The paper deals with simulation modeling of shell structures of mirror antennas in the evaluation of accuracy, stiffness, and stability. The relevance of the study lies in the fact that it is the loads on thin-walled elements of mirror antennas that determine their strength and durability, depending on the purpose and method of operation of the device of which the antenna is an integral part. Attention is focused on loads that can be of different types and are determined mainly by mechanical effects and temperature conditions, as well as on the shell elements of DSA structures, which are made in the form of shells of complex shape. When assessing the strength and serviceability of shell structures, the problems of stress-strain state, stability, and bearing capacity in static and dynamic formulations are considered. The basic concepts, principles of satellite dishes functioning, and results of local stability study are considered. The method of realization of the formation of a reinforced composite material and the method of electric arc spraying of mesh material are described. Attention is focused on the mutual fixation of the sputtered material and thermal loads. The basic equations of the stressstrain state of the shells and the results of simulation modeling of wind loading are presented. A graphical representation of the reinforced composite material, an antenna with a diameter of 1.5 m and a mesh aperture of 5 mm is presented. Simulation modeling of the heterogeneity of the distribution of the safety factor for a 3 mm thick aluminum antenna and the safety factor for a reinforced composite parabolic antenna is carried out. Separately, the results of modeling deformation processes in an antenna made of a solid sheet of aluminum under wind load are presented, which is extremely important for natural factors affecting the stability, reliability, and durability of the means of transmitting and receiving signals from satellites, especially in war conditions.
The theory of shells has been developed quite fully and is presented in various works of
scientists [
Determination of the stress-strain state (SSS) (forces and strains) of shell elements is usually not sufficient for their limit states, which characterize the performance of these elements. Achievement of the limit states may be associated with different mechanisms of destruction of structural elements. These issues are related to the study of the entire process of deformation of the system in a physically and geometrically complicated nonlinear formulation with the determination of the limit or crestal loads in one sense or another, characterizing the limit state of the system.
It is important to take into account the real properties of the material: plasticity, viscoelasticity, and creep. This significantly complicates the formulation and methods for solving the problems of stress-strain state and boundary states of shell systems (compared to elastic ones).
Certain simplifications are achieved by adopting some idealized models. For example, the load-bearing capacity of elastoplastic structures is sometimes considered on the basis of a model of a perfectly plastic body. The corresponding section of the theory of plasticity, the theory of limit equilibrium, discusses effective methods for determining the bearing capacity that can be applied to the calculation of many real systems.
A satellite antenna is an antenna used to receive or transmit radio signals between ground stations and artificial Earth satellites [4, 5]. Mirror antennas are the most common class of satellite dishes. They are used in different bands, from decimeter waves to Kuband, and on different types of stations - from individual TV reception systems to space communication centers. Mirrored antennas are antennas in which the field in the opening is formed as a result of the reflection of an electromagnetic wave from the metal surface of a special reflector (mirror). The source of the electromagnetic wave is usually an antenna called a mirror irradiator or simply an irradiator. The mirror and the irradiator are the main elements of a mirror antenna. The surface of the mirror is shaped to form the desired radiation pattern (RP). The most common mirrors are in the form of a paraboloid of rotation, a truncated paraboloid, a parabolic cylinder, or a specially shaped cylinder. The irradiator is placed in the focus of the paraboloid or along the focal line of the cylindrical mirror [6] (Fig. 1).
From the definition of a parabola, its basic property follows: the sum of the distances from the focus F to any point A on the parabola and from this point A to point A' on the mirror's opening is constant FA AA' const . Due to this property, the optical paths from the focus to the opening points are identical, so that a spherical wave created by a point illuminator placed in the focus is converted into a plane wave after reflection from the paraboloid of rotation. Accordingly, a cylindrical wave produced by a linear irradiator is converted into a plane wave by a parabolic cylinder. In both cases, the field on the radiating surfaces of the parabolic antennas is in phase, which achieves high radiation directivity. There are two types of parabolic antennas: direct-focus (Fig. 2) and offset (Fig. 3).
