scholarly journals An Energy-Based Framework for Nonlinear Kinetostatic Modeling of Compliant Mechanisms Utilizing Beam Flexures

2021 ◽  
pp. 1-18
Author(s):  
Guimin Chen ◽  
Fulei Ma ◽  
Ruiyu Bai ◽  
Weidong Zhu ◽  
Spencer P Magleby ◽  
...  
Keyword(s):  
Strain Energy ◽  
Compliant Mechanisms ◽  
Modeling Framework ◽  
Geometric Nonlinearities ◽  
Complementary Strain ◽  
Constraint Model

Abstract Although energy-based methods have advantages over the Newtonian methods for kinetostatic modeling, the geometric nonlinearities inherent in deflections of compliant mechanisms preclude most of the energy-based theorems. Castigliano's first theorem and the Crotti-Engesser theorem, which don't require the problem being solved to be linear, are selected to construct the energy-based kinetostatic modeling framework for compliant mechanisms in this work. Utilization of these two theorems requires explicitly formulating the strain energy in terms of deflections and the complementary strain energy in terms of loads, which are derived based on the beam constraint model. The kinetostatic modeling of two compliant mechanisms are provided to demonstrate the effectiveness of the explicit formulations in this framework derived from Castigliano's first theorem and the Crotti-Engesser theorem.

A Framework for Energy-Based Kinetostatic Modeling of Compliant Mechanisms

2017 ◽  
Author(s):  
Guimin Chen ◽  
Fulei Ma ◽  
Ruiyu Bai ◽  
Spencer P. Magleby ◽  
Larry L. Howell
Keyword(s):  
Strain Energy ◽  
Compliant Mechanisms ◽  
Modeling Framework ◽  
Geometric Nonlinearities ◽  
Complementary Strain ◽  
Minimum Strain Energy ◽  
Constraint Model ◽  
Future Work

Although energy-based methods have advantages over the Newtonian methods for kinetostatic modeling, the geometric nonlinearities inherent in deflections of compliant mechanisms preclude most of the energy-based theorems. Castigliano’s first theorem and the Crotti-Engesser theorem, which don’t require the problem being solved to be linear, are selected to construct the energy-based kinetostatic modeling framework for compliant mechanisms in this work. Utilization of these two theorems requires explicitly formulating the strain energy in terms of deflections and the complementary strain energy in terms of loads, which are derived based on the beam constraint model. The kinetostatic modeling of two compliant mechanisms are provided to demonstrate the effectiveness of using Castigliano’s first theorem and the Crotti-Engesser theorem with the explicit formulations in this framework. Future work will be focused on incorporating use of the principle of minimum strain energy and the principle of minimum complementary strain energy.

Modeling Large Planar Deflections of Flexible Beams in Compliant Mechanisms Using Chained Beam-Constraint-Model1

2015 ◽  
Vol 8 (2) ◽  
Author(s):  
Fulei Ma ◽  
Guimin Chen
Keyword(s):  
Strain Energy ◽  
Large Deflection ◽  
Compliant Mechanisms ◽  
Research Community ◽  
New Method ◽  
Flexible Beam ◽  
Large Deflections ◽  
Flexible Beams ◽  
Versatile Tool ◽  
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. The approaches for determining the strain energy stored in a deflected beam and the stress distributed on it are also presented within the framework of CBCM. Several typical examples were analyzed and the results show CBCMs capabilities of modeling various large deflections of flexible beams in compliant mechanisms. Generally, CBCM can serve as an efficient and versatile tool for solving large deflection problems in a variety of compliant mechanisms.

Structural Topology Optimization Using Frame Elements Based on the Complementary Strain Energy Concept for Eigen-Frequency Maximization

2005 ◽  
Author(s):  
Akihiro Takezawa ◽  
Shinji Nishiwaki ◽  
Kazuhiro Izui ◽  
Masataka Yoshimura
Keyword(s):  
Topology Optimization ◽  
Cross Sections ◽  
Strain Energy ◽  
Optimization Procedure ◽  
Optimization Method ◽  
Structural Topology ◽  
Energy Concept ◽  
Conceptual Design Phase ◽  
Complementary Strain ◽  
Topology Optimization Method

This paper discuses a new topology optimization method using frame elements for the design of mechanical structures at the conceptual design phase. The optimal configurations are determined by maximizing multiple eigen-frequencies in order to obtain the most stable structures for dynamic problems. The optimization problem is formulated using frame elements having ellipsoidal cross-sections, as the simplest case. Construction of the optimization procedure is based on CONLIN and the complementary strain energy concept. Finally, several examples are presented to confirm that the proposed method is useful for the topology optimization method discussed here.

The Behaviour of Multiple Loops in Torsion

1988 ◽  
Vol 15 (3) ◽  
pp. 149-156 ◽  
Author(s):  
R. A. Cavina ◽  
N. E. Waters
Keyword(s):  
Energy Method ◽  
Strain Energy ◽  
Vertical Axis ◽  
Experimental Measurements ◽  
Complementary Strain ◽  
Radial Axis ◽  
Strain Energy Method ◽  
Good Agreement

The angular stiffness of a multiple looped span, subject to rotation about a vertical axis (torsion) and also to rotation about a horizontal or radial axis (mesio-distal tilt), have been derived using the complementary (strain) energy method. Experimental measurements on enlarged models were in good agreement with the values calculated from the theoretical relationships obtained. The variations in angular stiffness resulting from changes in the loop height, width, and position of clinical sized loops are discussed.

