Восточно-Сибирский государственный университет технологий и управления
Институт машиноведения имени А.А. Благонравова РАН Уханьский текстильный университет (Китай)
Иркутский государственный университет путей сообщения Братский государственный университет
ПРОБЛЕМЫ МЕХАНИКИ СОВРЕМЕННЫХ МАШИН
Материалы
VII Международной научной конференции 25 – 30 июня 2018 г.
Том 3
Улан-Удэ
Издательство ВСГУТУ
2018
2
УДК 621.01 ББК 34.41
П 781
Печатается по решению редакционно-издательского совета Восточно- Сибирского государственного университета технологий и управления
Редакционная коллегия И.Г. Сизов, г. Улан-Удэ
В.К. Асташев, г. Москва
В.С. Балбаров – ответственный редактор, г. Улан-Удэ А.Д. Грешилов, г. Улан-Удэ
С.В. Елисеев, г. Иркутск
С.К. Каргапольцев, г. Иркутск П.М. Огар, г. Братск
П 781 Проблемы механики современных машин: материалы VII Междунар. науч. конф. – Улан-Удэ: Изд-во ВСГУТУ, 2018. – Т. 3.
– 68 с.
ISBN 978-5-6041868-3-1 (Т.3) ISBN 978-5-6041868-0-0
ББК 34.41
© ВСГУТУ, 2018
3
Ministry of Education and Science of the Russian Federation Russian Academy of Sciences
East Siberia State University of Technology and Management Mechanical Engineering Research Institute
of the Russian Academy of Sciences (IMASH RAN) Wuhan Textile University (People's Republic of China)
Irkutsk State Transport University Bratsk State University
ISSUES ON MODERN MACHINES MECHANICS Proceedings of the VII International scientific Conference
June 25 – 30, 2018 Ulan-Ude, Baikal
Tom 3
Ulan-Ude
Publishing Department of East Siberia State University of Technology and Management
2018
4
УДК 621.01 ББК 34.41
П 781
Editorial Staff I.G. SIZOV, Ulan-Ude V.K. ASTASHEV, Moscow
V.S. BALBAROV, Ulan-Ude – Science editor A.D. GRESHILOV, Ulan-Ude
S.V. ELISEEV, Irkutsk
S.K. KARGAPOLTSEV, Irkutsk P.M. OGAR, Bratsk
П 781 Issues on Modern Machines Mechanics: Proceedings of the VII International scientific Conference. – Ulan-Ude: Publishing Department of East Siberia State University of Technology and Management, 2018. – T.3. – 68 p.
ISBN 978-5-6041868-3-1 (Т.3) ISBN 978-5-6041868-0-0
ББК 34.41
© ВСГУТУ, 2018
5 УДК 62
AERODYNAMIC EVALUATION OF BIBLADE PROPELLER Erdenebat T., Ganbat D., Batzorigt E.
Mongolian University of Science and Technology, Mongolia jonon@must.edu,.mn ganbatda@must.edu.mn, ebatzorigt@hotmail.com
A small scale blade configuration with fixed pitch propeller for unmanned aerial vehicle (UAV) is used at lowReynolds number.Biblade propeller design aims at achieving high propulsive efficiency at low levels of energy, usually with double lift surface. In order to obtain good aerodynamics under low-Reynolds number conditions, we built double aerodynamic surface. biblade significantly reduces flow separation due to position of surfaces at low Reynolds number. This paper investigates the influence of the two-dimensional effect to the clark Y airfoils using the biplane concept.
Keywords: biblade propeller; double surfaces propeller.
Introduction
Nowadays, Propellers for small scale unmanned aerial vehicle are improving and developingat low Reynolds numbers. Many scientists developed by the use of new materials, new airfoil models and new propeller configurations. The optimum design methods is developed for the blade design configurations. In this study, an optimum blade design feature is calculated to design the two-dimensional Ansys fluentanalysis. Therefore,propeller induced velocity is calculated based on low Reynolds number.
One of typical problem of lowReynolds number air flow moving a blade is flow separation. Many studies about aerodynamic quality in the blade have been carried out to understanding how to decrease flow separation. It has been understanding that a flow separation point plays an important problem to increase the aerodynamic quality.
The present some study aims to clarify three-dimensional flow characteristics around the propeller blade due to low-aspect ratio and due to low-Reynolds number in order to develop a propeller with the high efficiency [3]. The leading edge vortex has spanwise velocity component due to low-aspect ratio of the airfoil and the spanwise velocity component enhance the stability of the leading edge vortex [3].Other scientists, the influences of the blade planform are depending on skewed blade. The blade skew affects the tip vortex structures.
