DESIGN AND CHARACTERISTICS OF a NOVEL COMPLIANT PLANAR SPRING CAPABLE OF a POSITIVE STIFFNESS FOR a COMPACT VIBRATION ISOLATOR THIẾT kế và đặc TÍNH của lò XO PHẲNG mềm mới CHO bộ TÁCH RUNG ĐỘNG

Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV DESIGN AND CHARACTERISTICS OF A NOVEL COMPLIANT PLANAR SPRING CAPABLE OF A POSITIVE STIFFNESS FOR A COMPACT VIBRATION ISOLATOR THIẾT KẾ VÀ ĐẶC TÍNH CỦA LÒ XO PHẲNG MỀM MỚI CHO BỘ TÁCH RUNG ĐỘNG Thanh-Phong Dao1,2a, Shyh-Chour Huang3b Division of Computational Mechatronics (DCME), Institute for Computational Science (INCOS), Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences 415 Chien-Kung Road, Kaohsiung 80778, Taiwan R.O.C a a daothanhphong@tdt.edu.vn or daothanhphongck@yahoo.com; bshuang@cc.kuas.edu.tw ABSTRACT The quasi-zero stiffness (QZS) vibration isolator has been realized by using the conceptual design of coil springs Unlike previous studies, this paper proposes a novel design of compliant planar spring capable of a positive stiffness for a compact QZS vibration isolator instead of employing coil springs The static and dynamic characteristics of the proposed compliant planar spring are then analyzed by using finite element method (FEM) in ANSYS software Finally, an applicable model of QZS vibration isolator is designed via using the compliant planar spring with the positive stiffness The deformation of the isolator is analyzed using the FEM Compared with a linear isolator, the proposed QZS isolator has a better isolating capacity The results showed that the stiffness of the proposed spring is low and can be tuned to achieve a desired stiffness It is also useful for vibration isolators in a compact space Keywords: compliant mechanism, planar spring, quasi-zero stiffness, vibration isolator, finite element method TÓM TẮT Bộ tách rung động độ cứng gần không (QZS) công nhận dùng lò xo cuộn Không giống nghiên cứu trước đây, báo đề nghị thiết kế lò xo phẳng mềm có độ cứng dương cho tách rung động QZS kích thước nhỏ Đặc tính tĩnh học động lực học phân tích phương pháp phần tử hửu hạn (FEM) phần mềm ANSYS Cuối cùng, mô hình ứng dụng QZS dựa lò xo phẳng mềm thiết kế Biến dạng mô hình phân tích FEM So sánh với tách rung động tuyến tính, mô hình có khả tách rung động tốt Kết lò xo phẳng mềm có độ cứng thấp độ cứng điều chỉnh Thiết bị hữu ích cho tách rung động không gian nén Từ khóa: cấu mềm, lò xo phẳng, độ cứng gần không, tách rung động, phương pháp phần tử hửu hạn INTRODUCTION A linear vibration isolator can effectively perform if its natural frequency is much than the excitation frequency This is obtained by decreasing the stiffness of the vibration isolator; however, it can cause an undesired large deflection A nonlinear isolator with quasi-zero 795 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV stiffness (QZS) has been proposed to achieve a large static but low dynamic stiffness Kovacic et al [1] combined the vertical linear spring and the two nonlinear pre-stressed oblique springs to form a QZS isolator The vertical linear spring proposes a positive stiffness while the two oblique springs possess a negative stiffness Wu et al [2] employed a compression spring to realize the positive stiffness while the negative stiffness achieved using magnet spring Zhou and Liu [3] proposed a vibration isolator with a tunable high static but low dynamic stiffness to the QZS at the equilibrium position In their study, the flexible beam is used as a mechanical spring coupled with the permanent magnet spring Le and Ahn [4] suggested the QZS vibration isolator for vehicle seat Compliant mechanism has been received a great interest of high resolution manipulators that offer advantages such as non-friction, lubrication-free, easy to fabricate, allowing for a monolithic manufacturing, and a minimal maintenance In contrast, in vibration isolator systems, there has been a little attention to the use of compliant mechanism Platus [5] used flexible beam in the horizontal direction to provide the negative stiffness for the QZS isolator Also in Ref [3] the flexible beam was utilized as a mechanical spring However, the stiffness compliant planar spring with a positive in the vertical direction has been rarely explored and investigated As shown in Fig 1a, a traditional vibration isolator uses coil springs The applied force Fn and Fp in the horizontal and vertical directions, respectively are utilized to adjust the compression or tension deflections of the springs so that the isolator will return back at the equilibrium position Unlike previous studies, this study proposes the compliant planar spring with the positive stiffness k p and the negative stiffness k n for a compact QZS nonlinear vibration isolator, as seen in Fig 1b The vertical spring is aimed to design and analyze in this paper This paper aims to propose a design of novel compliant planar spring capable of a positive stiffness for the compact QZS nonlinear vibration isolator instead of utilizing coil springs This study focuses on the static and modal characteristics of the proposed spring A model of compact QZS vibration isolator is then constructed as an application of compliant mechanism Figure Diagram of a QZS isolator: (a) Traditional isolator, (b) proposed isolator DESIGN OF PROPOSED COMPLIANT PLANAR SPRING Successful design of the compliant planar spring requires that the positive stiffness k p can be easily tuned to achieve a desired stiffness for various isolators Fig 2a shows a 3-D model of the spring Fig 2b presents a 2-D view of the spring Table shows the dimensions of the spring The proposed spring is designed by combining between the rigid links with the in-of-plane thickness h i (i = 1, 2, 3) and the flexible hinges with the in-of-plane thickness t i (i = 1, 2, 3) In order to decrease the stiffness of the spring, each rigid link and each flexible hinge are coupled as a lever type of amplification mechanism Because the lever type of amplification mechanism can amplify the displacement or deflection 796 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Figure (a) A 3-D model and (b) 2-D view of the compliant planar spring capable of a positive stiffness Table Design parameters of the proposed spring Parameter Value Unit h1 6.00 mm h2 8.00 mm h3 10.00 mm t1 0.40 mm t2 0.40 mm t3 0.39 mm θ1 100.00 Degree θ2 130.00 Degree From the mentioned discussion, if a various stiffness of the spring is required, the out of plane thickness w of flexure hinge or the in of plane thickness t of flexure hinge must also be changed A new stiffness of the isolator can be found by adjusting w or t for each spring in the following equation k p2 k p1 w2t23 = w1t13 (1) where k p is the positive stiffness and subscript and indicate the original and new parameters, respectively To achieve a new specific stiffness k p2 , the value of t or w must be changed accordingly so that the right side of Eq (1) equals to the left side STATIC ANALYSIS The spring made of Titanium alloy material, which has the high yield strength of 930 MPa A static finite element analysis using ANSYS 13 is conducted to achieve the displacement of the spring; and then the stiffness of spring can be calculated The automatic method is applied for meshing both the spring Next, the model is refined for flexible hinges to achieve a good meshing quality according to Skewness criterion in ANSYS The meshed model is given in Fig 3a The applied force F is exerted on the top of the spring and the 797 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV bottom is fixed The force-to-displacement curve of the proposed spring is obtained to measure the stiffness, as shown in Figs 3b The average stiffness of the spring is 20.93 N/mm The maximum displacement of the spring is d = 1.1939 mm when the applied load F of 25 N is located at the top of the spring, as depicted in Fig 4a The stress-displacement curve is shown in Fig 4b The results indicated that the compliant planar spring have linear characteristics similar to as a coil spring Therefore, the planar spring is adopted for the proposed vibration isolator They also have applications where require more compact spaces The number of components and complex assembly are reduced via using the compliant planar springs Figure (a) Meshed model and (b) force-displacement curve of the proposed spring Figure (a) Schematic diagram of the deformation and (b) stress-displacement curve of the spring MODAL ANALYSIS Modal analyses are performed to extract the eight six modes of the spring with the applied force F of 25 N The first, second, and fifth shape modes represents the rotation about the z-axis The third mode is the translational motion along the y-axis The fourth mode is the rotation about the x-axis The sixth mode represents the translational motion along the z-axis All modes are shown in Fig The values of first natural frequencies are presented in Table 798 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Figure Shape modes of the compliant planar spring Table The natural frequencies of proposed spring Mode shapes Frequency (Hz) 1st 64.717 2nd 166.78 3rd 282.08 4th 428.14 5th 680.85 6th 804.47 AN APPLICATION FOR A COMPACT QZS VIBRATION A QZS vibration isolator is developed by using the compliant planar spring with the positive stiffness, as shown in Fig 6a The isolator includes eleven components as: (1) Base, (2) inner ring, (3) support, (4) horizontal adjuster, (5) negative stiffness compliant planar spring, (6) compliant planar spring with positive stiffness, (7) isolated object, (8) compliant planar spring with negative stiffness, (9) horizontal adjuster, (10) support, (11) vertical adjuster 799 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV The excitation is applied on the bottom of the base (1) In order to illustrate the deformation of the isolator, for example, a finite element analysis ANSYS 13 software is used The based (1) is fixed and the applied fore F of 25 N is exerted on the top of the isolated object (7) The deformation result is indicated as in Fig 6b It indicated that the vertical spring is capable of subjecting a weight isolator load while the horizontal spring makes low dynamic stiffness for the isolator To demonstrate the concept of high static stiffness but low dynamic stiffness for the proposed isolator, it would be proved later Figure (a) A physical model and (b) the deformation of the QZS vibration isolator The high-static-low-dynamic stiffness concept for an anti-vibration isolator is that it can increase its isolating capacity via lowering the natural