimental results of the coupled added mass coefficients μ22 + μ24. Within the given frequency range, the maximum value of the dimensionless coupled added mass coefficients is 1.12. Fig. 21 shows the experimental results of the coupled damping coefficients λ22 + λ24. The dimensionless coupled damping coefficients show a nonlinear characteristic, the value first increases and then decreases with the dimensionless frequency. Fig. 22 shows the experimental results of the coupled added mass coefficients μ44 + μ42. The dimensionless coupled added mass coefficient varies in the range of 0.18–0.20. Fig. 23 shows the experimental results of the coupled damping coefficients λ44 + λ42. When the dimensionless frequency is 0.31, the maximum coupling damping coefficient is 0.018.
Acknowledgement The authors gratefully acknowledge the financial support from the 57
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National Natural Science Foundation of China (NSFC) under Project Number 51679097&51609090&51709120, and the financial support from the China Postdoctoral Science Foundation under Project Number 2018M632867.
maneuvering in ice experiments using a planar motion mechanism. In: 17th International Symposium on Ice, Saint Petersburg, Russia. Lee, S.K., Joung, T.H., Cheon, S.J., Jang, T.S., Lee, J.H., 2011. Evaluation of the added mass for a spheroid-type unmanned underwater vehicle by vertical planar motion mechanism test. Int. J. Nav. Architect. Ocean. Eng. 3, 174–180. Luo, M.L., Mao, X.F., Wang, X.X., 2011. CFD based hydrodynamic coefficients calculation to forced motion of two-dimensional section. J. Hydrodyn. 26 (4), 509–515. Millan, D., Thorburn, P., 2010. A planar motion mechanism (PMM) for Ocean Engineering studies. In: Conference (NECEC’10), St. John's, Newfoundland. Park, J.Y., Kim, N., Shin, Y.K., 2016. Experimental study on hydrodynamic coefficients for high-incidence-angle maneuver of a submarine. Int. J. Nav. Architect. Ocean. Eng. 9, 100–113. Rhee, K.P., Yoon, H.K., Sung, Y.J., Kim, S.H., Kang, J.N., 2000. An experimental study on hydrodynamic coefficients of submerged body using planar motion mechanism and coning motion device. In: International Workshop on Ship Maneuverability, pp. 1–20. Ross, A., Hassani, V., Selvik, Ørjan, Ringen, E., Fathi, D., 2015. Identification of nonlinear manoeuvring models for marine vessels using planar motion mechanism tests. In: Proceedings of the 34th International Conference on Ocean, Offshore and Arctic Engineering, St. John's, Newfoundland, Canada, 2015. Sheng, Z.B., Liu, Y.Z., 2004. Principles of Ship (Part 2). Shanghai JiaoTong University Press, Shanghai, pp. 278–279. Shin, H.K., Choi, S.H., 2011. Prediction of maneuverability of KCS using captive model test. J. Soc. Nav. Architect. Korea 48 (5), 465–472. Stewart, D., 1965. A platform with six degrees of freedom. Proc. Inst. Mech. Eng. 180 (1), 371–386. Sun, J., Tu, H., Chen, Y., Xie, D., Zhou, J., 2016. A study on trim optimization for a container ship based on effects due to resistance. J. Ship Res. 60, 30–47. Tu, H., Yang, Y., Zhang, L., Xie, D., Lyu, X., Song, L., Guan, Y., Sun, J., 2018. A modified admiralty coefficient for estimating power curves in EEDI calculations. Ocean Eng. 150, 309–317. Yun, K.H., Kim, Y.G., 2012. Study on the maneuverability of barge by captive model test. J. Naviga. Port Res. 36 (8), 613–618. Zhang, X.G., Zou, Z.J., 2013. Estimation of the hydrodynamic coefficients from captive model test results by using support vector machines. Ocean Eng. 73, 25–31.