The direct-focus antenna is a classical rotation parabolic antenna. This allows for a more accurate orientation to the selected satellite. Typically, such antennas are used to receive a signal in the C-band, which is weaker than the signal in the Ku-band. However, it is also possible to receive a signal in the Ku-band, as well as a combined signal.
The offset antenna is the most common in individual satellite TV reception. An offset antenna is an elliptical paraboloid (in the cross-section of an ellipse). The focus of such a segment is located below the geometric center of the antenna, which eliminates shadowing of the antenna's useful area by the irradiator and its supports, which increases its efficiency with the same mirror area as an axisymmetric antenna. Typically, offset antennas are used to receive Ku-band signals (in linear and circular polarization). However, it is also possible to receive a signal in the C-band, as well as a combined signal [7].
In the course of operation, under the influence of mechanical loads or temperature field, structural elements may acquire imperfections: imperfect shapes, changes in material characteristics, such as hardening, etc. Such imperfections also occur during the manufacturing process of structures. They must be taken into account when calculating the stress-strain state and especially the ultimate loads, since imperfections resulting from various causes can reduce the load-bearing capacity of the structure. At the same time, special types of imperfections (e.g., for the shell - if they do not coincide with the form of exhaustion of its bearing capacity) do not reduce and, in principle, can slightly increase the bearing capacity of some shell elements. Therefore, the issue of accounting for initial imperfections is of great importance. It becomes even more complicated for elastic-plastic shell systems, where not only the effect of changes in the initial shape of the shell is important, but also the effect of the loading history (in particular, changes in material properties during loading). Some issues of studying the shape imperfections and residual stresses on the bearing capacity of shells are considered in the paper.
Below is the basic information about mechanical loads acting on thin-walled elements of the SFA in various cases of their operation. Some problems of thermal conductivity are considered, which allow determining the temperature field in shells of complex shapes. The basic information of the theory of shells is presented, which is used to develop algorithms for calculating the stress-strain state of shells of complex shape, which are elements of thin-walled SAS.
It was noted above that in certain cases the performance of the antenna shell elements is determined by the loss of stability.
Methods for solving problems in the theory of shells are described in the abovementioned and other publications. Effective methods for solving the problems of stability and oscillations of shells were developed in [8].
The results of the study of the local stability (statics and dynamics) of one of the elements of the SAR: a mirror in the form of a spherical segment under an edge local impact are presented below.
Formation of shells by electric arc spraying and application of composite material. The formation of mirror antennas consists of several stages. The first stage is the deformation of a two-dimensional mesh and the fixing of nodes by electric arc spraying with aluminum [9] (Fig. 4).
The mesh blank 3 is placed on the working surface of the punch and its edges are secured with a clamp 4. A pre-made rim 5 is placed on the surface of the mesh blank along the contour of the punch and moved, for example, with elastic clamps 6, in a direction that ensures that the blank is crimped on the working surface of the punch. In the manufacture of large-sized shells, one or more additional mesh blanks with larger meshes and made of wire of a larger diameter are applied to the first blank with small meshes. The workpiece made of mesh material formed from a wire of small diameter d (Fig. 5), applied to the working surface of the punch, practically under the action of its own weight, takes the form of the working surface of the punch, as free mutual displacement of wires 7 and 8 in nodes 9 is ensured. The next step is the operation of coating the workpiece, which is performed, for example, using an electric arc or plasma sprayer 10 (Fig. 4). In this case, the end of the outlet nozzle of the sprayer is set at a distance h to the surface of the workpiece, and the axis of the arc torch is normal to the surface of the workpiece. The mutual movement of the punch and the nozzle of the spray gun or several spray guns during their parallel operation (for example, rotation with a variable angular velocity n of the punch 2 together with the workpiece 3 and radial feeding S of the spray gun 10 with its corresponding angular displacementϕ depending on the coordinate of the spraying zone) is taken to ensure uniformity of the coating layer over the entire surface of the workpiece. The shape of the torch and the nature of the distribution of the sprayed material and temperature over its volume (which can be controlled, for example, by changing the air pressure P that forms the torch and the wire drawing speed when using electric arc spraying), the linear speed of the relative movement of the sprayer relative to the workpiece, the distance h to the workpiece and the number of passes of one spraying zone determine the time during which the mutual fixation of the sprayed material 11 (Fig. 7) of the wires 7 and 8 in all the same type nodes 9 of the mesh workpiece and the specified values a1 and b1 of the average mesh sizes (Fig. 6) or the time for complete closure of the mesh meshes and formation of a continuous surface of the shells are achieved.
