Control of a 3d flexible catilever beam

ĐẠI HỌC QUỐC GIA TP.HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA VŨ NGUYỄN TRÍ GIANG ĐIỀU KHIỂN VỊ TRÍ VÀ DAO ĐỘNG CHO CƠ CẤU THANH MỀM CÔNG XÔN TRONG KHÔNG GIAN CONTROL OF A 3D FLEXIBLE CANTILEVER BEAM Chuyên ngành : KỸ THUẬT CƠ ĐIỆN TỬ Mã số: 8520114 LUẬN VĂN THẠC SĨ TP HỒ CHÍ MINH, tháng 09 năm 2020 Cơng trình hồn thành tại: Trường Đại học Bách Khoa – ĐHQG-HCM Cán hướng dẫn khoa học : PGS TS Nguyễn Quốc Chí (Ghi rõ họ, tên, học hàm, học vị chữ ký) Cán chấm nhận xét 1: PGS TS Trương Đình Nhơn (Ghi rõ họ, tên, học hàm, học vị chữ ký) Cán chấm nhận xét 2: PGS TS Lê Mỹ Hà (Ghi rõ họ, tên, học hàm, học vị chữ ký) Luận văn thạc sĩ bảo vệ Trường Đại học Bách Khoa, ĐHQG Tp HCM Ngày 03 tháng 09 năm 2020 Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm: (Ghi rõ họ, tên, học hàm, học vị Hội đồng chấm bảo vệ luận văn thạc sĩ) PGS TS Nguyễn Tấn Tiến PGS TS Trương Đình Nhơn PGS TS Lê Mỹ Hà TS Lê Thanh Hải TS Đoàn Thế Thảo Xác nhận Chủ tịch Hội đồng đánh giá LV Trưởng Khoa quản lý chuyên ngành sau luận văn sửa chữa (nếu có) CHỦ TỊCH HỘI ĐỒNG PGS TS Nguyễn Tấn Tiến TRƯỞNG KHOA PGS TS Nguyễn Hữu Lộc ĐẠI HỌC QUỐC GIA TP.HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM Độc lập - Tự - Hạnh phúc NHIỆM VỤ LUẬN VĂN THẠC SĨ Họ tên học viên: VŨ NGUYỄN TRÍ GIANG MSHV: 1870422 Ngày, tháng, năm sinh: 09/11/1996 Nơi sinh: TP Hồ Chí Minh Chuyên ngành: Kỹ thuật Cơ điện tử Mã số : 8520114 I TÊN ĐỀ TÀI: Điều khiển vị trí dao động cho cấu mềm công xôn không gian Control of a 3D flexible cantilever beam II NHIỆM VỤ VÀ NỘI DUNG: - Phân tích động lực học cho cấu dầm mềm có giảm chấn khơng gian chiều (1D) - Thiết kế điều khiển khử dao dộng cho cấu dầm mềm 1D - Kiểm chứng mơ hình tốn 1D điều khiển mơ thực nghiệm - Phân tích động lực học cho cấu dầm có giảm chấn không gian ba chiều (3D) - Kiểm chứng mô hình tốn dầm 3D mơ III NGÀY GIAO NHIỆM VỤ: 19/08/2019 IV NGÀY HOÀN THÀNH NHIỆM VỤ: 07/06/2020 V CÁN BỘ HƯỚNG DẪN: PGS.TS Nguyễn Quốc Chí, BM Cơ điện tử, Khoa Cơ Khí Tp HCM, ngày tháng 09 năm 2020 CÁN BỘ HƯỚNG DẪN CHỦ NHIỆM BỘ MƠN ĐÀO TẠO PGS TS Nguyễn Quốc Chí PGS TS Nguyễn Quốc Chí TRƯỞNG KHOA (Họ tên chữ ký) PGS TS Nguyễn Hữu Lộc ACKNOWLEDGEMENT First and foremost, I would like to praise and thank God, the almighty, who has granted countless blessings, knowledge, and opportunity to the writer, so that I have been finally able to accomplish the thesis I would first like to thank my thesis advisor Assoc Prof Nguyen Quoc Chi of the Department of Mechatronics at Ho Chi Minh City University of Technology for the continuous support of my master study and research, for his patience, motivation, enthusiasm, and immense knowledge His guidance helped me in all the time of research and writing of this thesis I thank my colleagues in Think Alpha Co., Ltd for the stimulating discussions, for theoretical and experimental, and for all the fun we have had Finally, I must express my very profound gratitude to my parents and to my sister for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis This accomplishment would not have been possible without them Thank you Vu Nguyen Tri Giang i ABSTRACT A study of a flexible beam fixed on a translating hub is presented in this thesis The primary focus is to develop a dynamic model of a flexible beam and reduce the residual vibration of the flexible beam produced by the hub’s motion The dynamic model is, however, a discrete model using the Galerkin discretization method to predict the behaviors of the system in the experimental process A damping mechanism is introduced to apply to the undamped discrete model by the Galerkin method First, a 1D flexible beam will be investigated The input shaping control method is used to suppress the vibration of the system where various input shapers including simple and advanced shapers are applied The simulation and experiment are conducted to show the qualitative agreement between the numerical and experimental results From the understanding obtained from previous works, a 3D flexible beam – vibrations in 3D coordinates is modeled with decoupled relationships among these 3D vibrations is presented TĨM TẮT Luận văn trình bày nghiên cứu dầm công xôn mềm cố định bệ gá tịnh tiến Trọng tâm phát triển mơ hình động lực học giảm độ rung dầm chuyển động bệ gá gây Mơ hình động lực học rời rạc xây dựng phương pháp Galerkin để dự đoán chuyển động hệ thống trình thực nghiệm Mơ hình giảm chấn áp dụng cho mơ hình rời rạc không giảm chấn Đầu tiên, dầm công xôn mềm chiều khảo sát Phương pháp điều khiển input shaping sử dụng để triệt tiêu rung động hệ thống với việc áp dụng input shaper khác từ đơn giản tới nâng cao Mô thực nghiệm thực thấy thống kết số thực nghiệm Từ hiểu biết thu được, dầm công mềm ba chiều – với dao động tọa độ ba chiều mơ hình hóa với mối quan hệ tách rời rung động 3D trình bày ii LỜI CAM ĐOAN Tơi xin cam đoan tất số liệu nội dung trình bày luận văn trung thực khơng chép cơng trình nghiên cứu cá nhân hay tổ chức Tơi xin đảm bảo thực nghiêm