Journal of Applied Mathematics and Computational Mechanics 2016, 15(1), 25-38
www.amcm.pcz.pl p-ISSN 2299-9965
DOI: 10.17512/jamcm.2016.1.03 e-ISSN 2353-0588
RE-DERIVATION OF LAPLACE OPERATOR ON CURVILINEAR COORDINATES USED FOR THE COMPUTATION OF FORCE
ACTING IN SOLENOID VALVES
Robert Goraj private means Erlangen, Germany robertgoraj@gmx.de
Abstract. This article presents two mathematical methods of derivation of the Laplace operator in a given curvilinear co-ordinate system. This co-ordinate system is defined in the area between the armature and the yoke of a high-speed solenoid valve (HSV). The Laplace operator can further be used for the numerical solving of the Laplace’s equation in order to determine the electromagnetic force acting on the armature of the HSV. In further steps the author derived an expression for the gradient and the vector surface element of the armature side surface in this co-ordinate system. The solution of the derivation was compared with one other solution derived in the past for the computational investigations on HSVs.
Keywords: Laplace operator, solenoid valve, armature, electromagnetic force
1. Introduction
On account of the growing globalization and the rising competition in the indus- try, the enterprises must develop economically and use sophisticated calculation algorithms [1]. Electromagnetic HSVs can be pre-calculated with suitable mathe- matical models already at an early time of the construction phase. Solenoid valves (SV) are used in fluid power pneumatic and hydraulic systems to control cylinders, fluid power motors or larger industrial valves. Domestic washing machines and dishwashers use SVs to control water entry into the machine. SVs can be used for a wide array of industrial applications, including general on-off control, calibration and test stands, pilot plant control loops and process control systems. They are also set very widely in the automotive industry. A number of numerical algorithms concerning computation of high-speed solenoid valves were published in [1].
The aim of the paper is a re-derivation and further development of some of them.
The special interest lies in the derivation of the Laplace operator. This operator can be used for numerical computation of an electromagnetic force (EMF) acting on the armature of the HSV. In [2] the authors carried out research of key factors
on EMF of HSV. Further EMF is an input to the computation of e.g. armature eccentricity, which is needed for building models of HSV similar to
ies on HSVs assumed that the armature is concentrically positioned in the sleeve.
Under this assumption the transv
zero. Using the derived Laplace operator one can compute the armature eccentricity as a function of the sleeve thickness or as a function of hydraulic clearance between the armature and the sleeve. After fi
permeance of the radial air gap, which has a direct impact on the drop of the magnetomotive force and finally influences the driving component of the magnetic force. The presided determination of the EMF is also u
of HSVs. In [4] the authors described the method of closed loop control for the closure time and hold current, wh
2.
Preface to derivationsThe region ௫௬ (see
Laplace’s differential equation that
differences. The inner border defined by the function solenoid valve armature where:
tion is the inner side of the magnet yoke. It
transformations are valid only for the case of no contact between the armature and the inner side of the magnet yoke
ferential position . The room between the borders has the p to the permittivity of a vacuum
Fig. 1. Computation room of the
Further EMF is an input to the computation of e.g. armature eccentricity, which is needed for building models of HSV similar to [3].
ies on HSVs assumed that the armature is concentrically positioned in the sleeve.
Under this assumption the transversal component of the magnetic force is equal to zero. Using the derived Laplace operator one can compute the armature eccentricity as a function of the sleeve thickness or as a function of hydraulic clearance between the armature and the sleeve. After finding the eccentricity one can compute the permeance of the radial air gap, which has a direct impact on the drop of the magnetomotive force and finally influences the driving component of the magnetic The presided determination of the EMF is also useful for the controlling
the authors described the method of closed loop control for the closure time and hold current, which strongly depends on the EMF.
Preface to derivations
(see Fig. 1) is the computation domain for the solution to the Laplace’s differential equation that is going to be solved using the method of finite differences. The inner border defined by the function is the contour of the solenoid valve armature where: 0 2 . The outer border given by the fun
is the inner side of the magnet yoke. It should be noticed that following transformations are valid only for the case of no contact between the armature and the inner side of the magnet yoke neither at 0, nor at any other
. The room between the borders has the permittivity compared to the permittivity of a vacuum.
