3D Finite Element Thermal and Structural Analysis of the RT-70 Full-Circle Radio Telescope

Transcription

1 3D Finite Element Thermal and Structural Analysis of the RT-70 Full-Circle Radio Telescope Alexey I. Borovkov Denis V. Shevchenko Alexander V. Gaev Alexander S. Nemov Computational Mechanics Laboratory, St.Petersburg State Polytechnical University, Russia Abstract: In the current paper finite element modeling and analysis of deformed state of the radio telescope RT-70 main reflector is performed. The construction is loaded by gravity body force and temperatures originating in the main reflector under the action of solar heat. Radio telescope RT-70 is one of the biggest radio telescopes in the world; diameter of the main reflector is 70 m, its height is 50 m. The hierarchical sequence of FE models was developed to describe in details and analyze temperature and displacement fields originating in the radio telescope RT-70 by means of direct FE modeling. With use of ANSYS software finite element models were developed that include three main construction elements of the radio telescope: - Main reflector (simulated by 1188 thin shell elements number of that is equal to the number of facets in the radio telescope); - Structural frame (simulated by means of ~ beam elements); - Counter-reflector (simulated by means of thin shells and 3D solid elements). Developed mathematical and 3D finite element models enabled to compute displacements and temperatures in the points of RT-70 radio telescope main reflector for various climate conditions (Yevpatoria, Crimea; Suffa plateau, Uzbekistan). To verify the results of simulation the comparison with experimental data obtained as a result of experimental analysis of RT-70 telescope located in Yevpatoria, Crimea was fulfilled. The importance of results obtained and high degree of finite element results correspondence with experimental data show that computer simulation is essential during radio telescopes design. Such simulation could be used to estimate displacements and temperatures originating in the radio telescope under various external loads. Mathematical and 3D finite element models developed are applied for analysis and design of RT-70 radio telescope that is being built at Suffa plateau, Uzbekistan. Introduction One of the most urgent problems arising during design and creation of unique antennas including RT-70 radio telescope is a detailed analysis of temperature, strain and stress fields originating in operating regime. Solution of similar problems taking into account all peculiarities of the construction can be obtained only by means of finite element (FE) modeling [7]. For a long time the analysis of the behavior of radio telescopes designed and produced in Russia was performed by experimental methods. This way does enable to analyze construction behavior under influence of various external factors, but it is very expensive, time-consuming and in some situations inapplicable. At present time the design process of competitive high-tech production at world leading companies requires full-scale FE modeling substituting expensive full-scale experiment with much cheaper numerical

2 simulation. FE modeling enables to analyze behavior of spatial constructions under the action of several external factors. Today s level of hardware enables to solve complex spatial thermo-mechanical problems with contact interaction, physical (temperature-dependent material properties, radiation, plastic material properties, etc.) and geometrical (large displacements, strains and rotations) non-linearities and obtain a solution in several hours. It should be noted that full-scale experiments as a rule enable to measure values in dozens or hundreds points, but in FE simulation the number of these points is increasing to hundreds of thousands and even more. Among the examples of successful application of FE modeling during the design process of antennas we can mark out the creation of telescope for Peking Astronomical Observatory (diameter 50 m) [5]. Fullscale FE modeling of the telescope with use of ANSYS FE software [6] was done in China before the realization of this project. FE modeling enabled to analyze deformed state and estimate stress concentration zones in the construction elements, evaluate natural frequencies and corresponding modes of the construction. As another example of application of modern methods of computer simulation based on FE analysis we can present design development of Maksutov Cassegrain space telescope [5]. During this project implementation the 3D model of space telescope was created in FE software ANSYS. This model allowed to analyze thermal stresses and evaluate temperature fields originating in the telescope on the lower Earth orbit. Analysis of telescope vibrations observed during orbiting was also carried out. One of the usages of RT-70 is radio-astronomy and ultra space communications with space apparatus. Maximum efficiency of RT-70 operation will be reached if it works in millimeter-wave band. Nevertheless, at operating in this regime the noise becomes significant originating due to solar heating of radio telescope main reflector and deformations of the mirror due to various external factors. Analysis of these factors and their influence on the correct operating of the radio telescope requires detailed analysis of stressed and strained states, temperature and thermal strains fields originating under various loadings. In order to solve this problem it is necessary to carry out analysis for various loading types such as: 1. Structural analysis of the construction under gravity loading; 2. Steady and transient thermal analysis; 3. Thermal stresses and strains analysis; 4. Modal analysis; 5. Structural analysis of the construction under wind loading; 6. Structural analysis of the construction under all types of the above loading types; 7. Dynamic analysis of the construction under seismic loading. In the current paper with use of ANSYS software FE modeling and analysis of 3D deformed state of RT-70 radio telescope caused by the action of gravity and solar heating of RT-70 main reflector is carried out. Finite element modeling and analysis of temperature fields of the radio telescope RT-70 main reflector Differential equation of heat conduction in heterogeneous anisotropic media. Initial and boundary conditions Consider temperature field T(r, t) in heterogeneous anisotropic body V in orthogonal Cartesian coordinate system 0x 1 x 2 x 3 ; r =x 1 e 1 + x 2 e 2 + x 3 e 3 = x k e k vector-radius of some point (Einstein rule about summing over repeating index is used to write down vector r in a compact way), t time. Differential equation of thermal conductivity defining connection between alteration of temperature in space and time in heterogeneous anisotropic body is the following [1]: ( K ( r) T ) + q (r, t = ρ ct& ) v

3 where = e k Hamilton derivative operator; K (r) symmetrical 2 nd rank conductivity tensor, x T K = K ; (, t) k q v r internal heat generation per unit volume; ρ ( r); c( r) density and specific heat of. T material; symbol " " defines derivative with respect to time T& = ; symbol " " defines convolution. t For transient processes of heat conduction it is necessary to define initial distribution of temperature in the body (, t) = Τ( r, ) ( r) Τ t 0 0 = Τ r = 0 that have effect on the temperature field only at the initial stage of transient process if we consider finite body. Consider boundary conditions at the surface of the body S= S 1 S 2 S 3 : ( r, t) = T ( r, t) T S 1 s I type of boundary conditions (Dirichlet conditions) temperature values are prescribed at S 1 surface; q = n K( r) T q ( r t) II type of boundary conditions (Neumann conditions); q s (, t) n S S = s, 2 2 heat flux per unit area at S 2 surface; n unit vector normal to the surface; ( T T ( r, )) r n q ( S = n K r) T (, t) t 3 S = α 3 S r S 3 III type of boundary conditions (conditions of Cauchy or Newton-Riehmann); T reference temperature; α s heat transfer coefficient at S 3 surface. Remark: Third type boundary conditions can be used when considering heating or cooling of bodies by radiation (radiation heat transfer). Stefan-Boltzmann law states that radiational heat transfer between two surfaces is equal to ( r, t, T )( T T ( r, )) 4 4 qs ( r, t) = σ 0ε T Tα = α s S t S 3 α where σ 0 = 5, W/(m 2 K 4 ) Stefan- 3 Boltzmann constant, ε emissivity factor of the surface, T α absolute temperature of the heated surfase, α s radiant heat exchange coefficient: α 2 2 ( r, t, T ) σ ε ( T + T )( T T ) s = S α S α Finite element thermal analysis To perform FE analysis the full-scale 3D FE model of the full-circle antenna was developed. This model includes three main construction elements of the real physical structure: main reflector consisting of facets (thin shells); structural frame of the radio telescope and counter-reflector. Structural frame is simulated by beam finite elements (2-node LINK33 elements [6] with one degree of freedom temperature in the node). Main reflector is simulated by 4-node shell finite elements SHELL57 [6] with one degree of freedom temperature in the node. Counter-reflector is also simulated by 4-node shell finite elements SHELL57 and 8-node solid finite elements SOLID70 [6] with one degree of freedom temperature in the node. In Figures 1, 2 FE model is shown in comparison with operating radio telescope P-2500 that is similar to RT-70 and situated in Yevpatoria, Crimea (Ukraine) [3,4].

4 Figure 1. Operating radio telescope P-2500 located in Yevpatoria, Crimea (Ukraine) Figure 2. FE model of RT-70 radio telescope

