Technical Brief

Bi-BCM: A Closed-Form Solution for Fixed-Guided Beams in Compliant Mechanisms

[+] Author and Article Information
Fulei Ma

School of Electro-Mechanical Engineering,
Xidian University,
Xi'an 710071, Shaanxi, China

Guimin Chen

School of Electro-Mechanical Engineering,
Xidian University,
Xi'an 710071, Shaanxi, China
e-mail: guimin.chen@gmail.com

1Corresponding author.

Manuscript received March 10, 2016; final manuscript received October 4, 2016; published online November 23, 2016. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 9(1), 014501 (Nov 23, 2016) (8 pages) Paper No: JMR-16-1065; doi: 10.1115/1.4035084 History: Received March 10, 2016; Revised October 04, 2016

A fixed-guided beam, with one end is fixed while the other is guided in that the angle of that end does not change, is one of the most commonly used flexible segments in compliant mechanisms such as bistable mechanisms, compliant parallelogram mechanisms, compound compliant parallelogram mechanisms, and thermomechanical in-plane microactuators. In this paper, we split a fixed-guided beam into two elements, formulate each element using the beam constraint model (BCM) equations, and then assemble the two elements' equations to obtain the final solution for the load–deflection relations. Interestingly, the resulting load–deflection solution (referred to as Bi-BCM) is closed-form, in which the tip loads are expressed as functions of the tip deflections. The maximum allowable axial force of Bi-BCM is the quadruple of that of BCM. Bi-BCM also extends the capability of BCM for predicting the second mode bending of fixed-guided beams. Besides, the boundary line between the first and the second modes bending of fixed-guided beams can be easily obtained using a closed-form equation. Bi-BCM can be immediately used for quick design calculations of compliant mechanisms utilizing fixed-guided beams as their flexible segments (generally no iteration is required). Different examples are analyzed to illustrate the usage of Bi-BCM, and the results show the effectiveness of the closed-form solution.