Computation room of the Laplace’s differential equation
Further EMF is an input to the computation of e.g. armature . Most stud- ies on HSVs assumed that the armature is concentrically positioned in the sleeve.
ersal component of the magnetic force is equal to zero. Using the derived Laplace operator one can compute the armature eccentricity as a function of the sleeve thickness or as a function of hydraulic clearance between nding the eccentricity one can compute the permeance of the radial air gap, which has a direct impact on the drop of the magnetomotive force and finally influences the driving component of the magnetic seful for the controlling the authors described the method of closed loop control for the
) is the computation domain for the solution to the is going to be solved using the method of finite is the contour of the . The outer border given by the func-
should be noticed that following transformations are valid only for the case of no contact between the armature and nor at any other circum-
ermittivity compared
Re-derivation of Laplace operator on curvilinear coordinates
The room ௫௬ is supposed to be discretized in a particular manner. Mesh lines in a radial direction are
outer border. The number of mesh lines in a radial direction is kept constant ind pendently from the distance
ferential position 0
alise the capacity of the transformation this distance at in Figure 1 to a very small value. In general there is:
cretizing method has the advantage that regions w
contributes dominantly to the sought force) are meshed more densely than regions with big . That means that
one can avoid an unnecessary fine mesh in the case of big n
The density of mesh lines is variable both in the radial and peripheral direction.
This method of discretizing allows for increase of computation precision with a simultaneous reduction of the mesh node numbers.
Fig. 2. Computation room of the Laplace’s differential equation
The aim of the work is to re
system in which the computation domain gets a rectangle The derivation of the Laplace operator in
the following transformation:
Functions and in
isomorphic and bijective in the whole domain validity of the function
done using the transformation shoal
derivation of Laplace operator on curvilinear coordinates …
is supposed to be discretized in a particular manner. Mesh lines in a radial direction are distorted in such a way that they exactly fit the inner and outer border. The number of mesh lines in a radial direction is kept constant ind pendently from the distance between the borders. That means that at the circu
0 there is still a positive distance 0 0. In order to vis alise the capacity of the transformation this distance at 0 was purposely set
1 to a very small value. In general there is: ⋁ஸఝழଶగ
cretizing method has the advantage that regions with small (in which the solution contributes dominantly to the sought force) are meshed more densely than regions . That means that - in the difference to equidistant meshing methods one can avoid an unnecessary fine mesh in the case of big nonmagnetic gaps.
The density of mesh lines is variable both in the radial and peripheral direction.
This method of discretizing allows for increase of computation precision with ction of the mesh node numbers.
Computation room of the Laplace’s differential equation in , co-ordinate system
the work is to re-derive the Laplace operator to a co system in which the computation domain gets a rectangle ఈ shown in
The derivation of the Laplace operator in , co-ordinate system was done using the following transformation:
ିଵ
in (1) are restricted only to functions which allow
isomorphic and bijective in the whole domain ఈ. Furthermore, the range of function is restricted to: 0 1. In [1] this transformation was done using the transformation shoal
ఈ ଶ ଶఈఈ 2
27
is supposed to be discretized in a particular manner. Mesh lines distorted in such a way that they exactly fit the inner and outer border. The number of mesh lines in a radial direction is kept constant inde-
between the borders. That means that at the circum- . In order to visu- was purposely set
0. This dis- (in which the solution contributes dominantly to the sought force) are meshed more densely than regions in the difference to equidistant meshing methods - onmagnetic gaps.
The density of mesh lines is variable both in the radial and peripheral direction.
This method of discretizing allows for increase of computation precision with
derive the Laplace operator to a co-ordinates in Figure 2.
was done using
(1) (2) are restricted only to functions which allow being Furthermore, the range of this transformation was
ఈ (3)
with the operators:
∇ܽ = ܽ݁+ݎିଵܽఝ݁ఝ (4)
∇ߙ = ݎିଵߙఝ݁ఝ (5)
∆ܽ = ݎିଵܽ+ܽ+ݎିଶܽఝఝ (6)
∆ߙ = ݎିଶߙఝఝ (7)
The re-derivation of the Laplace operator will now be done in two ways: using the differential operators and using the differential geometry. In Table 1 the over- view of derivation ways is presented.
