三质体振动筛动力学特征仿真

Simulation Analysis of Dynamic Characteristics on Three-Body Vibrating Screen Chuan Guang Ding a, Fang Zhen Song b, Bo Song, Xiu Hua Men School of Mechanical Engineering, University of Jinan, Jinan, 250022, China aujnding02@163.com, bme_songfz@ujn.edu.cn(Corresponding Author), Keywords. Three-Body Vibrating Screen, Dynamic Characteristics, Motion simulation. Abstract. However three-body vibrating screen is widely used in engineering practice, many dynamics of it have not well been addressed. For studying three-body vibrating screen dynamics, the vibrating screen dynamic model and the virtual model were established. Through deriving the dynamic equations and simulating the vibrating screen start - running process, the analytical equations and the trajectories were obtained. Studies have shown that the VBVS vibration in the Z direction is complex and the X direction which perpendicular to the axis of exciters is the sensitive direction for the installation error. In addition, the SBD’s trajectories in the direction which does not exist exciting force are superimposed by the trajectories of USB and LSB. Introduction With the development of vibration technology, the number of vibrating bodies of vibrating screen (VBVS) tends to increasing. Now, the three-body vibrating screen has been used for engineering production, but dynamic analysis are more concentrated in single-body and double-body vibrating screen[1-5]. Many dynamics of three-body vibrating screen have not been well addressed, so the dynamic characteristics analysis has a practical engineering significance. Dynamics Model Fig. 1 Virtual Model Fig. 2 Dynamic model Fig. 3 Trajectories in the Y direction The vibrating screen structure includes four parts: the upper screen box (USB), the lower screen box (LSB), the second damping bearing (SDB) and the base, as shown in Fig.1. The SDB is bearing the LSB and the SDB and located in the base the same time. Two pair of vibrators, which have the same parameters, are installed on the LSB and the SDB and drive the vibrating screen [6]. The springs between the USB and SDB are the up springs (US).The springs between the LSB and SDB are the middle springs (MS). The springs between SDB and the base are the low springs (LS). The vibrating screen dynamic model is established according to the structure, as shown in Fig.2. Applied Mechanics and Materials Vols. 105-107 (2012) pp 311-315 Online available since 2011/Sep/27 at www.scientific.net © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.105-107.311 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 124.128.158.35-29/09/11,02:23:01) [ ] [ ] [ ]{ } { }FxKxCxM =+      +       ••• (1) [ ]K =       ×× × 66126 1212 DB BA T (4) [ ]M = [ ]Tzyxzyxzyx JmJmJmJmJmJmJmJmJmI 3333332222221111111818× (2) { }x = [ ]Tzyxzyxzyx zyxzyxzyx 333333222222111111 θθθθθθθθθ (3) [ ]A = [ ]Tzzyyxxzzyyxx kkkkkkkkkkkkI ϕϕϕϕϕϕ 2222221111111212× (6) [ ] [ ] [ ]21 BBB += (5) [ ]1B = [ ]Tzzyyxx kkkkkkI ϕϕϕ 11111166 −−−−−−× (7) [ ]2B = [ ]Tzzyyxx kkkkkkI ϕϕϕ 22222266 −−−−−−× (8) [ ]D = [ ]Tzzzzzzyyyyyyxxxxxx kkkkkkkkkkkkkkkkkkI ϕϕϕϕϕϕϕϕϕ 32132132132132132166 ++++++++++++× (9) { }F = [ ] ( )tfTzzyyxxzzyyxx ⋅⋅ ωδαδαδαδαδαδα sin000000222222111111 (10) Among them, the 1m , xJ1 , yJ1 and zJ1 are the mass and the moment of inertia of the USB; the 2m , xJ 2 , yJ 2 and zJ 2 are the mass and the moment of inertia of the LSB; the 3m , xJ3 , yJ3 and zJ3 are the mass and the moment of inertia