The second stage after spraying on the punch is to obtain a surface with a high temperature and apply a composite material such as polystyrene. Polystyrene is a polymeric material used in the manufacture of various products due to its unique properties. As a result, we obtained a shell surface that can be called a reinforced composite material [10, 11].
Let's consider the temperature fields of the fuel cells made as shells. The most typical influence that causes such fields is solar radiation. Some works indicate methods for determining temperature fields in structures [12, 13, 14].
The heat balance equation has the form: cqsF в T 4Fв (1) radiation; в is the total surface of own heat radiation.
where с, в,
are the absorption (emission) coefficients of solar and own thermal radiation, radiation of an absolutely black body; is the solar constant, = 1400Вт/ м2; 0 is the projection of the illuminated part of the surface to the direction of solar 1 T c qs Fо 4
в
Fв
It should be noted that a number of papers contain information on the spectral nature of all radiation characteristics, the nature of wave propagation at the interface between two media, reflection and absorption of waves from different media (metals, dielectrics, etc.).
The parameters of thermal fields depend on the refractive, absorptive, and emissive coefficients (the degree of blackness of the material). They are interrelated and depend on the electrical and magnetic constants of the material. It should be noted that the temperature fields in shell structures arbitrarily oriented with respect to the direction of the rays have not been studied.
Assuming that the antenna mirror is made in the form of a shell, consider a rotating increase in thermal energy (enthalpy) is zero: shell on which a parallel stream of solar radiation falls ( ) .
According to the heat balance equation for a solid volume element (a consequence of the first law of thermodynamics, the law of conservation of energy), the change in the total heat flux in the volume element due to heat conduction , radial transfer , and an divqс qr cpTt 0 (3) where and are the specific heat capacity and density of the medium; and are described above.
When calculating the radiation transfer, individual monochromatic electromagnetic waves of a given frequency are usually considered . It is convenient to operate with the frequency rather than the wavelength l, since the latter changes its value when radiation passes from one medium to another. Denoting the spectral absorption coefficient ( ) as the inverse of the average free path length of a photon = 1 , consider an optically thick (opaque) medium when the optical thickness 0 =
The radiating medium can be considered as a continuum of photons. In this case, it is assumed that each element of the medium, as in the case of molecular conductivity, is influenced only by neighboring elements. Under such conditions, the transfer of radiation energy in the medium can be equated to diffusion transfer [15].
The spectral flux of radiation in this case is represented as:
4 3x( ) qr
Ib (T ) where ( ) is described by the Planck radiation formula: (2) (4) Ib (T ) co2 exp h( ) 1 lim dFdcEos dd dt
KT Here = 1,38 ⋅ 10−32 J/K is the Boltzmann constant: h= 6,625 ⋅ 10−34 J/s is Planck's constant; ( ) is the refractive index of the medium; is the speed of light in a vacuum; T is the absolute temperature; is a gradient symbol.
The equations describing the deformation of a shell under load include equilibrium equations, relations that relate deformations and displacements, and physical relations. The first two groups of equations are applicable to describe the state of various shells regardless of the model of shell behavior used.