Table 1 Overview of derivation methods of Laplace operator
Author Vogel Goraj
derivation using differential operators ݊ = ܽ, ߙ = ߮ ݊ = ݊ሺܽሻ, ߙ = ߙሺ߮ሻ derivation using the transformation shoal ݊ = ݊ሺܽሻ, ߙ = ߙሺ߮ሻ not done derivation using the differential geometry not done ݊ = ݊ሺܽሻ, ߙ = ߙሺ߮ሻ
3. Derivation of Laplace operator using differential operators
The Laplace operator in polar co-ordinate system is given by [5]:߂ = ߲+ݎିଵ߲+ݎିଶ߲ఝఝ (8) The differential operator ߲ can be split using rules of partial differentiation:
߲=߲ܽ+ߙ߲ఈ (9)
For the differential operator ߲ can be written:
߲=߲ሺ߲ሻ = ߲ᇣᇧᇧᇤᇧᇧᇥሺ߲ܽሻ
+߲ᇣᇧᇧᇤᇧᇧᇥሺߙ߲ఈሻ
(10) The application of the product rules on the term A yields:
ܣ = ߲ሺܽሻ߲+߲ܽሺ߲ሻ (11) For the partial differential operator ߲ can be written with the use of (9)
߲ሺ߲ሻ = ߲߲ܽ+ߙ߲ఈ߲=߲ܽ+ߙ߲ఈ (12)
Re-derivation of Laplace operator on curvilinear coordinates … 29
With the use of (12) the formula (11) can be simplified to:
ܣ = ߲ܽ+ܽଶ߲+ܽߙ߲ఈ (13) Analogously to A hold for B:
ܤ = ߲ሺߙሻ߲ఈ+ߙ߲ሺ߲ఈሻ (14) For the partial differential operator ߲ఈ one can use (9) again, which gives:
߲ሺ߲ఈሻ = ߲ܽሺ߲ఈሻ + ߙ߲ఈሺ߲ఈሻ = ߲ܽఈ+ߙ߲ఈఈ (15) The use of (15) in (14) yields:
ܤ = ߲ሺߙሻ߲ఈ+ߙ߲ܽఈ+ߙଶ߲ఈఈ (16) The summation of A and B yields the operator ߲
߲ =߲ܽ+ߙ߲ఈ+ 2ߙ߲ܽఈ+ܽଶ߲+ߙଶ߲ఈఈ (17) Analogously to (17), one obtains ߲ఝఝ:
߲ఝఝ =ܽఝఝ߲+ߙఝఝ߲ఈ+ 2ߙఝܽఝ߲ఈ+ܽఝଶ߲+ߙఝଶ߲ఈఈ (18) The unknowns in equations (17) and (18) are the first and the second derivative of ܽ and ߙ in ݎ and ߮ – direction. From the definition (2) one obtains:
ߙ = 0 (19)
ߙ = 0 (20)
The desolation of (1) for r leads to:
ݎ = ߞሺ߮ሻ + ݊ሺܽሻℎሺ߮ሻ (21)
The application of ߲ in (21) gives:
1 =݊ܽℎ (22)
The application of ߲ in (22) yields:
0 =݊ܽଶℎ +݊ܽℎ (23) The desolation of (22) for ܽ and the desolation of (23) for ܽ results in:
ܽ =݊ିଵℎିଵ (24)
ܽ= −݊݊ିଷℎିଶ (25)
The application of ߲ఝ in (21) yields:
