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    Procedia Earth and Planetary Science 1 (2009) 776–784 www.elsevier.com/locate/procedia=The 6thInternational Conference on Mining Science reliability; hyperstatic net-beam structure; finite element; dynamic optimization 1. Introduction The structural strength and stiffness needs to be improved with the maximization of screening machines. The enlargement of structure of vibrating screen will lead to the increase of vibration mass and exciting force. Dynamic load on the vibrating screen is also increased which will lead to a greater deformation, tear of the side plate, fracture of the crossbeam, and thus affects the service life of the vibrating screen seriously[1-3]. Conventional static strength calculation and analogy method was used in the design of vibrating screen in our country at present which neglect the dynamic characteristic influence of higher modal frequencies on the vibrating screen[4]. High exciting force is likely to lead to fatigue damage of vibrating screen. Engineering experience indicates that the screen can also be ∗Corresponding author. Tel.: +86-516-83590092. bJã~áä=~ÇÇêÉëëW [email protected] 187 - /09/$– See front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.pro .2009.09.1238 5220epsProcedia Earth and Planetary Science damaged even if the static strength meet performance requirement well. Thus the performance, structure and strength need to be analyzed dynamically in the design process. New type structure of vibrating screen was developed to fit the maximization requirement of screening machines, improve the reliability and extend service life through dynamic design. 2. Structure of large vibrating screen with hyperstatic net-beam The hyperstatic net-beam structure, as shown in Fig 1[5], was applied in the design of new types vibrating screen based on the characteristics of hyperstatic structure[6]. A hyperstatic net-beam device was set between two side plates of the vibrating screen which was composed of six beams, two exciter beams and three static plates. The beams distributed on the support plates according to the requirement of exciting force. The vibration exciters were set on both sides of static plates and the exciting transmission shaft connected to synchronous vibration generator. Thus big vibration generators were replaced by series connected small vibration generators. The girder steel or rectangular steel were not needed in the hyperstatic net-beam structure. So the volume of supporting parts was reduced, the vibrating mass was lowed, the reliability and comprehensive performance of the vibrating screen was enhanced which was able to meet the technological requirements of vibrating screen whose width of the screen surface is larger than 3.6m. The vibrating screen improved production effect of single machine and is suitable for medium drainaging of raw coal, ore and other materiel. The structure of CWS3675 large vibrating screen with hyperstatic net-beam structure is shown in Fig 2. Fig. 1. Hyperstatic net-beam structure Fig. 2. Model of vibrating screen 3. Dynamic characteristic analysis of large vibrating screen Dynamic characteristic of the vibrating screen includes structural natural characteristic and dynamic response[7-8]. The FEM was used to analyze the dynamic characteristic of the vibrating screen in the design process, in order to realize dynamic characteristic of the vibrating screen and modify the structure. PKNK bëí~ÄäáëÜãÉåí=çÑ=ÑáåáíÉ=ÉäÉãÉåí=ãçÇÉä======777/ Procedia Earth and Planetary Science 1 (2009) 776–784Z. Yue-min et al. = =The structure and load on the vibrating screen should be considered according to actual operating conditiong6632 to make the finite element model accurate and effective. The main structure adopted solid element SOLID95 and SOLID92.The spring used COMBINE14 and the vibration generator was simplified as lumped mass element MASS21.The imaginary beam was introduced to replace the flange, considering the stiffness of flanges and ignoring its mass. The bear was simplified as sleeve and the exciting force distributed on the inner face of the sleeve uniformly. The finite element model was finished in ANSYS, as shown in Fig 3. Fig. 3. Finite element model of vibrating screen PKOK k~íìê~ä=ÅÜ~ê~ÅíÉêáëíáÅ=~å~äóëáë=Natural characteristic is composed of natural frequency, natural vibration modes and other modal parameters. One purpose of natural characteristic analysis is to avoid resonance and harmful vibration mode. The results is also used to provide necessary basis for dynamic response analysis[8]. Block Lanczos method with high precision and computation speed was used solve natural characteristic. The first twelve modal results are shown in Table 1. Table 1. Modal calculation results of vibrating screen Set Natural frequency Ñ=(Hz) Natural modes of vibration 1 