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.

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.

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.

Closed-Form Elliptic Integral Solution of Initially-Straight and Initially-Curved Small-Length Flexural Pivots

2014 ◽  
Author(s):  
Ashok Midha ◽  
Raghvendra Kuber
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Small Length ◽  
Body Model ◽  
Rigid Body Model ◽  
Elliptic Integral Method ◽  
Integral Solutions ◽  
Made In

Compliant mechanisms gain some or all of their mobility from the deflection of their flexible members. The pseudo-rigid-body model (PRBM) concept allows compliant mechanisms to be modeled using existing knowledge of rigid-body mechanisms, thereby, simplifying the design process. A pseudo-rigid-body model represents a compliant segment with two or more rigid-body segments, connected by pin joints or characteristic pivots. A compliant segment that is small in length, compared to the relatively rigid segments between which it is affixed, is termed a small-length flexural pivot (SLFP). This paper presents closed-form deflection solutions using the elliptic integral method for initially-straight and initially-curved SLFPs. The assumptions made in modeling the small-length flexural pivots in a PRBM are validated by means of the elliptic integral solutions.

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.

Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms

1994 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Spring Stiffness ◽  
Model Concept ◽  
Analysis And Design ◽  
Body Model ◽  
Rigid Body Model ◽  
Body Joints

Abstract Compliant mechanisms gain some or all of their mobility from the flexibility of their members rather than from rigid-body joints only. More efficient and usable analysis and design techniques are needed before the advantages of compliant mechanisms can be fully utilized. In an earlier work, a pseudo-rigid-body model concept, corresponding to an end-loaded geometrically nonlinear, large-deflection beam, was developed to help fulfill this need. In this paper, the pseudo-rigid-body equivalent spring stiffness is investigated and new modeling equations are proposed. The result is a simplified method of modeling the force/deflection relationships of large-deflection members in compliant mechanisms. Flexible segments which maintain a constant end angle are discussed, and an example mechanism is analyzed. The resulting models are valuable in the visualization of the motion of large-deflection systems, as well as the quick and efficient evaluation and optimization of compliant mechanism designs.

Parametric Deflection Approximations for Initially Curved, Large-Deflection Beams in Compliant Mechanisms

1996 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Curved Beam ◽  
Analysis And Design ◽  
Body Model ◽  
Rigid Body Model ◽  
Fundamental Type ◽  
Applied Forces ◽  
Flexible Member

Abstract The analysis of systems containing highly flexible members is made difficult by the nonlineararities caused by large deflections of the flexible members. The analysis and design of many such systems may be simplified by using pseudo-rigid-body approximations in modeling the flexible members. The pseudo-rigid-body model represents flexible members as rigid links, joined at pin joints with torsional springs. Appropriate values for link lengths and torsional spring stiffnesses are determined such that the deflection path and force-deflection relationships are modeled accurately. Pseudo-rigid-body approximations have been developed for initially straight beams with externally applied forces at the beam end. This work develops approximations for another fundamental type of flexible member, the initially curved beam with applied force at the beam end. This type of flexible member is commonly used in compliant mechanisms. An example of the use of the resulting pseudo-rigid-body approximations in compliant mechanisms is included.

Nonlinear Mechanics of Interlocking Cantilevers

2017 ◽  
Vol 84 (12) ◽  
Author(s):  
Joseph J. Brown ◽  
Ryan C. Mettler ◽  
Omkar D. Supekar ◽  
Victor M. Bright
Keyword(s):  
Large Deflection ◽  
Elliptic Integral ◽  
Contact Point ◽  
Reaction Force ◽  
Insertion Force ◽  
Element Analysis ◽  
Fast Analysis ◽  
Analysis And Design ◽  
Physical Measurements ◽  
Compliant Systems

The use of large-deflection springs, tabs, and other compliant systems to provide integral attachment, joining, and retention is well established and may be found throughout nature and the designed world. Such systems present a challenge for mechanical analysis due to the interaction of contact mechanics with large-deflection analysis. Interlocking structures experience a variable reaction force that depends on the cantilever angle at the contact point. This paper develops the mathematical analysis of interlocking cantilevers and provides verification with finite element analysis and physical measurements. Motivated by new opportunities for nanoscale compliant systems based on ultrathin films and two-dimensional (2D) materials, we created a nondimensional analysis of retention tab systems. This analysis uses iterative and elliptic integral solutions to the moment–curvature elastica of a suspended cantilever and can be scaled to large-deflection cantilevers of any size for which continuum mechanics applies. We find that when a compliant structure is bent backward during loading, overlap increases with load, until a force maximum is reached. In a force-limited scenario, surpassing this maximum would result in snap-through motion. By using angled cantilever restraint systems, the magnitude of insertion force relative to retention force can vary by 50× or more. The mathematical theory developed in this paper provides a basis for fast analysis and design of compliant retention systems, and expands the application of elliptic integrals for nonlinear problems.

Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms

1996 ◽  
Vol 118 (1) ◽  
pp. 126-131 ◽  
Author(s):  
L. L. Howell ◽  
A. Midha ◽  
T. W. Norton
Keyword(s):  
Rigid Body ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Spring Stiffness ◽  
Model Concept ◽  
Analysis And Design ◽  
Body Model ◽  
Rigid Body Model ◽  
Body Joints

Compliant mechanisms gain some or all of their mobility from the flexibility of their members rather than from rigid-body joints only. More efficient and usable analysis and design techniques are needed before the advantages of compliant mechanisms can be fully utilized. In an earlier work, a pseudo-rigid-body model concept, corresponding to an end-loaded geometrically nonlinear, large-deflection beam, was developed to help fulfill this need. In this paper, the pseudo-rigid-body equivalent spring stiffness is investigated and new modeling equations are proposed. The result is a simplified method of modeling the force/deflection relationships of large-deflection members in compliant mechanisms. The resulting models are valuable in the visualization of the motion of large-deflection systems, as well as the quick and efficient evaluation and optimization of compliant mechanism designs.

A Method for a More Accurate Calculation of the Stiffness Coefficient in a Pseudo-Rigid-Body Model (PRBM) of a Fixed-Free Beam Subjected to End Forces

2014 ◽  
Author(s):  
Ashok Midha ◽  
Raghvendra S. Kuber ◽  
Vivekananda Chinta ◽  
Sushrut G. Bapat
Keyword(s):  
Rigid Body ◽  
Inflection Point ◽  
Compliant Mechanisms ◽  
Load Factor ◽  
Stiffness Coefficient ◽  
Body Angle ◽  
Analysis And Design ◽  
Body Model ◽  
Body Parameter ◽  
Rigid Body Model

The pseudo-rigid-body model (PRBM) concept allows compliant mechanisms to be modeled using existing knowledge of rigid-body mechanisms, thereby considerably simplifying their analysis and design. The PRBMs represent the compliant segments with two or more rigid-body segments, connected using pin joints (characteristic pivots). The beam compliance is modeled using a torsional spring placed at the characteristic pivot, whose spring constant K is evaluated using a pseudo-rigid-body parameter termed as the beam stiffness coefficient. This paper presents a method to more accurately calculate the beam stiffness coefficient for a fixed-free compliant beam subjected to a combination of horizontal and vertical forces. The improved stiffness coefficient (KΘ) expressions are derived as a function of the pseudo-rigid-body angle, Θ and the load factor, n. To exemplify the application of the improved results, the expressions derived are successfully implemented in modeling a fixed-guided beam with an inflection point, allowing it to be modeled as two fixed-free beams pinned at the inflection point.

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