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

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.

Download Full-text

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

Volume 5A: 40th Mechanisms and Robotics Conference ◽

10.1115/detc2016-59063 ◽

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.

Download Full-text

Scattering by hard and soft parallel half-planes

Canadian Journal of Physics ◽

10.1139/p81-248 ◽

1981 ◽

Vol 59 (12) ◽

pp. 1879-1885 ◽

Cited By ~ 14

Author(s):

R. A. Hurd ◽

E. Lüneburg

Keyword(s):

Plane Wave ◽

Closed Form ◽

Half Plane ◽

Diffraction Problem ◽

Closed Form Solution ◽

Normal Derivative ◽

Form Solution ◽

The Other ◽

Total Field

We consider the diffraction of a scalar plane wave by two parallel half-planes. On one half-plane the total field vanishes whilst on the other its normal derivative is zero. This is a new canonical diffraction problem and we give an exact closed-form solution to it. The problem has applications to the design of acoustic barriers.

Download Full-text

DIVISIBLE LOAD DISTRIBUTION IN A NETWORK OF PROCESSORS

Journal of Interconnection Networks ◽

10.1142/s021926590800214x ◽

2008 ◽

Vol 09 (01n02) ◽

pp. 31-51 ◽

Cited By ~ 1

Author(s):

SAMEER BATAINEH

Keyword(s):

Closed Form ◽

Load Distribution ◽

Closed Form Solution ◽

Form Solution ◽

The Other ◽

Optimum Number ◽

Sequencing Problem ◽

Divisible Load ◽

Finish Time ◽

Optimum Sequence

The paper presents a closed form solution for an optimum scheduling of a divisible job on an optimum number of processor arranged in an optimum sequence in a multilevel tree networks. The solution has been derived for a single divisible job where there is no dependency among subtasks and the root processor can either perform communication and computation at the same time. The solution is carried out through three basic theorems. One of the theorems selects the optimum number of available processors that must participate in executing a divisible job. The other solves the sequencing problem in load distribution by which we are able to find the optimum sequence for load distribution in a generalized form. Having the optimum number of processors and their sequencing for load distribution, we have developed a closed form solution that determines the optimum share of each processor in the sequence such that the finish time is minimized. Any alteration of the number of processors, their sequences, or their shares that are determined by the three theorems will increase the finish time.

Download Full-text

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

10.1115/detc2021-69028 ◽

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.

Download Full-text

Spontaneous potential log response expressed as convolution

Geophysics ◽

10.1190/1.1441395 ◽

1982 ◽

Vol 47 (9) ◽

pp. 1335-1337

Author(s):

E. A. Nosal

Keyword(s):

Closed Form ◽

Signal Analysis ◽

Closed Form Solution ◽

Form Solution ◽

The Other ◽

Individual Contribution ◽

Spontaneous Potential ◽

The Earth ◽

The Individual ◽

Special Case

A special case of spontaneous potential (SP) logging, which has a closed‐form solution, will be expressed as a convolutional operation. Such a formal demonstration serves two purposes. First, it separates the individual contribution of the tool from that of the earth. Second, it places this logging device within the mathematical context of signal analysis. The special case for which a closed‐form solution is known is that where all resistivities are equal. Fourier analysis applied to this solution leads to a product of two functions, of which one is identified as the contribution of the earth and the other of the tool.

Download Full-text

An Inverse Force Analysis of a Tetrahedral Three-Spring System

Journal of Mechanical Design ◽

10.1115/1.2826136 ◽

1995 ◽

Vol 117 (2A) ◽

pp. 286-291 ◽

Cited By ~ 6

Author(s):

P. Dietmaier

Keyword(s):

Closed Form ◽

Closed Form Solution ◽

Form Solution ◽

The Other ◽

Force Analysis ◽

Equilibrium Configurations ◽

Spring System ◽

The Common ◽

The Stability ◽

Inverse Force

A tetrahedral three-spring system under a single load has been analyzed and a closed-form solution for the equilibrium positions is given. Each of the three springs is attached at one end to a fixed pivot in space while the other three ends are linked by a common pivot. The springs are assumed to behave in a linearly elastic way. The aim of the paper at hand was to find out what the maximum number of equilibrium positions of such a system might be, and how to compute all possible equilibrium configurations if a given force is applied to the common pivot. First a symmetric and unloaded system was studied. For such a system it was shown that there may exist a maximum of 22 equilibrium configurations which may all be real. Second the general, loaded system was analyzed, revealing again a maximum of 22 real equilibrium configurations. Finally, the stability of this three-spring system was investigated. A numerical example illustrates the theoretical findings.

Download Full-text

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

Journal of Mechanisms and Robotics ◽

10.1115/1.4052514 ◽

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.

Download Full-text

Diffraction by an infinite set of soft/hard parallel half-planes: the Riemann approach

Canadian Journal of Physics ◽

10.1139/p82-153 ◽

1982 ◽

Vol 60 (8) ◽

pp. 1125-1138 ◽

Cited By ~ 2

Author(s):

E. Lüneburg

Keyword(s):

Plane Wave ◽

Boundary Value Problem ◽

Closed Form ◽

Closed Form Solution ◽

Normal Derivative ◽

Form Solution ◽

The Other ◽

Bragg Angle ◽

Total Field ◽

Infinite Set

We consider the diffraction of a plane wave by an infinite set of parallel equidistant half-planes. On each plate the total field vanishes on one side and the normal derivative vanishes on the other side. A closed-form solution for Bragg angle incidence is obtained by reducing the boundary value problem to a solvable Riemann problem.

Download Full-text