References Avila, P.J., Adamowski, J.C., 2011. Experimental evaluation of the hydrodynamic coefficients of a ROV through Morison's equation. Ocean Eng. 38, 2162–2170. Burgess, M., Dell, S., Sullivan, M., 2011. A New Horizontal Planar Motion Mechanism (PMM) for the NRC-IOT Towing Tank and Ice Tank. Advanced Model Measurement Technology for EU Maritime Industry-AMT’11. Chen, P., Huang, C., Fang, M., 2006. An inverse design approach in determining the optimal shape of bulbous bow with experimental verification. J. Ship Res. 50 (1), 1–14. Feldman, J., 1987. Straightline and Rotating Arm Captive-model Experiments to Investigate the Stability and Control Characteristics of Submarines and Other Submerged Vehicles. David Taylor Research Center, Bethesda MD, Ship Hydromechanics Department. Feng, A., Chen, Z.-M., Price, W., 2015. A desingularized rankine source method for nonlinear wave–body interaction problems. Ocean Eng. 101, 131–141. Gough, V.E., 1957. Contribution to discussion of papers on research in automobile stability and control and in tyre performance by Cornell staff. Proc. Inst. Mech. Eng. 171, 392–395. Hassani, V., Ross, A., Selvik, Ørjan, Fathi, D., Sprenger, F., Berg, T.E., 2015. Time domain simulation model for research vessel Gunnerus. In: Proceedings of the 34th International Conference on Ocean, Offshore and Arctic Engineering, St. John's, Newfoundland, Canada, 2015. Kramer, J.A., Steen, S., Savio, L., 2016. Experimental study of the effect of drift angle on a ship-like foil with varying aspect ratio and bottom edge shape. Ocean Eng. 121, 530–545. Lau, M., Liu, J.C., Ahmed, D.A., Williams, F.M., 2004. Preliminary results of ship
58
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Performing captive model tests with a hexapod a
a
a,b,c
Haiwen Tu , Lei Song , De Xie a b c
a
a
a,b,c,∗
, Zeng Liu , Zhengyi Zhang , Jianglong Sun
T
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), Wuhan, Hubei, 430073, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, Shanghai, 200240, China Hubei Key Laboratory of Naval Architecture & Ocean Engineering Hydrodynamics (HUST), Wuhan, Hubei, 430073, China
ARTICLE INFO
ABSTRACT
Keywords: Captive mode test Hydrodynamic coefficient Six-DOF motion platform
In this study, a hexapod, six-degree of freedom (DOF) platform has been implemented to perform the captive model tests. With this novel device, a physical ship model can be forced to produce independent or coordinated motions in six-DOF that cannot be accomplished by a traditional one. This paper starts with the demonstration of the principle of the novel captive model test device. The motion relations between the driving legs of the testing platform and the physical ship model have been derived, based on which the mathematical model of hydrodynamic force has been established. Next, the testing procedures have been given by performing hydrodynamic coefficients tests of a rectangular frame and two ship models using the novel test device. Finally, the test results are compared with the theoretical values to verify the reliability of the device. The test device proposed in this paper provides a novel alternative to the traditional captive model test devices. It can identify the complex hydrodynamic coefficients of the mathematical model simultaneously in a single device, which is of great significance for the efficient prediction of hydrodynamics performance of given ships and offshore structures.
1. Introduction Experimental test is a classical and effective method to study ship's hydrodynamic performance as well as to validate and improve the theoretical models (Chen et al., 2006; Feng et al., 2015; Sun et al., 2016; Tu et al., 2018). In order to predict the ship's hydrodynamic performance, it is necessary to perform captive model test where the ship model is forced to move with the motions of the testing devices (Lau et al., 2004; Shin and Choi, 2011; Yun and Kim, 2012; Zhang and Zou, 2013). Forces and moments on the hull are measured during these forced movements, and data is collected and then used to estimate hydrodynamic coefficients in mathematical modeling. Generally, the Rotating Arm (RA) and the Planar Motion Mechanism (PMM) are the two main mechanisms employed by the traditional captive model tests (Park et al., 2016). Through the RA, vertical PMM, and horizontal PMM, the physical model can be forced to produce rotary motion, independent or coordinated motion of heave and pitch, and independent or coordinated motion of sway and yaw, respectively. The RA is an early test device of captive model (Feldman, 1987). In 1944, the Davidson basin in the United States built the first RA to study the maneuverability of submarines, other countries subsequently built similar RA devices. The RA device played a significant role in the 1950s. However, it was very rare to build the RA device in recent years due to
the larger size, higher cost and the test limitation. Instead, the PMM was selected to carry out the captive model tests. In 1952, Goodman first developed the PMM to determine the stability derivative of the aircraft at Langley wind tunnel. In 1957, Goodman and Gertler installed the PMM to the tank carriage to determine the hydrodynamic coefficients of the vertical plane of submarines in Taylor tank. Since then, the PMM was developed from the vertical PMM which could only be used to study the hydrodynamic performance of submarines to the horizontal PMM which could be used to study the hydrodynamic performance of surface ship. In the 1960s, the Danish tank, the University of Delft, the University of California, the University of Tokyo and the British naval tank successively built the PMM devices. In 1972, Denmark and America built the large amplitude PMM, which could measure the nonlinear hydrodynamic coefficients of the model in the horizontal plane. In the early 1980s, the advent of computer-controlled tank carriage led to further development of the captive model device. The PMM was considered to be the most effective device to study the ship hydrodynamic performance. Rhee et al. (2000) adopted the PMM and the coning motion device in model test to obtain the hydrodynamic coefficients of a submerged body. Lee et al. (2011) evaluated the added mass of a spheroid-type unmanned underwater vehicle by vertical PMM test, and the vertical PMM test results agreed well with the theoretical calculations and the CFD results. Millan and Thorburn. (2010) and Burgess et al. (2011) described a new
∗ Corresponding author. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), Wuhan, Hubei, 430073, China. E-mail address: [email protected] (J. Sun).