Shell behavior models are accounted for by the third group of equations - physical relations. These models are determined by the properties of the material from which the shell is made (elasticity, plasticity, viscoelasticity, creep, etc.).
The equations describing the deformation of shells differ depending on the class of problems considered (stress-strain state, stability, fracture, etc.), the class of shells (shape, structure), the type of loading (statics, dynamics, temperature effects), and accuracy requirements [16].
Let us consider a deformable solid body and introduce a curved orthogonal coordinate system for it . The total displacement of a point of the body is characterized by a vector whose projections on the coordinate lines are denoted by , , . Let us write the general relations expressing deformations and displacements in terms of the so-called Lamé coefficients (i = 1, 2, 3), which relate the increase in the arcs of the coordinate lines to the increase in the corresponding curved coordinates: e e e e e e 1 1 H1 u , H1H2 1 1 H2 u , H2H3 1 1 H3 u , H1H3 H1 1 u H2 H1
1 H1, u H H 1 3 1 H2, u H H 1 2 1 H3, u H H
2 3 H2 1 u ,
H1 H2 H2 1 H3 H2 (7)
Deformations , , characterize the elongation of linear elements parallel to the axes ; deformations , , - shifts - changes in the initially right angle between linear elements. Shell - a body with two dimensions much larger than the third. Consider a single-layer shell. Let's introduce a median surface, the position of a point on which is determined by the Gaussian curvilinear coordinates , along the thickness of the shell the z coordinates. When using the equation of coordinate (7) in the theory of shells, the
The theory of shells is traditionally based on a number of pleading hypotheses. According to the Kirchhoff-Love hypothesis, straight fibers of a shell normal to the median surface remain straight and normal to it after deformation. According to the Timoshenko hypothesis, the first condition is weakened: it is assumed that after deformation, these fibers remain straight and not perpendicular to the median surface, while the second condition remains the same. The introduction of the first of these hypotheses greatly simplifies the relation (7). In this case:
The displacements of any of the shell layers located at a distance z from the median surface are determined as follows: (8) (9) (10) (11)
Here , , , , , surface, presented in the form: - deformations and changes in the curvature of the median
e e e 0 u u z ,u v z ,u e e z , e e z , e 2z where , = , = are the angles of rotation of the median surface of the plane , = − 1 2 + 2, = −1, 1 + 1, = − 1 Ri are the principal radii of curvature.
Relationships (10) are written on the assumption that elongations and displacements can be neglected compared to unity.
The deformations of these shell layers are defined as follows [17]: A1A2
The equilibrium equations and boundary conditions are obtained on the basis of the variational principle of possible displacements of La Grange, according to which for equilibrium systems the sum of the work of external and internal forces is zero, i.e.
A1 A2 0 where ∃ is the variation of the work of internal forces; 1 is the work of external forces applied to the shell surface 2 is the variation of the work of external contour forces.
Let the shell be bounded by an edge contour L, consisting of parts L₁ and L₂, for which = and . are subjected to edge normal 0 shear 0 forces, intersecting forces 0 and bending moments 0 (Fig. 8). In addition, the displacements of the points of the median surface 0, 0, 0 and the angle of rotation 0 can be set on . The surface of the shell S bounded by the contour is subjected to the loads distributed by the surface 1, 2, . The normal and tangential stresses in the cross-sections of the shell are denoted by , , . Then ∃, 1, 2 will be expressed as follows [18]: A1 q1 u q2 qz A1A2d ,
F A2 T0 u T0 Q0 M01 A2d
1 T0 u T0 Q0 M0 2 A1d ,
2 e e e H1H2d 2dz
V (14)
Instead of voltages, let's introduce their corresponding integral characteristics-forces: normal: tangents: shear forces: bending and torsional moments:
T 1 zk2 dz, T 1 zk1 dz, T 1 zk2 dz, T 1 zk1 dz; T 1 kz2 dz, T 1 zk1 dz; Q 1 zk2 dz,
Q 1 zk1 dz; M 1 zk2 zdz, M 1 zk1 zdz, M 1 zk2 zdz,
M 1 zk1 zdz;
Here, the integration is carried out within the range (-h/2, h/2); values of zk; less than one are neglected.