of the SDB. The xk1 , yk1 , zk1 , xk ϕ1 , yk ϕ1 and zk ϕ1 are the stiffness and the rotational stiffness of the US; xk2 , yk2 , zk2 , xk ϕ2 , yk ϕ2 and zk ϕ2 are the stiffness and the rotational stiffness of the MS; xk3 , yk3 , zk3 , yk ϕ3 , xk ϕ3 and zk ϕ3 are the stiffness and the rotational stiffness of the LS. the x1α , y1α , z1α , x1δ , y1δ and z1δ are the force coefficient and torque coefficient of the USB; the x2α , y2α , z2α , x2δ , y2δ and z2δ are the force coefficient and torque coefficient of the LSB; the [ ]C and [ ]K have the same form and [ ]C was omitted[7]. Response Analysis According to actual conditions, set { }X = { } ( )tx ⋅⋅ ωsin [7]. The response amplitude of the harmonic excitation can be obtained from the Eq. 1: [ ]X = [ ] { }FMK 12 −−ω (11) 1x = ( )( )[ ]{ } ( ) ( )( )( ) ( ) 221213232112122221222 22113 2 3212 2 2 2 2 xxxxxxxxxx xxxxxxxxx kmkmkkkmkmkkmk fkkmkkkmkk ωωωωω ααωω −+−++−−−− ⋅−−++−− (12) xϕθ1 = ( )( )[ ]{ } ( ) ( ) ( )( )( )xxxxxxxxxxxxxx xxxxxxxxxxx JkkkJkJkkJkJkk fkkJkJkkkk 3 2 3212 2 21 2 1 2 12 2 21 2 1 2 2 21212 2 23 2 321 2 2 ωωωωω δδωω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕϕϕϕ −++−−−−+− ⋅−−−++− (13) 1y = ( )( )[ ]{ } ( ) ( )( )( ) ( ) 221213232112122221222 22113 2 3212 2 2 2 2 yyyyyyyyy yyyyyyyyy kmkmkkkmkmkkmk fkkmkkkmkk ωωωωω ααωω −+−++−−−− ⋅−−++−− (14) yϕθ1 = ( )( )[ ]{ } ( ) ( ) ( )( )( )yyyyyyyyyyyyyy yyyyyyyyyyy JkkkJkJkkJkJkk fkkkJkJkkk 3 2 3212 2 21 2 1 2 12 2 21 2 1 2 2 2211 2 22 2 23 2 321 ωωωωω δδωω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕϕϕϕ −++−−−−+− ⋅−+−−++− (15) 1z = ( )( )[ ]{ } ( ) ( )( )( ) ( ) 221213232112122221222 2211 2 23 2 3212 2 2 zzzzzzzzz zzzzzzzzz kmkmkkkmkmkkmk fkkkmkkkmk ωωωωω ααωω −+−++−−−− ⋅−+−++−− (16) 312 Vibration, Structural Engineering and Measurement I zϕθ1 = ( ) ( ) ( )( )( ) ( ) ( )[    −−−+−−−++ −+− 2 2 2 2 1 2 11 2 1 2 1 2 21 2 12 2 23 2 111 2 1 21 2 121212 2 2211 mkkkmkkkJkmkJkkkk mkkkkmkkkk xzxxzxzzxzzzzx xxzzxxxzzx ωωωωω αωαω ϕϕϕϕϕϕϕ ϕϕϕϕ ( )( )( ) ( ) ( )( )22212121323211211213211122 1 mkmkkmkkkJkmkJkkkk xxzxxxzzxzzzzx ωωωωωω ϕϕϕϕϕ −−−++−−−−++− ( )( )( )( )( )]2221211213232132321 1 mkJkmkJkkkmkkk xzzxzzzzxxx ωωωωω ϕϕϕϕ −−−−++−+++ ( ) ( )( ) fJkkkJkk Jkkk Z zzzzzzz zzzz ⋅     −++−− −++ − 1 3 2 3211 2 1 2 1 3 2 321 δ ωω ω ϕϕϕϕϕ ϕϕϕ (17) 2x = ( )( )[ ]{ } ( ) ( )( )( ) 3 2 3212 2 21 2 1 2 21 2 1 2 2 12 2 21 2 1112 mkkkmkmkkmk fkmkmkkk xxxxxxx xxxxxxx ωωωω αωωα −++−−−− ⋅−−−−− (18) xϕθ2 = ( )( )[ ]{ } ( ) ( ) ( )( )( )xxxxxxxxxxxxxx xxxxxxxxxxx JkkkJkJkJkkkJk fkJkkkJkkk 3 2 3212 2 21 2 12 2 2 2 1 2 21 2 1 2 2 13 2 3211 2 1112 ωωωωω δωωδ ϕϕϕϕϕϕϕϕϕ ϕϕϕϕϕϕϕϕ −++−−+−−− ⋅−−++−−− (19) 2y = ( )( )[ ]{ } ( ) ( )( )( ) ( ) 221213232112122221222 1212 2 13 2 3211 2 1 yyyyyyyyy yyyyyyyyy kmkmkkkmkmkkmk fkkkmkkkmk ωωωωω ααωω −+−++−−−− ⋅−+−++−− (20) yϕθ2 = ( )( )[ ]{ } ( ) ( ) ( )( )( )yyyyyyyyyyyyyy yyyyyyyyyyy JkkkmkJkkJkJkk fkkkJkJkkk 3 2 3212 2 21 2 1 2 12 2 21 2 1 2 2 1212 2 11 2 13 2 321 ωωωωω δδωω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕϕϕϕ −++−−−−+− ⋅−+−−++ (21) 2z = ( )( )[ ]{ } ( ) ( )( )( ) ( ) 221213232112122221222 1212 2 13 2 3211 2 1 zzzzzzzzz zzzzzzzzz kmkmkkkmkmkkmk fkkkmkkkmk ωωωωω ααωω −+−++−−−− ⋅−+−++−− (22) zϕθ2 = ( )( ) ( ) ( ) ( )( )( ) ( )( )( )[    −−−++−−−−++ −−+−− zzxzzzzxzzxzzzzx xzzzxxxxzzzx JkmkJkkkkJkmkJkkkk JkkmkkmkJkkk 1 2 11 2 13 2 111 2 21 2 12 2 23 2 111 2 1 21 2 1 2 21 2 1212 2 21 2 1 2 21 ωωωωωω αωωαωω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕ ( ) ( ) ( ) ( )( )222121213232122221211212122 