https://doi.org/10.1016/j.oceaneng.2018.10.037 Received 10 March 2018; Received in revised form 21 September 2018; Accepted 23 October 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
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PMM, capable of both hydrodynamic maneuvering studies and ice maneuvering studies. It paved the way for an expanded NRC-IOT research program related to in-ice maneuvering. Avila and Adamowski (2011) undertook forced oscillation tests using a PMM to evaluate the hydrodynamic coefficients of full-scale ROV, and the experimental results were compared with the ones estimated by system identification method. Park et al. (2016) conducted an experimental study on hydrodynamic coefficients for high-incidence-angle maneuver of a submarine. The horizontal PMM was used to carry out pure sway, pure yaw, and combined sway/yaw tests, while pure heave, pure pitch, and combined heave/pitch tests was completed by the vertical PMM device. Although the classic RA method and the PMM have been widely implemented to carry out the captive model tests in the past decades, they still face some limitations and challenges, especially for physical models with complex motions. Both RA and the PMM have motion limitations. In particular, the RA can only generate rotary motion, the vertical PMM can only generate pure heave, pure pitch, and combined heave/pitch motions, the horizontal PMM can only generate pure sway, pure yaw, and combined sway/yaw motions. When measuring the hydrodynamic coefficients of model with complex motions, the RA, vertical PMM, and horizontal PMM need to be implemented in conjunction with each other to complete the captive model tests. It makes the experiment complex, inefficient and costly. Fortunately, the six-degree of freedom (DOF) mechanism has been emerged as a promising solution to the aforementioned issue. By using a six-DOF mechanism, it is possible to produce independent or coordinated movement in six-DOF in a single device, which is of great importance for the effective measurement of hydrodynamic coefficients for physical models with complex motions. The six-DOF motion mechanism was first proposed by Gough (1957), and was formally presented by Stewart (1965). Recently, the six-DOF motion mechanism had gained widely attention and application. It was applied to flight simulator, ship motion simulator, submarine simulator and vehicle driving simulation system. The multipurpose carriage in MARINTEK's towing tank includes a 6 degrees-offreedom hexapod motion platform, allowing forced motion tests, including large-amplitude PMM tests for maneuvering studies. The hexapod is mounted on transverse rails and a pivot axis, making it capable of large sway and yaw motions. A series of PMM tests were carried out on the scaled model of Gunnerus by the 6 DOF hexapod motion platform (Ross et al., 2015; Hassani et al., 2015). Kramer et al. (2016) used the 6 degrees-of-freedom hexapod motion platform to study the effect of drift angle on a ship-like foil with varying aspect ratio and bottom edge shape at the MARINTEK towing tank facilities. At present, to the authors' best knowledge there is no relevant literature on the usage of the 6 DOF motion mechanism in captive model testing. The objective of this study is to propose a novel captive model testing device, which employed a hexapod, six-DOF platform to provide independent or coordinated movement in six-DOF in a single device. Through this device, the efficiency is improved for the determination of the hydrodynamic coefficients for given physical models of ships and offshore structures. The rest of the paper is organized as follows. First, the principle of the novel captive model test device is shown. The motion relations between the driving legs of the testing platform and the physical ship model have been derived, based on which the mathematical model of hydrodynamic force has been established. Next, the testing procedures have been given by performing hydrodynamic coefficient tests of a rectangular frame and two ship models using the proposed testing device. Finally, the test results are compared with the theoretical values to verify the reliability of the device.
Fig. 1. The structure of the testing device of captive model.