frequency of the isolator, while maintaining the same static load bearing capacity QZS is proposed in this paper to suppress vertical vibrations with a high-static-low-dynamic stiffness when the base is excited causing vibration, which is transmitted to the isolating system The vibration level of the isolated object W depends on the dynamic stiffness of the vibration isolator The isolator is supported by a frame whose displacement is represented by z The frame is excited via a sinusoidal displacement z with magnitude Z and frequency w to describe external disturbance, as given in Fig The damping effect on the isolator is illustrated using a viscous damper added with the vertical spring in parallel so that the equation of the motion for the vibration isolator system under harmonic excitation accounts for dissipative terms Figure Schematic diagram of dynamic model The governing equation of the dynamics of the isolator is achieved using Lagrange’s formulation First, the kinetic energy T and potential energy V of the isolator are as follows: = T ( ) 1 m d + z , and V = kd 2 (2) where m is the mass of the isolated load W, and k is the stiffness of the isolator system 800 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV The dissipation function C can be described as follows: C= 2 cd (3) where c is the damping coefficient of the damper The dynamic equation of the proposed isolator is derived by combining Eqs (2) and (3) The Lagrange L is formed by taking the difference of the scalar quantities of kinematic energy T and potential energy V of the system, L = T- V Using Lagrange’s formulation, the equation of the motion for the QZS system is expressed in non-dimensional form as: dˆ ′′ + 2ξ dˆ ′ + kˆQZS dˆ =Ω Zˆ cos Ωτ (4) by introducing the non-dimensional parameters as follows: = ξ ˆ ˆ c Z w ˆ k ˆ ′′ d , d= ˆ ′ d ,= ,= Zˆ ,Ω = , kQZS = ,= τ w0t , d= dˆ d L0 2mw0 L0 w0 kp w0 w0 where w0 = kp m (5) is the un-damped natural frequency of the system without negative stiffness The numerical simulation of displacement with respect to time is realized through approximately dynamic equation Eq (4), as indicated in Fig The input signal is consinusoid with amplitude zˆ of 10, the excitation w = 1.5 rad/s and the phase φ of zero Eq (4) is solved by using the Ode45 package in MATLAB The physical parameters are used in the numerical simulation, given as follows: α = 1, ξ = 0.03, m = kg, and k p = 1.6 N/mm Fig shows a comparison between the proposed QZS isolator and a linear system with respect to ξ = 0.05 It can be observed that the effective frequency of the proposed isolator is smaller than that of the linear system, and the peak transmissibility of the proposed isolator is less than that of the linear system Another important observation is that the transmissibility of the proposed isolator is lower than that of the linear isolator in the low frequency range In terms of low frequency vibration isolators, the proposed QZS vibration isolator system is superior to a linear system Figure Displacement response with respect to time history 801 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Figure Comparison between the proposed isolator and a linear CONCLUSION A compliant planar spring capable of a positive stiffness for a compact quasi-zero stiffness vibration isolator has been developed in this paper The proposed compliant planar spring has been designed based on the concept of compliant mechanism The static and dynamic characteristics have been studied using the finite element method in ANSYS software The results revealed that the stiffness of the compliant planar spring can be easily adjusted by changing the in-of-plane thickness or the out-of-plane thickness A virtual prototype of the QSZ vibration isolator has been built in this paper The deformation of the proposed isolator has been monitored via using ANSYS software By combining between the positive stiffness compliant planar and the negative stiffness compliant planar spring, the isolator can obtain a quasi-zero stiffness property with respect to high static stiffness but low dynamic stiffness The results indicated that the QZS isolator has the isolating capacity better than that of the linear isolator Future work will investigate on complete design of the QZS vibration isolator Simulations and experiments will be performed to evaluate the effective performances of the proposed compact QZS vibration isolator REFERENCES [1] Kovacic, I., Brennan, M J., Waters, T P., A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic Journal of Sound and Vibration, 2008, Vol 315, p 700-711 [2] Wu, W J., Chen, X D., Shan, Y H., Analysis and experiment of a vibration isolator using a novel magnetic spring with negative stiffness Journal of Sound and Vibration, 2014, Vol 333, p 2958-2970 [3] Zhou, N., Liu, K., A tunable high-static–low-dynamic stiffness vibration isolator, Journal of Sound and Vibration, 2010, Vol 329, p 1254-1273 [4] Le, T D., Ahn, K K., A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat Journal of Sound and Vibration, 2011, Vol 330, p 6311-6335 [5] Platus, D., Negative-stiffness-mechanism vibration isolation systems Proceedings of the SPIE’s International Symposium on Vibration control in Microelectronics, Optics and Metrology, 1991 802