The aerodynamics of the propeller can be clarified by assuming section of the blade.Wherethrough the blade section is an airfoil its aerodynamics can be studied in the same way, using the same terms. The cross- section of biblade propeller may be comparable to the wing of anBusemann biplane airfoil. A basic blade configuration includes a double surface, attached together to the central hub. There is double thrust and double drag.
propeller design
The biblade propeller is a two-bibladed propeller which is a fixed pitch propeller. There is two aerodynamic surface in the biblade propeller. One of the bigger blade is primary surface. And another blade is additional aerodynamic surface. The cross-section of blade consists of clark-Y airfoil near the tip with low Reynolds number, build to form a sharp leading edge blade configuration, as shown in Fig 1.
Fig. 1.3D printed biblade printed propeller
6 Airfoil model
In order to absorb the best results, the 2D Ansys fluent analysis was performed. The biblade propeller airfoil used in this study, as shown in Figure 2. The appropriate size and position of the biblade airfoil was determined from reference file. The 2D fluint simulation has been determined clark-Y airfoil with 1 m chord line. All simulation has been installed on 8-120 AOA.
Fig 2.
Fig. 3. 3D model of Biblade propeller Results
The numerical analysis results were compared aerodynamic evolution of basic blade airfoil and additional blade airfoil. Accurate prediction of 2D airfoils' aerodynamics at stall conditions even at low Reynolds numbers is still a challenging problem.
Fig. 4. Busman biplane airfoils
In the Busman biplane airfoils, airseparation was high at the upper surface of top blade and air density was high at the lower surface of top blade butair flow was accelerating between blades.
7
Fig. 5. Upper and trailing addition blade
Flow separation region was small at the top blade and also reduce the flow separation between two blades. There is a good aerodynamic performance (Fig. 4).
Fig. 6. Additional blade is lower and trailing position
Flow separation was decreased in the upper surface of bottom blade but separation was kept and high separation region in upper surface of main blade.
Fig. 7.Additional blade is upper and leading position
Flow separation was decreased in the upper surface of main blade but air density was extremely high at the upper surface of main blade. Separation was significantly high in the upper surface of auxiliary blade.
Сonclusion
In this study, the transient calculations of low Reynolds number flows are carried out over anclark Y airfoil in two- dimensions. Flow separation region are calculated in each case with the available constant velocity.Upper and trailing addition blade position is better flow simulation than other position. Air flow is very smoothly and flow separation region is too low.
Bibliography
1. Sereez M., Abramov N.B.. Computational Simulation of Airfoils Stall Aerodynamics at Low Reynolds Numbers // Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia.
2. Batzorigt E., Dashhuyag G. Bi-blade high efficiency propeller // ICMAR Perm, Russia. –2016. –P.
22-23.
3. Yonezawa K., Shigeru S. Experimental and Numerical Investigations of Three-Dimensional Flows around Propellers in Low-Reynolds Number Flows // Trans. JSASS Aerospace Tech. Japan. – 2014. – Vol.
12, No. 29. – P. 65-70.
4. Kutty H.A., Rajendran P. 3D CFD Simulation and Experimental Validation of Small APC Slow Flyer Propeller Blade // MDPI, Basel, Switzerland, 2017.
8
5. Hartman E.P., Biermann D. Aerodynamics of full-scale propellers having 2, 3 and 4 blade of clark Y and R. A. F. 6 airfoil sections // NASA report No.640.
УДК 62
COUPLED ANALYSIS AND LATTICE-BOLTZMANN METHOD VEHICLE MOTION OF A PIURS XW20 CAR IN CROSSWIND CONDITION
Tumur E., Batzorigt E., Lochin K., Tumur-Ochir E., Baasanbold B.
Mongolian University of Science and Technology, Mongolia, Ulaanbaatar enkhbold@must.edu.mn, e.batzorigt@hotmail.com, khenmed@must.edu.mn,
jonon@must.edu.mn, Baasandorj.space@must.edu.mn
Mongolians countryside road traffic occurring fatal accident many times in the years. Therefore sedan car accident concern 92.5 percent in total fatal accidents, tractor heavy vehicle accident concern 3.7 percent, bus accident concern 1.5 percent. Fatal accident occurring in countryside road 395 last year, metropolitan road 112 last year. Cause of fatal accident is depending on driver’s control, road condition, car’s technical problem, crosswind condition. This paper concerning crosswind condition car fatal accidents. Prius series car lighter weight for improving power performance and reducing fuel consumption, a prius with higher-speed and lighter weight will lead to the prius more sensitive to the crosswind, which will affect the stability and drivability of the prius. This paper employs the fully-coupled method to investigate a prius subjected crosswind with consideration of the car motion.
Lattice-Boltzmann and dynamic mesh is adopted to investigate the unsteady aerodynamic, and the vehicle is treated as a three-freedom-system and driver’s control is considered to investigate the vehicle dynamic.
Crosswind stability and drivability of high-speed prius has attracted more and more attention during the last decades, in order to promote vehicle aerodynamic design and technology to a higher level. Crosswind may threaten the safety of the running vehicle due to the unsteady aerodynamic loads (Cai 2015). Many reports (Wang et al. 2014) show that the crosswind is a significant factor causing the vehicle accidents.