1 mkmkkmkkkmkkkmkkk xxzxxxxzxxzx ωωωωω ϕϕϕ −−−++−−−−+ ( )( )( )( )( )]( )zzxzzxzzzzxxx JkmkJkmkJkkkmkkk 2222221211213232132321 1 ωωωωωω ϕϕϕϕϕ −−−−−++−+++ ( )( )[ ]( ) ( ) fJkJkJkkkJkk kk zz Z zzzzzzzzz Zzz ⋅     − + −−++−− − 2 2 2 2 2 2 23 2 3211 2 1 2 1 121 ω δ ωωω δ ϕϕϕϕϕϕϕ ϕϕ (23) 3x = ( ) ( ){ } ( ) ( )( )( )3232122212122121 221 2 1112 2 2 mkkkmkmkkmk fkmkkmk xxxxxxx xxxxxx ωωωω αωαω −++−−−− ⋅−−−− (24) xϕθ3 = ( ) ( ){ } ( ) ( ) ( )( )( )xxxxxxxxxxxxxx xxxxxxxx JkkkJkJkJkkkJk fkJkkJk 3 2 3212 2 21 2 12 2 2 2 1 2 21 2 1 221 2 1112 2 2 ωωωωω δωδω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕ −++−−+−−− ⋅−−−− (25) 3y = ( ) ( ){ } ( ) ( )( )( ) ( ) 212223232112122222121 112 2 2221 2 1 yyyyyyyyy yyyyyy kmkmkkkmkmkkmk fkmkkmk ωωωωω αωαω −+−++−−−− ⋅−+−− (26) Applied Mechanics and Materials Vols. 105-107 313 yϕθ3 = ( ) ( ){ } ( ) ( ) ( )( )( )yyyyyyyyyyyyyy yyyyyyyy JkkkmkJkkJkJkk fkJkkJk 3 2 3212 2 21 2 1 2 12 2 21 2 1 2 2 221 2 1112 2 2 ωωωωω δωδω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕ −++−−−−+− −−−− (27) 3z = ( )[ ] ( ){ } ( ) ( )( )( ) ( ) 212223232112122222121 12 2 21221 2 1 zzzzzzzzz zzzzzz kmkmkkkmkmkkmk fmkkkmk ωωωωω αωαω −+−++−−−− ⋅−−−− (28) zϕθ3 = ( ) ( ) ( ) ( ) ( )( )( ) ( )( )( )[    −−−++−−−−++ −−+−− zzxzzzzxzzxzzzzx xxzzzxxxzzzx JkmkJkkkkJkmkJkkkk mkkJkkmkkJkk 1 2 11 2 13 2 111 2 21 2 12 2 23 2 111 2 1 21 2 121 2 1212 2 221 2 11 ωωωωωω αωωαωω ϕϕϕϕϕϕϕϕϕ ϕϕϕϕ ( ) ( ) ( ) ( )( )222121213232122221211212122 1 mkmkkmkkkmkkkmkkk xxzxxxxzxxzx ωωωωω ϕϕϕ −−−++−−−−+ ( )( )( )( )( )]2221211213232132321 1 mkJkmkJkkkmkkk xzzxzzzzxxx ωωωωω ϕϕϕϕ −−−−++−+++ ( )( ) fJkkkJkk k Z zzzzzzz z ⋅     −++−− − 1 3 2 3211 2 1 2 1 1 δ ωω ϕϕϕϕϕ ϕ (29) From the equations, the vibration and swing of the VBVS in the X or Y direction is affected by the component of forces and torques in the X or Y direction. However, the swing in the Z direction is affected except by the torque in Z direction also affected by the component of forces in X direction. For example, the four forces, namely fx ⋅1α , fx ⋅2α , fz ⋅1δ and fz ⋅2δ , affect the swing of the LSB in the Z direction. Three-Body Linear Vibrating Screen’s Motion Simulation When y1α = y2α =1 and other force and torque coefficient are equal to 0, the vibrating screen is the linear vibrating screen and the dynamic structure is shown in Figure 2. However, due to the installation errors and the drives’ small differences, the vibrating screen will produce some smaller forces in the Y and Y direction, which will cause the vibration and swing in the two directions [3]. The vibrating screen’s virtual model (as shown in Fig.1) was built and imported into ADAMS[4,8-9]. In modeling, the focus of the VBVS is away from equilibrium position the same distance (1 mm) in the X and Y direction in order to obtain the effect about the installation errors which influence the vibration trajectories [5]. Through the simulation of the start –running process, the running trajectories were obtained. The vibration trajectories are shown in Fig. 3-6 and the stable operation double amplitudes are shown in Table 1. Fig.4 Trajectory in the X direction Fig. 5 Trajectory in the Z direction Fig. 6 The local Enlargement of Fig.5 Table 1 The simulation double amplitude VBVS In the X direction (mm) In the Y direction (mm) In the Z direction (mm) USB 0.0812 1.811 0.0201 LSB 0.0506 2.120 0.0370 SDB 0.0398 0.632 0.0165 314 Vibration, Structural Engineering and Measurement