1.66 Rigid motion along z axis 2 2.88 Rigid motion along x axis 3 2.92 Rigid motion along y axis 4 3.43 Rigid rotation around y axis 5 3.94 Rigid rotation around x axis 6 4.18 Rigid rotation around z axis 7 14.85 Bending and swing of discharge end along y axis 8 16.92 Twist of side plates along y axis and swing of feed end along z axis 9 18.98 Twist of screen frame along y axis 10 23.54 Reversal around x axis and bending of screen along z axis 11 33.41 Bending and twist of feed end along x axis 12 36.45 Flexural oscillations of middle and back of the screen along z axis Table 1 show that the natural vibration modes include rigid motion and bending deflection. The first six vibration modes are rigid modal and modal frequency is lower which depends on the vibration mass and stiffness of spring. The seventh to twelfth vibration modes are deformable modes and modal frequency lies on structural stiffness of the vibrating screen.The seventh to tenth vibration modes are shown in Fig 4. From Table 1 and Fig 4 we know that there are two elastic modal around the working frequency, that are the seventh and the eighth modal which take a central part to the total deformation of the vibrating screen and are easy to lead to resonance. The seventh modal frequency is 14.85Hz, the bending and swing amplitude of the discharge plate is greater, of which the maximum deformation reaches 4.25mm. The result shows that the intensity of the discharge end is weaker and the structure is easy to break. Thus the structure of discharge end needs to be modified. The eighth modal frequency is 16.92Hz, the deformation of connection part of two side plates and the feed end is a ====== ===778 / Procedia Earth and Planetary Science 1 (2009) 776–784Z. Yue-min et al. little larger and this part is easy to crack. So the structural stiffness of the parts needs to be strengthened. The result that the deformation of middle parts of two side plates and two exciting beam is small shows that the vibrating screen owns greater stiffness. This is due to the application of hyperstatic net-beam structure which enhance structural stiffness and stability. Fig. 4. The seventh to tenth vibration modesPKPK aóå~ãáÅ=êÉëéçåëÉ=~å~äóëáë=The side plate and cross beam will generate larger dynamic stress when the vibrating screen works. The dynamic response of the vibrating screen at any time can be obtained through finite element analysis and the result is used to judge whether the intensity meets the deformation requirements and to verify whether the designed structure is able to overcome harmful effect, caused by resonance and fatigue[9]. Fig 5 shows the total and z directional displacement of vibrating screen under working frequency 12.17Hz. Fig 6 presents the dynamic stress distribution of hyperstatic net-beam structure and side plate. Fig. 5. Total and ò directional displacement ====== 779/ Procedia Earth and Planetary Science 1 (2009) 776–784Z. Yue-min et al. = =Fig. 6. Distribution of dynamic stress (a); hyperstatic net-beam; (b) side plate The dynamic characteristic of CWS675 large vibrating screen shows that adding stiffening angle and longitudinal stiffeners on two side plates can reduce transverse deformation of the side plate, ameliorate the twist deformation of the screen frame and improve the stiffness of vibrating screen. The maximum transverse swing displacement of the vibrating screen is 0.13 mm, the maximum difference in vibration amplitude of corresponding points is 0.44 mm and the maximum dynamic stress is 16.63 MPa while dynamic stress of other parts are lower than 2 MPa which accord with national standards[10]that the maximum transverse displacement is less than 1mm, the maximum difference in vibration amplitude of corresponding points is less than 0.5 mm and dynamic stress is less than 24.50 MPa. The maximum transverse displacement and dynamic stress happens at the connection parts of discharge plate and side plate. Thus support stiffeners can be added to the discharge end to increase structural stiffness. 4. Dynamic optimal design of vibrating screen The initial force mainly distributes on the side plate and cross beams when the vibrating screen works. Then it makes a rigid motion and elastic deformation which generates a greater dynamic stress on the side plate and leads to fatigue crack of the side plate[11]. Adding stiffeners to base structure[12]is an effective method to control stress level of mechanical structure. But the structural size and positional distribution of stiffeners on the side plate is a difficult problem for the design of side plate. The position, size and thickness of stiffeners were established according to the experience of engineers. So the design method lacks theory basis. Most of the present research works[13-14]on the optimization of vibrating screen are about technological parameters. The augmented