Design of a bistable mechanism with B spline profiled beam for versatile switching forces Accepted Manuscript Title Design of a bistable mechanism with B spline profiled beam for versatile switching f[.]
Accepted Manuscript Title: Design of a bistable mechanism with B-spline profiled beam for versatile switching forces Authors: I-Ting Chi, Tien Hoang Ngo, Pei-Lun Chang, Ngoc Dang Khoa Tran, Dung-An Wang PII: DOI: Reference: S0924-4247(19)30560-6 https://doi.org/10.1016/j.sna.2019.05.028 SNA 11400 To appear in: Sensors and Actuators A Received date: Revised date: Accepted date: April 2019 29 April 2019 16 May 2019 Please cite this article as: Chi I-Ting, Ngo TH, Chang P-Lun, Tran NDK, Wang D-An, Design of a bistable mechanism with B-spline profiled beam for versatile switching forces, Sensors and amp; Actuators: A Physical (2019), https://doi.org/10.1016/j.sna.2019.05.028 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Design of a bistable mechanism with B-spline profiled beam for versatile switching forces SC RI PT I-Ting Chi1, Tien Hoang Ngo1, Pei-Lun Chang1, Ngoc Dang Khoa Tran2, Dung-An Wang1* Graduate Institute of Precision Engineering, National Chung Hsing University, Taichung 40227, Taiwan, ROC Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, 12 Nguyen Van Bao, Ward 4, Go Vap District, Ho Chi Minh City, Vietnam * Corresponding author: Tel.:+886-4-22840531 ext 365; fax:+886-4-22858362 N U E-mail address: daw@dragon.nchu.edu.tw (D.-A Wang) EP TE D M A Graphical Abstract A CC Highlights ‧ A bistable mechanism composed of B-spline curved beams for versatile switching forces in the forward and backward directions is proposed ‧An analytical model to solve for the nonlinear force-displacement characteristics of the B-spline profiled beams is developed ‧Smoother force-displacement curve is achieved compared to existing design Abstract A compliant bistable mechanism composed of B-spline curved beams for design of switching forces in the forward and backward directions is developed The parametric B- SC RI PT spline curve has five control points to give a high design freedom in the output force of the beam-type compliant mechanisms An analytical model is developed to provide an efficient tool to obtain the force-displacement characteristics of the B-spline profiled bistable mechanism B-splined profiled bistable mechanisms with various ratios of the switching forces in the forward and backward motions are designed The results are U confirmed by experiments The developed bistable mechanism with high force versatility N has applications in devices where precise geometric activation and quantifiable load M A bearing capacity are desired TE Introduction D Keywords: compliant mechanism; B-spline curve; switching force EP A compliant bistable mechanism (CBM) with desired characteristics of force and displacement output is desired in its industrial applications Aerospace industries need a CC reliable release mechanism to launch deployable structures, such as antennae, satellites and solar panels CBM’s high reliability, low vibration sensibility in force response, and less A susceptibility to temperature variation have led them to space application [1] Precise geometric activation and quantifiable load bearing capacity of CBMs make them suitable for emerging applications in automotive, building and biomedical industries [2] In automotive industry, CBMs can be employed in trunk lid design to compensate to lid weight during opening/closing operation [3] Transition of a robotic end effector can be achieved with controllable bistable positions of CBMs [4] Precise control of the forcedisplacement behavior of CBMs is a prerequisite for their successful applications in SC RI PT switches [5], projection displays [6], and nonvolatile memory elements [7], etc For applications in nonvolatile memory devices, nearly equal switching forces in back and forth directions of CBMs may facilitate two logical levels “1” and “0” corresponding to the two stable states The equal switching force of CBMs can see its use in threshold accelerometers to achieve two sensing directions along one sensing axis [8] U Design for force/displacement output of CBMs has attracted attention in recent N years Li and Chen [9] proposed a rational function to model the force-displacement (F-d) A characteristics of several CBMs Their function with a cubic polynomial numerator and M quadratic polynomial denominator can capture key features of the F-d curve, where regression analyses are required to obtain the polynomials to describe the relations between D design parameters and the F-d characteristics of the CBMs Huang et al [10] developed TE an optimization based method for design of CBMs with specific switching forces The By EP presented CBMs have cosine curved beams with multiple reinforced segments modifying the length, width, thickness, element number and position of the reinforced CC segments, the desired switching forces of the CBMs can be achieved The F-d curves of their designs with nearly equal switching forces in the forward and backward directions A might not be smooth Smooth force spectrum may be advantageous to avoid chatter vibration during operation Palathingal and Ananthasuresh [11] proposed a shape optimization approach to obtain the