Các thông số tối ưu của bộ hấp thụ động lực cho các trục quay với vận tốc góc thay đổi

Nội dung text: Các thông số tối ưu của bộ hấp thụ động lực cho các trục quay với vận tốc góc thay đổi

1. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY OPTIMUM ABSORBER PARAMETERS FOR ROTATING SHAFTS WITH VARIABLE ANGULAR VELOCITY CÁC THÔNG SỐ TỐI ƯU CỦA BỘ HẤP THỤ ĐỘNG LỰC CHO CÁC TRỤC QUAY VỚI VẬN TỐC GÓC THAY ĐỔI Nguyen Duy Chinh ABSTRACT [1-8] only considered the rotating shaft with constant The rotating shaft is used to transmit torque from a driving device, such as a angular velocity. motor or engine. The rotating shaft can carry pulleys, gears, etc to transmit To the best knowledge of the authors, there have been rotary motion via belts, and chains, mating gears. But the shaft is not always no studies based on the maximum equivalent viscous rotating at constant angular velocity but due to unstable current or due to resistance method to determine the optimum parameters sudden acceleration or deceleration. The rotating shaft with variable angular of the TMD for the rotating shaft with variable angular velocity. Therefore, this paper studies to determine the optimal parameter of the velocity. So, to overcome the limitations and develop the tuned mass damper to reduce the torsional vibration for the rotating shaft with research results in references [1-4]. In this paper, the author variable angular velocity by using the maximization of equivalent viscous continues to find the optimal parameters of the TMD to resistance method. reduce torsional vibration for the shaft, in which the Keywords: The rotating shaft, torsional vibration, equivalent viscous resistance, rotating shaft with variable angular velocity by using the optimum absorber parameter. maximum equivalent viscous resistance method according to the reference [9]. TÓM TẮT Trục quay được sử dụng để truyền mô-men xoắn từ thiết bị truyền chuyển 2. SHAFT MODELLING AND VIBRATION EQUATIONS động, chẳng hạn như motor hoặc động cơ. Trục quay có thể mang ròng rọc, bánh Figure 1 shows a pendulum type TMD attached to a răng, để truyền chuyển động quay qua dây đai, dây xích hoặc bánh răng phối shaft. The symbols of the shaft and TMD are summarized in ghép. Nhưng không phải lúc nào trục cũng quay với vận tốc góc không đổi mà do Appendix. dòng điện không ổn định hoặc do tăng giảm tốc đột ngột. Trục sẽ quay với vận j1 tốc góc thay đổi. Do đó, bài báo này trình bày nghiên cứu xác định các tham số tối Q m ưu của bộ hấp thụ động lực để giảm dao động xoắn cho trục quay có vận tốc góc j 2 m biến đổi theo phương pháp cực đại lực cản nhớt tương đương. t km Từ khóa: Trục quay, dao động xoắn, lực cản nhớt tương đương, tham số tối ưu c của bộ hấp thụ động lực. L m N Faculty of Mechanical Engineering, Hung Yen University of Technology and Education k Email: duychinhdhspkthy@gmail.com t Received: 04/01/2021 M Revised: 20/02/2021 j Accepted: 26/02/2021 Figure 1. Shaft model attached with a TMD From [4], we have 1. INTRODUCTION [3M 2 2 ( m L 2 3mL 2 )] j Absorber is a tuned-mass damper (TMD), or dynamic t 1 (1) vibration absorber (DVA), is found to be an efficient, 2 2 2mL()()()t 3mL j2 3k t j 1 j 3Mt 0 reliable, and low-cost suppression device for the technical constructions and mechanical devices [1-10]. In [5-8] 2 2 2 2 2 ()()2mLt 6mL j1 2 2mL t 6mL j 2 6cL j 2 3k m j 2 0 (2) studied to find the optimal parameter of the DVA to reduce torsional vibration for the shaft. When designing absorbers In this paper, the author considers the cause of torsional to reduce vibration for the main system, the shape of the vibration for the system is because the rotating shaft with absorbers is quite rich, depending on the type of structure variable angular velocity, so we have: to be installed. So, in [1-4] studied and determined the j j(t )  j  j ( t ); j  j ( t ); j  j ( t ) optimal parameters of the TMD to reduce torsional 1 1 1 1 (3) vibration for the shaft. However, the studies in references M() t 0 Website: Vol. 57 - No. 1 (Feb 2021) ● Journal of SCIENCE & TECHNOLOGY 43
