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Citation: Zhao, B.; Sun, J.; Zhang, D.;
Zhu, K.; Jiang, H. Dynamic Analysis
of Underwater Torpedo during
Straight-Line Navigation. Appl. Sci.
2023,13, 4169. https://doi.org/
10.3390/app13074169
Academic Editor: Francesca
Scargiali
Received: 2 March 2023
Revised: 18 March 2023
Accepted: 22 March 2023
Published: 24 March 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
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4.0/).
applied
sciences
Article
Dynamic Analysis of Underwater Torpedo during
Straight-Line Navigation
Bowen Zhao 1,2, Jiyuan Sun 2, Dapeng Zhang 1, * , Keqiang Zhu 3and Haoyu Jiang 4
1Ship and Maritime College, Guangdong Ocean University, Zhanjiang 524088, China
2Ocean College, Zhejiang University, Zhoushan 316021, China
3Faculty of Maritime and Transportation, Ningbo University, Ningbo 315211, China
4School of Electronics and Information Engineering, Guangdong Ocean University, Zhanjiang 316021, China
*Correspondence: zhangdapeng@gdou.edu.cn
Abstract:
Torpedoes play an irreplaceable role in naval warfare; therefore, it is significant to study
the dynamic response of the direct navigation of torpedoes. In order to study the dynamic response
of torpedoes under different Munk moment coefficients, the dynamic equation of torpedoes is
established based on the momentum theorem and the momentum moment theorem. The linear
motion mathematical model of torpedoes is obtained. The relationship between the torpedo and
the Munk moment coefficient is derived. The straight-line motion model of the torpedo under
different Munk moments is established, and the dynamic properties of the space motion of the
torpedo are analyzed. It is found that the Munk moment coefficient increase will lead to an increase
in the deflection of the torpedo’s direct motion on each degree of freedom, and the Munk moment
coefficient is related to the additional mass matrix. During the design of the torpedo, the added mass
should be reduced by changing the shape of the torpedo as much as possible so as to reduce the pitch
moment, yaw, and roll moments of the torpedo.
Keywords: torpedo; dynamic response; Munk moment; lumped-mass method
1. Introduction
Since the 21st century, underwater vehicles such as torpedoes have gradually become
a significant research goal [
1
,
2
]. Underwater vehicles with marine resources exploration,
marine environmental observation, and autonomous navigation can be used for underwater
military reconnaissance missions [
3
,
4
]. In recent years, people vigorously developed
underwater vehicles, such as torpedoes. With the development of ocean engineering and
technology, torpedoes play an irreplaceable role in naval warfare. Its mathematical model
has an important impact on its dynamic performance and accurate control, navigation,
and guidance [
5
–
8
]. Different mathematical models or numerical methods have been
proposed to study the hydrodynamic characteristics of underwater vehicles. Wang et al. [
9
]
developed a mathematical model for an autonomous underwater vehicle based on the
computational fluid dynamics (CFD) method and strip theory, which could simulate an
underwater vehicle’s dynamics and precise control. Piotr Szymak et al. [
10
] studied the
problem of mathematical modeling of an underwater vehicle with undulating propulsion.
The model can be used for the initial tuning of a control system of new underwater vehicles.
Xu et al. [
11
] proposed a support vector machine (SVM)-based identification method for
modeling the nonlinear dynamics of underwater vehicles and validated it by using a typical
torpedo-shaped autonomous underwater vehicle. The SVM was suitable for nonlinear
function regression, and the simulation results showed its high generalization performance.
At present, research on the force of underwater vehicles in straight-line and rotary
motion is sufficient [
12
,
13
]. Most of the research is based on the CFD method. The applica-
tion of CFD in the marine industry is growing. This numerical method has high accuracy,
Appl. Sci. 2023,13, 4169. https://doi.org/10.3390/app13074169 https://www.mdpi.com/journal/applsci
Appl. Sci. 2023,13, 4169 2 of 12
making it possible to apply them to the force and moment calculations of underwater vehi-
cles [
14
–
18
]. Tyagi and Sen [
19
] studied the fluid drag coefficients and moment coefficients
of underwater vehicles in transverse flows. These hydrodynamic coefficients were used
to predict the maneuvering motion of underwater vehicles. The CFD method provided a
new way to solve hydrodynamic coefficients. The research also showed that the changes
in hydrodynamic force and moment is nonlinear. Thanh-Long and Duc-Thong [
20
] also
used the CFD method to investigate the hydrodynamic properties of a torpedo-shaped
underwater glider. Profiling equations were used to establish the physical model of the
underwater glider. The CFD results depicted that the nose of the underwater glider was
under great pressure when moving in the water. The drag and lift coefficients of the under-
water glider were influenced by the combined effects of velocity and the angle of attack.