Key words: LBM, Coupled analysis, fatal accident.
Introduction
Ldttice-Boltzmann’s equations
–collision integral, –distribution function.
Ω
The target passenger car in the present study is a simplified passenger car model which is based on the real commercial priusXW20, with its length, width and height of 4,450mm, 1,725mm, and 1,490mm, respectively. Fig. 1 shows the side view of the simplified passenger car geometry.
Fig.1. Prius XW20
9
Fig.2. Coordinate of a PriusXW20’s body and tire
Fig.3. Drag, lift coefficient three dimensional analysis wolfram matematica program Conclusion
An investigation of unsteady aerodynamic of a prius XW20 crosswind coupled with passenger car motion have been performed in this study, which employs the fully-coupled, Lattice-Boltzmann methods. For the vehicle dynamics simualtion, the passenger car is simplified as a 3 DOF system and the driver’s control is considered. Lattice-Boltzmann methods, the aerodynamic loads are obtained firstly and applied on the vehicle dynamics model. However, the aerodynamic loads is obtained as the priusXW20 motion in fully- coupled simulation.
The result of Lattice-Boltzmann is more similar to reality than the result of the fully-coupled method simulation. The results of the fully-coupled, quasi-steady and LB method indicate that unsteady aerodynamics loads have siginificant influences to the passenger car motion of the priusXW20. Large difference could be found by comparison of the results of the two methods. The peak of the lateral displacement in LB method is smaller than in fully-coupled simulation, the peak vaule in LB method, quasi- steady simulation is 0.72m, 0.79m and 0.83m, respectively. The change tendency of the lateral force, yaw angle, yaw angular velocity and steering angle in LB method is quite different with the change tendency in fully-coupled and LB method. Because the hysteresis effect of the flow, the change of results of the fully coupled simulation is more obviously than LB method. These results clearly indicated the importance of the unsteady aerodynamic in moving analysis of passenger car.
10 Bibliography
1. Zhu H., Zhigang Y. Fluid-structure interaction study of three-dimensional vehicle model under crosswind. // Advances mechanical engineering. – 2015. – Vol.7 (6), No. 1-10.
2. Choi H., Lee J., Park H. Aerodynamics of Heavy Vehicles. Fluid mechanics 2014.16.441-468.
3. Baker C. J. High sided articulated road vehicles in strong cross winds. //J. Wind Eng. Ind. Aerodyn.
– 1988. – Vol. 31, No.1.– P. 67-85.
4. Bakker E., Pacejka H. B., Lidner L. A new tire model with an application in vehicle dynamics studies. // SAE paper. – 1989.
5. Cai C. S., Hu J., Chen S., Han Y., Zhang W. and Kong X. A coupled wind-vehicle-bridge system and its applications: a review. // Wind Struct. – 2015. – Vol. 20, No. 2. – P. 117-142.
6. Carrarini A. Reliability based analysis of the crosswind stability of railway vehicles. // J. Wind Eng.
Ind. Aerodyn.– 2007. – Vol. 95, No. 7. – P. 493-509.
7. Chu C., Chang C., Huang C., Wu T., Wang C. and Liu M. (2013). Windbreak protection for road vehicles against crosswind. // J. Wind Eng. Ind. Aerodyn. – 2013. – Vol. 116, No. 5. – P. 61-69.
8. Cui T., Zhang W. and Sun B. Investigation of train safety domain in cross wind in respect of attitude change. // J. Wind Eng. Ind. Aerodyn. – 2014. – Vol. 130. – P. 75-87.
9. Gu Z. Q., Huang T. M., Chen Z., Zong Y. Q. and Zeng W. large eddy simulation of the flow-field around road vehicle subjected to pitching motion. // J. Appl. Fluid Mech. – 2016. –Vol. 9, No. 6. – P. 2731- 2741
10. Gulyasa A., Bodorb A., Regertb T. and Jánosic I. M. PIV measurement of the flow past a generic car body with wheels at LES applicable Reynolds number. // J. Heat Fluid Flow. – 2013. Vol. – 43. – P. 220-232
11. Hucho, W. and Sovran G. Aerodynamics of road vehicles. // Annual Reviews of Fluid Mechanics. – 1993. – Vol. 25. – P. 485-537.
12. Jonathan, M., Erik F., Gregory R., Rajan K., Kunihiko T., Farrukh A., Yoshihiro Y. and Kei M. Drag reduction on a flat-back ground vehicle with active flow control. // J. Wind Eng. Ind. Aerodyn. – 2015. – Vol. 145. – P. 292-303.
13. Krajnovic, S., Bengtsson A. and Basara B. Large eddy simulation investigation of the hysteresis effects in the flow around an oscillation ground vehicle. //Journal of fluids engineering – 2011. – Vol. 133, Issue 12. – P. 121103-1-121103-9.
