2 Checking a Reachability of Manoeuvring Air Targets by a Missile on Pursuit Courses (An Attack from a Back Hemisphere)
Tải bản đầy đủ - 0trang
140
D. Gapiński and Z. Koruba
xW = 1000 m, yW = 2000 m, zW = 50 m. The distance to a target in the time of its
detection by OGSS amounted to r = 2236 m. The distance to the target at the time
of missile launch amounted to r = 2822 m (Figs. 21, 22, 23 and 24).
Fig. 21 Flight-path of a target and a missile on a pursuit courses
Fig. 22 Flight-path of missile and target from Fig. 21 presented in vertical and horizontal position
Analysis of Reachability Areas of a Manoeuvring …
141
Fig. 23 Change of distance between missile and target for simulation presented in Fig. 21
Fig. 24 Cross-sectional overload of a missile during its flight on the basis of trajectory presented
in Fig. 21
4.2.2
Simulation of Homing for an Example 6
A simulation conducted for a velocity of target flight vC = 300 m ̸s. The location of
a target towards rocket launcher in the time of its detection by OGSS amounted to
xW = 1150 m, yW = 2000 m, zW = 50 m. The distance to a target in the time of its
detection by OGSS amounted to r = 2307 m. The distance to the target at the time
of missile launch amounted to r = 2928 m (Figs. 25, 26, 27 and 28).
142
D. Gapiński and Z. Koruba
Fig. 25 Flight-path of missile and target after crossing the border of reachability
Fig. 26 Missile velocity during its flight on the basis of trajectory presented in Fig. 25
Fig. 27 Cross-sectional overload of a missile during its flight on the basis of trajectory presented
in Fig. 25
Analysis of Reachability Areas of a Manoeuvring …
143
Fig. 28 Angular velocities of a missile during its flight on the basis of trajectory presented in
Fig. 25
5 Summary and Conclusion
The results of a simulation show that the developed minimal-time algorithm after
taking into account all delays T that influence missile launch and after taking into
account the tactical and technical parameters of a modernised ZU-23-2MRE system
is able to effectively determine the possibilities of reaching a manoeuvring air target
with a satisfactory accuracy for homing conditions. The conducted analysis also
revealed that minimum target detecting spheres performed by OGSS for missiles
coincide with the destruction zone for artillery ﬁre unit (double cannons 2A14). In
case of using the system complementarity of these zones enables effective use of ﬁre
in the airspace almost in the whole sector (in the distance from 5500 m from the
system). From the research data presented in point 4.1 it can be concluded that the
closer border of destruction zone for missile systems during shooting on rendezvous
courses is mainly limited by cross-sectional overload. A further border of
destruction zone on rendezvous courses will be limited to, above all, a target
radiation, for which the homing head of a missiles is still able to detect and track it.
The results presented in 4.2 revealed that in case of pursuit courses the closer border
of destruction zone will be limited mainly by maximum angular velocity which
allows effective tracking the target by the head and cross-sectional missile overload.
Further destruction zone border on pursuit courses will be limited mainly by
controlled flight range rockets, limited working hours of drive system and the
powder pressure battery, as well as meeting the minimum allowable speed missile
with the aim of enabling activation of the fuse of a head.
The graphs presented on Figs. 7, 9, 15 and 17 show that the developed algorithm
effectively imposes restrictions resulting from the aerodynamic missile capabilities,
144
D. Gapiński and Z. Koruba
shaping the maximum angular velocity of the missile at a level that the
cross-sectional overloads does not exceed the limit values.
Due to the fact that the proposed algorithm has no time-consuming complete
missile dynamics equations, it can be used for reachability analysis of the so-called
air targets at real-time mode. In the results of the study, time constant T1 includes
the delay resulting from the processing carried out by a number of mathematical
operations and it has no signiﬁcant impact on the overall analysis process.