0 =ߞఝ+݊ܽఝℎ +݊ℎఝ (26)
The application of ߲ఝ in (26) yields:
0 =ߞఝఝ+݊ܽఝଶℎ +݊ܽఝఝℎ + 2݊ܽఝℎఝ+݊ℎఝఝ (27) The desolation of (26) for ܽఝ and the desolation of (27) for ܽఝఝ results in:
ܽఝ = −ߞఝ+݊ℎఝ
݊ℎ (28)
ܽఝఝ = − 1
݊ℎ ൫ߞఝఝ+݊ܽఝଶℎ + 2݊ܽఝℎఝ+݊ℎఝఝ൯ (29) Setting of (28) in (29) yields:
ܽఝఝ= − 1
݊ℎ ൭ߞఝఝ+݊ቆߞఝ+݊ℎఝ
݊ℎ ቇ
ଶ
ℎ − 2݊ቆߞఝ+݊ℎఝ
݊ℎ ቇ ℎఝ+݊ℎఝఝ൱ (30) Setting of (9) and (17) in (19) and (20) yields:
߲ =߲ܽ (31)
߲ =߲ܽ+ܽଶ߲ (32)
After the replacement of ܽ and ܽ from (24) and (25) in (31) and (32), one obtains the transformed differential operators ߲ and ߲:
߲ =݊ିଵℎିଵ߲ (33)
߲ = −݊݊ିଷℎିଶ߲+ሺ݊ℎሻିଶ߲ (34) Setting now the relations (28) and (30) in (18) one gets the transformed differential operator ߲ఝఝ:
߲ఝఝ= − 1
݊ℎ ൭ߞఝఝ+݊ቆߞఝ+݊ℎఝ
݊ℎ ቇ
ଶ
ℎ − 2݊ቆߞఝ+݊ℎఝ
݊ℎ ቇ ℎఝ
+݊ℎఝఝቇ ߲+ߙఝఝ߲ఈ− 2ߙఝ
ߞఝ+݊ℎఝ
݊ℎ ߲ఈ
+ቆߞఝ+݊ℎఝ
݊ℎ ቇ
ଶ
߲+ߙఝଶ߲ఈఈ
(35)
Re-derivation of Laplace operator on curvilinear coordinates … 31 After setting of the operators (33) to (35) in (8) one obtains - under the usage of (1) - the Laplace operator in ܽ, ߙ co-ordinate system:
߂ = ቌ 1
ሺߞ + ݊ℎሻ݊ℎ− ݊
݊ଷℎଶ
− 1
ሺߞ + ݊ℎሻଶ݊ℎ ൭ߞఝఝ+݊
ℎ ቆ
ߞఝ+݊ℎఝ
݊ ቇ
ଶ
− 2ߞఝ+݊ℎఝ
ℎ ℎఝ+݊ℎఝఝቇቍ߲+ ߙఝఝ
ሺߞ + ݊ℎሻଶ߲ఈ
+൭ 1
ሺ݊ℎሻଶ+ቆ ߞఝ+݊ℎఝ
ሺߞ + ݊ℎሻ݊ℎቇ
ଶ
൱ ߲+ ߙఝଶ
ሺߞ + ݊ℎሻଶ߲ఈఈ
− 2ߙఝ
ߞఝ+݊ℎఝ
ሺߞ + ݊ℎሻଶ݊ℎ ߲ఈ
(36)
The operator (36) is identical to the one derived in [1].
4. Derivation of Laplace operator using differential geometry
In differential geometry, the Laplace operator can be generalized to operate on functions defined on surfaces in Euclidean space. The more general operator is called the Laplace-Beltrami operator. This operator of a scalar function in any cur- vilinear co-ordinate system can be expressed using Einstein notation [6-8]:
ߘߘܸ = 1
ඥ߲݃൫ඥ߲ܸ݃݃൯ (37)
݃ in (37) is here the contravariant metric tensor of the second rank. Its general covariant form is [8]:
݃=߲ܺ
߲ݍ ߲ܺ
߲ݍ (38)
The ݇ in (38) is the summation index. The position vector ܺ is defined by (39):