switching forces and the distance between equilibrium states of CBMs numerically For improved accuracy of their method, several mode shapes should be used for approximation of the beam profile of the CBMs Han et al [12] proposed a CBM with both tensural segments and compresural segments The combination of the tensural and compresural segments tailors their CBM for different design SC RI PT requirements, such as switching forces and equilibrium positions Gao et al [13] presented a design process for a CBM with required switching forces and the distance between equilibrium states Pre-compression and reinforced segments are key design parameters of their design A lookup table representing the variation of snap-through properties with the design parameters, such as pre-compression length and length of reinforced segment, needs U to be constructed in their approach Finite element analyses are required to solve for the Li and Hao [14] N F-d characteristics for their pre-compressed bistable structures A investigated a generic double-slider four-bar linkage with spring and established its F-d M formulation for design of a mechanism with expected nonlinear characteristics, including a bistable behavior with desired switching forces D The beam profiles of traditional CBMs can be straight line [15,16,17] or cosine TE curve [11,18,19] The commonly used straight line and cosine curve profiles may restrict EP the degree of freedom in design of CBMs A pre-load operation may bring the beams of CBMs into desired initial configuration to control their switching forces [20] Design CC complexity can be introduced by the additional control scheme for the pre-compression of the beams Antagonistic pre-shaped beams can be utilized to obtain desired force output A of CBMs [21] A preload is still necessary to operate their device Beams of CBMs with parametric curve profiles may offer higher degree of freedom in design In this investigation, we develop a double beam type CBM with a nonrational Bspline profile for design of switching forces and the distance between equilibrium positions The beams of the CBM have a reinforced segment at its center A model is developed to evaluate the F-d characteristics of the CBMs An optimal design of the CBM is sought by a multiobjective optimization algorithm Experiments are carried out to verify the F-d SC RI PT output of the CBM Several designs of the CBMs with various design targets are obtained to demonstrate the high design freedom of the B-spline profiled CBM Design 2.1 Design U Fig 1(a) schematically shows a CBM It is comprised of a shuttle mass and four N segmented beams The center of the shuttle mass is located at the symmetric axis of the A CBM The beams have a nonrational B-spline profile [22] and are divided into three M segments A Cartesian coordinate system is also shown in the figure It is assumed that the CBM remains parallel to the underlying plane Due to the chevron configuration of the D CBM, the stiffness in X direction is much larger than that in the Y direction Therefore, TE the salient direction of the motion of the CBM is along the Y direction Due to geometry EP symmetry, a quarter model can be utilized to analyze the F-d characteristics of the CBM Fig 1(b) shows a quarter model of the CBM The left end of the beam of the quarter model CC can be represented by a fixed boundary condition The symmetric plane of the quarter model can be represented by a roller boundary condition The B-spline profile of the beam A is indicated by a center line in the figure The beam is dissected into three segments as numbered in the figure The width of segment 2, v2 , is much larger than the widths of segment and segment 3, v1 and v3 , respectively The span of segment 1, segment and segment are denoted by D1 , D2 and D3 , respectively The span and apex height of the B-spline curved beam are represented by D and H , respectively The shuttle mass has a width of Dm and a height of vm (see Fig 1(a)) The CBM has a thickness of T A typical F-d curve of a CBM is shown in Fig When the shuttle mass is displaced SC RI PT in the Y direction, see Fig 1(a), the CBM moves from its first stable equilibrium position S1 , passes its unstable equilibrium position Q , then reaches its second stable equilibrium position S The force experienced by the shuttle mass in this displacement range has a maximum value Fmax and a minimum value Fmin When the shuttle mass is moved in the U Y direction, the mechanism follows the F-d curve reversely in the displacement N controlled mode of motion The insets of Fig show two stable equilibrium states of the A CBM M Huang et al [10] presented a cosine curved beam with multiple reinforced segments to achieve desired force output of a CBM Dissimilar to the existing segment-reinforced D CBM design of Huang et al [10], the beam of the CBM has a parametric curve as its profile TE and one reinforced segment to achieve high design freedom in switching forces and the distance between equilibrium positions In this investigation, an open uniform nonrational EP B-spline (OUNBS) curve is selected as the beam profile of the CBM The OUNBS curve CC is determined by a five-point polygon B1B2 B3 B4 B5 as shown in Fig The OUNBS curve defined by a nonglobal basis allows the degree of the curve to be