túc việc trích dẫn tài liệu tham khảo sử dụng luận văn TP Hồ Chí Minh, ngày 25 tháng 09 năm 2020 Vũ Nguyễn Trí Giang iii CONTENTS ACKNOWLEDGEMENT i ABSTRACT ii LỜI CAM ĐOAN iii CONTENTS iv LIST OF FIGURE vi LIST OF TABLE viii CHAPTER 1: INTRODUCTION .1 1.1 Previous work 1.2 Research objectives .4 1.3 Thesis outline CHAPTER 2: ONE-DIMENSIONAL DYNAMIC MODEL 2.1 Problem formulation 2.1.1 Kinetic energy 2.1.2 Potential energy 2.1.3 Work done 2.2 The continuous model 2.2.1 The variation with respect to variable h(t) 1.1.2 The variations with respect to w( z, t ) 2.3 The discrete model .10 CHAPTER 3: BASIC INPUT SHAPING THEORY .14 3.1 Basic theory 14 3.2 Multi-mode input shaping 17 iv CHAPTER 4: SIMULATION AND EXPERIMENT OF THE ONE – DIMENSIONAL SYSTEM 19 4.1 Experimental testbed 19 4.2 Parameter determination 20 4.2.1 Direct method 20 4.2.2 Experimental modal analysis 24 4.3 Simulation and experiment 28 4.3.1 Model verification 28 4.3.2 Input shaping verification 30 CHAPTER 5: THREE DIMENSIONAL DYNAMIC MODEL 36 5.1 Modeling 36 5.1.1 Inextensional beam 37 5.1.2 Three-dimensional beam theory .38 5.1.3 Coordinate projection .38 5.1.4 The damped system 41 5.2 Simulation 42 CHAPTER 6: CONCLUSION 47 6.1 Conclusions 47 6.2 Future work 48 REFERENCES 49 v LIST OF FIGURE Figure a) Industrial robot arm; b) Satellite; c) MEMS sensor Figure Flexible cantilever beam system with a moving hub .6 Figure Block diagram of input shaping .14 Figure Basic concept of input shaping .14 Figure Experimental testbed 19 Figure The cross-section of the experimental beam 20 Figure The excitation signal [0 10] Hz in: a) Time-domain; b) Frequency domain .21 Figure The tip response of the beam corresponding to [0 20] Hz excitation in: a) Time-domain; b) Frequency domain .22 Figure The excitation signal [0 8] Hz in: a) Time-domain; b) Frequency domain 23 Figure 10 The tip response of the beam corresponding to [0 8] Hz excitation in: a) Time-domain; b) Frequency domain .23 Figure 11 Free vibration of the tip in: a) Time-domain; b) Frequency domain .24 Figure 12 The modal parameter estimation procedure .26 Figure 13 FRF of the tip of the beam with Lin1 excitation 26 Figure 14 FRF of the tip of the beam with Exp5 excitation .27 Figure 15 FRF of the tip of the beam with Exp10 excitation .27 Figure 16 FRF of the tip of the beam with Exp15 excitation .27 Figure 17 An S-curve displacement 29 Figure 18 Simulation verification of log-decrement and EMA methods 29 Figure 19 Comparison of the results of EMA and log-decrement methods .31 vi Figure 20 ZV shaper for the first mode 32 Figure 21 ZVD and ZVDD shapers experiment for the first mode 33 Figure 22 Two-mode ZV shaper experiment 33 Figure 23 Two-mode ZVD and ZVDD shapers experiments .34 Figure 24: A schematic of a vertical cantilever beam with axial, lateral deformations .36 Figure 25: Deformation of a beam element 37 Figure 26 The hub motion in a planar in simulation 42 Figure 27 The transverse vibration along with X-axis in simulation 5.1 .43 Figure 28 The transverse vibration along with Y-axis in simulation 5.1 .43 Figure 29.The longitudinal vibration in the simulation 5.1 44 Figure 30 A beam with a circular cross-section 44 Figure 31 The hub motion in a planar in simulation 5.2 .45 Figure 32 The transverse vibration along with X-axis in simulation 5.2 .46 Figure 33 The transverse vibration along with Y-axis in simulation 5.2 .46 Figure 34 The longitudinal vibration in simulation 5.2 46 vii Science & Technology Development Journal – Engineering and Technology, 3(2):416-424 Figure 1: The cantilever beam with translational hub in three-dimensional space  0   K=    0 K11 K21 Kn1 K12 K22 Kn2  K1n    K2n  ,    (6) Knn     α 1      q     0  1       q   , u = f (t) △ =  2 , F =          .        qn (7) The matrix element Mi j and Ki j in Eqs (5) and (6) are given as: Mi j = ρ ∫L φi φ j dz + M φi (L)φ j (L), Ki j = EI ∫L ′′ ′′ φi φ j dz (9) In this section, an input shaping control with a twoimpulse sequence is designed The control goal is to cancel oscillations of the cantilever beam while the hub moves to the desired position Input shaping is feedforward control, in which a shaped command is achieved by convolving a sequence of impulses The impulse response of any vibratory order system can be indicated as a series of second-order functions [ 29 , pp 1071] Its expression in the time domain is: 1−ζ e−ζ ω0 (t−t0 ) ] A = √amp ( )2 ( )2 ∑Nj=1 B j cos ϕ j + ∑Nj=1 B j sin ϕ j where ϕ1 = ω √ (1 − ζ )t j (11) (12) (8) INPUT SHAPING CONTROL METHOD y(t) [ = A √ ω0 where A and t are the amplitude and the occurrence time of the impulse, respectively The principle of input shaping technique is illustrated in Figure When the first impulse is exerted on the system, it produces a vibration with amplitude (solid line) After a one-half period, it is superposed to cause an opposite vibration (dash line) As a result, two impulses cancel each other to obtain the vibration-free system (dot line) From Eq (10), the vibration amplitude for a multipleimpulse