УДК 62
SPARK PLASMA SINTERING METHOD ON THE FABRICATION OF TUNGSTEN BASED HARD ALLOY
Telmenbayar L., Yadamragchaa T., Delgermaa М.
School of mechanical engineering and transportation, MUST, Ulaanbaatar, Mongolia telmenbayar@must.edu.mn, Deegii_m@yahoo.com, Ydmaa1008@gmail.com
In this work, the effect of the sintering temperature, pressure, holding time and WC:Co weight ratio on optimization of tungsten carbide based hard alloys were investigated. And alloys with good properties were successfully prepared by spark plasma sintering method. The result show that sintering temperature, pressure, holding time and WC:Co weight ratio are critical factors affecting the densification and mechanical properties of the alloys. Pure tungsten carbide (99.99%) and cobalt (99.99%) powder mixed 24 h with mixing speed of 60 rpm. Mixtures heated up 1000-12000С, under pressure of 30-70 MPa, during 15- 30 min in the Spark plasma sintering furnace. ). Sample is measured by Rockwell hardness testing machine and density measurement method. When the sintered for 30 min under 70 MPa, the nearly fully densified alloys are produced. And when the sintering temperature increases to 10000C, alloys have the best hardness.
Therefore, the WC-Co cemented carbide sintered at 10000C for 30 min under 70 MPa with weight ratio of 87.5:12.5 exhibits the optimum densification and mechanical properties.
Key words: hard alloy, temperature, pressure, holding time, hardness, weight ratio, sps
11 Introduction
Conventional cemented carbide is matrixed of refractory carbides (e.g. WC, TiC, TaC) with ductile metal Co as the binder phase, and prepared by powder metallurgy technology [1]. WC-Co cemented carbides with high hardness and high temperature strength, good fracture toughness and suitable wear resistance, have been extensively used in cutting, machining, drilling, mining tools, wear resistant parts, etc [1-2]. There is large amount application of hard alloy in metal processing field in Mongolia, but all companies import hard alloy from abroad. Hard alloy is used for drilling bit and machining process on many kinds of material. It is very hard and wear resistant material. There are over 400 drilling companies, plenty of metal processing workshops and laboratories in Mongolia. For instance, only “Tanan Impecs” LLC utilize 112 pieces of hard alloy in one month and it costs about 520 million tugrugs [11-12]. For that reason, fabricating hard alloy in Mongolia is demanded. Furthermore, there are some tungsten deposits in Mongolia and tungsten is main element for hard alloy [13-14]. Therefore, study for fabricating hard alloy in Mongolia is started.
Spark plasma sintering (SPS) is a newly developed sintering method, which enables a powder compact to be sintered by Joule heat by high pulsed electric current through the compact. During the past few years, this kind of sintering method has been described for sintering different kind of materials [3, 5-9] and WC-Co cemented carbides also included.
In this paper, various amounts of WC-Co powders were consolidated to full density by SPS. Effect of sintering temperature, pressure, time and holding time were investigated.
The goal of this study was to evaluate the applicability of SPS method to fabricating tungsten based hard alloy.
Materials and experimental methods
Pure tungsten carbide (99.99%) and cobalt (99.99%) powder were used as raw materials. Powders mixed for 24 h with speed of 60 rpm. Following mixture rule was used for mixing the powders.
(1) Volume fraction and density of cobalt and tungsten carbide is calculated by above formula. Mixtures pressed without binder into cylindrical pellets compact of 30 mm diameter. The compact was transferred to SPS machine with graphite crucible. The chamber was evacuated with a vacuum pressure of 30-70 MPa for 15-30 min at 1000-12000C under argon gas. Nominal parameters for experiment are shown in Table 1.
Table 1. Sintering condition, theoretical density, experimental density and hardness of the investigated alloys
№
Tungsten carbide content, %
Cobalt content,
%
Pressure , MPа
Temper ature,
0С
Holding time, min
Preliminary calculated density, g/сm3
Density after sintering , g/сm3
Hardness, HRC
1 95 5 40 1200 20 15.26 11.99 73.2
2 92.5 7.5 70 1000 25 15.08 7.81 86.9
3 87.5 12.5 70 1000 30 14.76 12.18 85.3
4 87.5 12.5 70 1000 25 14.76 8.04 73.1
5 87.5 12.5 70 1000 20 14.76 10.3 44
6 87.5 12.5 70 1000 15 14.76 7.19 74
7 87.5 12.5 30 1000 25 14.76 6.09 57.3
8 80 20 40 800 20 14.26 6.79 36.7
9 72.5 27.5 70 1000 25 13.75 8.74 31.6
After sintering, hydraulic pressing machine is used for release samples from graphite mold. The densities of the sintered samples were measured according to Archimedes method. For mechanical tests, hardness was measured using a Vickers indenter on the polished surface with a 60 kg load.
