A Methodology for Compliant Mechanisms Design: Part II — Shooting Method and Application

1992 ◽  
Author(s):  
I. Her ◽  
A. Midha ◽  
B. A. Salamon
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Calculation Method ◽  
Shooting Method ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Shooting Methods ◽  
Beam Structures ◽  
Major Task ◽  
A Chain

Abstract A major task envisioned in the design of compliant mechanisms entails analyzing a large-deflection elastica for its deformed configuration, while being subjected to given displacement and force boundary conditions. The accuracy and convergence behavior of a chain calculation method for predicting nonlinear deflections in beam structures is examined. Three shooting methods are implemented to reduce boundary condition closure errors. These methods are evaluated by considering two example problems in compliant mechanisms.

A Methodology for Compliant Mechanisms Design: Part I — Introduction and Large-Deflection Analysis

1992 ◽  
Author(s):  
A. Midha ◽  
I. Her ◽  
B. A. Salamon
Keyword(s):  
Large Deflection ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Companion Paper ◽  
Computational Techniques ◽  
Shooting Methods ◽  
Mechanism Synthesis ◽  
Calculation Algorithm ◽  
Deflection Analysis ◽  
Kinematic Properties

Abstract A broader research proposal seeks to systematically combine large-deflection mechanics of flexible elements with important kinematic considerations, in yielding compliant mechanisms which perform useful tasks. Specifically, the proposed design methodology will address the following needs: development of the necessary nomenclature, classification and definitions, and identification of the kinematic properties; categorization of mechanism synthesis types, both structurally as well as by function; development of efficient computational techniques for design; consideration of materials; and application and validation. Contained herein, in particular, is an introduction to the state-of-the-art in compliant mechanisms, and the development of an accurate chain calculation algorithm for use in the analysis of a large-deflection, cantilevered elastica. Shooting methods, which permit specification of additional boundary conditions on the elastica, as well as compliant mechanism examples are presented in a companion paper.

scholarly journals A Pseudo-Rigid-Body Model for Large Deflections of Fixed-Clamped Carbon Nanotubes

2010 ◽  
Vol 2 (3) ◽  
Author(s):  
Larry L. Howell ◽  
Christopher M. DiBiasio ◽  
Michael A. Cullinan ◽  
Robert M. Panas ◽  
Martin L. Culpepper
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Rigid Body ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Molecular Simulations ◽  
Design Parameters ◽  
Body Model ◽  
Initial Design ◽  
Rigid Body Model

Carbon nanotubes (CNTs) may be used to create nanoscale compliant mechanisms that possess large ranges of motion relative to their device size. Many macroscale compliant mechanisms contain compliant elements that are subjected to fixed-clamped boundary conditions, indicating that they may be of value in nanoscale design. The combination of boundary conditions and large strains yield deformations at the tube ends and strain stiffening along the length of the tube, which are not observed in macroscale analogs. The large-deflection behavior of a fixed-clamped CNT is not well-predicted by macroscale large-deflection beam bending models or truss models. Herein, we show that a pseudo-rigid-body model may be adapted to capture the strain stiffening behavior and, thereby, predict a CNT’s fixed-clamped behavior with less than 3% error from molecular simulations. The resulting pseudo-rigid-body model may be used to set initial design parameters for CNT-based compliant mechanisms. This removes the need for iterative, time-intensive molecular simulations during initial design phases.

Large Deflection of Various Functionally Graded Beam Using Shooting Method

2011 ◽  
Vol 110-116 ◽  
pp. 4705-4711 ◽  
Author(s):  
Ali Soleimani
Keyword(s):  
Exact Solution ◽  
Elastic Modulus ◽  
Loading Condition ◽  
Shooting Method ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Longitudinal Direction ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Arbitrary Loading

The equation of large deflection of functionally graded beam subjected to arbitrary loading condition is derived. In this work assumed that the elastic modulus varies by exponential and power function in longitudinal direction. The nonlinear derived equation has not exact solution so shooting method has been proposed to solve the nonlinear equation of large deflection. Results are validated with finite element solutions. The method will be useful toward the design of compliant mechanisms driven by smart actuators. Finally the effect of different elastic modulus functions and loading conditions are investigated and discussed.

Numerical Study of Large Defection of Functionally Graded Beam with Geometry Nonlinearity

2011 ◽  
Vol 403-408 ◽  
pp. 4226-4230
Author(s):  
A. Soleimani ◽  
M. Saadatfar
Keyword(s):  
Elastic Modulus ◽  
Shooting Method ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Numerical Study ◽  
Longitudinal Direction ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Arbitrary Loading ◽  
Geometry Nonlinearity

The equation of large deflection of functionally graded beam subjected to arbitrary loading condition is derived. In this work assumed that the elastic modulus varies by power function in longitudinal direction. The nonlinear derived equation has not exact solution so shooting method has been proposed to solve the nonlinear equation of large deflection. Results of shooting method are validated with finite element solutions. The method will be useful toward the design of compliant mechanisms driven by smart actuators. Finally the effect of different elastic modulus functions and loading conditions are investigated and discussed.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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