Ling et al. [
21
] performed a regression analysis of the hydrodynamic coefficients of a bare
submarine structure in linear motion relative to the free surface at different submergence
depths. They also described the hydrodynamic coefficients related to the forward speed
and submergence depth as a segmented function. The results showed that the functions
had their own set of regression coefficients. The submergence depth could control the am-
plitude of the curve. These regression coefficients generally decreased with the exponential
decay of power law and submergence depth. Zhang et al. [
22
] investigated the hydrody-
namic properties of an underwater vehicle suspended with a torpedo. The effects of wave
parameters were studied. The results showed that the heave motion and moving speed of
the torpedo were strongly affected by the wave motion. The fluid environment was directly
related to the hydrodynamic performance of the towing system of underwater vehicles
and torpedoes. Kilavuz et al. [
23
] investigated the flow characteristics of an unmanned
underwater vehicle (UUV) with a commonly used Myring profile using PIV and CFD. The
utilized CFD approach especially yielded excellent agreement with the PIV measurements
with discrepancy.
Underwater vehicles, such as torpedoes, will produce unstable pitch and yaw moments
when sailing, namely the Munk moment, which will increase in proportion to the square of
the speed and bring about motion stability problems for torpedoes. The Munk moment
refers to the two equal and opposite forces generated by an object moving at a certain angle
of attack (or drift angle) in a steady, straight line in an ideal fluid motion in the front and rear
halves [
24
]. Hakamifard and Rostami [
25
] compared the difference between the numerical
simulations and analytical formulas in calculating the additional mass coefficients of a
torpedo under Munk moments. The results showed that the numerical simulations were
more convenient for the calculation of the Munk moment. It was also found that the Munk
moment seriously affected the motion stability of the torpedo [
26
]. According to various
pieces of literature, despite the influence of hydrodynamic coefficients on the hydrodynamic
properties of torpedoes and underwater vehicles being reported, there are few studies on
the effects of Munk moments. The analysis of the Munk moment is mainly based on the
analysis of other hydrodynamic coefficients. For instance, Tyagi and Sen [
19
] provided
a possible explanation for the different hydrodynamic coefficients between the CFD and
semi-empirical methods that, at small angles of attack where the inviscid nature of the
flow dominates, the transverse force is generally small and the moment is predominantly
a Munk moment. Anderson and Chhabra [
24
] also believed that a major contribution
to a hydrodynamic coefficient M
w
was a Munk moment. The above investigations are
conjectures about the Munk moment. Although they have been proven to be correct,
there is still a lack of specific influence of the Munk moment on the dynamic properties of
navigation vehicles. This is also the main goal of this article.
In this paper, the torpedo is simplified as a rigid body with six degrees of freedom. The
research on Munk moments is to simplify Munk moments to dimensionless coefficients and
then add Munk moment coefficients to the motion equations of the torpedo. Hydrodynamic
force and other forces are taken as external forces. The kinematics equations are established
according to the transition matrix, and the dynamic equations of the torpedo are established
based on the theorem of momentum. The linear motion mathematical model of torpedoes
Appl. Sci. 2023,13, 4169 3 of 12
is synthesized, and the relationship between the traditional torpedo and the Munk moment
coefficients is derived. The simulation platform used is the OracFlex software. The direct
navigation model of the torpedo under different Munk moments is established, and the
dynamic response of the space motion of the torpedo is analyzed.