mechanism and six-component balance. Among them, the six-DOF motion mechanism is composed of fixed platform, universal joint, six driving legs and moving platform. The installing frame is rigidly fixed on the carriage beam along the longitudinal direction of the towing tank. The fixed platform of the six-DOF motion mechanism is rigidly connected with the installing frame. The moving platform of the sixDOF motion mechanism is rigidly connected with the ship model by the six-component balance that installed at the center of gravity of the model. Between the moving platform and the six-component balance, there is a cylinder used for connection. Different lengths of cylinders are used to accommodate different loading conditions or different ship models. The fixed platform is connected with the moving platform by six telescopic driving legs and universal joints. The position and posture of the moving platform is controlled by telescopic movement of the driving legs. The testing device is controlled by computer. The servo motors drive the electric cylinders to produce telescopic movement once the parameters are input by the control system. Then, the moving platform and the ship model will generate displacement and rotation motions in different directions. The device can drive the ship model to produce independent or coordinated motions in six-DOF. The forces and moments are measured by the six-component balance during these motions, and the data are used to estimate hydrodynamic coefficients for mathematical modeling. The performance indicators of the testing device are shown in Table 1, including the maximum amplitude of displacement, velocity and acceleration of the six-DOF motions. 2.2. Motion analysis The testing device forces the ship model to generate the specified motion. It is driven by the driving legs of the six-DOF motion mechanism. The motion analysis is to establish the motion relation between the six driving legs and the ship model, including displacement, velocity and acceleration. To describe the motions of the device and the ship model, we introduce two right-handed Cartesian coordinate systems. One is the fixed coordinate system O-XYZ, its origin is the geometry center of the fixed Table 1 Performance indicators of the testing device of captive model.
2. Description of mechanism 2.1. Description of structure The structure of the test device is shown in Fig. 1. The test equipment consists of control system, installing frame, six-DOF motion 50
Motions
Displacement
Velocity
Acceleration
Frequency (rad/s)
Sway Surge Heave Roll Pitch Yaw
150.0 mm 150.0 mm 100.0 mm 20.0° 15.0° 15.0°
250.0 mm/s 250.0 mm/s 250.0 mm/s ± 15.0°/s ± 15.0°/s ± 15.0°/s
0.3g 0.3g 0.3g ± 20.0°/s2 ± 20.0°/s2 ± 20.0°/s2
0.628–6.28 0.628–6.28 0.628–6.28 0.628–6.28 0.628–6.28 0.628–6.28
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The length of the driving leg can be calculated by the follow formula:
Lix2 + Liy2 + Liz2
Li = | L i | =
(5)
The median length of each driving leg is denoted as |Li |m . Then the flexible quantity of each driving leg is expressed as follows:
Si = |Li |
(6)
|Li |m
By taking a derivative with respect to time for formula (1), the velocity vector VPi of each hinged point Pi can be written as: (7)
VPi = TPi + R
By taking a derivative with respect to time for T = TZ TY TX , we can get the following formula: (8)
T = Sk TZ TY TX + TZ Sj TY TX + TZ TY Si TX , Fig. 2. Coordinate systems and the relations of space vectors.
where Si, Sj and Sk are anti-symmetric matrices which can be written as:
platform. The other is the moving coordinate system O1-X1Y1Z1, its origin is the center of the ship model gravity. The positive direction is according to the right-handed system with positive X1 in the ship bow, positive Y1 in the port, and positive Z1 upward. In the initial state, the OZ axis and the O1Z1 axis overlap, the OX axis is parallel to the O1X1 axis, and the OY axis is parallel to the O1Y1 axis, as shown in Fig. 2. The hinged point of the driving leg and the fixed platform is Bi (i = 1,2, …,6), the hinged point of the driving leg and the moving platform is Pi (i = 1,2, …,6). The position vectors of O1, Bi and Pi in the fixed coordinate system are R, Bi and Pi, respectively, the position vector of Pi in the moving coordinate system is Pi , the length vector of driving leg from Bi to Pi in the fixed coordinate system is Li. The geometrical relation of the vectors is shown in Fig. 2. Pi in the moving coordinate system can be transformed into Pi in the fixed coordinate system by coordinate transformation, the formula is as follows:
0 0 Si = 0 0 0 1
cos sin 0
sin cos 0
1 0 TX = 0 cos 0 sin
0 0 , TY = 1
cos 0 sin
0 sin 1 0 0 cos
cos cos sin
Bi = TPi + R
Bi
(9)
(11)
where I3 is 3 × 3 identity matrix, X = ( , , , x , y , is the velocity vector of ship model. The velocity vector of the six hinged points Pi can be written as VP = (VP1, VP2, VP3, VP4, VP5, VP6)T . The relation of velocity vectors between the six hinged points Pi and the ship model can be expressed as follows:
z )T
(12)
VP = J1 X ,
Where J1 =
TZ TY Si TX P1 TZ Sj TY TX P1 Sk TZ TY TX P1
I3
TZ TY Si TX P2 TZ Sj TY TX P2 Sk TZ TY TX P2
I3
TZ TY Si TX P3 TZ Sj TY TX P3 Sk TZ TY TX P3 I3
.