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Howell, L. L. , 2001, Compliant Mechanisms, Wiley, New York.
Masters, N. D. , and Howell, L. L. , 2003, “ A Self-Retracting Fully Compliant Bistable Micromechanism,” J. Microelectromech. Syst., 12(3), pp. 273–280. [CrossRef]
Chen, G. , and Ma, F. , 2015, “ Kinetostatic Modeling of Fully Compliant Bistable Mechanisms Using Timoshenko Beam Constraint Model,” ASME J. Mech. Des., 137(2), p. 022301. [CrossRef]
Wilcox, D. L. , and Howell, L. L. , 2005, “ Fully Compliant Tensural Bistable Micromechanisms (FTBM),” J. Microelectromech. Syst., 14(6), pp. 1223–1235. [CrossRef]
Howell, L. L. , DiBiasio, C. M. , Cullinan, M. A. , Panas, R. , and Culpepper, M. L. , 2010, “ A Pseudo-Rigid-Body Model for Large Deflections of Fixed-Clamped Carbon Nanotubes,” ASME J. Mech. Rob., 2(3), p. 034501. [CrossRef]
Hao, G. , 2015, “ Extended Nonlinear Analytical Models of Compliant Parallelogram Mechanisms: Third-Order Models,” Trans. Can. Soc. Mech. Eng., 39(1), pp. 71–83.
Hao, G. , and Li, H. , 2015, “ Nonlinear Analytical Modeling and Characteristic Analysis of a Class of Compound Multibeam Parallelogram Mechanisms,” ASME J. Mech. Rob., 7(4), p. 041016. [CrossRef]
Lott, C. D. , McLain, T. W. , and Harb, J. N. , and Howell, L. L. , 2002, “ Modeling the Thermal Behavior of a Surface-Micromachined Linear-Displacement Thermomechanical Microactuator,” Sens. Actuators A, 101(1), pp. 239–250. [CrossRef]
Wittwer, J. W. , Baker, M. S. , and Howell, L. L. , 2006, “ Simulation, Measurement, and Asymmetric Buckling of Thermal Microactuators,” Sens. Actuators A, 128(2), pp. 395–401. [CrossRef]
Holst, G. L. , Teichert, G. H. , and Jensen, B. D. , 2011, “ Modeling and Experiments of Buckling Modes and Deflection of Fixed-Guided Beams in Compliant Mechanisms,” ASME J. Mech. Des., 133(5), p. 051002. [CrossRef]
Lyon, S. M. , and Howell, L. L. , 2002, “ A Simplified Pseudo-Rigid-Body Model for Fixed-Fixed Flexible Segments,” ASME Paper No. DETC2002/MECH-34203.
Shoup, T. E. , 1972, “ On the Use of the Nodal Elastica for the Analysis of Flexible Link Devices,” J. Eng. Ind., 94(3), pp. 871–875. [CrossRef]
Zhao, J. , Jia, J. , He, X. , and Wang, H. , 2008, “ Post-Buckling and Snap-Through Behavior of Inclined Slender Beams,” ASME J. Mech. Des., 75(4), p. 041020. [CrossRef]
Kimball, C. , and Tsai, L. W. , 2002, “ Modeling of Flexural Beams Subjected to Arbitrary end Loads,” ASME J. Mech. Des., 124(2), pp. 223–235. [CrossRef]
Kim, C. , and Ebenstein, D. , 2011, “ Curve Decomposition for Large Deflection Analysis of Fixed-Guided Beams With Application to Statically Balanced Compliant Mechanisms,” ASME J. Mech. Rob., 4(4), p. 041009. [CrossRef]
Zhang, A. , and Chen, G. , 2013, “ A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021006. [CrossRef]
Awtar, S. , Slocum, A. H. , and Sevincer, E. , 2007, “ Characteristics of Beam-Based Flexure Modules,” ASME J. Mech. Des., 129(6), pp. 625–639. [CrossRef]
Awtar, S. , Shimotsu, K. , and Sen, S. , 2010, “ Elastic Averaging in Flexure Mechanisms–A Three-Beam Parallelogram Flexure Case Study,” ASME J. Mech. Rob., 2(4), p. 041006. [CrossRef]
Sen, S. , 2013, “ Beam Constraint Model: Generalized Nonlinear Closed-Form Modeling of Beam Flexures for Flexure Mechanism Design,” Ph.D dissertation, the University of Michigan, Ann Arbor, MI.
Awtar, S. , and Sen, S. , 2010, “ A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Load-Displacement Formulation,” ASME J. Mech. Des., 132(8), p. 081008. [CrossRef]
Awtar, S. , and Sen, S. , 2010, “ A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation,” ASME J. Mech. Des., 132(8), p. 081009. [CrossRef]
Ma, F. , and Chen, G. , 2016, “ Modeling Large Planar Deflections of Flexible Beams in Compliant Mechanisms Using Chained Beam-Constraint-Model (CBCM),” ASME J. Mech. Rob., 8(2), p. 021018. [CrossRef]
Chen, G. , and Bai, R. , 2016, “ Modeling Large Spatial Deflections of Slender Bisymmetric Beams in Compliant Mechanisms Using Chained Spatial-Beam-Constraint-Model,” ASME J. Mech. Rob., 8(4), p. 041011. [CrossRef]
William, M. F. , 1996, “ A Geometric Interpretation of the Solution of the General Quartic Polynomial,” Am. Math. Mon., 103(1), pp. 51–57. [CrossRef]
Shamshirasaz, M. , and Asgari, M. B. , 2008, “ Polysilicon Micro Beams Buckling With Temperature-Dependent Properties,” Microsyst. Technol., 14(7), pp. 975–961.


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Fig. 3

Free body diagrams for the two element of a fixed-guided beam end remains zero

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Fig. 2

Schematic of a fixed-guided beam

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Fig. 1

Illustration of different compliant mechanisms that employ fixed-guided beams: (a) a bistable mechanism, (b) a thermomechanical in-plane microactuator, (c) a compound compliant parallelogram mechanism, and (d) a double parallelogram mechanism

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Fig. 4

Illustration of two possible deflected configurations corresponding to the second mode bending (configuration I corresponding to solution (a) and configuration II corresponding to solution (b))

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Fig. 5

Schematic of a fixed-guided beam in a bistable mechanism

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Fig. 8

Comparison of the two deflected configurations obtained by Bi-BCM solution and the experimental results in Ref. [10]

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Fig. 9

Schematic of a fixed-guided beam taken from a compliant parallelogram mechanism

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Fig. 6

The force–deflection curves of the bistable mechanism

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Fig. 7

The boundary between first and second mode bending according to Eq. (42) for the fixed-guided beam. Several deflected beam configurations are also shown.

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Fig. 12

The force–displacement relationship of the thermomechanical in-plane microactuator

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Fig. 10

The force–deflection curves of the compliant parallelogram mechanism

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Fig. 11

Schematic of a deflected TIM leg



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