2. Derivation of Munk Moment
A torpedo sailing in an unsteady current will undergo an unstable moment called
the Munk moment. In order to calculate the Munk moment, it is necessary to make some
assumptions. The torpedo is regarded as a 6 DOF body, and the mass and mass distribution
remain unchanged during navigation. The shape of the torpedo is a revolving body,
symmetrical in the middle-longitudinal section and middle-transverse section. The product
of inertia is not taken into account, which means J
xy
=J
yx
=J
xz
=J
zx
=J
yz
=J
zy
= 0. During
torpedo navigation, the density and pressure of the fluid remain unchanged.
The position coordinates and linear velocity are defined as follows:
r=[x0,y0,z0]T,v=vx,vy,vzT,v0=vx0,vy0,vz0T(1)
The attitude angle and angular velocity are defined as follows:
Ω=[θ,ψ,φ]T,ω=ωx,ωy,ωzT(2)
The rudder angle and trajectory angle are defined as follows:
δ=[δe,δr,δd]T,∠=[Θ,Ψ,Φc]T(3)
The motion equations of the torpedo include kinetic equations and kinematic equa-
tions. According to the theorem of momentum, the kinetic equation of the torpedo can
be obtained, as depicted in Equations (4)–(9) (the torpedo conducts small maneuvering
motion, neglects the second-order terms of the torpedo’s motion parameters, and regards
the position of the center of mass as a small first-order quantity).
(m+λ11).
vx=T−CxS
1
2ρv2S−∆Gsin θ(4)
(m+λ22).
vy+(mxc+λ26 ).
ωz+mvxωz=1
2ρv2SCα
yα+Cδe
yδe+Cωz
yωz−∆Gcos θcos φ(5)
(m+λ33).
vz−(mxc+λ35 ).
ωy−mvxωy=1
2ρv2SCβ
zα+Cδr
zδr+Cωy
zωy+∆Gcos θsin φ(6)
(Jxx +λ44 ).
ωx−mvxycωy+zcωz=1
2ρv2SLmβ
xβ+mδr
xδr+mδd
xδd+mωx
xωx+
mωy
xωy+Gcos θ(ycsin φ+zccos φ)+∆Mxp
(7)
Jyy +λ55.
ωy−(mxc−λ35 ).
vz+mxcvxωy=1
2ρv2SLmβ
yβ+mδr
yδr+mωx
yωx+
mωy
yωy−G(xccos θsin φ+zcsin θ)(8)
(Jzz +λ66).
ωz+(mxc+λ26 ).
vy+mxcvxωz=1
2ρv2SLmα
zα+mδe
zδe+mωz
zωz+
G(ycsin θ−xccos θcos φ)(9)
The kinematic equations can be obtained based on the transition matrix, as depicted
in Equations (10)–(21). .
θ=ωysin φ+ωzcos φ(10)
Appl. Sci. 2023,13, 4169 4 of 12
.
ψ=ωysec θcos φ−ωzsec θsin φ(11)
.
φ=ωx−ωytan θcos φ+ωztan θsin φ(12)
.
x0=vxcos θcos ψ+vy(sin ψsin φ−sin θcos ψcos φ)+vz(sin ψcos φ−sin θcos ψsin φ)(13)
.
y0=vxsin θ+vycos θcos φ−vzcos θsin φ(14)
.
z0=−vxcos θsin ψ+vy(cos ψsin φ+sin θsin ψcos φ)+vz(cos ψcos φ−sin θsin ψsin φ)(15)
v2=v2
x+v2
y+v2
z(16)
α=−arctanvyvx(17)
β=arctanvzqv2
x+v2
y(18)
sin Θ=sin θcos αcos β−cos θcos φsin αcos β−cos θsin φsin β(19)
sin Ψcos Θ=sin ψcos θcos αcos β+cos ψsin φsin αcos β+
sin ψsin θcos φsin αcos β−cos ψcos φsin β+sin ψsin θsin φsin β(20)
sin Φccos Θ=sin θcos αsin β−cos θcos φsin αsin β+cos θsin φcos β(21)
The meaning of the above symbols is shown in Abbreviations.
The full kinematic parameters for solving the spatial motion of the torpedo can be
obtained from the above 21 equations. The Munk moment is added to the torpedo dynamics
equations in the form of dimensionless coefficients. The force of the ideal fluid on the
torpedo can be divided into three parts: the force of the fluid, the moment generated by the
steady motion of the torpedo with constant rotation and steady motion, and the moment
generated by the unsteady motion of the torpedo. The Munk moment is such a moment
generated by the unstable motion of the torpedo.