ܺ = ൫݊ሺܽሻℎሺ߮ሻ + ߞሺ߮ሻ൯ ቀcos߮
sin߮ቁ (39)
The variables of the position vector (39) are ݍଵ=ܽ and ݍଶ=ߙ. The derivatives of the position vector (39) are:
ܺ=݊ℎቀcos߮
sin߮ቁ (40)
ܺఈ =ߙఝିଵቆ൫݊ℎఝ+ߞఝ൯ cos ߮ − ሺ݊ℎ + ߞሻ sin ߮
൫݊ℎఝ+ߞఝ൯ sin ߮ + ሺ݊ℎ + ߞሻ cos ߮ቇ (41) The use of (38) gives the components of the covariant metric tensors:
݃ଵଵ=ሺ݊ℎሻଶ (42)
݃ଵଶ=݊ℎఝ+ߞఝ
ߙఝ ݊ℎ (43)
݃ଶଵ=݃ଵଶ (44)
݃ଶଶ =൫݊ℎఝ+ߞఝ൯ଶ+ሺ݊ℎ + ߞሻଶ
ߙఝଶ (45)
The determinant of the metric tensors is equal to:
݃ =ሺ݊ℎሻଶሺ݊ℎ + ߞሻଶ
ߙఝଶ (46)
The contravariant metric tensor is defined as [6]:
݃ =݃ିଵ (47)
The components of the metric tensor in the contravariant form are:
݃ଵଵ= 1
ሺ݊ℎሻଶ+ቆ ݊ℎఝ+ߞఝ
݊ℎሺ݊ℎ + ߞሻቇ
ଶ
(48)
݃ଵଶ= −ߙఝ
݊ℎఝ+ߞఝ
݊ℎሺ݊ℎ + ߞሻଶ (49)
݃ଶଵ=݃ଵଶ (50)
݃ଶଶ= ߙఝଶ
ሺ݊ℎ + ߞሻଶ (51)
The Laplace operator simplifies in the considered case to:
∆=ቀ߲൫ඥ݃݃ଵଵ൯ + ߲ఈ൫ඥ݃݃ଶଵ൯ቁ ߲
ඥ݃+ቀ߲ఈ൫ඥ݃݃ଶଶ൯ + ߲൫ඥ݃݃ଵଶ൯ቁ ߲ఈ
ඥ݃
+݃ଵଵ߲+݃ଶଶ߲ఈఈ+ 2݃ଵଶ߲ఈ
(52)
Re-derivation of Laplace operator on curvilinear coordinates … 33
The multiplier of ݃ିଵ/ଶ߲ఈ can be expressed as:
߲ఈ൫ඥ݃݃ଶଶ൯ + ߲൫ඥ݃݃ଵଶ൯ = ߲ఈ
݊ℎߙఝ
݊ℎ + ߞ ᇣᇧᇧᇤᇧᇧᇥ
−߲
݊ℎఝ+ߞఝ
݊ℎ + ߞ ᇣᇧᇧᇧᇤᇧᇧᇧᇥ
(53)
The first term of (53) is equal to:
ܣ =݊ℎఝ߮ఈߙఝሺ݊ℎ + ߞሻ + ݊ℎߙఝఝ߮ఈሺ݊ℎ + ߞሻ − ൫݊ℎఝ+ߞఝ൯߮ఈ݊ℎߙఝ
ሺ݊ℎ + ߞሻଶ (54)
The second term of (53) is equal to:
ܤ =݊ℎఝሺ݊ℎ + ߞሻ − ݊ℎ൫݊ℎఝ+ߞఝ൯
ሺ݊ℎ + ߞሻଶ (55)
The subtraction of the terms (54) and (55) yields for (53):
߲ఈ൫ඥ݃݃ଶଶ൯ + ߲൫ඥ݃݃ଵଶ൯ = ݊ℎߙఝఝ
ߙఝሺ݊ℎ + ߞሻ (56)
The first part-multiplier of ݃ିଵ/ଶ߲ becomes:
߲൫ඥ݃݃ଵଵ൯ = ߲
݊ℎ + ߞ
݊ℎߙఝ
ᇣᇧᇧᇤᇧᇧᇥ
+߲
൫݊ℎఝ+ߞఝ൯ଶ
݊ℎߙఝሺ݊ℎ + ߞሻ ᇣᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇥ
(57)
The first term of (57) is equal to:
ܥ = 1 ߙఝ
−ሺ݊ℎ + ߞሻ݊
ℎߙఝ݊ଶ (58)
The second term of (57) is equal to:
ܦ = 2 ݊ℎఝ+ߞఝ
ℎߙఝሺ݊ℎ + ߞሻℎఝ−൫݊ℎఝ+ߞఝ൯ଶ݊
ℎߙఝ݊ଶሺ݊ℎ + ߞሻ− 1
ߙఝቆ݊ℎఝ+ߞఝ
݊ℎ + ߞ ቇ
ଶ
(59)
The second part-multiplier of ݃ିଵ/ଶ߲ becomes:
߲ఈ൫ඥ݃݃ଶଵ൯ = − ߲ఈ
݊ℎఝ+ߞఝ
݊ℎ + ߞ ᇣᇧᇧᇧᇤᇧᇧᇧᇥ
ா
(60)