Results and discussion
In order to determine the densification behavior of WC-Co materials, firstly sintering studies were carried out in the temperature range 800-12000C. The sintering conditions and densification results are given in table 1. Nearly fully dense tungsten carbide samples were obtained at 10000C, under a pressure of 70 MPa for 30 min. When the holding time further increased, the density of sample were decreased.
12
Fig.1. Thermal influence for hardness and density of alloys
Temperature affect for sample were shown in Fig 1. When temperature increased more than 10000C, hardness of samples was decreased. Density of sample 3 with weight ratio of WC:Co=87.5:12.5, was heated at 10000C under 70 MPa during 30 min was nearer to preliminary calculated density. Other sample’s density was lower than sample 3, it seems holding time and pressure influenced to density of samples. Sintering at higher temperature (12000C) was influenced negatively to the properties of hard alloy. For that reason, temperature should be below 11000C in this sintering method. When pressure below 70 MPa, density of alloy was low, thus pressure should be above 70 MPa.
Hardness of sample 2 with weight ratio of WC:Co=92.5:7.5 was 86.9 HRC, higher than other samples.
And hardness of sample 3 was 85.3 HRC. On the other hand, hardness of samples 5, 8 and 9 were lower than others. It seems that related with less pressure, holding time and amount of tungsten carbide.
Hardness of standard tungsten based hard alloy was 88-92 HRC [14]. Thus, hardness of fabricated hard alloy was slightly lower than standard alloy.
Conclusion
In this study, the sintering behavior and mechanical properties of WC-Co alloys sintered by SPS and the effect of temperature, holding time, pressure and WC:Co weight ratio on optimization of tungsten carbide based hard alloy were investigated using SPS. Almost full densification of WC-Co powders was achieved by SPS. Optimal conditions for sintering process were identified to be temperature below 11000С, pressure above 70 MPa, holding time of 25 min and WC:Co=87.5:12.5 weight ratio which resulted in tungsten based hard alloy with hardness of 86.9 HRC and density of 12.18 g/cm3.
Bibliography
1. Gao Y., Luo B.-H., He K., Jing H, Bai Z., Chen W., Zhang W.-W. Mechanical properties and microstructure of WC-Fe-Ni-Co cemented carbides prepared by vacuum sintering // Vacuum. –2017. – Vol.
143. – P. 271-282.
2. Fang Z.Z., Eso O.O. Liquid phase sintering of functionally graded WC-Co composites // Scr. Mater.
– 2005. – Vol. 52. – P. 785-791.
3. Yaman B., Mandal H, Spark plasma sintering of Co-WC cubic boron nitride composites // Materials Letters. – 2009. – Vol. 63. – P. 1041-1043.
4. Enayati M.H., Aryanpour G.R., Ebnonnasir A. Production of nanostructured WC–Co powder by ball milling // Int. Journal of Refractory Metals & Hard Materials.– 2009. – Vol. 27. – P. 159–163.
5. Siwak P., Garbiec D., Microstructure and mechanical properties of WC−Co, WC−Co−Cr3C2 and WC−Co−TaC cermets fabricated by spark plasma sintering // Trans. Nonferrous Met. Soc. China. – 2016. – Vol. 26. – P. 2641−2646.
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6. Kim H.-C., Oh D.-H., Shon I.-J., Sintering of nanophase WC–15vol.%Co hard metals by rapid sintering process // International Journal of Refractory Metals & Hard Materials. – 2004.– Vol. 22. – P. 197–
203.
7. Sivaprahasam D., Chandrasekar S.B., Sundaresan R. Microstructure and mechanical properties of nanocrystalline WC–12Co consolidated by spark plasma sintering // International Journal of Refractory Metals & Hard Materials. – 2007. – Vol. 25. – P. 144–152.
8. Zhao J, Holland T, Unuvar C., Munir Z.A. Sparking plasma sintering of nanometric tungsten carbide // Int. Journal of Refractory Metals & Hard Materials. – 2009. – Vol. 27. – P. 130–139.
9. Hulbert D.M., Jiang D., Dudina D.V., Mukherjee A.K. The synthesis and consolidation of hard materials by spark plasma sintering // Int. Journal of Refractory Metals & Hard Materials. – 2009.– Vol. 27.
– P. 367–375.
10. Eriksson M., Radwan M., Shen Z. Spark plasma sintering of WC, cemented carbide and functional graded materials // Int. Journal of Refractory Metals and Hard Materials. – 2013. – Vol. 36. – P. 31–37.
11. Delgermaa M., Yadamragchaa T, Zolbayar О., Murayama Y. Study for pressure on the hard
alloy produced by powder metallurgy. – 2016.
12. Telmenbayar L. Preparation of tungsten carbide from natural minerals with assistance of
mechanical activation, thesis. – 2016.
13.