lagrangian method was used in the dynamic optimal design of large vibrating screen to confirm the best position of stiffeners on the side plate, gain the lower dynamic stress with the least number of stiffeners and improve the reliability of the vibrating screen. We made an optimal design of the vibrating screen based on multiple frequencies constraints and took the mass of side plate as optimization object. An adaptive optimization criterion was given to optimize the structural size of vibrating screen. QKNK _~ëáÅ=ëíêìÅíìêÉ=çÑ=ëáÇÉ=éä~íÉ=The vibrating screen with hyperstatic net-beam structure was taken as research object and the original basic structure of side plate is shown as Fig 7. Fig. 7. Original structure of side plate. 1. longitudinal stiffener ; 2. longitude stiffener ; 3. longitudinal stiffener ; 4. static plate; 5. side plate; 6. stiffening angle; 7. peripheral stiffener; 8. rivet hole ======780 / Procedia Earth and Planetary Science 1 (2009) 776–784Z. Yue-min et al. QKOK bëí~ÄäáëÜãÉåí=çÑ=çéíáã~ä=é~ê~ãÉíÉêë=The elements on the connection place of peripheral stiffener and side plate will become distortional because of interactive solving which leads to the suspension of optimization program. The structure of side plate should be simplified properly in order to gain a good optimization effect. The structural frequency will decrease, to some extent, which will be considered in optimization process. The simplified structure of side plate is shown in Fig 8. 1ê2ê3êFig. 8. Simplified structure of side plate Section widths of three longitudinal stiffeners ê1, ê2, ê3and the thickness of stiffeners Ü are taken as design parameters in the structural size optimal process and the initial values of ê1, ê2, ê3and Ü are 100 mm. The upper and lower limits of design parameters are listed as follows. 1mm mmSM ê NQM≤≤ ,2mm mmSM ê NQM≤≤ , 3mm mmSM ê NQM≤≤ , mm mmSM Ü NPM≤≤ . The dynamic characteristic of vibrating screen was studied in order to meet the reliability of mechanical structure. The first three elastic modal frequencies Ñ1, Ñ2, Ñ3, which are close to working frequency and have a great impact on the vibration modes are regarded as constraints. The upper and lower limits of state variables are as follows, considering the influence of structure simplification. 1Ñ ≤TKNT Hz; 2Ñ ≥NTKNTHz; 3Ñ ≥NVKPMHz. The mass of side plate ttis taken as object function and the optimal result is to obtain the minimum mass and low manufacturing cost. The original mass of side plate is 2352.80 kg. QKPK léíáãáò~íáçå=ÅêáíÉêáçå=For multi-frequency constraints, the constraint equation can be expressed as follows: 220() ( ) 0àààÖñ Ñ Ñα=− ≤ ENFWhere 20àÑαis the square of expected à=set frequency. The characteristic equation is shown as follows: { } { } { } { }2[] [ ] 0qqàààà àhÑjΨΨΨΨ−= EOF Where [K] is the global structural stiffness matrix; [M] is the global structural mass matrix; { à} is the j eigenvector. The element stiffness matrix and element mass matrix can be expressed as follows[15], considering structural nonlinear characteristic. (0) (1) 2 2[][] [] []ÉÉ É Éááááááhh ñhñh=+ + + EPF (0) (1)[][] []ÉÉ Éáááájj ñj=+ EQF From formula (2) to (4), we got the parameter áàd which can be expressed as: { } { }(1) (2) 2 (1)([ ] 2 [ ] [ ] )qáà à á á á à á àdhñhÑjΨΨ=+−, where { }àΨis the standardized modal vector. The lagrange’s equation are built according to optimal parameters. ====== 781/ Procedia Earth and Planetary Science 1 (2009) 776–784Z. Yue-min et al. = =22011(, ) ( )åãááá à à àáàiñ ñä Ñ Ñζρ ζα===− −¦¦ERF Where ñáis design variable; äáis the element length; !á=is density; àis lagrange operator. Differentiated ñá=in formula (5) and obtained the minimum of i. The optimization criterion could be described as follows. 11(1,2,)ãáàààáádáåäζρ===¦ESF Supposing that1ãáàáàà áádäφζρ==¦, the recursion formula can be expressed as: 11() ()1 ( 1)áê áê áêññîφ+ª º=+−« »¬ ¼ETF Where=ê=is recurrent number, î is step factor. QKQK léíáã~ä=êÉëìäíë=~å~äóëáë=çÑ=ëáÇÉ=éä~íÉ=Iterative computation is accomplished according to the recurrence formula and the optimization process iterated twelve times. The optimum result was the sequence ten and ê1is 60.29 mm, ê2 is 60.31 mm, ê3 is 61.10 mm, Ü=is 60.36 mm, Ñ1 is 4.17 Hz, Ñ2is 15.30 Hz, Ñ3is 17.66 Hz, ttis 2158.30 kg. Table 2 listed the comparative results of parameters before and after optimization. Table 2. Optimal parameters of side plate Parameters Before optimization After optimization Change range (%) ê1 (mm) 100.00 60.29 39.71 ê2 (mm) 00. 60.31 39.69 ê3 (mm) 100.00 61.10 38.90 Ü (mm) 00. 60.36 39.64 Ñ1(Hz) 4.24 4.17 1.65 Ñ2(Hz) 15.04 15.30 1.73 Ñ3(Hz) 17.16 17.66 2.91 tt(kg) 2352.80 2158.30 8.27 From Table 2 we know that the mass of side plate is decreased by 194.50kg, the first modal frequency is rigid motion frequency and the value
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