Differentiation of (60) gives:
ܧ = 1 ߙఝ
݊ℎఝఝ+ߞఝఝ
݊ℎ + ߞ − 1
ߙఝቆ݊ℎఝ+ߞఝ
݊ℎ + ߞ ቇ
ଶ
(61)
Addition of the terms ܥ (58) and ܦ (59) together with the subtraction of terms ܧ (61) results in the multiplier of ݃ିଵ/ଶ߲. This multiplier equals:
߲൫ඥ݃݃ଵଵ൯ + ߲ఈ൫ඥ݃݃ଶଵ൯
= 1 ߙఝ
−ሺ݊ℎ + ߞሻ݊
ℎߙఝ݊ଶ + 2 ݊ℎఝ+ߞఝ
ℎߙఝሺ݊ℎ + ߞሻℎఝ
−൫݊ℎఝ+ߞఝ൯ଶ݊
ℎߙఝ݊ଶሺ݊ℎ + ߞሻ− 1 ߙఝ
݊ℎఝఝ+ߞఝఝ
݊ℎ + ߞ
(62)
Finally, inserting of (56), (62) and (46)-(51) in (52) yields to the Laplace operator:
∆= ߙఝఝ
ሺ݊ℎ + ߞሻଶ߲ఈ
+൭ 1
ሺ݊ℎ + ߞሻ݊ℎ− ݊
ℎଶ݊ଷ+ 2 ݊ℎఝ+ߞఝ
݊ℎଶሺ݊ℎ + ߞሻଶℎఝ
−൫݊ℎఝ+ߞఝ൯ଶ݊
ℎଶ݊ଷሺ݊ℎ + ߞሻଶ− ݊ℎఝఝ+ߞఝఝ
݊ℎሺ݊ℎ + ߞሻଶቇ ߲
+൭ 1
ሺ݊ℎሻଶ+ቆ ݊ℎఝ+ߞఝ
݊ℎሺ݊ℎ + ߞሻቇ
ଶ
൱ ߲+ ߙఝଶ
ሺ݊ℎ + ߞሻଶ߲ఈఈ
− 2ߙఝ
݊ℎఝ+ߞఝ
݊ℎሺ݊ℎ + ߞሻଶ߲ఈ
(63)
The operator (63) is identical to (36) and to the one derived in [1].
5. Gradient
The gradient of a scalar function ܸ in any curvilinear co-ordinate system is a covariant vector defined as [7]
ሺߘܸሻ=߲݃ܺ
߲ݍ ߲ܸ
߲ݍ (64)
Re-derivation of Laplace operator on curvilinear coordinates … 35 The derivatives of the position vector (39) are given by (40) and (41). With the use of the new basis with unit vectors:
݁=݁ (65)
݁ఈ=ቀݎଶ+൫݊ℎఝ+ߞఝ൯ଶቁିଵ/ଶቀ൫݊ℎఝ+ߞఝ൯݁+ݎ݁ఝቁ ߙఝ (66) and the use of (1) one can write these derivatives in more compact manner:
ܺ=ݎ݁ (67)
ܺఈ=ቀݎଶ+൫݊ℎఝ+ߞఝ൯ଶቁଵ/ଶ݁ఈ (68) The gradient (64) can be expressed by its components as follows:
ሺߘሻଵ =݃ଵଵݔ߲+݃ଵଶݔ߲ఈ (69) ሺߘሻଶ=݃ଶଵݔఈ߲+݃ଶଶݔఈ߲ఈ (70) The elements of the contravariant metric tensor (48) to (51) can also be written in a shorter way:
݃ଵଵ= 1
ݎଶ+ቆ൫݊ℎఝ+ߞఝ൯ ݎݎ ቇ
ଶ
(71)
݃ଵଶ= −ߙఝ
൫݊ℎఝ+ߞఝ൯
ݎݎଶ (72)
݃ଶଵ=݃ଵଶ (73)
݃ଶଶ=ߙఝଶ
ݎଶ (74)
The sum of (69) and (70) together with (67), (68) and with relations (71)-(74) yields:
ߘ =݁
ݎଶ൭ݎଶ+൫݊ℎఝ+ߞఝ൯ଶ
ݎ ߲−ߙఝ൫݊ℎఝ+ߞఝ൯߲ఈ൱ +݁ఈ
ݎଶටݎଶ+൫݊ℎఝ+ߞఝ൯ଶቆߙఝ߲ఈ−൫݊ℎఝ+ߞఝ൯ ݎ ߲ቇ
(75)
The derivatives of (1) in ܽ direction are:
ݎ=݊ℎ (76) Setting of (1) and (76) in (75) results in the nabla operator in ܽ, ߙ co-ordinate system.