5 To describe in details and analyze temperature fields originating in the radio telescope RT-70 by means of direct FE modeling the hierarchical sequence of FE models was developed: I-level 3D model the surface of main reflector consists of connected to each other surfaces. The number of surfaces is equal to the number of facets in the real construction (1188 facets), structural frame is simulated by means of thermal conductive beam elements. Beam model enables to simulate all pipes forming the structural frame (~ ), but doesn t simulate heat exchange between beams and environment; II-level 3D model facets are not connected to each other, i.e. heat exchange through the lateral boundary of the facet is considered; III-level 3D model (model of the highest adequacy to the real physical object among all described above) structural frame is simulated by means of shell elements, i.e. this model enables to take into consideration heat exchange between beams and environment through the lateral boundary of beam elements of the construction; IV-level 3D model is applied to obtain more precise results in some part of the II or III level models with use of the submodeling method. Submodeling method enables to obtain more exact results is some specific parts of the model (macro-model). To do it one should create the FE model of this part of interest (submodel) so that FE mesh is fine enough to get accurate results. Temperatures obtained, as the result of macro-model analysis, should be applied as boundary conditions at the cut-boundary of sub-model. In the present paper the result of FE modeling and analysis with use of I- and II-level models are presented. Thermo-physical properties of the materials are the following: facets, counter-reflector aluminum: k 1 = 237 W/(m K) conductivity; structural frame steel: k 2 = 43 W/(m K). Parameters of the 3D I and II level FE models: number of FE NE S = 378 3D solid elements; NE Sh = shell elements; NE B = beam elements. At Figure 3 scheme of heat transfer between main reflector and environment is presented: q c solar heat flow; q k heat flow originating as the result of convective heat transfer between reflector and environment; α = 10.6 W/(m 2 K) coefficient of convective heat transfer; T 0 = 23 0 С reference temperature. k Figure 3. Scheme of heat transfer between RT-70 radio telescope main reflector and environment At Figure 4 the steady-state temperature field originating in the radio telescope RT-70 is presented (I level model). It should be noted that maximum values of temperature originate in the counter-reflector and central part of the mirror, T max = С.

6 Figure 4. Steady temperature field (I level model) In Figure 5 steady temperature field is presented that is originating in radio telescope (2 nd level model). Maximum values of temperature are observed in the eighth band of the radio telescope main reflector, T max = С. Figure 5. Steady temperature field (II level model) In Figure 6 distribution of heat flow originating in radio telescope (II level model) is shown. Maximum values of heat flow are observed in the central part of the mirror.

7 Figure 6. Heat flow distribution (II level model) In table 1 the results of FE simulation with use of I -and II-level models and experimental data temperature values in control points on the internal side of the main reflector facets are listed to compare TEXP Т FEA (See Figure 7). In brackets the value of relative error ε = 100% is presented. T EXP Figure 7. Positions of sensors

8 It follows from table 1 that error is exceeding 25% for the I-level model, and application of the II-level model enables to make the results more accurate and decrease the error to 7,6%. Such significant elaboration of the results obtained with use of II-level FE model in comparison with I-level FE model confirms necessity to consider the convective heat transfer with environment through the side boundary of the facet. It should be also noted that temperature in the center of main reflector is the closest value to experimental data. It refers that it is necessary to consider variation of convection heat transfer coefficient at the surface of the main reflector. Table Т EXP, C 30,1 30,6 31,0 31,9 32,8 31,9 31,7 32,8 T I, C FEA ( ε %) 39,5 (31,2) 40,1 (31,0) 40,5 (30,6) 40,9 (28,2) 41,2 (25,6) 41 (28,5) 40,8 (28,7) 41,3 (25,9) T II, C FEA ( ε %) 32,4 (7,6) 32,3 (5,5) 32,8 (5,8) 32,7 (2,5) 32,9 (0,3) 32,7 (2,5) 32,4 (2,2) 31,9 (2,7) Finite element modeling and analysis of 3D deformed state of RT-70 radio telescope main reflector under action of gravity Differential equations of equilibrium. Boundary conditions. Determinative relations. Quasi-static problem of the theory of elasticity in terms of displacements for heterogeneous anisotropic media is consisting in the solution of three differential equations of equilibrium and determination of displacement vector components: 4 ( u) + f = 0 C(r), where r = x kek = x1e1 + x 2e2 + x3e3 vector-radius of some point; u = u kek displacement vector; 4 = e k Hamilton derivative operator; C(r) = Cijkleie je kel fourth rank tensor of elastic moduli; x k V fvkek f = body force vector; symbol "٠" defined convolution. In addition to system of equilibrium that determine behavior of body point, boundary conditions should be considered on the surrounding the body surface S: u S = us(r) kinematical boundary conditions, where u S = uskek prescribed on the boundary displacement vector; 4 ( C(r) u) fs n = S statical boundary conditions, where f S = fskek prescribed on the boundary vector of surface load; n = n e k k unity normal vector to the surface of the body. 4 ( C(r) u) = fs S = S1 S2 u mixed boundary conditions. = u S S (r) ; n ; 1 S2 V