I Fig. 3 shows the vibration trajectory in the Y direction and we can find that the smaller installation errors which didn’t affect the performance of vibrating screen. It can be obtained from the Fig. 4 that the trajectories of the VBVS in the X direction are much similar than the trajectories in the Y direction and larger than Z direction. Because the X direction is perpendicular to the axis of vibrators, the power in the X direction comes from the installation errors and slight differences in the parameters of vibrators and is smaller than the exciting forces and is much large than the forces in the Z direction which parallel to the axis of vibrators. From the results, it can be obtained that the smaller installation errors didn’t affect the performance of vibrating screen and the vibration amplitude in the X direction is still several times in the Z direction even the same installation error, as shown in the table 1. So the X direction is the sensitive direction for the installation error. According to the structure of the vibrating screen and Eq.24-29, the forces driving the SBD’s vibration come from the USB and LSB and the SBD’s vibration trajectories are affected and superimposed by the trajectories of USB and LSB, as shown in the Fig. 4-5 and the table 1. Conclusions (1)According to the structure of three-body vibrating screen, the dynamic model has been established and the vibration displacement formulas were derived. (2)The VBVS vibration in the Z direction is complex and affected by the component force in the Z direction and by the torque in the other direction. (3)The X direction which perpendiculars to the axis of exciter is the sensitive direction for the installation error. (4)The smaller installation error will not affect the vibrating screen’s work. (5)The SBD’s trajectories in the direction which does not exist exciting force are superimposed by the trajectories of USB and LSB. References [1] J.T. LIU, J. LIU, X.H. LI, L. DAI and H.Y. LIU: Journal of Vibration and Shock. Vol. 28 (2009) No.9, p.109. (In Chinese) [2] L.Q. MENG, Z.W. WANG, Y.N. LI, C.F. DUAN, M. ZHANG and L.H. BO: Coal Mine Machinery. Vol. 31 (2005) No.4, p.57. (In Chinese) [3] H.X. ZHANG: The Dynamics Analysis and Study of Large-Sized Linear Vibration Screener(MS., Qingdao University of Science and Technology, China),p.22. [4] J.C. WEI, M.L. ZHANG and F. WANG: Journal of Beijing University of Chemical Technology. Vol. 31 (2004) No.2, p.91. (In Chinese) [5] J.G. Yu and H.R. Dong. Petroleum Drilling Techniques. Vol. 37 (2009) No.4, p.76. (In Chinese) [6] T.X. ZHANG, X.Y. E and B.C. WEN. Journal of Northeastern University. Vol. 24 (2003) No.9, p.0839. (In Chinese) [7] X.XIAO, D.C. ZHANG and X. LI: Journal of University of Science and Technology Liaoning. Vol. 32 (2009) No.8, p.369. (In Chinese) [8] W.B. ZHU and J.J. YAN. Drilling & Production Technology. Vol. 28 (2005) No.3, p.77. (In Chinese) [9] L.D.ZHU, J.S. SHI, G.Q.CAI, W.S. WANG: Journal of Northeastern University(Natural Science). Vol. 28 (2007) No.10, p.1473. (In Chinese). [10] J. Wang and J. Li: Journal of Liaoning Technical University (natural science edition). Vol. 19 (2000) No.3, p.278. (In Chinese) Applied Mechanics and Materials Vols. 105-107 315 Vibration, Structural Engineering and Measurement I doi:10.4028/www.scientific.net/AMM.105-107 Simulation Analysis of Dynamic Characteristics on Three-Body Vibrating Screen doi:10.4028/www.scientific.net/AMM.105-107.311