input is obtained as follows 20 (10) ) ( √ sin ω0 − ζ (t − t0 ) , Vibration suppression requires Aamp is zero when the input completes By solving these conditions, the vibration-free solution could be obtained The basic shaper is called as zero vibration (ZV):  [ ]  K Aj = 1+K 1+K  (13) tj 0.5T d From Eq (13), the ZV shaper requires the system’s natural frequency and damping, which can be obtained by experiment or computation In this research, these two parameters of the flexible beam are derived from experiments (see Section for details) The vibration suppression technique above is effective only if system parameters are exactly known Some constraints have been made to make the shaper insensitive to modeling error The key idea in 20 is to take vibration error into an account to achieve robustness with respect to modeling error of the natural frequency Let the derivative of vibration error equals to 418 Science & Technology Development Journal – Engineering and Technology, 3(2):416-424 Figure 2: Schematic diagram of the input shaping method zero, the ZVD shaper is obtained as follows:  [ ]  2K K2 Aj =  JZV D JZV D JZV D  , tj 0.5Td Td (14) the amplitude of the vibration is significantly reduced Finally, the ZVDD shaper obtained the best performance with zero vibration after 0.4 seconds Table 1: System Parameters where JZV D = + 2K + K Symbol Definition Values L Beam length 0.45 m SIMULATION AND EXPERIMENTAL RESULTS h Beam height of area 28.8 x 10−3 m b Beam width of area 0.88 x 10−3 m The model of the flexible beam Eqs (4)-(9) established in Section is used to simulate the dynamics of the beam, whose parameters are listed in Table These parameters are both used in simulation processes and experiments It should be noted that, in this research, the tip mass is neglected in the control design of input shapers The behaviors of the flexible beam are simulated with three commands: (i) The unshaped command is an Scurve profile with a displacement of 50 mm in 0.1s; (ii) ZV, ZVD, and ZVDD shapers are implemented to compare the performance of each shaper As shown in Figure 3, the unshaped command causes a large vibration of the tip, which takes a long time for suppression In the case of shaped command , the ZV shaper achieves the settling time is 0.5 seconds when the tolerance band 5% is applied ZVD shaper shows better performance compared to the ZV shaper when ρ Beam linear density 0.2 kg.m−1 EI Beam flexural rigidity 332.714 x 10−3 Nm2 m Mass of hub 1.5 kg 419 The experiments are carried out in this research The testbed and data acquisition are shown in Figure The system consists of one single translational DOF, whose link is a flexible aluminum beam One beam’s end is fixed rigidly to a brushless linear motor while the other end is free In the experiment system, an LM series linear servo motor from Yokogawa performs movements directly This feature brings many advantages, the motor’s velocity and the ratio of thrust to weight are significantly large (0.8 m/s, and 7m/s2 , respectively) Also, the positioning precision of the fixed end’s link is equal to the Science & Technology Development Journal – Engineering and Technology, 3(2):416-424 Figure 3: Simulation results of the tip displacement of the flexible beam servo motor’s precision (0.5 µ m) due to backlash absence These factors contribute to high performance of the whole system The motor is powered by TM series servo amplifier from Yokogawa The command is fed by a motion control unit, which is Turbo Com® pact UMAC CPU from Delta Tau With Motorola DSPs Processor, the Turbo CPU is a powerful motion controller that provides a packed combination of motion and I/O control for a complex automatic system The integrated features include trajectory generation, servo, commutation, compensation, motion program, and so forth The damping ratio is obtained by logarithmic decrement δ which is defined as the natural log of the ratio of any two successive peaks This method is conducted through the result of the experiment on the underdamped vibration of the beam (see Figure 5) The logarithmic decrement δ is presented as ) ( ) ( xi 33.19 δ = ln = ln = 0.047 (15) n xi+n 31.67 The damping ratio, then, determined by the relationship with logarithmic decrement as follows: δ ζ=√ = 0.0075 δ + 4π (16) The natural frequency is determined by plotting the frequency spectrum (Figure 6) of the free vibration of the beam with initial disturbance The experimental results are shown in Figure 7, it is shown that experimental data is consistent with predicted results With the unshaped command, the vibrations of the flexible beam occur for a long time Meanwhile, the ZV, ZVD, and ZVDD shapers eliminate vibrations significantly The simple ZV shaper reduces a large vibration of 9% compared to unshaped command while the vibration of ZVD and ZVDD shapers are both lower than 2% in relation to unshaped command DISCUSSIONS AND CONCLUSION In this paper, three input shapers ZV, ZVD, and ZVDD for the simultaneous position