ߘ = ݁
ሺ݊ℎ + ߞሻଶ൭ሺ݊ℎ + ߞሻଶ+൫݊ℎఝ+ߞఝ൯ଶ
݊ℎ ߲−ߙఝ൫݊ℎఝ+ߞఝ൯߲ఈ൱
+ ݁ఈ
ሺ݊ℎ + ߞሻଶටሺ݊ℎ + ߞሻଶ+൫݊ℎఝ+ߞఝ൯ଶቆߙఝ߲ఈ
−൫݊ℎఝ+ߞఝ൯
݊ℎ ߲ቇ
(77)
The operator (75) can also be expressed by means of the unit vectors of the ݎ, ߮ basis:
ߘ = ߲݁
ݎ
+݁ఝ
ݎ ቆߙఝ߲ఈ−൫݊ℎఝ+ߞఝ൯
ݎ ߲ቇ (78)
Now the operator (78) can be inspected by means of the operator (79) in polar co-ordinate system [5]:
∇=߲݁+ݎିଵ݁ఝ߲ఝ (79)
For the operator ߲ఝ can be written by means of rules of partial differentiation:
߲ఝ=ܽఝ߲+ߙఝ߲ఈ (80)
Substitution of ܽఝ from (28) yields:
߲ఝ= −ߞఝ+݊ℎఝ
݊ℎ ߲+ߙఝ߲ఈ (81)
After the setting of (19) and (24) in (9) one obtains the operator ߲:
߲ = ߲
݊ℎ (82)
Finally, the usage of (81), (82) and (21) in (79) yields to
∇= ݁
݊ℎ ߲+ ݁ఝ
ߞ + ݊ℎ ቆߙఝ߲ఈ−ߞఝ+݊ℎఝ
݊ℎ ߲ቇ (83)
Re-derivation of Laplace operator on curvilinear coordinates … 37 Under the use of (21), (76) one can see that the operators (83) and (78) are identical to each other.
6. Vector surface element of the armature side surface
For the parameterization of the armature side surface the position vector (39) must be extended in the axial direction:
ܺ = ݎ݁+ݖ݁௭ (84)
The vector surface element can be obtained from the cross product of partial derivatives of (84) in the ߙ and ݖ direction:
݀ܣ = ሺܺఈ×ܺ௭ሻ݀ߙ݀ݖ (85)
The derivatives of (84) are:
ܺఈ =ߙఝିଵቀ൫݊ℎఝ+ߞఝ൯݁+ݎ݁ఝቁ (86)
ܺ௭ =݁௭ (87)
Setting (86) and (87) in (85) yields the vector surface element of the armature side surface
݀ܣ = ߙఝିଵቀሺߞ + ݊ℎሻ݁−൫ߞఝ+݊ℎఝ൯݁ఝቁ
ୀ݀ߙ݀ݖ (88)
7. Conclusions
The Laplace operator in the curvilinear co-ordinate system used for the numeri- cal computation of electromagnetic force acting on the armature of high-speed solenoid valves was derived in three different ways: transformation shoal in [1], differential operators and differential geometry. All these three Laplace operators are identical to each other.
References
[1] Vogel R., Numerische Berechnung der Ankerreibung eines elektromagnetischen Schaltventils, Studienarbeit, Universität Dortmund, Dortmund 2006.
[2] Peng L., Liyun F., Qaisar H., De X., Xiuzhen M., Enzhe S., Research on key factors and their interaction effects of electromagnetic force of high-speed solenoid valve, The Scientific World Journal 2014, p. Article ID 567242.
[3] Huber B., Ulbrich H., Modeling and experimental validation of the solenoid valve of a common rail diesel injector, SAE Technical Paper 2014, 2014-01-0195.
[4] Shahroudi K., Peterson D., Belt D., Indirect adaptive closed loop control of solenoid actuated gas and liquid injection valves, SAE Technical Paper 2006, 2006-01-0007.
[5] Bronstein I.N., Semendjajew K.A., Musiol G., Mühlig H., Taschenbuch der Mathematik, Edition Harri Deutsch, Berlin 2000.
[6] Epstein M., Differential Geometry, Basic Notions and Physical Examples, International Publishing: Springer, 2014.
[7] McInerney A., First Steps in Differential Geometry, Riemannian, Contact, Symplectic, Springer, New York 2013.
[8] Nguyen-Schäfer H., Schmidt J.-P., Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Springer, Berlin-Heidelberg 2014.