9 Specifying combined boundary conditions is also possible, when of three conditions that musty be defined in every point of the surface S, one (two) are formulated in terms of displacements and two (one) in terms of forces. Finite element analysis of 3D deformed state of RT-70 radio telescope under action of gravity To perform FE analysis the 3D FE model of full-circle antenna RT-70 was developed that considers three main structural elements of the real construction: main reflector, consisting of facets (thin shells); structural frame of the radio telescope and counter-reflector. Structural frame is simulated by means of 2-node beam elements BEAM189 [6] (Timoshenko beam) with 6 degrees of freedom in every node three components of displacement vector u x, u y, u z and 3 rotations r x, r y, r z. Beam model considers all tubes of structural frame (~13 000). To simulate reflector 4-node shell finite elements SHELL63 [6] with 6 degrees of freedom in every node three components of displacement vector u x, u y, u z and 3 rotations r x, r y, r z are applied. Surface of main reflector consists of non-connected surfaces, number of that is equal to the number of facets in real construction (1 188 facets). Counter-reflector is also simulated with use of 4-node shell elements SHELL57 and 8-node solid elements SOLID45 [6] with 3 degrees of freedom in every node three components of displacement vector u x, u y, u z. α angle defining orientation of main reflector relative to Earth surface, four values of angle are considered in the current paper α = 0, 30 0, 60 0, 90 0 (See Figure 8) [2,4]. Figure 8. Orientation of main reflector relative to Earth surface Physical and mechanical properties of materials: facet, counter-reflector aluminum: E 1 = 70 GPa Young s modulus, ν = 0.31 Poisson s coefficient; structural frame steel: E 1 = 210 GPa, ν = Parameters of 3D FE model: number of FE NE S = 378 3D elements; NE SH = D shell elements; NE B = D beam elements. Structural frame is constrained at two supporting rings. In Figures 9-12 displacement vector modulus fields are presented that originate in RT-70 main reflector at α = 0, 30 0, 60 0, 90 0.

10 Figure 9. Field of displacement vector modulus in RT-70 main reflector at α = 0 Figure 10. Field of displacement vector modulus in RT-70 main reflector at α = 30 0

11 Figure 11. Field of displacement vector modulus in RT-70 main reflector at α = 60 0 Figure 12. Field of displacement vector modulus in RT-70 main reflector at α = 90 0 In table 2 maximum values of displacement vector modulus in RT-70 main reflector are listed depending on orientation angle of main reflector relative to Earth surface. It follows from table 2 that maximum value of displacement vector modulus in RT-70 main reflector is observed at orientation angle equal to 60 0.

12 Table 2 α, max u, см 1,2 2,9 3,89 3,86 Conclusions 1. Thermo-mechanical problems of the full-circle radio telescope RT-70 are classified; 2. Mathematical and 3D finite element models are developed to analyze temperature fields in the radio telescope RT-70 main reflector; 3. The hierarchical sequence of FE models was developed to describe in details and analyze temperature fields originating in the radio telescope RT-70 by means of direct FE modeling; 4. 3D FE heat conduction analysis of RT-70 structure is performed; 5. It is shown that 3D FE results obtained by mean of II-level model are in good agreement with experimental results for the structures similar to RT-70; 6. Mathematical and 3D finite element models are developed to structural analysis of the construction under gravity loading; 7. 3D FE structural analysis of the construction under gravity loading analysis of RT-70 structure is performed References 1. Borovkov A.I., Zubov A.V. Finite element solutions of stationary and non-steady heat conduction non-linear thermal problems for heterogeneous anisotropic media. Moscow. VINITI publishing. No. 897-В p. (in Russian) 2. Borovkov A.I., Shevchenko D.V., Nemov A.S. Finite element modeling and analysis of 3D deformed state of RT-70 radio telescope main reflector under action of gravity // Proc. V Int. Conf. «Scientific and technical problems of prognostication and durability of structures and solution methods». St.Petersburg. SPbSPU pp Borovkov A.I., Shevchenko D.V., Gaev A.V. Finite element modeling and analysis of temperature fields in RT-70 radio telescope main reflector // Proceedings of V Int. Conf. «Scientific and technical problems of prognostication and durability of structures and solution methods». St.Petersburg. SPbSPU pp Borovkov A.I., Shevchenko D.V., Gimmelman V.G., Machuev Yu.I., Gaev A.V. Finite-Element Modeling and Thermal Analysis of the RT-70 Radio Telescope Main Reflector // Proc. IVth Int. Conf. "Antenna Theory and Techniques". Sevastopol. Ukraine pp. 5. Proc. 10 th Int. ANSYS'2002 Conf. "Simulation: Leading Design into the New Millennium". Pittsburgh. USA ANSYS theory reference. Eleventh edition. SAS IP, Inc., Zienkiewicz O.C., Taylor R.L. The Finite Element Method. Butterworth-Heinemann, 2000.