Mαix =(λ22 −λ33 )vyvz−λ32v2
y+λ23v2
z+λ21vz−λ31 vyvx
Mαiy =(λ33 −λ11)vzvx−λ13v2
z+λ31v2
x+(λ32vx−λ12 vz)vy
Mαiz =(λ11 −λ22)vxvy−λ21v2
x+λ12v2
y+λ13vy−λ23 vxvz
(22)
Mαix =(λ22 −λ33 )vyvz
Mαiy =(λ33 −λ11)vzvx
Mαiz =(λ11 −λ22)vxvy
(23)
As the angle of attack
α=−arctan vx
vy
, the sideslip angle
β=−arctan vz
qv2
x+v2
y
, assuming
that the angle of attack and the sideslip angle are both small, the Munk moment can be
rewritten as the following:
[M]=1
2v2
sin 2β0 0
0 sin 2β0
0 0 sin 2α
·
λ22 −λ33 0 0
0λ33 −λ11 0
0 0 λ11 −λ22
(24)
Appl. Sci. 2023,13, 4169 5 of 12
3. Validation Cases
In order to validate the correctness of the above derivation and OrcaFlex software, we
carried out three case studies. The first case is the calculation of the Munk moment of a
semi-ellipsoid. The dimensions of the semi-ellipsoid are as follows: the length of the long
half-axis is 1 m, the length of the short transverse half-axis is 0.1 m, and the length of the
short vertical half-axis is 0.1 m. The added mass of the semi-ellipsoid is m
11
= 0.445 kg
and m
22
= 20.906 kg. According to the results of Reference [
27
], using a lift-free model,
the Munk moment of the semi-ellipsoid with a sideslip angle of 5
◦
was calculated under
inviscid conditions, and the result was 1.687 N
·
m. The Munk moment, calculated using the
above-derived formula, is 1.794 N·m. Both results are very close, with an error of 6.34%.
The second case is the calculation of the Munk moment of a Wigley ship. The Wigley
ship is a mathematical model commonly used in ship research internationally. The mathe-
matical definition of the standard Wigley ship is as follows:
y=2B"1
4
−x
Lpp 2#1−z
D2(25)
where L
pp
is the length between the perpendiculars, Bis the beam of the ship, Dis the
draft of the ship, and x,y, and zrepresent the three-dimensional point coordinates of the
ship-type value. In this paper, Lpp = 2 m, B= 0.2 m, and D= 0.1 m.
The added mass of the Wigley ship is m
11
= 0.335 kg and m
22
= 24.767 kg. According
to the results of Reference [
27
], using a lift-free model, the Munk moment of the Wigley
ship with a sideslip angle of 5
◦
was calculated under inviscid conditions, and the result was
2.035 N
·
m. The Munk moment, calculated using the above-derived formula, is 2.198 N
·
m.
Both results are very close, with an error of 8.01%.
The third case is the calculation of the Munk moment of a trimaran with a square
stern. The length, beam, and draft of the main hull of the trimaran are 7 m, 0.56 m, and
0.28 m, respectively. The length, beam, and draft of the side hull of the trimaran are 2.5 m,
0.12 m, and 0.1 m, respectively. The distance from the side hull to the main hull is 0.673 m,
and the longitudinal distance from the side hull is 2.25 m. The added mass of the trimaran
is
m11 = 10.855 kg
and m
22
= 713.369 kg. According to the results of Reference [
27
], using
a lift-free model, the Munk moment of the Wigley ship with a sideslip angle of 5
◦
was
calculated under inviscid conditions, and the result was 52.896
·
m. The Munk moment,
calculated using the above-derived formula, is 56.389 N
·
m. Both results are very close, with
an error of 6.61%.
The above three cases have proved the reliability of the Munk moment equation
derived in this paper.
4. Numerical Model
4.1. The Establishment of Simulation Model
The torpedo in this paper is modeled by the commercial software, OrcaFlex. The
simulated scene is in a lentic environment, ignoring the effects of the wind, waves, and
current, and the depth of water is 100 m. The torpedo model has been simulated by an
OrcaFlex 6D buoy with a total length of 13.2 m. The model was divided into four sections,
with the length from the head to the tail being 0.5 m, 0.7 m, 10.5 m, and 1.5 m, respectively.
The outer diameters are, respectively, 1.0 m, 1.5 m, 2.0 m, and 1.0 m. An elastoplastic solid
fairing with an axial and normal stiffness of 100 KN/m
3
was arranged on the head of a
torpedo. The overall centroid coordinates of the torpedo are (0 m, 0 m, and 0 m). In the
tail of a torpedo, there are four tail wings with a wingspan of 1.5 m and a chord length of
1.5 m. The local coordinate origin of the tail fin in the torpedo body coordinate system is,
respectively, (
−
6 m, 0 m,1.6 m), (
−
6 m, 1.6 m, 0 m), (
−
6 m,
−
1.6 m, 0 m), (
−
6 m,
−
1.6 m,
0 m), and (
−
6 m,
−
1.6 m, 0 m). The lift and damping coefficients of the tail wings have
a low impact on the dynamic response of the torpedo. In order to save calculation time,
these two coefficients are set to 0. The coordinate of the initial position of the torpedo
Appl. Sci. 2023,13, 4169 6 of 12
in the global coordinate system is (0 m, 0 m,
−
5 m). The fixed thrust T = 50 KN in the
x
-direction is applied at the origin of the mine coordinate system, the normal and axial
drag force coefficients of the mine body are 1.0 and 0.1, respectively, and the additional
mass coefficients are both 1.0. The total simulation time is 100 s. The torpedo model is
shown in Figure 1.
Figure 1. Schematic diagram of torpedo model.
4.2. Results of Torpedo Direct Navigation under Munk Moment of 0
With a fixed thrust T = 50 KN, the torpedo’s Munk moment coefficients are changed to 0,
0.02, 0.04, 0.06, 0.08, and 0.10, respectively, and the torpedo-related motion parameters can
be achieved by simulating the torpedo direct sailing motion and observing the change of its
sailing state. Under the simulation condition that the Munk moment coefficient is 0, then the
velocity diagram in the
x
-direction, horizontal displacement diagram in the
x
-direction, the
depth-fixed navigation diagram and the lateral displacement diagram of the model are drawn
to verify the effectiveness of the torpedo direct navigation motion model.
It can be seen from Figure 2that during the simulation time, the speed of the torpedo
reached a stable state, and the steady speed was about 20.76 m/s, which is consistent with
the sailing speed of general torpedoes. The sailing distance of the torpedo reaches about
2000 m in the x-direction and basically shows a linear growth. The torpedo can maintain a
stable navigation depth at this speed, which conforms to the actual working situation of the
torpedo. The lateral displacement of the torpedo is basically 0 within the simulated time of
100 s. That is to say, the torpedo basically maintains straight-line navigation in the
x0O0y0
plane. The above conclusions well verify the effectiveness of the torpedo direct navigation
model; therefore, the following simulation can be carried out based on this model.
Appl. Sci. 2023,13, 4169 7 of 12
Figure 2.
Validation results: (
a
) torpedo’s speed in the x-direction; (
b
) horizontal displacement in the
x-direction; (c) torpedo’s navigation depth; (d) torpedo’s lateral displacement.
5. Results and Discussion
5.1. Navigation Depth and Pitch Angles under Different Munk Moment Coefficients
As can be seen from Figure 3, under the same Munk moment coefficient, the navigation
depth shows an increasing trend in the time domain. At the same time, with the increase
of the Munk moment coefficient, the torpedo will increase its navigation depth, and
the deviation from the initial navigation depth will increase gradually. The two curves
of the 0 and 0.02 Monk moment coincide. In Figure 4, the pitch angle also shows an
increasing trend in the time domain. When the Munk moment coefficients are 0.06, 0.08,
and 0.1, the navigation depth of the torpedo will suddenly drop from 50 s. The larger the
Munk moment, the faster the torpedo will drop. The increase in pitch angle precedes the
decrease in navigation depth. When the Munk moment coefficient is 0.10, the torpedo
has already touched the seabed at about 81.7 s. This phenomenon can be observed in
Figures 3and 4
, where the curve presents obvious nonlinearity. The reasonable explanation
for this phenomenon is that the pitch moment is strongly affected by the Munk moment
and increases with the increase of the Munk moment coefficient. The increase in the pitch
moment will lead to an increase in the pitch angle, and the thrust Tacts in the negative
direction of the x-axis, thus increasing the navigation depth.
Appl. Sci. 2023,13, 4169 8 of 12
Figure 3. Navigation depth under different Munk moment coefficients.
Figure 4. Pitch angle under different Munk moment coefficients.
5.2. Lateral Displacement and Yaw Angles under Different Munk Moment Coefficients
It can be seen from Figure 5that in the time domain, the lateral displacement under
the same Munk moment coefficient shows an increasing trend. The two curves of the 0 and
0.02 Monk moment coincide. At the same time, with the increase of the Munk moment
coefficient, the lateral displacement of the torpedo will increase, and the distance from the
initial motion direction will gradually increase. It can be seen from Figure 6that in the time
domain, the yaw angle under the same Munk moment coefficient also shows an upward
trend. At the same time, with the increase of the Munk moment coefficient, the yaw angle
of the torpedo will increase. In other words, the deviation angle from the initial movement
direction will gradually increase. When the Munk moment coefficient is 0.10, the torpedo
produces an abnormal yaw angle at about 81 s because the torpedo contacts the seabed.
The reason for the above phenomenon is that the yaw moment is strongly affected by the
Munk moment coefficient and increases with the increase of the Munk moment coefficient.
It makes the torpedo yaw. More importantly, due to the yaw moment of the torpedo, the
yaw angle of the torpedo increases and the lateral displacement increases.
Appl. Sci. 2023,13, 4169 9 of 12
Figure 5. Lateral displacement under different Munk moment coefficients.
Figure 6. Yaw angles under different Munk moment coefficients.
5.3. Roll Angles under Different Munk Moment Coefficients
It can be seen from Figure 7that the roll angle presents an increasing trend under
the same Munk moment coefficients in the time domain. The two curves of the 0 and
0.02 Monk moment coincide. At the same time, with the increase of the Munk moment
coefficient, the amplitude of the roll angle of the torpedo will increase, and the amplitude
of the rotation angle of the mine body around the x-axis will gradually increase. This is
because the rolling moment is affected by the Munk moment coefficient and increases with
the increase of Munk moment coefficients. The roll moment is the cause of the roll angle of
the torpedo, and thus, the roll angle increases.
Figure 7. Roll angles under different Munk moment coefficients.
Appl. Sci. 2023,13, 4169 10 of 12
6. Conclusions
(1) Based on the above derivation of the Munk moment formulas in classical torpedo nav-
igation mechanics and the comparison of Munk moment expression in the software,
OrcaFlex, it can be seen that the Munk moment coefficient mentioned in this paper
is the product of the additional mass matrix and the mass of the displacement of the
torpedo. Thus, the Munk moment coefficient is related to the additional mass matrix
of the torpedo.
(2)
Under the same Munk moment coefficient, the navigation depth shows an increasing
trend in the time domain. At the same time, the navigation depth of the torpedo will
increase with the increase of the Munk moment coefficient, and the deviation from
the initial navigation depth will gradually increase. Under the same Munk moment
coefficient, the pitch angle also increases in the time domain. This is because the pitch
moment is affected by the Munk moment coefficient and increases with the increase
of the Munk moment coefficient.
(3)
Under the same Munk moment coefficient, the lateral displacement increases in
the time domain. At the same time point, with the increase of the Munk moment
coefficient, the lateral displacement of the torpedo will increase numerically, and the
distance from the initial motion direction will increase gradually. Under the same
Munk moment coefficient, the yaw angle increases in the time domain. This is because
the yaw moment is affected by the Munk moment coefficient and increases with the
increase of the Munk moment coefficient.
(4) Under the same Munk moment coefficient, the roll angle increases in the time domain.
At the same time, the amplitude of the roll angle of the torpedo will increase with
the increase of the Munk moment coefficient, and the amplitude of the rotation angle
of the mine body around the x-axis gradually increases. This is because the rolling
moment is affected by the Munk moment coefficient and increases with the increase
of the Munk moment coefficient.