2. KHOA H ỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 Introduce the parameters Matrix Δ is a solution of Lyapunov algebraic equation in m m k 3k Eq. (12).  ,,,,  t 2 t  2 m t D2 d 2 ΨΔ + ΔΨTT +Y Ζ Ζ = 0 (12) MM M 2() 3m m L a a a t (4) 3c where  ,,,  d DL    D 2() 3m mt  d 11    12 13 14 Substituting Eqs. (3, 4) into Eqs. (1, 2). The matrix 0 equation of the system can be rewritten as 21 22  23  24 0 ΨΖ , a (13) 1 11    31     12 13 14  0 32 33 34 0 j221 22  23  24 j 2 0 41 42  43  44 j()t (5) Substituting Eq. (13) into Eq. (12), the matrix Δ can be  31 32  33  34  1 determined as: j2 41 42  43  44 j 2 0 11 12 13 14 in which Δ 21 22 23 24 (14) 2 2  110;;;;;;  21 0   31 D   41 D 12 0  22 0 31 32 33 34 2 22 2 2 2 2 2()() 3  tDD 3    6 2 t 41 42 43 44 32 ;;  42 3 3 (6) in which    131;;;;;; 23 0 33 0  43 0  14 0  24 1 12 1 2 2 2 2 3[(  )  ] Y 4()() 3    2 3 6  2     t a  tDD;  t 3 2 (15) 34 344 3 32 2 2 ()3 t   D 3. DETERMINATION OF OPTIMAL PARAMETERS OF THE 1 1 1 2 3 4 TMD (())  t  8 2 3 After short modification the Eqs. (1-3), we obtain 3Ya 21 2 1 2 1 1 1 2 2    2( ( ) )(   ( ) ) 2 2 2 3t 2 4 2 3 t M  kt  k m j 2 2cL j2 M j() t (7) (16) 33 ()3  2   So, the torque equivalent of the TMD set on the rotating t D shaft is 2 2 2 2 1 Ya () 2 t  6  3 9 (17) 2 34 M eqv k m j 2 2cL j 2 (8) 72 1 2 () t    D According to [9], the equivalent resistance coefficient of 3 the TMD on the primary structure is obtained as Using Eqs. (4, 11, 15-17) gives 1 1 2 2 M  8() m mt  D  eqv 2 6 ceqv (18) ceqv (9) 2 1 1 4 4 2(())()8   2 12   2 2  3t 3 t 1 (())16  4 2 16  2 6 2 4 4 Substituting Eq. (8) into Eq. (9), this becomes t 3 2 2  2 2 2cLj2  km j 2  4 2 1 ceqv (10) 2  Maximization of the equivalent resistance coefficient are expressed as If the primary system is excited by the angular ceqv 0 acceleration of the rotating shaft, j()t , is assumed as white (19)  MEVR opt noise, has the spectral density Ya, then the average value of Eq. (10) are the components of the matrix Δ in Eq. (12). We c eqv have 0 (20)  MEVR opt  2cL2 k c 34 m 32 (11) Combining Eq. (18) with Eqs. (19-20), we obtain the eqv 33 optimal parameters as follows 44 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 57 - Số 1 (02/2021) Website:
3. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY 3 From the parameters in Tables 1, 2 and case 2, Maple MEVR (21) opt 2 2 software is usedto simulate the torsional vibration of the 2t  6  3 rotating shaft is shown in Figure 3. 3   MEVR t 4.3. Case 3: Initial torsional vibration θ0 = 0.03(rad)and opt (22) 2   2 • 6 12 4 t initial angular velocity of θ0 = 1.0 (rad/s) Eqs. (21, 22) represent the optimal parameters of the From the parameters in Tables 1,2 and case 3.The TMD to reduce the torsional vibration of the rotating shaft author uses Maple software to simulate the torsional with variable angular velocity by using the maximization of vibration of the rotating shaft is shown in Figure 4. equivalent viscous resistance method. 4. NUMERICAL SIMULATION To evaluate the reliability of the optimal parameters are determined by Eqs. (21, 22). The author simulates the vibration of the system with the input parameters of the rotating shaft and TMD are given in Table 1. Table 1. The input parameters of the rotating shaft and TMD Parameter M kt mt m L µ  Value 450kg 1.2m 106Nm/rad 14kg 11kg 1.0m 0.035 0.83 From Eqs. (4, 21, 22) and Table 1, we infer the optimal Figure 4. The vibration of the shart in the case with initial torsional vibration parameters of the TMD in Table 2. • θ0 = 0.03(rad)and initial angular velocity of θ0 =0,() 03 rad Table 2. The optimal parameters of the TMD From Figures 2, 3 and4. We find that the optimal MEVR MEVR Parameter c km opt opt parameters of the TMD are defined in this paper has a good effect for reducing torsional vibration of the rotating shaft. Value 0.954 0.107 126.08Ns/m 43996.26Nm/rad 5. CONCLUSION AND DISCUSSION 4.1. Case 1: Initial torsional vibration θ = 0.03(rad) 0 The main objective of this paper is to find the optimal From the parameters in Tables 1 to 2 and case 1, using parameters of the tuned mass damper to reduce torsional Maple software simulates the torsional vibration of the vibration for the shaft in the case of the rotating shaft with rotating shaft is shown in Figure 2. variable angular velocity. The optimal parameters of the tuned mass damper are determined by the maximization of equivalent viscous resistance method are expressed according to equations (21, 22). To evaluate the effect of reducing vibration, the author uses Maple software to simulate the torsional vibration of the whole system. Through