Kurlov A.S., Gusev A.I., Tungsten carbide structure, properties and application inhardmetals. // Springer International Publishing. – 2013. – 242 p. – ISBN 978-3-319-00523-2.
14.
Upadhyaya G.S.Cemented tungsten carbides // William Andrew. – 1998. – 420 p.
УДК 539.3
ДВУМЕРНАЯ МОДЕЛЬ МНОГОСЛОЙНЫХ ТОНКОСТЕННЫХ КОНСТРУКЦИЙ Абидуев П.Л.*, Дармаев Т.Г.**
*Бурятская государственная сельскохозяйственная академия, Россия, Улан-Удэ, apl087@yandex.ru
** Бурятский государственный университет, Россия, Улан-Удэ, dtg@bsu.ru
Получены уравнения для многослойных тонкостенных конструкций в предположении, что многослойная пластина рассматривается как единое тело и модули упругости представляются как кусочно-однородные функции по толщине.
Ключевые слова: Многослойная тонкостенная пластина, напряженно-деформированное состояние, тангенциальные усилия, моменты, модули упругости.
TWO-DIMENSIONAL MODEL OF MULTILAYER THIN-WALLED DESIGNS
Abiduev P.L. *, Darmaev T. G. **
* Buryat state agricultural academy, Russia, Ulan-Ude, apl087@yandex.ru
** Buryat state university, Russia, Ulan-Ude, dtg@bsu.ru
The equations for multilayer thin-walled designs in the assumption are received that the multilayer plate is considered as a uniform body and elastic moduli are represented as patch homogeneous functions on thickness.
Key words: Multilayer thin-walled plate, intense strained state, tangential efforts, moments, elastic moduli.
Введение
Принципы сведения трехмерных уравнений упругости для многослойных тонкостенных конструкций к двумерным уравнениям теории пластин и оболочек давно известны [1-3]. Они сводятся к методам, основанным на применении кинематических гипотез, либо использование метода разложения по толщине, а также асимптотическим методам. В работе построение и
14
упрощение уравнений трехмерной теории упругости производится на основе представления многослойной конструкции, как тела с однородным напряженно-деформированным состоянием произвольной по толщине структуры. Данный подход является известным [3] и применялся при анализе уравнений слоистых пластин и оболочек. В данной работе рассматриваются уравнения без учета поперечных касательных напряжений. Определены приведенные упругие постоянные, т.е.
многослойную пластину рассматриваем как единое тело и модули упругости представляются как кусочно-однородные функции по толщине x3.
Вывод уравнений
Исходные уравнения выглядят следующим образом:
,j i 0
ij F
(i,j1,2,3) (1)
( )
1
33 22 11
1
11
E 12 12
2 1
( )
1
33 11 22
2
22
E 13 13
2 1
(2)
( )
1
22 11 33
3
33
E 23 23
2 1
ij
ui,j uj,i
2
1
(i, j1,2,3) (3) Поскольку для тонкостенных пластин и оболочек мы учитываем только моменты нулевого и первого порядков, то уравнения равновесия (1) проинтегрируем дважды: сначала только по толщине многослойной пластины, а затем предварительно умножив уравнения (1) на x3. Имеем
0 ))
( )
(
(
3 3 00
,j i i i
ij h h F
0 ))
( )
(
(
3 3 10 3 1
,j i i i i
ij h h h F
, i1,2,3; j1,2 (4) где
3
0 d x
h
h ij
ij
, 1 x3 dx3
h
h ij
ij
, Fi0 Fdx3
h
h
i
, Fi1 x3Fdx3
h
h
i
(5) Далее разложим в ряд по x3 напряжения:
n n
ijn
ij x x x B x1 x2 x3
0 3 2
1, , ) ( , )
(
(6) Из (5) для определения коэффициентов Bijn имеем:
1 1 0
1 (
1 n k
)
n k ijn ijk nn k
h h
B
, k0,1 (7) Введем обозначения:0 ij
Tij , Nij i03 , Mij ij1 (i, j1,2) (8) И уравнения (4) с учетом этих обозначений, можем записать в виде:
0 ))
( )
(
(
3 3 0,j i i i
ij h h F
T
0 ))
( )
(
(
33 33 30, h h F
Nii (9)
0
)) ( )
(
(
3 3 1,j i i i i
ij N h h h F
M , i, j1,2
Поскольку мы не учитываем поперечные касательные напряжения, т.е.
15
33
0
23
13
и напряженно-деформированное состояние реализуется тангенциальными усилиями T11,T12,T21,T22 и моментами M11,M12=M21,M22. И надо определить 11,12,22- однородные относительно x1,x2 и зависящие от x3.