(5)
According to classical torpedo navigation mechanics, the Munk moment is composed
of pitch, roll, and yaw moments. With the expression of the Munk moment in OrcaFlex,
it can only be concluded that the combined moment increases with the increase of the
Munk moment coefficient, yet, the increase or decrease of each partial moment cannot
be determined. However, according to conclusions (2), (3), and (4), the three moments
can all increase with the increase of the Munk moment coefficient.
(6)
In conclusion, the increase of the Munk moment coefficient will lead to an increase
in deflection of the torpedo’s direct sailing motion at each degree of freedom, which
has an adverse effect on the torpedo’s accurate strike. According to conclusion (1),
the Munk moment coefficient is related to the additional mass matrix. Therefore, in
the design of the torpedo, we should reduce the additional mass of the torpedo by
changing the shape of the torpedo so as to reduce the Munk moment coefficient of the
torpedo and reduce the pitch, yaw, and roll moments of the torpedo.
Author Contributions:
Conceptualization, D.Z. and B.Z.; methodology, D.Z. and H.J.; software, K.Z.;
validation, D.Z. and K.Z.; formal analysis, D.Z. and B.Z.; investigation, K.Z. and B.Z.; resources, D.Z.;
data curation, K.Z.; writing—original draft preparation, D.Z.; writing—review and editing, D.Z.;
visualization, K.Z. and J.S.; supervision, D.Z. and J.S.; funding acquisition, D.Z. and H.J. All authors
have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the Program for Scientific Research Start-up Funds of
Guangdong Ocean University, grant number 060302072101, the Zhanjiang Marine Youth Talent
Project—Comparative Study and Optimization of Horizontal Lifting of Subsea Pipeline, grant num-
ber 2021E5011, and the National Natural Science Foundation of China, grant number 62272109.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Appl. Sci. 2023,13, 4169 11 of 12
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
Symbol Meaning
mThe torpedo mass
LThe length of the torpedo
λThe additional mass
.
vx,.
vy,.
vzThe acceleration of the torpedo along the x-, y-, z-directions in the torpedo body
coordinate system
TThrust
CxS Resistance factor with maximum cross section Sas characteristic area
ρDensity of fluid
vTorpedo speed
SThe maximum cross-sectional area of the torpedo
∆GNegative buoyancy of torpedo, ∆G=G−B
xc,yc,zcCentroid position coordinates
.
ωx,.
ωy,.
ωzAngular acceleration of torpedo along x-, y-, z-directions in torpedo body
coordinate system
Cα
y,Cδe
yPosition derivative of torpedo lift factor with respect to the angle of attack αand
position derivative with respect to horizontal rudder angle δe
Cβ
z,Cδr
zPosition derivative of side force factor to sideslip angle βand position derivative
to vertical rudder angle δr
Cωz
yThe dimensionless factor of the rotational derivative of diagonal velocity ωzof lift
Cωy
zThe dimensionless factor of rotation derivative of angular velocity ωyof side force;
ωDimensionless angular velocity
mβ
x,mβ
yPosition derivative of the rolling moment factor and yaw moment factor of the
torpedo to sideslip angle β
mα
zPosition derivative of pitching moment Factor of the torpedo to attack
angle α
mδr
x,mδd
xPosition derivative of torpedo rolling moment factor with respect to vertical
rudder angle δrand differential rudder angle δd
mδr
yPosition derivative of torpedo yaw moment factor to vertical rudder angle δr
mδe
zPosition derivative of torpedo pitching moment factor to horizontal rudder angle δe
mωx
x,mωy
xRotation derivative of rolling moment factor of ωxand ωy
mωx
y,mωy
yRotational derivative of yaw moment factor of ωxand ωy
mωz
zRotation derivative of pitch moment factor of ωz
∆Mxp Unbalanced moment possibly generated by a single propeller and an unbalanced
counter-rotating propeller
.
θPitch angular velocity
.
ψYaw rate
.
x0,.
y0,.
z0The velocity of the torpedo in the x-, y-, z-directions in the ground coordinate system
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