and vibration control of a flexible cantilever beam excited by a moving hub are proposed The design of input shapers is based on the ODE model of the flexible beam system, which is derived by using the Galerkin method The validity of the proposed input shapers is verified by both simulations and experiments ACKNOWLEDGEMENT This work was supported T-CK-2018-07 LIST OF ABBREVIATIONS USED PDE: Partial differential equation ODE: Ordinary differential equation ZV: Zero vibration ZVD: Zero vibration derivative ZVDD: Zero vibration derivative derivative PPF: positive position feedback SRF: strain rate feedback COMPETING INTERESTS The authors declare that they have no conflicts of interest 420 Science & Technology Development Journal – Engineering and Technology, 3(2):416-424 Figure 4: Experimental testbed Figure 5: Free vibration of a viscous damped cantilever beam 421 Science & Technology Development Journal – Engineering and Technology, 3(2):416-424 Figure 6: The experimental frequency spectrum of the beam Figure 7: Experimental results shows the displacements of the flexible beam in uncontrolled case and with controlled cases (with ZV, ZVD, and ZVDD shaper) 422 Science & Technology Development Journal – Engineering and Technology, 3(2):416-424 AUTHOR CONTRIBUTION Nguyen Tri Giang Vu has deisgned the controller for simulation and experiment purposes and has written the manuscript Phuong Tung Pham has investigated the dynamic model and has examined the initial experiment Quoc Chi Nguyen has provided research ideas, has guided the research and has edited the manuscript REFERENCES Paranjape AA, Guan J, Krstic M PDE boundary control for flexible articulated wings on a robotics aircraft IEEE Transactions on Robotics 2013;29(3):625–640 Available from: https: //doi.org/10.1109/TRO.2013.2240711 Yang 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beam: an adaptive feedforward control method Smart Materials 2017;26(4) Available from: https://doi.org/10.1088/1361-665X/aa64c6 Sahinkaya MN Input shaping for vibration-free positioning of flexible systems Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2001;215(5):467–481 Available from: https://doi.org/10.1177/ 095965180121500504 PI - Solution for precision motion and positioning,” Physik Instrumente (PI) GmbH & Co KG, [Online] [Accessed 20 January 2020];Available from: https://www.physikinstrumente.com/en/applications/ automation/smarter-motion-positioning/?utm_medium= Video&utm_source=PI-Youtube&utm_campaign=Input% 20Shaping%20for%20Vibration%20Free%20Motion Pham PT, Nguyen QC Dynamic model of a three-dimensional flexible cantilever beam attached a moving hub Proceedings of 11th Asian Control Conference, Gold Coast, Queensland, Australia 2017;p 2744–2749 Available from: https: //doi.org/10.1109/ASCC.2017.8287611 Ray BE, George TL CRC Handbook of Tables for Applied Engineering Science, Boca Raton, Florida,: CRC Press, Inc 1973; Tạp chí Phát triển Khoa học Cơng nghệ – Kĩ thuật Công nghệ, 3(2):416-424 Bài Nghiên cứu Open Access Full Text Article Điều khiển hệ dầm công xơn mềm có bệ gá di động phương pháp input shaping Vũ Nguyễn Trí Giang1 , Phạm Phương Tùng2 , Nguyễn Quốc Chí1,* TĨM TẮT Use your smartphone to scan this QR code and download this article Giới thiệu: Dầm công xôn kết cấu phổ biến kỹ thuật, với đầu gắn cứng Kết cấu dùng để mơ tả cánh tay robot, với độ cứng lớn để đảm bảo độ cứng vững ổn định hệ thống Một dầm công xôn mềm cung cấp kết cấu nhẹ hiệu chi phí, gây dao động chuyển động tốc độ cao Trong báo này, hướng đến việc điều khiển dao động dầm mềm gắn bệ gá có chuyển động thẳng Phương pháp: Mơ hình tốn học dầm công xôn mềm biểu diễn phương trình vi phân đạo hàm riêng (PDE-partial differential equation) Mơ hình xấp xỉ phương pháp Galerkin với kết mơ hình biểu diễn hệ phương trình vi phân thơng thường (ODE-ordinary differential equation) Phương pháp thực nghiệm sử dụng để xác định thơng số chưa biết hệ thống Mơ hình ODE tạo điều kiện cho thiết kế ba giải thuật điều khiển input shaping: (i) Zero-Vibration, (ii) Zero-Vibration-Derivative, (iii) Zero-Vibration-Derivative-Derivative, dùng để di chuyển dầm công xôn mềm đến vị trí mong muốn giảm thiểu dao động Kết kết luận: Mơ hình động lực hệ thiết lập với phương trình vi phân thông thường Các thông số chưa biết xác định thực nghiệm Các điều khiển khác thiết kế để khử dao động dầm Quá trình mơ dự đốn đáp ứng động lực học để xác minh độ hiệu điều khiển phương pháp số Thực nghiệm hợp lệ mơ hình tốn thơng qua thống liệu mô thực nghiệm hiệu điều khiển với hệ thống thực Những điều khiển có ưu điểm như: không cần gắn thêm thiết bị; điều khiển chuyển động không bị tác động nên điều khiển áp dụng với nhiều hệ thống có sẵn Từ khố: Dầm cơng xơn mềm, phương pháp Galerkin, input shaping, khử dao động Bộ môn Cơ điện tử, Trường Đại học Bách khoa - Đại học Quốc gia Thành phố Hồ Chí Minh Khoa kỹ thuật Cơ khí, Đại học Quốc gia Pusan Liên hệ Nguyễn Quốc Chí, Bộ mơn Cơ điện tử, Trường Đại học Bách khoa - Đại học Quốc gia Thành phố Hồ Chí Minh Email: nqchi@hcmut.edu.vn Lịch sử • Ngày nhận: 01-10-2019 • Ngày chấp nhận: 30-01-2020 • Ngày đăng: 16-8-2020 DOI : 10.32508/stdjet.v3i2.605 Bản quyền © ĐHQG Tp.HCM Đây báo công bố mở phát hành theo điều khoản the Creative Commons Attribution 4.0 International license Trích dẫn báo này: Giang V N T, Tùng P P, Chí N Q Điều khiển hệ dầm cơng xơn mềm có bệ gá di động phương pháp input shaping Sci Tech Dev J - Eng Tech.; 3(2):416-424 424 2020 20th International Conference on Control, Automation and Systems (ICCAS 2020) Oct 13~16, 2020; BEXCO, Busan, Korea Parameter identification of a flexible cantilever beam with a moving hub Nguyen Tri Giang Vu1, Quoc Chi Nguyen1 and Van Thuat Nguyen1 Department of Mechatronics, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam (trigiang1996@gmail.com) Vietnam (vanthuatme@gmail.com)* Corresponding author Abstract: This paper presents a parameter identification of a flexible beam attached to a translating hub A dynamic model of the beam-hub system is obtained by applying the Euler-Bernoulli beam theory, in which dynamic coupling between the beam and the hub is shown The continuous model is discretized for utilizing identification purposes by the Galerkin method Then, the identification process is conducted in the frequency domain along with modal analysis to determine unknown parameters of the system, where a vision system is applied to record the transverse vibrations of the beam Finally, the validity of the identified parameter is verified by comparison of the measured vibrations of defined points of the beam-hub system with the simulation results Keywords: Flexible system Identification, , vibration, Euler-Bernoulli beam INTRODUCTION A cantilever is a well-known structure in mechanical, aerospace, and civil engineering due to its advantages of compactness and simplicity The cantilevers can be assumed to be rigid or flexible The rigid models are valid in the case of the large stiffness of the cantilevers, which usually appear in the heavy structures Meanwhile, in the light-weight structures, the flexible models are employed It is well-known that the light-weight structures are compact and consume low expenditure Therefore, the flexible cantilever structures receive a large number of applications, which can be found in marine risers [1] flexible manipulators [2-3], computer disk drives [4], etc In contrast to rigid structures, flexible structures undergoing high-speed motions yield significant mechanical vibrations, which affect their performance Therefore, vibration analysis of the flexible cantilever structures is desirable From the engineering point of view, beam models can be used to descript the flexible cantilevers A number of researches have been investigated dynamics of the flexible cantilever dynamics, which can be classified into two cases: (i) Stationary case [5-7] (ii) Moving case including rotating beams [8, 9] and translating beams [10-12] For Case (i), the beams are clamped at one end, while it is free at the other end In Case (ii), the beams are fixed on moving hubs, which generates rotational or translational motions These motions not only drive the hubs but also govern the vibrations of the beams Therefore, it is noticeable that the dynamics of the moving hubs should be coupled with the dynamics of the beam systems, in which the work on this issue is rare Euler-Bernoulli and Timoshenko beam theories have been applied to develop the dynamic models for flexible cantilever beams In the Euler-Bernoulli beam theory [13], the beam is subjected to bending moment only, which provides an essential solution for common engineering problems, especially for beam systems made of isotropic materials Timoshenko [14] considered the effects of shear deformation and cross-section rotation to the Euler-Bernoulli beam Therefore, the Timoshenko beam model is advanced but yields complication for dynamic analysis Han et al [15] found that if the slenderness ratio is large enough, the differences between the Euler-Bernoulli model and the others are not significant Therefore, the Euler-Bernoulli beam model is still valid in the dynamic analysis of the beam structures Pham and Nguyen [16] developed the Euler-Bernoullibased translating beam system, where the dynamics of the moving hub is included The Euler-Bernoulli beam systems are usually represented by partial differential equations (PDEs) Since PDEs are infinite-order systems, they result in the complexity of obtaining the solution and consequently, the difficulty of the dynamic analysis Therefore, approximation methods such as the Galerkin decomposition method [16] and the finite element method [17] are used to tackle the issues Instead of the PDE models, these approximations convert PDEs into a set of ordinary differential equations (ODEs), which is a finite-order system It should be noted that the accuracy of the approximation models depends on the choice of order of the ODEs There are unknown parameters in the flexible beam systems: For instance, it is