TZ TY Si TX P4 TZ Sj TY TX P4 Sk TZ TY TX P4 I3 TZ TY Si TX P5 TZ Sj TY TX P5 Sk TZ TY TX P5 I3 TZ TY Si TX P6 TZ Sj TY TX P6 Sk TZ TY TX P6 I3
cos cos
18 × 6
The relation of the velocity vectors between each driving leg and each hinged point Pi can be expressed as follows:
Li =
cos sin cos + sin sin sin cos cos cos sin
Li VP = eiT VPi, |Li | i
(13)
sin sin (3)
where eiT is the unit direction vector of the driving leg. The velocity vector of the six driving legs can be written as L = (L1 , L 2 , L3 , L4 , L5 , L6 )T . The relation of the velocity vectors between the six driving legs and the six hinged points Pi can be expressed as follows:
The position vector R, Bi, Pi and the rotation matrix T can be obtained once the geometric parameters of the six-DOF motion mechanism and the posture of the ship model are given. Using the relation between the space vectors in Fig. 2, the length vector Li (i = 1,2, …,6) of the driving leg can be expressed as:
Li = Pi
1 0 0 0
VPi = (TZ TY Si TX Pi , TZ Sj TY TX Pi , Sk TZ TY TX Pi , I3)3 × 6 X ,
(2)
cos sin sin sin sin sin sin + cos cos sin
0 0
Then the relation of the velocity vectors between each hinged point Pi and ship model can be expressed as follows:
,
0 sin cos
cos sin
0 1 0 0 0 , Sk = 1 1 0 0 0
(10)
Then, the rotation matrix T can be obtained by the following formula.
T = TZ TY TX =
0 0
VPi = (TZ TY Si TX Pi , TZ Sj TY TX Pi , Sk TZ TY TX Pi )( , , )T + (x , y , z )T
where T is the rotation matrix from the moving frame to the fixed frame. The posture change from the moving coordinate system to the fixed coordinate system requires rotation of γ angle around the Z axis first, and then rotation of β angle around the Y axis, and then rotation of α angle around the X-axis. According to the rotation formula of the coordinate system, the rotation matrix around each axis can be expressed as:
TZ =
0
1 , Sj =
By substituting formula (8) into (7), the velocity vector can be further derived as:
(1)
Pi = TPi + R,
0
(4)
L = J2 VP, 51
(14)
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e1T 01 × 3
where J2 =
01 × 3 01 × 3 e2T
01 × 3 01 × 3 0
0
0
0
01 × 3 e3T
0
0
0
0
e4T 01 × 3
3×9
01 × 3 01 × 3 e5T
.
Fig. 4. The coordinate systems.
01 × 3
displacement history of the ship in surge, sway, heave, roll, pitch and yaw motions are shown in Fig. 3. The largest error of linear displacement is 1.24 mm. The largest error of angular displacement is 1.56 e−03 rad.
3×9 01 × 3 01 × 3 e6T By substituting formula (12) into (14), the relation of the velocity vectors between the six driving legs and the ship model can be given as:
2.3. Mathematical model of hydrodynamic force
(15)
L = J2 J1 X
By taking a derivative with respect to time for formula (15), the relation of the accelerations between the six driving legs and the ship model can be given as:
The ship is subjected to various hydrodynamic forces while it is moving in the water. To predict the navigation performance of the ship, the mathematical model of hydrodynamic force should be established. Three right-handed Cartesian coordinate systems are introduced, including the fixed coordinate system, the moving coordinate system and the semi-fixed coordinate system (shown in Fig. 4). The fixed coordinate system and the moving coordinate system are defined in section 2.2. The semi-fixed coordinate system O2-xyz moves with the ship speed. It coincides with the moving coordinate system in equilibrium. As a note, the Y, Y1 is positive to starboard and Z, Z1 downwards in
(16)
L¨ = J2 J1 X¨ + J2 J1 X + J2 J1 X
Then, we have built the relations of displacement, velocity and acceleration between the six driving legs and the ship model. Once the theoretical motion of the ship model is given, the six driving legs can drive the ship model to achieve the specified movement. The comparisons between the actual value and the theoretical value of the
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. The displacement history of surge, sway, heave, roll, pitch and yaw motions. 52
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maneuvering. In maneuvering, the ship is subjected to unsteady hydrodynamic forces. Surge, sway, and yaw motions in the horizontal plane are considered in the moving coordinate system O1-X1Y1Z1. Under the ‘slow motion’ assumption, we describe the total hydrodynamic surge X and sway Y forces and yaw moment N acting on the ship as general functions of the state variables at a certain instant such as surge, sway, and yaw velocities (u, v, r, respectively) and accelerations (u , v , r , respectively) and rudder deflection angle δ such that (X , Y , N ) = f (u , v, r , u, v , r , ) . The general functions can be expanded by using the Taylor series expansion. Then, according to the momentum theorem of the rigid body, the linear equations of ship maneuvering motion can be expressed as:
mass coefficients, jk are the generalized damping coefficients. The subscript in turn represents the acting force and the direction of motion. For example, µ42 x¨2 represents the added inertial force caused by the ship model's swaying (k = 2) in the rolling (j = 4) direction, namely the corresponding component of the coupling force. Due to hull shape characteristics, the generalized added mass coefficients and damping coefficients can be expressed as:
{µjk } =
µ11 0 0 0 0 0 0 µ 22 0 µ 24 0 µ 26 0 µ33 0 µ35 0 0 , 0 µ42 0 µ44 0 µ46 µ µ 0 0 0 0 53 55 0 µ62 0 µ64 0 µ66
mu = Xu u + Xu u
11
(17)
m (v + u1 r ) = Yv v + Yr r + Yv v + Yr r + Y
{
jk }
0 0 0 0 0
=
Izz r = Nv v + Nr r + Nv v + Nr r + N where m is the ship's mass, Izz is the mass moment of inertia. u = u u1, u1 is the ship advance speed at the reference state. The simplified derivatives expressions, e.g., Xu = X / u , Xu = X / u and so on, are known as the hydrodynamic derivatives. In seakeeping, the ship is subjected to various hydrodynamic forces. It is assumed that the six-DOF motions of the ship are independent, the wave amplitude is small, and the ship motion amplitude is also small. The equation of ship oscillation motion can be expressed as:
µ 22 x¨2 + µ42 x¨2 + µ62 x¨2 +
jk
+ Fjk = 0
jk
+ Mjk = 0
0
44
0
53
0
55
62
0
64
0
26
0 46
0
(21)
66
22 x2
+ F22 = 0 x + M42 = 0 42 2 62 x2 + M62 = 0
(22)
(23)
2sin (
F2, µ sin ( t )
F2, cos ( t )
µ42 ( A
2sin (
t )) +
42 A cos ( t ) =
M4, µ sin ( t )
M4, cos ( t )
µ62 ( A
2sin (
t )) +
62
A cos ( t ) =
M6, µ sin ( t )
M6, cos ( t )
t )) +
22
A cos ( t ) =
(24) The added mass and damping of the sway motion can be written as:
µ 22 = 62
,
35
0
42
µ 22 ( A
F2, µ 2
, µ42 =
A M6, = A
M4, µ A
2
, µ62 =
M6, µ A
2
,
22
=
F2, , A
42
=
M4, , A (25)
Therefore, when the ship model is forced to make pure sway motion, the added mass and damping of the sway motion can be derived. The added mass and damping of the other motions can also be obtained by the same method.
(19)
where = 2 f , f is the frequency, A is the amplitude. xk , xk and x¨k are the displacement, velocity and acceleration of the ship model. For the ship model that performs forced oscillation in calm water, the hydrodynamic force is composed of two components, one is the added mass force that proportion to the acceleration, and the other is the damping force that proportion to the velocity. The forced motion equations can be expressed:
x¨k µjk + xk
0
0
By substituting formula (19) and (23) into (22), we can get the following formula:
The ship model is forced to make harmonic motion by using the captive model device. Its motion of the k-th mode can be expressed as:
x¨k µjk + xk
0 0
M62 = M6, µ sin ( t ) + M6, cos ( t )
3.1. Test principle
t)
33
24
F22 = F2, µ sin ( t ) + F2, cos ( t )
To study the test process and reliability of the hydrodynamic testing device, we carried out experiments to measure hydrodynamic coefficients of a rectangular frame and two ship models. The forced sway motion is performed for the rectangular frame. The test results are compared with the numerical results to verify the reliability of the testing device. The forced surge and sway motions are performed for the ship-I model, and a coupled forced surge and sway motion is performed for the ship-II model. The hydrodynamic coefficients of these two ship models are collected and analyzed.
2sin (
0
0
M42 = M4, µ sin ( t ) + M4, cos ( t )
3. Measurement tests of hydrodynamic coefficient
A
0 0
By using the least square method to fit the hydrodynamic force F22 and moment M42 , M62 , we can get the following formula:
where j = 1, 2, 3, 4, 5, 6 represent surge, sway, heave, roll, pitch and yaw motions, respectively. Mj is the inertia force coefficient, 2Nj is the damping force coefficient, Cj is the restoring force coefficient, and Fj is the wave exciting force.
xk = Asin ( t ), xk = A cos ( t ), x¨k =
22
Take the sway motion as an example, we will derive the expression of the added mass and damping. The linear equations of the sway motion can be written as follows:
(18)
Mj x¨j + 2Nj xj + Cj x j = Fj
0
3.2. Test model and equipment installation The three models used in the tests are shown in Fig. 5. The principal dimensions of the test models are summarized in Table 2. The equipment used in the experiment includes carriage system, six-DOF motion mechanism, six-component balance and ultrasonic range finder, as shown in Fig. 6. The performance parameters of the test equipment are shown in Table 3.
(20)
3.2.1. Installation of six-DOF motion mechanism First, the upper and lower parts of the installing frame are rigidly fixed on the carriage beam by bolts. Then, the inverted trapezoid key on the fixed platform of the six-DOF motion mechanism is aligned to the
where k, j = 1, 2, 3, 4, 5, 6 represent surge, sway, heave, roll, pitch and yaw motions, respectively Fjk and Mjk are the hydrodynamic force and the moment and are obtained by tests. For the heave, roll and pitch motions, the restoring force is considered. µjk are the generalized added 53
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(a)
(b)
(c) Fig. 5. Test models: (a) rectangular frame model, (b) ship-I model, (c) ship-II model. Table 2 Principal dimensions of the models. Main parameters
Length (m) Breadth molded (m) Depth molded (m) Design draft (m)
Models Rectangular frame
Ship-I
Ship-II
4.000 0.200 0.200 0.100
3.810 0.682 0.374 0.158
1.350 0.300 0.230 0.040
(a)
(b)
Fig. 7. The installation of the six-DOF motion mechanism. (c)
mechanism is rigidly fixed on the carriage system by the installing frame to guarantee the platform will not shake in the process of the test. The installation is shown in Fig. 7.
Fig. 6. Test equipment: (a) six-DOF motion mechanism, (b) six-component balance, (c) ultrasonic range finder.
3.2.2. Installation of six-component balance The position of the center of the ship model gravity should be found and marked. Then two solid blocks are installed in this position, and the steel base is fixed on the solid blocks by four bolts. The bottom of the six-component balance is connected to the steel base by the steel flange
keyway on the lower part of the installing frame. The fixed platform and the installing frame are further fixed by bolts. At last, the electromechanical control box is fixed on the installing frame and the electric lines are connected to the control system. The six-DOF motion Table 3 Performance parameters of the test equipment. No
Equipment
Type
Performance parameters
1 2 3 4
Carriage system Six-DOF motion mechanism Six-component balance Ultrasonic range finder
SIEMENS 6R24 Huster-SDMM FC-K6D68 UC2000-30 GM-IUR2-V15
Speed: 0.1～8.0 m/s; Accuracy: 0.1% See Table 1 Range: 5kN/50Nm; Accuracy: 0.1% Sensing distance: 80–2000 mm
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Fig. 10. The towing tank laboratory.
Fig. 8. The installation of the six-component balance.
to ensure rigid connection to the ship model. The vertical positions of the center of the ship model and the six-component balance gravity should be overlapped. The installation of the six-component balance is shown in Fig. 8. 3.2.3. Installation of ultrasonic range finder The displacement history of the ship model motion is measured by the ultrasonic range finder which is fixed on the carriage through an extended rod. A baffle is installed on the ship model along its width direction to insure that the sound waves are sent vertically upon it. The measured displacement is the absolute value of the ship model motion. The installation of the ultrasonic range finder is shown in Fig. 9. It can realize the displacement measurement of the surge motions. If we need to measure the displacement of sway and heave motions, only the positions of the ultrasonic range finder and the baffle need to be changed.
Fig. 11. Forced sway motion of the rectangular frame.
3.3. Test procedure
8
The experiment was performed at the towing tank laboratory of the School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), as shown in Fig. 10. The principle dimensions of the towing tank are 175 × 6 × 4 m. The motion parameters of the six-DOF motion mechanism are set before the test. In the process of the captive model tests, the force/displacement sensors are connected with the data acquisition system. The collection
6 4 F(N)
2 0 -2 -4 -6 -8
Ultrasonic range finder
0
5
10
15
20
25
t(s)
Baffle
Fig. 12. The horizontal force-time curve of sway motion with period T = 5s.
frequency of the system is set to 100 Hz. The data acquisition system continues to collect data until the model runs smoothly for 10 to 12 cycles. The second time data collection is carried out after the water surface is calm. 4. Experimental results Fig. 11 shows a forced sway motion of the rectangular frame. The amplitude of the motion is 0.04 m. The period of the motion is from 1s to 5s. Fig. 12 and Fig. 13 show the horizontal force-time curves of sway motion with period T = 5s and T = 1.25s, each curve has been performed a Fourier fit. The forces here represent the hydrodynamic part, and do not include the inertial part. The added mass coefficients are
Fig. 9. The installation of the ultrasonic range finder. 55
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160 120 80 F(N)
40 0 -40 -80 -120 -160
0
1
2
3
4
5
6
7
8
t(s) Fig. 16. The added mass coefficients of surge motion of the ship-I model.
Fig. 13. The horizontal force-time curve of sway motion with period T = 1.25s.
2.0 Numerical result
1.8
Experimental result
LDB
1.6 1.4 1.2 1.0 0.8 0.6
0.0
0.1
0.2
0.3
0.4
0.5
[
0.5
0.6
0.7
0.8 Fig. 17. Forced sway motion of the ship-I model.
Fig. 14. Comparison of sway added mass coefficient between the experimental result and the numerical result for the rectangular frame.
6s. The experimental results of the added mass coefficients are shown in Fig. 16. The horizontal axis represents the dimensionless frequency [b/(2g )]0.5 . Here is frequency and b is the width of the ship model. The vertical axis represents the added mass coefficient µ/ m . Here µ is the added mass and m is displacement of the ship model. According to the experimental results, the added mass coefficient varies in the range of 0.047–0.148. The added mass coefficient reaches a peak value near the dimensionless frequency of 0.4. Fig. 17 shows a forced sway motion of the ship-I model. The amplitude of the motion is 0.06 m. The period of the motion is from 2s to 6s. The experimental results of the added mass coefficients are shown in Fig. 18. According to the experimental results, the added mass coefficient presents strong nonlinear characteristic and varies in the range of 0.85–1.10. Fig. 19 shows a coupled forced motion of swaying and rolling of the ship-II model. The amplitude of the sway motion is 0.02 m. The
calculated by the method in section 3.1. The experimental results of the added mass coefficients are shown in Fig. 14. The horizontal axis represents the dimensionless frequency [B/(2g )]0.5 . Here is frequency and B is the width of the rectangular frame. The vertical axis represents the dimensionless added mass µ/ LBD . Here µ is the added mass, L is the length of the rectangular frame, D is the draft of the rectangular frame and is the water density. The added mass first increases and then decreases. The experimental results agreed well with the numerical results (Luo et al., 2011) except for a few points. It shows that the captive model device can measure the added mass coefficients accurately. Fig. 15 shows a forced surge motion of the ship-I model. The amplitude of the motion is 0.06 m. The period of the motion is from 2s to
Fig. 15. Forced surge motion of the ship-I model.
Fig. 18. The added mass coefficients of sway motion of the ship-I model. 56
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Fig. 19. The coupled forced motion of swaying and rolling of the ship-II model. Fig. 22. The experimental results of the coupled added mass coefficients μ44 + μ42.
Fig. 20. The experimental results of the coupled added mass coefficients μ22 + μ24. Fig. 23. The experimental results of the coupled damping coefficients λ44 + λ42.
It can be seen from the test results that the added mass coefficients of surge motion are within the range of 0.05–0.15, the added mass coefficients of sway motion are within the range of 0.9–1.2, and the added moment of inertia coefficients of roll motion are within the range of 0.05–0.35. These are in agreement with the descriptions in the Principles of Ship (Sheng and Liu, 2004). The reliability of the experiment is proved. 5. Conclusions This paper studies a hydrodynamic testing device of captive model. With this novel device, a physical ship model can be forced to produce independent or coordinated motions in six DOF that cannot be accomplished by a traditional one. First, the principle of the novel captive model test device is given, the test results show that the device can force the ship model to achieve the specified motion well. Next, the hydrodynamic coefficient tests of the rectangular frame and two ship models are carried out by the test device, and the test results agree well with the theoretical values, the reliability of the device is verified. The novel captive model test device can identify the complex hydrodynamic coefficients of the mathematical model simultaneously in a single device, which is of great significance for the efficient prediction of hydrodynamics performance of given ships and offshore structures. In future work, the captive model device will be used to carry out more tests to further verify and improve its accuracy.
Fig. 21. The experimental results of the coupled damping coefficients λ22 + λ24.
amplitude of the roll motion is 3 deg. The period of the motion is from 1.25s to 10s. Fig. 20 shows the experimental results of the coupled added mass coefficients μ22 + μ24. Within the given frequency range, the maximum value of the dimensionless coupled added mass coefficients is 1.12. Fig. 21 shows the experimental results of the coupled damping coefficients λ22 + λ24. The dimensionless coupled damping coefficients show a nonlinear characteristic, the value first increases and then decreases with the dimensionless frequency. Fig. 22 shows the experimental results of the coupled added mass coefficients μ44 + μ42. The dimensionless coupled added mass coefficient varies in the range of 0.18–0.20. Fig. 23 shows the experimental results of the coupled damping coefficients λ44 + λ42. When the dimensionless frequency is 0.31, the maximum coupling damping coefficient is 0.018.
Acknowledgement The authors gratefully acknowledge the financial support from the 57
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National Natural Science Foundation of China (NSFC) under Project Number 51679097&51609090&51709120, and the financial support from the China Postdoctoral Science Foundation under Project Number 2018M632867.
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