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.

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.

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.

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.

Direct Position Analysis of the 4–6 Stewart Platforms

1994 ◽  
Vol 116 (1) ◽  
pp. 61-66 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Although Stewart platforms have been applied in the design of aircraft and vehicle simulators and parallel robots for many years, the closed-form solution of direct (forward) position analysis of Stewart platforms has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral, the 3–6 and the 4–4 Stewart platforms, of which the forward position solutions were derived as an eighth or a twelfth degree polynomials with one variable in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more general case of the Stewart platform, the 4–6 Stewart platforms, in which two pairs of the upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of the 3–6 and 4–4 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

scholarly journals External Kinematic Calibration of Hybrid Kinematics Machine Utilizing Lower-DOF Planar Parallel Kinematics Mechanisms

2019 ◽  
Vol 21 (6) ◽  
pp. 995-1015 ◽  
Author(s):  
Abdur Rosyid ◽  
Bashar El-Khasawneh ◽  
Anas Alazzam
Keyword(s):  
Least Squares ◽  
Closed Form ◽  
Closed Form Solution ◽  
Nonlinear Least Squares ◽  
Form Solution ◽  
Geometric Parameters ◽  
Kinematic Calibration ◽  
Parallel Kinematics ◽  
Linear Least Squares ◽  
Hybrid Kinematics

AbstractThis paper presents the implementation of nonlinear least squares and iterative linear least squares algorithms for external kinematic calibration of a hybrid kinematics machine composed of two 3PRR planar parallel kinematics mechanisms by utilizing a laser tracker. First the hand-eye and robot-world transformations were obtained by a separable closed-form solution and refined by the nonlinear least squares. Subsequently, the geometric parameters of the machine’s mechanisms were estimated using the two algorithms. Due to the rank deficiency, we implemented the nonlinear least squares algorithm through a subset selection approach in which we performed the estimation in two steps. We iterated the closed-form solution of the linear least squares until the solution converges to the actual values. We have shown that the nonlinear least squares algorithm successfully refined the hand-eye and robot-world transformations and outperformed the iterative linear squares algorithm in the estimation of the geometric parameters of the mechanisms.

Direct Position Analysis of the 4-6 Stewart Platforms

1992 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Unknown Variable ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Abstract Although Stewart platforms have been applied in the designs of aricraft and vehicle simulators and parallel robots in many years, their closed-form solution of direct (forward) position analysis has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral Stewart and the 4-4 Stewart platforms, in which two pairs of both upper and lower joint centers are coincident. The former results in in an eighth degree polynomial and the latter results in an eighth and a twelfth degree polynomials for different cases. The single unknown variable is in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more genearl case of the Stewart platform, the 4-6 Stewart platforms, in which two pairs of upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of 4-4 and and 3-6 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

scholarly journals Closed-form solution for column buckling optimization

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Dimensional Analysis ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Elliptic Functions ◽  
Isoperimetric Inequalities ◽  
Form Solution ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Axial Forces

Abstract An optimization problem for a column, loaded by axial forces, whose direction and value remain constant, is studied in this article. The dimensional analysis introduces the dimensionless mass and rigidity factors, which simplicities the mathematical technique for the optimization problem. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental function. The principal results are the closed form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The proof of isoperimetric inequalities exploits the variational method and the Hölder inequality. The isoperimetric inequalities for Euler’s column are rigorously verified.

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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