difficult to obtain the exact values of Young’s modulus and the damping coefficient of the flexible beam by using measurements (even indirect method adopted) This problem is the challenge of analyzing dynamics and designing control systems To overcome this problem, the identification of unknown parameters is a viable solution Wei et al [18] presented the identification of natural frequency and damping ratio of a stationary beam, where the experiments were carried out Meanwhile, in Wang et al [19], stiffness and damping coefficient of a stationary beam was obtained by using machine learning-based method (support vector machine) For the rotating beams, a number of works have been carried out [20-22] Liu and Sun [20] proposed an identified model of a single-link flexible manipulator Nature frequencies of a rotating beam were obtained by using the Eigen realization algorithm in Xie et al [21] For the identification of a translating beam, it is very rare to research on this topic Yang et al [22] investigated vibrations of a machine tool modeled as a translating beam, where the particle swarm optimization algorithm was employed to obtain modal parameters of the beam Thus far, there has not been a research on parameter identification of a translating beam system under the dynamic effects of the moving hub In this paper, a dynamic model describing a flexible cantilever beam drove by a moving hub is developed by using the Euler-Bernoulli beam theory, in which the coupled dynamics between the beam system and the moving hub is considered After that, the continuous model is discretized by using the Galerkin method, in which the discrete model facilitates the use of the identification theory developed for ODE systems Then, the identification process that is based on frequency analysis and modal analysis is conducted to determine unknown nature frequencies Finally, simulations and experiments are carried out to verify the precision of the identification method DYNAMIC MODELS OF THE FLEXIBLE BEAM SYSTEM Fig illustrates a flexible cantilever beam system, where the solid line represents the deformed beam, and the dashed line indicates the neutral axis of the beam The deflections of the beam result from the motions of the hub, which moves on the X-axis of the world coordinate OXYZ The beam transverse displacement with the neutral axis is denoted by w(z, t) while b(z, t) and h(t) indicate the positions of the beam and of the hub, respectively The beam parameters are the length l, the mass per unit length r, Young’s modulus E, and the inertial moment I The external force f(t) exerts on the hub, where m is the mass of the hub The equations of motion are obtained by Hamilton’s principle: t1 t0 (1) where the kinetic energy T, the potential energy P, and the work done W, respectively are given as l (2) T = mh!2 (t ) + r ò éëh!(t ) + w! ( z, t ) ùû dz, l (3) P = EI ò wzz2 ( z, t )dz, (4) W = f (t )h(t ) Substituting Eqs (2) ~ (4) into Eq (1), the equations of motion are derived as follows [16] (5) !!( z, t ) + EIwIV ( z, t ) = 0, r h!!(t ) + w ( ) l !!( z, t ) ùûdz = f (t ), mh!!(t ) + r ò éëh!!(t ) + w with the boundary conditions w( z,0) = w0 ( z), wt ( z,0) = wt ( z,0), w(0, t ) = 0, wz (0, t ) = 0, EIwzzz (l , t ) = 0, wzz (l , t ) = Remark 1: It is well known that the dynamics of the beam ̈ (5) is affected by the acceleration of the moving hub ℎ(𝑡𝑡) It is obvious that the influence of the vibrations of the beam is also significant as shown in Eq (6), which represents the hub dynamics 2.2 The discrete model 2.1 The continuous model ò (d T + d W - d P ) dt = 0, Fig Flexible cantilever beam system with a moving hub (6) (7) (8) (9) To develop the discrete model, the approximation of the dynamic model Eqs (5) ~ (9) is carried out by using the Garlerkin method, where the transverse displacement of the beam can be represented by Galerkin decomposition: n w( z , t ) = å qr (t )j r ( z ), (10) r =1 where qr (t ) is the time varying variable, and jr (t ) denotes the r-th undamped linear mode shape function, which given as [23] jr ( z ) = ( cosh lr z - cos lr z ) (11) cosh lr l + cos lr l ( sinh lr z - sin lr z ) , sinh lr l + sin lr l with lr is the r-th solution obtained from the frequency equation, (12) + cosh ( lr L ) cos ( lr L ) = It should be noted that the linear mode shape functions are orthogonal Using Eq (10), Eq (6) can be rewritten as l n éỉ ù !! (13) m r l h ( t ) r + + ( ) ờỗ ũ ji ( z )dz ÷ q!!i (t ) ú = f (t ) å i =1 ë êè ø ûú By multiplying Eq (5) throughout by j j ( z ) , then integrating over [0, l], one obtains l l n ổ n ổ ử 2 ỗ r ị ji ( z )dz ÷q!!(t ) + å ç EI ị ji ÷q(t ) å i =1 è i =1 è 0 ø ø (14) l !! + r j h (t ) = ò j In matrix form, the Eq (13) ~ (14) can be rearranged, then added an extra term – damping matrix, and rewritten as (15) M!! y + Cy! + Ky = {F} , where, éJ êE ê M = ê E2 ê ê" êë En E1 M 11 E2 ! ! M 22 ! En 0 ù ú ú (16) ú, ú " " # " ú 0 ! M nn úû ! ù é0 ê0 K ! úú 11 ê (17) K = ê0 K 22 ! ú , ê ú " " # " ú ê" êë0 0 ! K nn úû éa ù ì f (t ) ü êq ú ï ï ê 1ú ïï ïï (18) y = ê q2 ú , { F } = í ý ê ú ï ï ! ê!