A novel design method of bistable structures with required snap-through properties
Introduction
Bistable structures are extensively used in industrial devices, such as switches [1,2] which sizes can vary from the macro-world to the micro-world [3], constant-force mechanisms [[4], [5], [6]] taking full advantage of negative stiffness characteristics of bistable configurations, energy harvesting devices [[7], [8], [9], [10], [11]] transforming ambient kinetic energy into electric power, actuators [[12], [13], [14], [15], [16]], and positioners [17]. Furthermore, the attractive characteristics of bistable structures include less power consumption [18], increased reliability, the precise positioning, structural simplicity [19], chaotic behavior [20,21], negative extensibility [22], etc.
Snap-through properties of bistable structures, mainly including the snapping force, the maintaining force and the snapping distance from one steady state snapping to another, are significantly dependent on many factors, such as the section shape, pre-compression [23], the actuation position [3], the apex height, boundary conditions [24] and the internal stress of the curved beams. Because of the complex coupling relationships between snap-through properties and influencing factors, it is often difficult to design a bistable structure with required snap-through properties. Thus, how to efficiently design and fabricate bistable structures with required properties has become an urgent problem in engineering applications.
Many efforts for designing bistable structures have been done, among them, one important type of bistable structures is based on the pre-shaped and/or pre-compressed beams [[23], [24], [25], [26], [27], [28], [29], [30], [31], [32]]. For examples, based on the relationships between the axial force, the axial displacement and the apex height in [23,25,26], pre-compressed bistable beams with uniform section are designed. Wu et al. [27] proposed a design criterion of a V-shaped bistable beam to investigate if the bistability can occur. Vangbo [28] theoretically analyzed bistable mechanics of a compressed bistable buckled beam for modeling the snap-through of a double-clamped bistable beam. Based on the generalized variation principle and meromorphic function method, Zhao et al. [29] modified the analytical solution of the snapping force in [28], improving the accuracy. Cazottes et al. [3] explored the influence of the actuation position on the snap-through of a pre-compressed bistable beam, and results showed a shifted actuation caused difficulties for a monolithic design. Huang et al. [30] modified local segment's geometry of a pre-shaped bistable beam for optimizing the ratio of the snapping force and the maintaining force. Aiming at the pre-shaped bistable configurations with the equal section, Qiu et al. [31] proposed the analytical solution of the lateral force and the center deflection in the case of neglecting higher modes, which was developed in [32]. Rafsanjani et al. [33] analyzed the effects of the snap-through instabilities on the tensile stress–strain curve of the proposed mechanical metamaterial comprising the bistable mechanism in [32]. The above contents illustrate that the snapping force, the maintaining force and the snapping distance are determined by the shape and the pre-stress distribution of the bistable structures, and it is a challenge to exactly control the pre-stress distribution in fabricating bistable structures. Besides, owing to the complex non-linear relationship between snap-through properties and the influence factors, it is also difficult to efficiently obtain bistable structures with required snap-through properties.
In this paper, a novel method is presented for designing and fabricating the bistable structures with required snap-through properties. The bistable structures are pre-compressed arch-shaped beams with local reinforcements, which are formed through compressing the corresponding straight beams into post-buckling state. The required shape and pre-stress distribution of the beams are exactly controlled through the post-buckling states. The post-buckling states are adjusted through the compressed length of the beam and the position and sizes of the local reinforcements. The rest of this paper is outlined as follows. In Section 2, the configuration of the bistable structure is proposed and the fabrication process is introduced. Then, in Section 3, the mechanical model of the proposed bistable structure is presented for analyzing the snap-through properties. In Section 4, the novel design method of bistable structures with required snap-through properties is put forward. In Section 5, the design and fabrication process of a design example is illustrated in detail, and the snap-through properties of the designed bistable beam are verified by the experiment. In the last section, some conclusions are summarized.
Section snippets
Configuration design and fabrication process
The configuration of the bistable structure proposed in this paper is a pre-compressed arch-shaped beam with local reinforcements, as shown in Fig.1. The snap-through properties can be adjusted through cooperatively controlling the position and sizes (length , width and thickness ) of the local reinforcements and the axial compressed length, namely, the pre-compressed length , shown in Fig. 2. Under the action of the lateral force F, the designed bistable beam can jump to the second
Mechanical model and control equation
According to Fig. 1, Fig. 2, the equilibrium equation of a pre-compressed bistable beam with local reinforcements satisfies:where is the deflection of the beam; is Dirac delta function; E is the Young modulus; is the span of the bistable beam and ; is the beam's inertial moment, written aswhere, and can be obtained by deformation coordination equations,
Design method
In order to obtain the bistable structures with desired snap-through properties, the dependency relationship of the snap-through properties on the key controlling parameters need to be determined, which can be described by the model spectrum like tables or figures representing the variation of snap-through properties with the control parameters. The model spectrums can be formed by using finite element simulations and experimental tests. If the required snap-through properties are given, the
Design example and experimental verification
In order to verify the proposed design method, the desired snap-through properties of a pre-compressed bistable beam in the design example are = 2.6 mm and = 3.5 N. The corresponding contours are shown in Fig. 6, which are extracted from Fig. 4.
Eight crossover points (A1 A8) can be obtained from Fig. 6, which meet design requirements of = 2.6 mm and = 3.5 N. Because the pre-compressed length of the point A1 is 0.43 mm, which is larger than that of crossover points and easy to be
Conclusions
Due to the complex coupling relationships between snap-through properties and the control parameters, it is difficult to realize the cooperative design of the required snapping force and snapping distance in quickly designing bistable structures. Therefore, a novel method is proposed for designing bistable structures with desired snap-through properties. By traversing the model spectrum, all design schemes meeting the required snap-through properties can be achieved, which can greatly enlarge
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11372063, 11572073, 11332004, 51575088) and the Fundamental Research Funds for the Central Universities.
Renjing Gao received the M.S. and Ph.D. degrees in electronic engineering from the Dalian University of Technology, Dalian, China, in 2002 and 2012, respectively. Since 1994, she has been with the Dalian University of Technology, where she is currently a Professor with the School of Automotive Engineering. She has performed research and published over 30 papers in a wide variety of design topics, including left-handed materials, mass sensors, digital control systems, and microelectromechanical
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Renjing Gao received the M.S. and Ph.D. degrees in electronic engineering from the Dalian University of Technology, Dalian, China, in 2002 and 2012, respectively. Since 1994, she has been with the Dalian University of Technology, where she is currently a Professor with the School of Automotive Engineering. She has performed research and published over 30 papers in a wide variety of design topics, including left-handed materials, mass sensors, digital control systems, and microelectromechanical systems. She holds eight Chinese patents. She has published one book on the subject of electronic technology.
Mingli Li is pursuing her PhD at the school of Dalian University of Technology, China. Her major research is on the mechanical properties of bistable structures and the design energy harvesters based on bistable structures. Her research interests include structural design, design method, mass sensors, actuators and multi-stable mechanisms.
Qi Wang received B.S. and Ph.D. degrees in engineering mechanics from Dalian University of Technology, Dalian, Liaoning, in 2010, and 2017, respectively. His major area of research is structural design in electromagnetics, and his areas of specialization include topology optimization, antennas, sensors, metamaterials and microwave communication.
Jian Zhao (M, 2009) received the B.S., M.S., and Ph.D degrees in 2003, 2006, and 2008, respectively. He is a professor at the school of automotive engineering, Dalian University of Technology, Dalian, China. He had been a research scientist in the mechatronics department of Johannes Kepler University Linz, Austria during 2008–2009. He was granted more than 20 Chinese Patents, and has published over 30 papers in a wide variety of MEMS fields, including intelligent structures, compliant mechanisms, shock sensors, and acceleration switches. His current research interests include intelligent structures, sensors, multi-stable mechanisms, and dynamic design methods.
Shutian Liu received B.S. and Ph.D. degrees in engineering mechanics from Dalian University of Technology, Dalian, Liaoning, in 1982, and 1994, respectively. During 1994–1996 and 1998-1999, he was a post-doc in the department of engineering mechanics in Dalian University of Technology and a post-doc associate at Wright State University. Since 1994, he has been at Dalian University of Technology, where he is currently a Professor of engineering mechanics. He was the recipient of the Excellent Young Teachers Awards of Ministry of Education, the State Science and Technology Advancement Award, Fellowship of Chinese Society for Composite Materials. He performed and published more than 100 papers, and holds 3 Chinese Patens. Professor Liu’s major research fields including material design, multi-scale analysis and design of composite materials, optimization, topology, sensors, machine design and band-gap materials.