changed without changing A the number of the defining polygon vertices The OUNBS curve is given by [22] n 1 P(t ) P i 1 n 1 Bi N i , k (t ) N i 1 (1) i,k (t ) where t is the parameter, n is the number of control points, and PBi the position vector of the point Bi N i , k (t ) is the i th normalized B-spline basis function of order k , and is defined as [22] 1 if X i t X i 1 N i ,1 (t ) otherwise 0 and (t X in ) N i , k 1 (t ) ( X in k t ) N i 1, k 1 (t ) N i , k (t ) X in k X in X in k X in1 SC RI PT (2) (3) 1 i k k 1 i n 1 n i n k 1 (4) N X in X in i k X in n k U The values of X in are elements of an open knot vector X on , which is given by [22] M A The parameter t in Eq (3) varies from tmin and tmax , where the values of tmin and tmax are those of the smallest element and largest element in the knot vector, respectively D The location of the control points is adjusted to achieve the high design freedom of TE the F-d characteristics of the CBM The B-spline profile of the beam is adjusted by EP allowing points B2 , B3 , B4 and B5 to move in the design space enclosed by the dashed rectangle in Fig Note that B5 is restricted to move along the Y direction The position CC of the point B1 is fixed at the origin of a Cartesian coordinate system as shown in Fig A The position of the point B5 is given by the span D and apex height H of the B-spline curved beam B2 and B4 are constrained to be below and above a dotted line connecting B1 and B5 , respectively (see Fig 3) The X coordinate of the centroid of the segment is specified as D / The desired F-d characteristics of the CBM can be achieved by an optimum design approach During the optimization design process, values of D , H , T , v1 , v2 and v3 are specified Force outputs, Fmax and Fmin , and the second stable equilibrium position S can be adjusted by varying the design variables of coordinates of B2 , B3 , B4 , H , D2 and the SC RI PT Y coordinate of the point B5 Note that the X coordinate of the point B5 , X B , is given by the specified span D , and the Y coordinate of the point B5 , YB5 , is the apex height H of the B-spline curved beam, where 3D / 110 YB5 D / 10 3D / 110 and X B1 X B2 X B4 X B5 The objective functions of the optimization problem are Min S2 1 1 (5) (6) (7) D d f2 U Fmin 1 N Min f1 A Fmax M Min TE where f1 and f are the target values of the output forces of the CBM, and d is the target EP value of the second equilibrium position 2.2 Model CC An analytical model that can accurately compute the F-d curve of the B-spline A profiled CBM is essential for the optimization design process F-d curves of straight beam based CBMs can be obtained by the chained beam constraint model (CBCM) developed by Ma and Chen [23] Based on the formulation of Awtar and Sen [24], Chen et al [25] developed discretization schemes to extend their CBCM for compliant mechanisms with initially curved beams In this investigation, a model to obtain the F-d relation of the CBM is developed based on the works of Chen and Ma [26] and Chen et al [25] An initial configuration and a deformed configuration of a quarter model of the CBM are schematically shown in Fig The beam is clamped at one end and the other SC RI PT end is modeled by a roller to represent the symmetry boundary condition The apex height of the beam is H A displacement A is applied on the roller In the deformed configuration, the apex height of the beam is H A Based on Eqs (1-4), the parametric B-spline curve of the beam with t ranging from to can be represented as ,0 t U t 7t 18t 12 t 3 2t t3 X f ( t ) X X X B4 B B 4 2 Y g1 (t ) t 7t 18t 12 YB t 3 2t YB t YB 4 (8) 2 t X 2 t 2t 1 X 2 t 7t 10t X t 13 X X f (t ) B2 B3 B4 B5 4 ,1 t 2 Y g (t ) 2 t Y 2 t 2t 1 Y 2 t 7t 10t Y t 13Y B2 B3 B4 B5 4 2 N M A In this investigation, X Y and x y represent the global and local Cartesian coordinate D system, respectively t and t refer to the parameter value of the left end and the TE right end of the beam, respectively Segment 1, and of the beam are discretized into 16, EP 32 and 16 elements, respectively The parameter values of the left end and the right end of the segment can be solved by their known X coordinates as the design variable D2 CC is given The coordinates ( X i ,Yi ) of the discretization points on the segments 1, and A are obtained by the 16, 32 and 16 equal increments of the parameter value The force and moment applied to the first, the i th, the (i 1) th and the final element of the beam are schematically shown in Fig 5(a-d), respectively, where q is the number of elements As seen in the figure, X i Yi represents a global Cartesian coordinate system for the i th element A local Cartesian coordinate system xi yi for each ... characteristics of the B- spline profiled bistable mechanism B- splined profiled bistable mechanisms with various ratios of the switching forces in the forward and backward motions are designed The results are... Graphical Abstract A CC Highlights ‧ A bistable mechanism composed of B- spline curved beams for versatile switching forces in the forward and backward directions is proposed ‧An analytical model.. .Design of a bistable mechanism with B- spline profiled beam for versatile switching forces SC RI PT I-Ting Chi1, Tien Hoang Ngo1, Pei-Lun Chang1, Ngoc Dang Khoa Tran2, Dung-An Wang1* Graduate