ú ï ï êë qn úû ỵï ỵù A necessary step to define a viscous damping system that leads to diagonal damping matrix is implementing Rayleigh damping [23], whose definition is: (19) C = a0 M + a1K, where 𝑎𝑎' and 𝑎𝑎( are the two Rayleigh damping coefficients The formula between them and the N modal damping factors is described as: ỉ a0 (20) + a1wr ÷ è wr ø These variables in Eqs (16) ~ (17) are given by: zr = ç J = m + rl , l Ei = ri ò ji ( z )dz, l M i = r ò ji2 ( z )dz , l K i = EI ị (jr¢¢( z ) ) dz , (21) i = 1, 2, , n SYSTEM IDENTIFICATION In this research, experimental modal analysis (EMA) is used as identification method by the measurements of the frequency response functions (FRFs) with fast Fourier transform (FFT) analyzer Then, modal parameters are extracted by a set of FRFs using curve fitting (see Fig 2) The Eq (11) can be represented with proportional damping and harmonic excitation as: (22) M!!y + Cy! + Ky = peiwt The mode-superposition of Eq (19) solution for steadystate response, based on the modes of the undamped system, is given by [24, Eq (18 2)]: N f fT p (23) y ss = å r r eiwt , K é ù r =1 r ổw w ờ1 - ỗ ữ ú + i 2x r wr êë è wr ø úû where ω is the forcing frequency in rad/s, which covers the frequency range of interest in the modal test The receptance FRF, which deals with displacement response at due to excitation, has the form [24, Eq (18 3)]: Fig Modal testing procedure frfrT p , K r (1 - rr ) + i ( 2x r rr ) r =1 N H ij ( f ) = å (24) where rr = w / w r is the frequency ratio for the r - th mode, and K r is the generalized stiffness The modal parameters can be estimated by curve fitting method such as the Rational fraction polynomial (RFP) method, whose model is [24, Eq (18 18)]: N ỉ Aijr Aijr* (25) H ij = ỗ + ữ, ỗ iw - lr* ữứ r =1 è iw - lr * where Aijr , Aijr are the modal residues, and lr , lr* = -z rwr ± iwr - z r2 = -z rwr ± iwdr , are corresponding poles (26) SIMULATIONS AND EXPERIMENTS Experiment and simulation are both performed to indicate the consistency between theoretical and actual beams A model expresses the transverse displacement of the beam, which have been verified by experiment using vision system In order to obtain the resonant frequencies of the flexible beam described in Fig 1, the flexible cantilever beam is considered with the system parameters listed in Table The experiment set-up (as shown in Fig 5) is used to verify the investigation cantilever beam An aluminum beam with the parameters as shown in Table is clamped on a linear motor Yokogawa, which generates the external force The linear motor system is controlled by a Table System parameters Parameter Beam material L h b ρ I E m Description Value – Aluminum Beam length Beam height of area Beam width of area Beam linear density Beam moment of inertia Beam Young modulus Mass of hub 0.45 m 0.029 m 0.001 m 0.1836 kg.m-1 8.896 ´ 10-11 m4 69 ´109 N.m-2 kg Fig Schematic diagram of the experimental set-up motion controller UMAC The motion program controlling the whole linear motor system is generated from accompanying software with UMAC controller (PeWin32Pro) The measured signals are collected via a PC with NI PCI motion card and processed in the LabVIEW program The input signal is encoder feedback pulse from linear motor drive and the beam vibration’s data in pixels is produced by a Keyence® high-speed camera, then transferred to the computer through Ethernet cable The laser sensor is used for calibration procedure in order to convert pixel to millimeter The whole system is synchronized with the clock generated from UMAC controller The experiment procedure is divided into two stages The first stage is configuration and calibration of the vision system A set of parameters is defined for the Fig Time response of the chirp excitation and output signal controller Keyence CX420 to enable the vision system capturing two selected points in Fig (the first point is located at 10 mm from the beam tip, the second point is at 225 mm from the beam tip) at the frequency of 100 Hz Afterward, calibration is taken to convert pixel values from the vision controller to millimeters The laser sensor is used as a reference for this procedure, then a number of data including the laser and the vision data is captured These pairs provide the pixel to mm conversion using Curve fitting tool of MATLAB® shown in Fig Fig Measured points in the beam Fig Curve fitting line converting the pixels value (from camera) to mm value (from laser sensor) The second stage is to apply excitation force, measure data, analysis to achieve the natural frequencies, and the transfer functions In the experiments, two types of excitation are exerted on the linear motor The first one is the chirp signal to obtain the natural frequencies as shown in Fig The chirp signal with a frequency varies from to 30 Hz was applied to the moving hub to have the chance of exciting the first two modes of the structure The amplitude