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

2016 ◽  
Author(s):  
Fulei Ma ◽  
Guimin Chen
Keyword(s):  
Closed Form ◽  
Compliant Mechanisms ◽  
Closed Form Solution ◽  
Form Solution ◽  
Boundary Line ◽  
Final Solution ◽  
Buckling Mode ◽  
Buckling Modes ◽  
Design Calculations ◽  
Constraint Model

A fixed-guided beam is one of 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 the BCM. Bi-BCM also extends the capability of the BCM for predicting the second buckling mode of fixed-guided beams. Besides, the boundary line between the first and the second buckling modes 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.

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

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
Fulei Ma ◽  
Guimin Chen
Keyword(s):  
Closed Form ◽  
Compliant Mechanisms ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Boundary Line ◽  
Final Solution ◽  
Second Mode ◽  
Design Calculations ◽  
Constraint Model

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.

Nonlinear Analysis of a Class of Inversion-Based Compliant Cross-Spring Pivots

2021 ◽  
Author(s):  
Shiyao Li ◽  
Guangbo Hao ◽  
Yingyue Chen ◽  
Jiaxiang Zhu ◽  
Giovanni Berselli
Keyword(s):  
Nonlinear Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Parameters ◽  
Contributing Factors ◽  
Characteristic Analysis ◽  
Center Shift ◽  
Axial Forces ◽  
Constraint Model

Abstract This paper presents a nonlinear model of an inversion-based generalized cross-spring pivot (IG-CSP) using the beam constraint model (BCM), which can be employed for the geometric error analysis and the characteristic analysis of an inversion-based symmetric cross-spring pivot (IS-CSP). The load-dependent effects are classified in two ways, including structure load-dependent effects and beam load-dependent effects, where the loading positions, geometric parameters of elastic flexures, and axial forces are the main contributing factors. The closed-form load-rotation relations of an IS-CSP and a non-inversion-based symmetric cross-spring pivot (NIS-CSP) are derived with consideration of the three contributing factors for analyzing the load-dependent effects. The load-dependent effects of IS-CSP and NIS-CSP are compared when the loading position is fixed. The rotational stiffness of the IS-CSP or NIS-CSP can be designed to increase, decrease, or remain constant with axial forces, by regulating the balance between the loading positions and the geometric parameters. The closed-form solution of the center shift of an IS-CSP is derived. The effects of axial forces on the IS-CSP center shift are analyzed and compared with those of a NIS-CSP. Finally, based on the nonlinear analysis results of IS-CSP and NIS-CSP, two new compound symmetric cross-spring pivots are presented and analyzed via analytical and FEA models.

CLOSED-FORM TRIGONOMETRIC SOLUTIONS FOR INHOMOGENEOUS BEAM-COLUMNS ON ELASTIC FOUNDATION

2004 ◽  
Vol 04 (01) ◽  
pp. 139-146 ◽  
Author(s):  
IVO CALIÒ ◽  
ISAAC ELISHAKOFF
Keyword(s):  
Closed Form ◽  
Elastic Foundation ◽  
Load Distribution ◽  
Flexural Rigidity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Material Density ◽  
Buckling Modes ◽  
Beam Columns ◽  
Closed Form Solutions

In this study, a special class of closed-form solutions for inhomogeneous beam-columns on elastic foundations is investigated. Namely the following problem is considered: find the distribution of the material density and the flexural rigidity of an inhomogeneous beam resting on a variable elastic foundation so that the postulated trigonometric mode shape serves both as vibration and buckling modes. Specifically, for a simply-supported beam on elastic foundation, the harmonically varying vibration mode is postulated and the associated semi-inverse problem is solved that result in the distributions of flexural rigidity that together with a specific law of material density, an axial load distribution and a particular variability of elastic foundation characteristics satisfy the governing eigenvalue problem. The analytical expression for the natural frequencies of the corresponding homogeneous beam-column with a constant characteristic elastic foundation is obtained as a particular case. For comparison the obtained closed-form solution is contrasted with an approximate solution based on an appropriate polynomial shape, serving as trial function in an energy method.

Closed-Form Critical Buckling Load of Simply Supported Orthotropic Plates and Verification

2019 ◽  
Vol 19 (12) ◽  
pp. 1950157 ◽  
Author(s):  
Zhao Jing ◽  
Qin Sun ◽  
Ke Liang ◽  
Jianqiao Chen
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Buckling Load ◽  
Coupling Coefficients ◽  
Orthotropic Plates ◽  
Buckling Mode ◽  
Simply Supported ◽  
Buckling Behavior ◽  
Torsional Coupling

The buckling mode is important to determine the critical load of specially orthotropic rectangular plates under axial compression with simply supported boundary. However, in classical laminated plate theory (CLPT), the critical buckling mode can only be obtained by iterative or numerical methods. This paper derives the critical buckling mode mathematically and presents the critical buckling load in closed form. By taking advantage of the derived closed-form solution, it is convenient to investigate the effects of aspect ratio, load ratio, and fiber orientation on the buckling load, and the parameters affecting the buckling mode can be easily obtained. The first-order shear deformation theory (FSDT)-based finite element method is developed to verify the closed-form solution. The bending-torsional coupling effects are analyzed and discussed to assess the approximation of the buckling behavior of specially orthotropic plates to general laminates. The obtained finite element solutions of general laminates are compared with the closed-form solutions of specially orthotropic plates. The accuracy of approximation of the buckling behavior of specially orthotropic plates to the general laminates increases as the bending-torsional coupling coefficients decrease. The closed-form solution can be applied to laminates with small bending-torsional coupling coefficients.

Nonlinear Analysis of a Class of Inversion-Based Compliant Cross-Spring Pivots

2021 ◽  
pp. 1-29
Author(s):  
Shiyao Li ◽  
Guangbo Hao ◽  
Yingyue Chen ◽  
Jiaxiang Zhu ◽  
Giovannni Berselli
Keyword(s):  
Nonlinear Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Parameters ◽  
Contributing Factors ◽  
Characteristic Analysis ◽  
Center Shift ◽  
Axial Forces ◽  
Constraint Model

Abstract This paper presents a nonlinear model of an inversion-based generalized cross-spring pivot (IG-CSP) using the beam constraint model (BCM), which can be employed for the geometric error analysis and the characteristic analysis of an inversion-based symmetric cross-spring pivot (IS-CSP). The load-dependent effects are classified in two ways, including the structure load-dependent effects and beam load-dependent effects, where the loading positions, geometric parameters of elastic flexures, and axial forces are the main contributing factors. The closed-form load-rotation relations of an IS-CSP and a non-inversion-based symmetric cross-spring pivot (NIS-CSP) are derived with consideration of the three contributing factors for analyzing the load-dependent effects. The load-dependent effects of IS-CSP and NIS-CSP are compared when the loading position is fixed. The rotational stiffness of the IS-CSP or NIS-CSP can be designed to increase, decrease, or remain constant with axial forces, by regulating the balance between the loading positions and the geometric parameters. The closed-form solution of the center shift of an IS-CSP is derived. The effects of axial forces on the IS-CSP center shift are analyzed and compared with those of a NIS-CSP. Finally, based on the nonlinear analysis results of IS-CSP and NIS-CSP, two new compound symmetric cross-spring pivots are presented and analyzed via analytical and FEA models.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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