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2 Checking a Reachability of Manoeuvring Air Targets by a Missile on Pursuit Courses (An Attack from a Back Hemisphere)

2 Checking a Reachability of Manoeuvring Air Targets by a Missile on Pursuit Courses (An Attack from a Back Hemisphere)

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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



+

Γ



+

Γ



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]







1

T 1⃗ = T ⃗ξ = ⇀  = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðı ⃗Shξ cos η + ȷ ⃗Chξ sin ηÞ

2

 gξ 

sin η + Sh2 ξ







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



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