vibration of the beam corresponds to the chirp is depicted in Fig The second one is a multi-sine wave signal to determine the transfer function The transfer function is verified by simulating another multi-sine wave signal and the first chirp signal The simulations of chirp signal (see Fig 10 Model validation at the second point The modal model can be determined by applying procedure in Fig as follows: (i) Estimating unknown parameter – damping ratio by applying Eq (25); (ii) Determining of the damping matrix (C) with Eq (20); (iii) Combining known mass (M) and stiffness (K) matrices with damping matrix (C), then the modal model is known Fig The two natural frequencies with a chirp input signal Fig 8) shows consistency of natural frequencies with experimental data (see Fig 7), where the first frequency is 567 Hz and the second frequency is nearly 22.3 Hz The vibration estimations at two pre-defined points indicate the validity of the transfer functions presented in Fig ~ 10 CONCLUSIONS Identification of a flexible beam attached on a translating hub has been implemented with a vision system A discrete model describing the vibration of the system under the hub’s motion, employing Rayleigh damping, has been demonstrated Two first modes of vibrations have been observed by the vision system, whose scheme has been shown The displacements of measured points of the beam have shown a strong consistency between experiment and simulation obtained from the model ACKNOWLEDGEMENTS This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number C2018-20-01 REFERENCES Fig Simulation frequency response of the system Fig Model validation at the first point [1] Z Zhao, X He, Z Ren and G Wen, "Boundary adaptive robust control of a flexible riser system with input nonlinearities," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol 49, no 10, pp 1971 - 1980, 2018 [2] Z Liu, J Liu and W He, 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flexible hub-beam based on different discretization methods of deformation fields," Archive of Applied Mechanics, vol 30, no 2, pp 291-304, February 2020 [18] S Wei, S Chen, Z Peng, X Dong and W Zhang, "Modal identification of multi-degree-of-freedom structures based on intrinsic chirp component decomposition method," Applied Mathematics and Mechanics, vol 40, pp 1741-1758, 2019 [19] Z Wang, Y Ding, W Ren, X Wang, D Li and X Li, "Structural dynamic nonlinear model and parameter identification based on the stiffness and damping marginal curves," Structural control and heath monitoring, vol 27, no 6, 2020 [20] K Liu and X Sun, "System identification and model reduction for a single-link flexible manipulator," Journal of Sound and Vibration, vol 242, no 5, pp 867-891, May 2001 [21] Y Xie, P Liu and G.-P Cai, "Frequency identification of flexible hub-beam system using control data," International Journal of Acoustics and Vibrations, vol 21, no 3, pp 257-265, September 2014 [22] M Yang, Y Dai, Q Huang, X Mao, L Li, X Jiang and Y Peng, "A modal parameter identification method of machine tools based on particle swarm optimization," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol 233, no 17, pp 6112-6123, 2019 [23] S R Singiresu, Mechanical Vibrations, London: Pearson, 2017, p 772 [24] R Craig and A Kurdila, Fundamentals of structural dynamics, New Jersey: John Wiley & Sons Inc, 2006 CURRICULUM VITAE Full Name: VU NGUYEN TRI GIANG Gender: Male Date of birth: November 11, 1996 Nationality: Vietnamese Marital status: Single Emails: trigiang1996@gmail.com 1870422@hcmut.edu.vn Cell phone: (84)-963155598 Address: 495/8/43 To Hien Thanh Street, Ward 12, District 10, Ho Chi Minh City EDUCATION: Degree Field M.Eng Mechatronics anticipated Engineering Institution Ho Chi Minh City Univ of Tech Vietnam Duration Sept.2018 Present Nov.2020 B.Eng Mechatronics Engineering (Honor program) Sept.2014 – Ho Chi Minh City Univ of Tech Vietnam 56 Sept 2018 PUBLICATION: [1] Vu, N T G., Nguyen, Q C., & Pham, P T (2020) Input shaping control of a flexible cantilever beam excited by a moving hub Science & Technology Development Journal - Engineering and Technology, 3(2), Online First https://doi.org/https://doi.org/10.32508/stdjet.v3i2.605 [2] Nguyen, V T., Vu, N T G., & Nguyen, Q C (2020) Parameter identification of a flexible cantilever beam with a moving hub 20th International Conference on Control, Robotics, and Systems, Busan, Korea (South) 57 ... input shaping control to minimize the vibration of a rotational beam Sadat-Hoseini et al [30] derived an optimal-integral feedforward control scheme to control vibrations of aircraft wings enabling... of motion of a rotating three-dimensional cantilever beam Vibration control of a flexible beam can be categorized into two types: open-loop and closed-loop controls In the field of feedback control, ... where a discrete mathematic model of a flexible beam attached on a moving hub (1-D) was developed with a single-mode input shaping controller was applied Anisotropic flexible beam with asymmetric