vibration simulation, we find that the vibration amplitude of the rotating shaft is suppressed when the tuned mass damper is installed. This confirms that the optimal parameters of the tuned mass damper are found in Figure 2. The vibration of the shart in the case with initial torsional vibration this paper are reliable. Helping scientists easily find the θ0 =0.03(rad) optimal parameters when applying to eliminate torsional vibration of the rotating shafts with variable angular 4.2. Case 2: Initial torsional vibration θ0 = 0(rad)and • velocity. initial angular velocity of θ0 = 1.0 (rad/s) ACKNOWLEDGEMENT This research is funded by Hung Yen University of Technology and Education under grand number UTEHY.L.2021.04. APPENDIX Notation m Concentrated mass at the top of the TMD km Torsional stiffness of spring of the TMD Figure 3. The vibration of the shart in the case with initial angular velocity of c Damping coefficient of damper 0 =1.0 (rad/s) L Length of a pendulum of the TMD Website: Vol. 57 - No. 1 (Feb 2021) ● Journal of SCIENCE & TECHNOLOGY 45
4. KHOA H ỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 [8]. K D Dien, V X Truong, N D Chinh, 2017. The fixed-points theory for shaft kt Torsion spring coefficient of shaft model by passive mass-spring-disc dynamic vibration absorber. Proceedings of The m Mass of pendulum rod t 2nd National Conference on Mechanical Engineering and Automation, ISBN 978- ρ Radius of gyration of primary system 604-95-0221-7, pp. 82-86. M Mass of of primary system [9]. N D Chinh, 2020. Vibration control of a rotating shaft by passive mass- j Angular displacement of the shaft spring-disc dynamic vibration absorber. Archive of Mechanical Engineering - AME, 67(3):279-297. doi: 10.24425/ame.2020.131693. φ1 Angular displacement of rotor [10]. N D Chinh, 2020. Optimal parameters of tuned mass dampers for an φ2 Relative torsional angle between TMD and rotor inverted pendulum with two degrees of freedom. Proc IMechE, Part K: J Multi-body Dynamics, DOI: 10.1177/1464419320971082. θ Torsional vibration of the primary system θ Initial condition of the torsional vibration 0 angle THÔNG TIN TÁC GIẢ µ Ratio between mass of the TMD and mass of primary system Nguyễn Duy Chinh Tuning ratio of the TMD Khoa Cơ khí, Trường Đại học Sư phạm Kỹ thuật Hưng Yên  Damping ratio of the TMD D Natural frequency of vibration of primary system d Natural frequency of vibration of the TMD  Frequency of angular acceleration of the rotating shaft  Ratio between length of pendulum and radius of gyration of rotor REFERENCES [1]. N D Chinh, 2019. Optimum design of the tuned mass damper to reduce the torsional vibration of the machine shaft subjected to random excitation. TNU Journal of Science and Technology, ISSN: 1859-2171, 203(10): 51-58. [2]. N D Chinh, 2019. Determining optimal parameters of the tuned mass damper to reduce the torsional vibration of the machine shaft by using the fixed- point theory. Journal of Science and Technology, P-ISSN 1859-3585, (55): 71-75. [3]. N D Chinh, 2019. Determination of optimal parameters of the tuned mass damper to reduce the torsional vibration of the shaft by using the principle of minimum kinetic energy. Proc IMechE, Part K: J Multi-body Dynamics, 233(2): 327-335. [4]. N D Chinh, 2020. Optimal parameters of tuned mass damper for machine shaft using the maximum equivalent viscous resistance method. Journal of Science and Technology in Civil Engineering (STCE) - NUCE, ISSN 1859-2996, 14(1) 127-135. [5]. N D Chinh, V X Truong, K DDien, 2017. Study on reduction for torsional vibration of shaft using minimization of kinetic energy. Journal of Structure Engineering and Contruction Technology, ISSN 1859-3194, Vol 25, pp. 5-12. [6]. V X Truong, N D Chinh, K D Dien, T V Canh, 2017. Closed-form solutions to the optimization of dynamic vibration absorber attached to multi-degree-of- freedom damped linear systems under torsional excitation using the fixed-point theory. Proc IMechE, Part K: J Multi-body Dynamics, 232(2): 237-252. [7]. V X Truong, K D Dien, N D Chinh,N D Toan, 2017. Optimal Parameters of Linear Dynamic Vibration Absorber for reduction of torsional vibration. Journal of Science and Technology (Technical Universities), Vol 119, pp.37-42. 46 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 57 - Số 1 (02/2021) Website: