An Elliptic Integral Solution to the Multiple Inflections Large Deflection Beams in Compliant Mechanisms

2014 ◽  
Vol 971-973 ◽  
pp. 349-352 ◽  
Author(s):  
Jiang Yong Song
Keyword(s):  
Euler Equations ◽  
Inflection Point ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Accurate Method ◽  
Integral Solution ◽  
Large Deflections ◽  
Cantilever Beams ◽  
Inflection Points

In this paper, a solution based on the elliptic integrals is proposed for solving multiples inflection points large deflection. Application of the Bernoulli Euler equations of compliant mechanisms with large deflection equation of beam is obtained ,there is no inflection point and inflection points in two cases respectively. The elliptic integral solution which is the most accurate method at present for analyzing large deflections of cantilever beams in compliant mechanisms.

A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Cantilever Beams in Compliant Mechanisms

2012 ◽  
Author(s):  
Aimei Zhang ◽  
Guimin Chen
Keyword(s):  
Large Deflection ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Accurate Method ◽  
Integral Solution ◽  
Large Deflections ◽  
Cantilever Beams ◽  
Complex Modes ◽  
Inflection Points ◽  
Moment Load

The elliptic integral solution is often considered as the most accurate method for analyzing large deflections of cantilever beams in compliant mechanisms. In this paper, by explicitly including the number of inflection points (m) and the sign of the end-moment load (SM) in the derivation, a comprehensive solution based on the elliptic integrals is proposed for solving the large deflection problems. The comprehensive solution is capable of solving large deflections of cantilever beams subject to any kind of load cases and of any kind of deflected modes. A few deflected configurations of complex modes solved by the comprehensive solution are presented and discussed. The use of the comprehensive solution in analyzing compliant mechanisms is also demonstrated by a few case studies.

A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms

2013 ◽  
Vol 5 (2) ◽  
Author(s):  
Aimei Zhang ◽  
Guimin Chen
Keyword(s):  
Large Deflection ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Accurate Method ◽  
Integral Solution ◽  
Large Deflections ◽  
Complex Modes ◽  
Inflection Points ◽  
Moment Load ◽  
Thin Beams

The elliptic integral solution is often considered to be the most accurate method for analyzing large deflections of thin beams in compliant mechanisms. In this paper, a comprehensive solution based on the elliptic integrals is proposed for solving large deflection problems. By explicitly incorporating the number of inflection points and the sign of the end-moment load in the derivation, the comprehensive solution is capable of solving large deflections of thin beams with multiple inflection points and subject to any kinds of load cases. The comprehensive solution also extends the elliptic integral solutions to be suitable for any beam end angle. Deflected configurations of complex modes solved by the comprehensive solution are presented and discussed. The use of the comprehensive solution in analyzing compliant mechanisms is also demonstrated by examples.

Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms

1995 ◽  
Vol 117 (1) ◽  
pp. 156-165 ◽  
Author(s):  
L. L. Howell ◽  
A. Midha
Keyword(s):  
Closed Form ◽  
Large Deflection ◽  
Numerical Solutions ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Cantilever Beams ◽  
Simple Method ◽  
Body Angle ◽  
Analysis And Design ◽  
Integral Solutions

Geometric nonlinearities often complicate the analysis of systems containing large-deflection members. The time and resources required to develop closed-form or numerical solutions have inspired the development of a simple method of approximating the deflection path of end-loaded, large-deflection cantilever beams. The path coordinates are parameterized in a single parameter called the pseudo-rigid-body angle. The approximations are accurate to within 0.5 percent of the closed-form elliptic integral solutions. A physical model is associated with the method, and may be used to simplify complex problems. The method proves to be particularly useful in the analysis and design of compliant mechanisms.

Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms

1992 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Closed Form ◽  
Large Deflection ◽  
Numerical Solutions ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Cantilever Beams ◽  
Simple Method ◽  
Body Angle ◽  
Analysis And Design ◽  
Integral Solutions

Abstract Geometric nonlinearities often complicate the analysis of systems containing large-deflection members. The time and resources required to develop closed-form or numerical solutions nave inspired the development of a simple method of approximating the deflection path of end-loaded, large-deflection cantilever beams. The path coordinates are parameterized in a single parameter, called the pseudo-rigid-body angle. The approximations are accurate to within 0.5% of the closed-form elliptic integral solutions. A physical model is associated with the method, and may be used to simplify complex problems. The method proves to be particularly useful in the analysis and design of compliant mechanisms.

A Simple and Accurate Method for Determining Large Deflections in Compliant Mechanisms Subjected to End Forces and Moments

1998 ◽  
Vol 120 (3) ◽  
pp. 392-400 ◽  
Author(s):  
A. Saxena ◽  
S. N. Kramer
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Elliptic Integral ◽  
Beam Equation ◽  
Large Deflections ◽  
Euler Beam ◽  
Body Model ◽  
Rigid Body Model ◽  
Analysis And Synthesis

Compliant members in flexible link mechanisms undergo large deflections when subjected to external loads. Because of this fact, traditional methods of deflection analysis do not apply. Since the nonlinearities introduced by these large deflections make the system comprising such members difficult to solve, parametric deflection approximations are deemed helpful in the analysis and synthesis of compliant mechanisms. This is accomplished by representing the compliant mechanism as a pseudo-rigid-body model. A wealth of analysis and synthesis techniques available for rigid-body mechanisms thus become amenable to the design of compliant mechanisms. In this paper, a pseudo-rigid-body model is developed and solved for the tip deflection of flexible beams for combined end loads. A numerical integration technique using quadrature formulae has been employed to solve the large deflection Bernoulli-Euler beam equation for the tip deflection. Implementation of this scheme is simpler than the elliptic integral formulation and provides very accurate results. An example for the synthesis of a compliant mechanism using the proposed model is also presented.

scholarly journals A Seminalytical Approach to Large Deflections in Compliant Beams under Point Load

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
N. Tolou ◽  
J. L. Herder
Keyword(s):  
Cantilever Beam ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Geometrical Nonlinearity ◽  
Adomian Decomposition Method ◽  
Analytical Form ◽  
Point Load ◽  
Large Deflections ◽  
Adomian Decomposition

The deflection of compliant mechanism (CM) which involves geometrical nonlinearity due to large deflection of members continues to be an interesting problem in mechanical systems. This paper deals with an analytical investigation of large deflections in compliant mechanisms. The main objective is to propose a convenient method of solution for the large deflection problem in CMs in order to overcome the difficulty and inaccuracy of conventional methods, as well as for the purpose of mathematical modeling and optimization. For simplicity, an element is considered which is a cantilever beam out of linear elastic material under vertical end point load. This can further be used as a building block in more complex compliant mechanisms. First, the governing equation has been obtained for the cantilever beam; subsequently, the Adomian decomposition method (ADM) has been utilized to obtain a semianalytical solution. The vertical and horizontal displacements of a cantilever beam can conveniently be obtained in an explicit analytical form. In addition, variations of the parameters that affect the characteristics of the deflection have been examined. The results reveal that the proposed procedure is very accurate, efficient, and convenient for cantilever beams, and can probably be applied to a large class of practical problems for the purpose of analysis and optimization.

Chained Beam-Constraint-Model (CBCM): A Powerful Tool for Modeling Large and Complicated Deflections of Flexible Beams in Compliant Mechanisms

2014 ◽  
Author(s):  
Fulei Ma ◽  
Guimin Chen
Keyword(s):  
Large Deflection ◽  
Compliant Mechanisms ◽  
Research Community ◽  
The Other ◽  
New Method ◽  
Flexible Beam ◽  
Large Deflections ◽  
Flexible Beams ◽  
Constraint Model

Modeling large deflections has been one of the most fundamental problems in the research community of compliant mechanisms. Although many methods are available, there still exists a need for a method that is simple, accurate, and can be applied to a vast variety of large deflection problems. Based on the beam constraint model (BCM), we propose a new method for modeling large deflections called chained BCM (CBCM), which divides a flexible beam into a few elements and models each element by BCM. It is demonstrated that CBCM is capable of modeling various large and complicated deflections of flexible beams in compliant mechanisms. In general, CBCM obtains accurate results with no more than 6 BCM elements, thus is more efficient than most of the other discretization-based methods.

Modeling of Flexural Beams Subjected to Arbitrary End Loads

2002 ◽  
Vol 124 (2) ◽  
pp. 223-235 ◽  
Author(s):  
Chris Kimball ◽  
Lung-Wen Tsai
Keyword(s):  
Rigid Body ◽  
Inflection Point ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
System Optimization ◽  
Beam Deflection ◽  
Beam Equation ◽  
Euler Beam ◽  
Single Degree Of Freedom ◽  
Rigid Body Model

The analysis of compliant mechanisms is often complicated due to the geometric nonlinearities which become significant with large elastic deflections. Pseudo rigid body models (PRBM) may be used to accurately and efficiently model such large elastic deflections. Previously published models have only considered end forces with no end moment or end moment acting only in the same direction as the force. In this paper, we present a model for a cantilever beam with end moment acting in the opposite direction as the end force, which may or may not cause an inflection point. Two pivot points are used, thereby increasing the model’s accuracy when an inflection point exists. The Bernoulli-Euler beam equation is solved for large deflections with elliptic integrals, and the elliptic integral solutions are used to determine when an inflection point will exist. The beam tip deflections are then parameterized using a different parameterization from previous models, which renders the deflection paths easier to model with a single degree of freedom system. Optimization is used to find the pseudo rigid body model which best approximates the beam deflection and stiffness. This model, combined with those models developed for other loading conditions, may be used to efficiently analyze compliant mechansims subjected to any loading condition.

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