References
1. Gapiński, D., Koruba, Z., Krzysztoﬁk, I.: The model of dynamics and control of modiﬁed
optical scanning seeker in anti-aircraft rocket missile. Mechanical System and Signal
Processing, Vol. 45 (2014), Issue 2, pp 433–447, ISSN 0888-3270
2. Gapiński, D., Krzysztoﬁk, I.: Software selection of air targets detected by the infrared skaning
and tracking seeker. Scientiﬁc Journal of Polish Naval Academy, Vol. 55 (2014), pp. 39–50,
PTMTS, ISSN 0860-889X
3. http://www.zmt.tarnow.pl/pl/oferta/systemy-przeciwlotnicze/23mm-przeciwlotniczy-morskizestaw-artyleryjsko-rakietowy-zu-23-2mr.html#1
4. Koruba, Z., Osiecki, J.W.: Structure, dynamics and navigation short-range missiles, Kielce
University of Technology, script no. 348, ISSN 0239-6386
5. Koruba, Z.: The dynamics and control of the gyroscope on board a flying object, Monographs,
Studies, Thesis 25, Kielce University of Technology (2001), PL ISSN 0239-4979
6. Krzysztoﬁk, I.: The Dynamics of the Controlled Observation and Tracking Head Located on a
Moving Vehicle. Solid State Phenomena, Vol. 180, pp. 313–322, Trans Tech Publications,
Switzerland, ISSN 1012-0394, (2012)
7. Milewski, S., Kobierski, J.W., Chmielewski, M.: Simulators Marine sets of rocket-artillery,
Scientiﬁc Journal of Polish Naval Academy, ROK LIII nr 3 (190) 2012, ss. 87–100, ISSN
0860-889X
8. Milewski, S., Kobierski, J.W.: Staff training of the marine reconnaissance-ﬁre systems with the
use of simulator TR ZU-23-2MR, Mechanics in Aviation, Vol. I, pp. 241–254, PTMTS ISBN
987-83-932107-2-5
9. Palumbo, N. F, Blauwkamp, R. A., Lloyd, J.M.: Basic Principles of Homing Guidance. Jons
Hopkins APL Technical Digest, Vol. 29, No. 1, 2010
10. Sonawane, H.R., Mahulikar, S.P.: Effect of Missile Turn Rate on Aircraft Susceptibility to
Infrared-Guided Missile. Journal of Aircraft Vol. 50, No. 2, 2013, DOI:10.2514/1.C031902
11. Stefański, K., Grzyb, M.: Comparison of Effectiveness of Antiaircraft Missile Homing
Controlled by Rotary Executive System, Problems of Mechatronics. Armament, Aviation,
Safety Engineering, Vol. 4, Nr 4(14), Warsaw, 2013, pp. 27–39, ISSN 2081-5891
12. General Staff of the Polish Army – Inspectorate Logistics: Antiaircraft missile Grom-I technical description and the operating instructions, Warsaw 1996
13. Yanushevsky, R.: Guidance of unmanned aerial vehicles, CRC Press Taylor & Francis Group,
U.S. 2011, ISBN 978-1-4398-5095-4
Angular Velocity and Intensity Change
of the Basic Vectors of Position Vector
Tangent Space of a Material System
Kinetic Point—Four Examples
Katica R. (Stevanović) Hedrih
Abstract Chapter starts from author’s previous published results about nonlinear
transformations of coordinate systems, from afﬁne space to functional-nonlinear
curvilinear coordinate system and corresponding geometrical and kinematical
invariants along nonlinear transformations of their coordinates from one to other
coordinate system. In a curvilinear coordinate system, coordinates of a geometrical
or kinematical point are not equal as coordinates of its’ corresponding position
vector. Expressions of basic vectors of tangent space of kinetic point vector position
in generalized curvilinear coordinate systems for the cases of orthogonal curvilinear
coordinate systems are derived and four examples are presented. Next, expressions
of change of basic vectors of tangent space of kinetic point vector position with
time, also, are done. In this chapter, new and original expressions of angular
velocity and velocity of dilatations of each of the basic vectors of tangent space of
kinetic point vector position, in four orthogonal curvilinear coordinate systems are
presented. List of these curvilinear coordinate systems are: three-dimensional
elliptical cylindrical curvilinear coordinate system; generalized cylindrical bipolar
curvilinear coordinate system; generalized elliptical curvilinear coordinate system,
and generalized oblate spheroidal curvilinear coordinate system.
1 Introduction
In author’s previously published paper [3–9] difference between linear and nonlinear transformation of coordinates is analyzed and discussed. Also, difference
between afﬁne, linear space and nonlinear, functional space with curvilinear
coordinate system are pointed out.
K.R. (Stevanović) Hedrih (✉)
Department of Mechanics, Mathematical Institute of Serbian Academy of Science
and Arts (SANU), Knez Mihailova 26/III, Belgrde, Serbia
e-mail: khedrih@sbb.rs; khedrih@eunet.rs
K.R. (Stevanović) Hedrih
University of Niš, ul Vojvode Tankosića 3/22, 18000 Niš, Serbia
© Springer International Publishing Switzerland 2016
J. Awrejcewicz (ed.), Dynamical Systems: Theoretical
and Experimental Analysis, Springer Proceedings
in Mathematics & Statistics 182, DOI 10.1007/978-3-319-42408-8_12
145
146
K.R. (Stevanović) Hedrih
In real three-dimensional coordinate system, position vectors of the material–kinetic points of a material system constrained by geometrical holonomic stationary
and nonstationary real constraints (see Refs. [1, 2, 10–13]), are denoted by ρ⃗ðαÞ ðqÞ,
α = 1, 2, 3, . . . , N and each as functions of generalized coordinated qiðαÞ ,
α = 1, 2, 3, . . . , N, i = 1, 2, 3, where N is the total number of material system mass
particles. Basic vectors of each position vector tangent space are denoted by g⃗ðαÞi ,
α = 1, 2, 3, . . . , N, i = 1, 2, 3, and can be expressed in the following form:
g⃗ðαÞi =
∂ρ⃗ðαÞ
, α = 1, 2, 3, . . . , N, i = 1, 2, 3
∂qiðαÞ
ð1Þ
or in the following form:
g⃗ðαÞi =
∂xðαÞ q1ðαÞ , q2ðαÞ , q3ðαÞ
∂qiðαÞ
ı+
⃗
∂yðαÞ q1ðαÞ , q2ðαÞ , q3ðαÞ
∂qiðαÞ
ȷ+
⃗
∂zðαÞ q1ðαÞ , q2ðαÞ , q3ðαÞ
∂qiðαÞ
k⃗
α = 1, 2, 3, . . . , N, i = 1, 2, 3
ð2Þ
Contravariant coordinates of position vectors ρ⃗ðαÞ ðqÞ, α = 1, 2, 3, . . . , N of each
of material system kinetic points are expressed in published Refs. [3–9].
In Refs. [3–9] the change of basic vectors of position vector tangent space of
kinetic point in three-dimensional spaces with curvilinear coordinate system are
derived. Without losing generality, let us list expressions for change of basic
vectors g⃗ðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3 of position vector ρ⃗ðαÞ ðqÞ,
α = 1, 2, 3, . . . , N in three-dimensional tangent space in curvilinear coordinate
system for one kinetic point of material system. For that reason, let us present
derivatives with respect to time of basic vectors g⃗ðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3 of
position vector tangent space, in the following forms:
dg⃗ðαÞ1
= g⃗ðαÞ1 Γ1ðαÞ11 q̇1ðαÞ + Γ1ðαÞ12 q̇2ðαÞ + Γ1ðαÞ13 q̇3ðαÞ +
dt
+ g⃗ðαÞ2 Γ2ðαÞ11 q̇1ðαÞ + Γ2ðαÞ12 q̇2ðαÞ + Γ2ðαÞ13 q̇3ðαÞ +
+ g⃗ðαÞ3 Γ3ðαÞ11 q̇1ðαÞ + Γ3ðαÞ12 q̇2ðαÞ + Γ3ðαÞ13 q̇3ðαÞ
dg⃗ðαÞ2 1
= ΓðαÞ21 g⃗ðαÞ1 + Γ2ðαÞ21 g⃗ðαÞ2 + Γ3ðαÞ21 g⃗ðαÞ3 q̇1ðαÞ +
dt
+ Γ1ðαÞ22 g⃗ðαÞ1 + Γ2ðαÞ22 g⃗ðαÞ2 + Γ3ðαÞ22 g⃗ðαÞ3 q̇2ðαÞ +
+ Γ1ðαÞ23 g⃗ðαÞ1 + Γ2ðαÞ23 g⃗ðαÞ2 + Γ3ðαÞ23 g⃗ðαÞ3 q̇3ðαÞ
ð3Þ
ð4Þ
Angular Velocity and Intensity Change of the Basic Vectors …
147
dg⃗ðαÞ3 1
= ΓðαÞ31 g⃗ðαÞ1 + Γ2ðαÞ31 g⃗ðαÞ2 + Γ3ðαÞ31 g⃗ðαÞ3 q̇1ðαÞ +
dt
ð5Þ
+ Γ1ðαÞ32 g⃗ðαÞ1 + Γ2ðαÞ32 g⃗ðαÞ2 + Γ3ðαÞ32 g⃗ðαÞ3 q̇2ðαÞ +
+ Γ1ðαÞ33 g⃗ðαÞ1 + Γ2ðαÞ33 g⃗ðαÞ2 + Γ3ðαÞ33 g⃗ðαÞ3 q̇3ðαÞ
Let us suppose that each position vector tangent space of each kinetic point is
three-dimensional and deﬁned in orthogonal curvilinear coordinates qiðαÞ ,
α = 1, 2, 3, . . . , N, i = 1, 2, 3, then is valid [3–9]
*
Â
Ã
dg⃗ðαÞi
= g⃗ðαÞi + ω⃗pðαÞi , g⃗ðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3
dt
ð6Þ
Then, on the basis (6), let us separate in expressions (3), (4), and (5) of the
dg⃗
corresponding derivatives dtðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3, of the basic vectors
g⃗ðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3 of position vector ρ⃗ðαÞ ðqÞ, α = 1, 2, 3, . . . , N
*
*
tangent space terms which correspond to terms of the relative derivatives g⃗ðαÞ1 , g⃗ðαÞ2
*
and g⃗ðαÞ3 , α = 1, 2, 3, . . . , N in the following forms:
*
g⃗ðαÞ1 = g⃗ðαÞ1 Γ1ðαÞ11 q̇1ðαÞ + Γ1ðαÞ12 q̇2ðαÞ + Γ1ðαÞ13 q̇3ðαÞ
*
g⃗ðαÞ2 = g⃗ðαÞ2 Γ2ðαÞ21 q̇1ðαÞ + Γ2ðαÞ22 q̇2ðαÞ + Γ2ðαÞ23 q̇3ðαÞ
*
g⃗ðαÞ3 = g⃗ðαÞ3 Γ3ðαÞ31 q̇1ðαÞ + Γ3ðαÞ32 q̇2ðαÞ + Γ3ðαÞ33 q̇3ðαÞ , α = 1, 2, 3, . . . , N
*
*
ð7Þ
*
These vector terms, g⃗ðαÞ1 , g⃗ðαÞ2 and g⃗ðαÞ3 , α = 1, 2, 3, . . . , N deﬁned by expressions (7) represent vectors of relative velocity of basic vectors extensions and is
possible to express in scalar forms as relative velocity of magnitude dilatation of
each of three basic vectors of each position vector tangent space of each material
point in following forms:.
dg⃗ðαÞ1 1
= ΓðαÞ11 q̇1ðαÞ + Γ1ðαÞ12 q̇2ðαÞ + Γ1ðαÞ13 q̇3ðαÞ
εðαÞ1 =
g⃗ðαÞ1 dt
dg⃗ðαÞ2 2
1
2
2
2
3
εðαÞ2 =
=
Γ
q̇
+
Γ
q̇
+
Γ
q̇
ð
α
Þ21
ð
α
Þ
ð
α
Þ22
ð
α
Þ
ð
α
Þ23
ð
α
Þ
g⃗ðαÞ2 dt
dg⃗ðαÞ31 3
= ΓðαÞ31 q̇1ðαÞ + Γ3ðαÞ32 q̇2ðαÞ + Γ3ðαÞ33 q̇3ðαÞ , α = 1, 2, 3, . . . , N
εðαÞ3 =
g⃗ðαÞ3 dt
ð8Þ
148
K.R. (Stevanović) Hedrih
Other terms in each of the expressions (3), (4), and (5) of the corresponding
dg⃗
derivatives dtðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3, of the basic vectors g⃗ðαÞi ,
α = 1, 2, 3, . . . , N, i = 1, 2, 3, of position vector ρ⃗ðαÞ ðqÞ, α = 1, 2, 3, . . . , N tangent
space are terms which represent the vector expressions of vector product between
angular velocity ω⃗pðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3 of corresponding basic vector
rotation and same basic vector g⃗ðαÞi , α = 1, 2, 3, . . . , N, i = 1, 2, 3. From these terms
it is easier to express angular velocities of the basic vectors of position vector
tangent space during motion of the corresponding kinetic point. These expressions
are in the following forms:
Ã
ω⃗pðαÞ1 , g⃗ðαÞ1 = g⃗ðαÞ2 Γ2ðαÞ11 q̇1ðαÞ + Γ2ðαÞ12 q̇2ðαÞ + Γ2ðαÞ13 q̇3ðαÞ +
+ g⃗ðαÞ3 Γ3ðαÞ11 q̇1ðαÞ + Γ3ðαÞ12 q̇2ðαÞ + Γ3ðαÞ13 q̇3ðαÞ
Â
Ã
ω⃗pðαÞ2 , g⃗ðαÞ2 = g⃗ðαÞ1 Γ1ðαÞ21 q̇1ðαÞ + Γ1ðαÞ22 q̇2ðαÞ + Γ1ðαÞ23 q̇3ðαÞ +
+ g⃗ðαÞ3 Γ3ðαÞ21 q̇1ðαÞ + Γ3ðαÞ22 q̇2ðαÞ + Γ3ðαÞ23 q̇3ðαÞ
Â
Ã
ω⃗pðαÞ3 , g⃗ðαÞ3 = g⃗ðαÞ1 Γ1ðαÞ31 q̇1ðαÞ + Γ1ðαÞ32 q̇2ðαÞ + Γ1ðαÞ33 q̇3ðαÞ +
+ g⃗ðαÞ2 Γ2ðαÞ31 q̇1ðαÞ + Γ2ðαÞ32 q̇2ðαÞ + Γ3ðαÞ33 q̇3ðαÞ , α = 1, 2, 3, . . . , N
Â
ð9Þ
In previously presented expressions, denotations ω⃗pðαÞ1 , ω⃗pðαÞ2 and ω⃗pðαÞ3 present
angular velocities of the basic vectors of a position vector tangent space during
material point motion. In previous expressions (9), ΓkðαÞij , α = 1, 2, 3, . . . , N,
i, j, k = 1, 2, 3 are Christoffel’s symbols of the second kind, and ΓðαÞij, k ,
α = 1, 2, 3, . . . , N, i, j, k = 1, 2, 3 Christoffel’s symbols of the ﬁrst kind in corresponding curvilinear coordinate system of vector position tangent space of corresponding material kinetic point. These Christoffel’s symbol, ﬁrst and second kind,
are expressed by corresponding covariant gðαÞij ðqÞ, α = 1, 2, 3, . . . , N, i, j, k = 1, 2, 3
or contrvariant gklðαÞ ðqÞ, α = 1, 2, 3, . . . , N, i, j, k = 1, 2, 3 metric tensor of corresponding position vector tangent space, in the following relations [1, 2, 11, 13]:
!
1 ∂gðαÞik ∂gðαÞjk ∂gðαÞij
ΓðαÞij, k =
+
− k
, α = 1, 2, 3, . . . , N, i, j, k = 1, 2, 3 ð10Þ
2 ∂qðjαÞ
∂qiðαÞ
∂qðαÞ
ΓkðαÞij = gklðαÞ ΓðαÞij, l , α = 1, 2, 3, . . . , N, i, j, k = 1, 2, 3
ð11Þ
À
Á
gðαÞik = g⃗ðαÞi , g⃗ðαÞk
ð12Þ
Angular Velocity and Intensity Change of the Basic Vectors …
149
2 Three-Dimensional Elliptical Cylindrical Curvilinear
Coordinate System
For ﬁrst example, let us determine the change of basic vectors g⃗ðαÞi ξðαÞ , ηðαÞ , zðαÞ ,
α = 1, 2, 3, . . . , N, i = 1, 2, 3 in three-dimensional elliptical cylindrical curvilinear
coordinate system with curvilinear coordinates ξðαÞ , ηðαÞ , zðαÞ , α = 1, 2, 3, . . . , N of
kinetic point NðαÞ , α = 1, 2, 3, . . . , N in three-dimensional elliptical cylindrical
curvilinear system deﬁned as NðαÞ ξðαÞ , ηðαÞ , zðαÞ , α = 1, 2, 3, . . . , N, and with
corresponding position by position vector ρ⃗ðαÞ ξðαÞ , ηðαÞ , zðαÞ , α = 1, 2, 3, . . . , N
with cotravariant coordinates ρiðαÞ ξðαÞ , ηðαÞ , zðαÞ , α = 1, 2, 3, . . . , N, i = 1, 2, 3. By
using previous considerations and derived expressions (see Fig. 1a*) it is valid to
write the following expressions, without index ðαÞ, and without losing generalities
(see Refs. [3–9]) for position vector
⃗ yðξ, η, zÞȷ +
⃗ zðξ, η, zÞk ⃗
ρðξ, η, zÞ = xðξ, η, zÞı +
⇀
ρðξ, η, zÞ = ı ⃗Chξ cos η + ȷ ⃗Shξ sin η + zk ⃗ = ρi ðξ, η, zÞg⃗i ðξ, η, zÞ
⇀
ð13Þ
and corresponding expressions for covariant basic vectors g⃗ðαÞi ξðαÞ , ηðαÞ , zðαÞ ,
α = 1, 2, 3, . . . , N, i = 1, 2, 3 of position vector tangent space in three-dimensional
three parabolic coordinate system:
Fig. 1 Presentation of the position vector of a kinetic point in different positions in
three-dimensional space, with corresponding basic vectors g⃗ðαÞi of position vector ρ⃗ðαÞ ðqÞ tangent
space (without index ðαÞ denotation of the order of point); a* in elliptic cylindrical coordinate
system with orthogonal curvilinear coordinates; b* in three-dimensional cylindrical bipolar
coordinate system with orthogonal curvilinear coordinates
150
K.R. (Stevanović) Hedrih
∂ρ⃗
= ı ⃗Shξ cos η + ȷ ⃗Chξ sin η
∂ξ
∂ρ⃗
⇀
⇀
g2 = gη =
= − ı ⃗Chξ sin η + ȷ ⃗Shξ cos η
∂η
∂ρ⃗
⇀
⇀
= k⃗
g3 = gz =
∂z
⇀
⇀
g1 = gξ =
ð14Þ
Intensities of the basic vectors in elliptical cylindrical coordinates are
⇀ ⇀ ∂ρ⃗ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g1 = gξ = = Sh2 ξ cos2 η + Ch2 ξ sin2 η
∂ξ
ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
⇀ ⇀ ∂ρ⃗ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g1 = gξ = = sin2 η + Sh2 ξ = Ch2 ξ − cos2 η
∂ξ
⇀ ⇀ ∂ρ⃗ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g2 = gη = = η2 + ξ = Ch2 ξ sin2 η + Sh2 ξ cos2 η
∂η
ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
⇀ ⇀ ∂ρ⃗ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g2 = gη = = sin2 η + Sh2 ξ = Ch2 ξ − cos2 η, g⇀3 = g⇀z = ∂ρ⃗ = 1
∂η
∂z
ð15Þ
Metric tensor of position vector tangent space in elliptical cylindrical coordinates is
0 2
sin η + Sh2 ξ
0
À Á
gij = g⃗i , g⃗j , g = gij = @
0
sin2 η + Sh2 ξ
0
0
À
Á
1
0
0A
1
ð16Þ
Scalar products between each two basic vectors of the position
tangent
vector
⇀
⇀
⇀
⇀
space in elliptical cylindrical coordinates are
g1 , g2 = gξ , gη = 0,
⇀
⇀
⇀
⇀
⇀
⇀
⇀
⇀
g1 , g3 = gξ , gz = 0 and g3 , g2 = gz , gη = 0. Then, we can conclude that
these three basic vectors of the position vector tangent space in elliptical cylindrical
coordinates are orthogonal.
Unit vectors in directions of the corresponding basic vectors of vector position of
a mass particle in elliptical cylindrical coordinates are [8–10]
⇀
gξ
1
T 1⃗ = T ⃗ξ = ⇀ = pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðı ⃗Shξ cos η + ȷ ⃗Chξ sin ηÞ
2
gξ
sin η + Sh2 ξ
⇀
gη
1
⇀
T 2⃗ = T ⃗η = ⇀ = pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð − ı ⃗Chξ sin η + ȷ ⃗Shξ cos ηÞ, T 3⃗ = gz = k ⃗
2
gη
sin η + Sh2 ξ
ð17Þ