Тогда из (2) имеем:
11 22
11
1
E 12 12
2
1
22 1
22 11
E 13
0
(10)
11 22
33
E 23
0
Далее из (3) имеем:
( , )1
3 2 1 1 22 12 11
1 x f x x
u E
x (x ,x )1
3 1 2 2 11 21 22
2 f
u E (11)
3
1 2
0
3 22 11
3
,
3
x x f E dx
u
x
Поскольку 13
0
и 23 0
, то ( , ) ( , ) 0
1
2 1 1 , 3 3 2 3 , 1 1 3 , 22
11
x f x x f x x
E
(12) x ( x , x ) ( x , x ) 0
1
2 1 2 , 3 3 1 3 , 2 2 3 , 11
22
f f
E
Тогда, можно сделать вывод, что производные от выражений в квадратных скобках по переменной x3 должны быть обязательно постоянными. Следовательно, из (10) имеем
11 22
11
1
E a1x3b1 (13)
22 11
22
1
E =a2x3b2 Далее, т.к. 2
1,2 2,1
2
1 u u
i
и по предположению, модули упругости представляются как кусочно-однородные функции по толщине x3, подставив i2 в четвертое соотношение в (10), имеем:
) , ( 2 3
2 ,
1 x x
f + f2,1(x1,x3)=
12
(14) Поскольку, по предположению правая часть в (14) есть функция только от x3, то каждое слагаемое в левой части есть тоже функция только от x3. Т.е.,
) , ( 2 3
2 ,
1 x x
f =G1
(
x3)
и f2,1(x1,x3) =G2(
x3)
Тогда,
) , (
2 31 x x
f U1
(
x3)
x2B1(
x3)
иf
2( x
1, x
3)
=U2(
x3)
x1B2(
x3)
(15) Из (12) и (15) имеем:1 3 3
1
(
x)
Сx CU ; U2
(
x3)
Сx3C2; B1(
x3)
S1x3S2; B2(
x3)
S3x3 S4 (16) где С,
C1,
C2,
S1,
S2,
S3,
S4 - постоянныеДалее, из (12), (15) и (16) получаем:
16
) , (
2 31 x x
f Сx3x2 C1x2 S1x3S2
;
f
2( x
1, x
3)
=Сx3x1C2x1 S3x3S4
(17)
) , (
1 23 x x
f ( )
2
1 2
2 2 2 1
1x a x
a Сx1x2 S1x1S3x2S5 Тогда здесь
L Сx
3
12 , ( )
2 1
2
1 С
C
L (18) Для перемещений имеем:
1 3 1 1
1 ax b x
u Сx3x2C1x2S1x3S2
2
u
a2x3 b2
x3Сx3x1C2x1S3x3S4 (19)
30
2 1 3 2 1 13 3
3
)
(
a a x b b dx Eu
x
)
2(
1 2
2 2 2 1
1x a x
a S1x1S3x2S5
Здесь
13 1
E ; (20) Постоянные С
,
C1,
C2,
S1,
S2,
S3,
S4 характеризуют здесь движение как движение жесткого целого.Из (13) и (18) следует, что
11 11
1 (0) e
b
22 22
2 (0) e
b (21)
12 12(0) e L
Т.е. b1,b2, L - тангенциальные деформации срединной плоскости.
Далее,
11 , 3 11 , 3 31 , 1
1 u u f
a ,
22 , 3 22 , 3 31 , 2
2 u u f
a (22)
12 , 3 12 , 3 31 , 2 32 ,
1 u u f
u
C Обозначим через
) , (
1 23 x x
f
w (23) - прогиб срединной поверхности.
Тогда из (13), (14), (22) и (23)
3 ,11 11
12
3 ,22 22
11
11 E x w e E x w e
3 ,11 11
11
3 ,22 22
12
22 E x w e E xw e
(24)
3 ,12 12
12 2 x w e
Здесь обозначены
11 2
1
E
E ; 12 2
1
E
E ; (25)
В (24) интегрируя от (h;h), имеем
12 , 12 11 , 11 12 12 11 11
11 A e A e K w K w
T
12 , 22 11 , 12 12 22 11 12
22 A e A e K w K w
T (26)
12 , 23 12
33
12 2A e 2K w
T
И также из (24) имеем выражения для моментов:
22 , 12 11 , 11 22 12 11 11
11 K e K e D w D w
M
22 , 22 11 , 12 22 22 11 12
22 K e K e D w D w
M (27)
17
12 , 33 12
33
12 2K e 2D w
M
Постоянные в формулах (26), (27) определяются:
dx3
E A
h
h ij
ij
, K E x3dx3
h
h ij
ij
, D E x32dx3
h
h ij
ij
,
3 12
33 dx
A
h
h
, K33 12x3dx3
h
h
, D33 12x32dx3
h
h
В этих формулах подынтегральные выражения представляют собой кусочно-однородные функции от координаты x3, что соответствует согласно предположению слоистой пластине.