Substituting expression (14) into (13), we obtain: T T S M M 2M A1A2dd
S
add d dd
S 1 S N10 N20 N3 A1A2qz dd,
S (16) (17) (18)
Substituting expressions (14), (15) into (13) and using (16), we obtain the variational equation after the transformations:
N1 A1A2qq 0, N2 A1A2q 0,
N3 A1A2qz 0 where
N10 T T0u S 2k2M M0
1 Q A12 M, Q0 M M0 1 A2d N1 A1A2q udd; S N20 T T0 S 2k1M M0 u
2 Q A11 M, Qo M M0 2 A1d N2 A1A2q dd; S N1 A2T A11 A12S A2,T k1A1M A1,k2M k1A1A2Q ;
1 N2 A1T A2 A22S A1,T k2A2M A2,k1M k2A1A2Q ;
N3 A2Q A1Q A1A2 k1T k2T ; Q A1A2 A2M A11 A12M A2,M A1A2 T1 S2;
Q A1A2 A1M A1 A22M A1,M A1A2T2 S1
2
Since , , , 1, 2 , are independent in the domain S and on the contours G₁, G₂, the equilibrium equations follow from the general variational equation:
T T0 , S 2k2M T0 , Q 1 A M , Q0 .
1
The Lamé coefficients 1, 2 and curvatures k₁, k₂ for the most commonly used shells as structural elements are as follows [19]: for a cylindrical shell: and boundary conditions: for a spherical shell: for conical shells: for the torus shell:
A1 1, A2 R, A2 d R sin ,
The normal wind load (Pa) can be determined in the first approximation by the formula
W=k×ρ×V2, where: k - drag coefficient (flow); ρ - air density, kg/m3 (ρ = 1.225 kg/m3 ); V is the wind speed, m/s.
According to the results of modeling (Figure 9) of the flow around a parabolic antenna with a diameter of 1.5 m [20] (made of a 3 mm thick aluminum sheet) by an incoming air flow with a speed of 20 m/s, a value of the normal wind load of ≈ 440 Pa was obtained, which corresponds to a flow coefficient k ≈ 0.9.
This is a 1.5-meter-diameter antenna model consisting of mesh material that has been electrodeposited and a composite layer overlaid on it Fig. 10.
To evaluate the stiffness-strength characteristics of antennas of solid and reinforced structures, we calculated their stress-strain state under the action of wind load and gravity Fig.11.
The calculation showed that despite the loss of stiffness of the reinforced structure antenna compared to the antenna made of a solid sheet of aluminum (105 kN/m vs. 664 kN/m), the safety factor of the reinforced structure antenna is 8.8 and remains sufficient, although it is significantly lower than that of the antenna made of a solid sheet of aluminum Fig. 12 and Fig.13.
The obtained thorough results on the evaluation of the accuracy, stiffness, and stability of the shell structures of mirror antennas using simulation modeling are the basis for their use in the instrumentation, radio engineering, and cyber-physical biosensor industries [21, 22, 23]. It should be noted that the rigidity of the antenna arrays affects the accuracy and quality of the transmitted information, but does not increase the speed.
In the course of the work, the methods of manufacturing antenna systems and manufacturing reflective surfaces based on new technological and design ideas are considered. The process of formation of shells from mesh material that can be used for the manufacture of axisymmetric and non-axisymmetric reflectors or individual elements of mirror antennas is investigated.
This is because the formation of this material consists of several stages. The first stage is deformation of the two-dimensional mesh and fixing of the nodes by electric arc spraying with aluminum. The second stage, after sputtering on the punch, is to obtain a surface with a high temperature and apply a composite material such as polystyrene.
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