Перемещения (19) можно представить таким образом:
1 , 3 10
1 u x w
u , u2 u20 x3w,2, u3wE13(1)w,11E23(1)w,22 E13(0)e11E23(0)e22 Здесь u10, u20 - тангенциальные перемещения срединной плоскости. Здесь приняты:
3 0
3 )
0 (
3
3
dx E E
x i
i
, 3 30 3 )
1 (
3
3
dx x E E
x i
i
(i1,2)Заключение
Уравнения (9), (26) и (27) представляют собой системы уравнений, с помощью которых можно описать связанные плоское и изгибное напряженное состояние слоистой пластины. Если имеет место симметрия относительно срединной плоскости, то постоянные Kij равны нулю и тогда плоское и изгибное напряженные состояния будут независимы.
Библиография
1. Касимов А.Т. К расчету напряженно-деформированного состояния многослойных ортотропных пластин в уточненной постановке // Пластины и оболочки. – Караганда: Тр. КарПТИ., 1991. – С.38-41.
2. Королев В.И. Слоистые анизотропные пластинки и оболочки из армированных пластмасс. – М.: Машиностроение, 1995. – 271с.
3. Зверяев Е. М., Макаров Г.И. Общий метод построения теорий типа Тимошенко // ПММ.–
2008.–Т.72,вып.2.– С.308-321.
УДК 631.362
КИНЕМАТИКА ВЗАИМОДЕЙСТВИЯ ЧАСТИЦ В ГРАВИТАЦИОННОМ ПОТОКЕ Алексеев А.А., Галсанова Э.Ц*, Задевалова Г.Э., Задевалов И.А.
Восточно-Сибирский государственный университет технологий и управления, Россия, Улан-Удэ *erjena_g@mail.ru
В статье рассмотрена кинематика гравитационного течения зерновых частиц круглой формы по наклонной поверхности.
Дано решение уравнения скорости движения частиц в гравитационном потоке адекватно определяемое параметрами движения.
Ключевые слова: зерновая частица, дифференциальное уравнение, гравитационный поток, кинематика гравитационного течения, сыпучая среда.
KINEMATICS OF PARTICLE INTERACTION IN A GRAVITATIONAL FLOW Alekseev A.A., Galsanova E.Z., Zadevalova G.E., Zadevalov I.A.
East-Siberia University of Technology and Management, Russia, Ulan-Ude
18
The kinematics of the gravitational flow of grain particles of circular shape along an inclined surface is considered in the article. The solution of the equation of the velocity of the particles in a gravitational flow is adequately determined by the parameters of the motion.
Key words: grain particle, differential equation, gravitational flow, kinematics of gravitational flow, loose medium.
В статье [1] нами рассмотрено движение зерновой смеси по наклонной поверхности с применением эффекта гравитационного течения расплескивающегося типа. Было дано решение уравнения обратной диффузии в виде системы нелинейных уравнений движения зерновых частиц внутри сыпучей среды.
В данной работе рассматривается кинематика гравитационного течения расплескивающегося типа потока круглых однородных частиц.
На Рис. 1 представлен ламинарный поток круглых частиц на наклонной плоскости, где h – высота потока, α – угол наклона рабочей поверхности, V – скорость частицы в потоке, τ – разделитель потока по размерам частиц.
V
Рис. 1. Схема движения частиц в ламинарном гравитационном потоке На рис. 2 приведено скалярное поле скоростей в координатах XOY.
V
XV
YV
Рис. 2. Скалярное поле скоростей частиц Скорость на плоскости V > 0, Vh > V0.
В первом приближении вектор gradV направлен по оси Y.
Примем
V grad
N . (1)
19 h
V N Vh 0
. (2)
С другой стороны, имеем:
2 2 2 2
y V x
V V
grad x y
. (3)
Однако эксперименты [2] дают выражение N в следующем виде:
2
0 ay
h V N Vh
. (4)
В итоге имеем уравнение
2 2
2 2 2
y ay V x
Vx y
. (5)
Определяющим фактором является параметр a, учитывающий размер частицы по высоте потока τ, отражающий распределение зерновых частиц внутри сыпучей среды.
Интегрирование (5) должно привести к некоторой зависимости
x
y bV
V . (6)
Далее введем влияние параметра a, которое усредненно связывает скорости:
cos
V
Vx , Vy V sin
, (7)
где β – угол отклонения вектора скорости от линии наклона поверхности.
Тогда уравнение (5) примет вид:
2 2
2 2
2
cos sin
y ay V x
V
. (8)
cos ) ,
(
22
2 ay
x y x
V
. (9)
4
12 1 cosa x
V
. (10)
Решение уравнения (9) показывает, что эпюра скоростей имеет форму параболы. Коэффициент а определяется экспериментально из уравнения (4) измерениями Vh ,V0 и h.
На рисунке 3 представлено графическое решение уравнения (10).
с м
Vh
2 /
, VО 1 , 5
м/
с, h0,2м/с.V x( ) 1 x 4 12
Рис. 3. График скоростей частиц в потоке