Modeling Large Spatial Deflections of Slender Bisymmetric Beams in Compliant Mechanisms Using Chained Spatial-Beam Constraint Model

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Guimin Chen ◽  
Ruiyu Bai
Keyword(s):  
Finite Element Analysis ◽  
Compliant Mechanisms ◽  
Research Community ◽  
Nonlinear Finite Element ◽  
Nonlinear Constraint ◽  
Element Analysis ◽  
Flexible Beams ◽  
Spatial Beams ◽  
Spatial Beam ◽  
Constraint Model

Modeling large spatial deflections of flexible beams has been one of the most challenging problems in the research community of compliant mechanisms. This work presents a method called chained spatial-beam constraint model (CSBCM) for modeling large spatial deflections of flexible bisymmetric beams in compliant mechanisms. CSBCM is based on the spatial-beam constraint model (SBCM), which was developed for the purpose of accurately predicting the nonlinear constraint characteristics of bisymmetric spatial beams in their intermediate deflection range. CSBCM deals with large spatial deflections by dividing a spatial beam into several elements, modeling each element with SBCM, and then assembling the deflected elements using the transformation defined by Tait–Bryan angles to form the whole deflection. It is demonstrated that CSBCM is capable of solving various large spatial deflection problems either the tip loads are known or the tip deflections are known. The examples show that CSBCM can accurately predict large spatial deflections of flexible beams, as compared to the available nonlinear finite element analysis (FEA) results obtained by ansys. The results also demonstrated the unique capabilities of CSBCM to solve large spatial deflection problems that are outside the range of ansys.

scholarly journals Reliability Based Analysis and Optimum Design of Laterally Loaded Piles

2017 ◽  
Author(s):  
Majid Movahedi Rad
Keyword(s):  
Limit Analysis ◽  
Strain Energy ◽  
Load Capacity ◽  
Optimization Procedure ◽  
Load Parameter ◽  
Plastic Limit ◽  
Laterally Loaded Piles ◽  
Plastic Limit Analysis ◽  
Complementary Strain ◽  
Laterally Loaded

In this study reliability based limit analysis is used to determine the ultimate capacity of laterally loaded piles.  The aim of this study is to evaluate the lateral load capacity of free-head and fixed-head long pile when plastic limit analysis is considered. In addition to the plastic limit analysis to control the plastic behaviour of the structure, uncertain bound on the complementary strain energy of the residual forces is also applied. This bound has significant effect for the load parameter. The solution to reliability-based problems is based on a direct integration technique and the uncertainties are assumed to follow Gaussian distribution. The optimization procedure is governed by the reliability index calculation.

scholarly journals Extension of Castigliano’s method for isotropic beams

2020 ◽  
Vol 231 (11) ◽  
pp. 4621-4640
Author(s):  
Juergen Schoeftner
Keyword(s):  
Shear Force ◽  
Strain Energy ◽  
Axial Stress ◽  
Bending Moment ◽  
Axial Direction ◽  
Present Contribution ◽  
Special Cases ◽  
Dummy Load ◽  
Complementary Strain ◽  
Castigliano’S Theorem

Abstract In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

scholarly journals Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation

2020 ◽  
Author(s):  
Arne Claus Hansen-Dörr ◽  
Jörg Brummund ◽  
Markus Kästner
Keyword(s):  
Crack Propagation ◽  
Phase Field ◽  
Strain Energy ◽  
Heterogeneous Media ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Jump Conditions ◽  
Modeling Framework ◽  
Material Interfaces ◽  
Mechanical Jump Conditions

Abstract In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the regularized crack. The key novelty is the combination of a strain energy split with a partial rank-I relaxation in the vicinity of the diffuse interface. The former is necessary to account for physically meaningful crack kinematics like crack closure, the latter ensures the mechanical jump conditions throughout the diffuse region. The model is verified by a convergence study, where a circular bi-material disc with and without a crack is subjected to radial loads. For the uncracked case, analytical solutions are taken as reference. In a second step, the model is applied to crack propagation, where a meaningful influence on crack branching is observed, that underlines the necessity of a reasonable homogenization scheme. The presented model is particularly relevant for the combination of any variational strain energy split in the fracture phase-field model with a diffuse modeling approach for material heterogeneities.

Multistable Behavior of Compliant Kaleidocycles

2015 ◽  
Author(s):  
Thomas A. Evans ◽  
Brett G. Rowberry ◽  
Spencer P. Magleby ◽  
Larry L. Howell
Keyword(s):  
Strain Energy ◽  
Stable Equilibrium ◽  
Compliant Mechanisms ◽  
Neutral Stability ◽  
Energy Characteristics ◽  
Continuous Rotation

We present an analysis of the compliant kaleidocycle, a mechanism which, unlike other compliant mechanisms, may undergo continuous rotation. We analyze the strain energy characteristics of this mechanism during its motion and show that by varying the stiffness and orientation of the flexures, kaleidocycles may be designed to achieve customizable multistable behavior. These devices may be designed to include up to four distinct stable equilibrium positions and may also include regions of neutral stability.

A Study of Conditions That Affect the Distribution of Compliance in Compliant Mechanisms

2008 ◽  
Author(s):  
Stephen L. Canfield ◽  
Daniel L. Chlarson ◽  
Alexander Shibakov ◽  
Joseph D. Richardson ◽  
Anupam Saxena
Keyword(s):  
Strain Energy ◽  
Compliant Mechanisms ◽  
Optimization Process ◽  
Objective Functions ◽  
Compatibility Analysis ◽  
Pareto Sets ◽  
Show Evidence ◽  
Final Design ◽  
Additional Constraints ◽  
Insight Into

Researchers in the field of optimal synthesis of compliant mechanisms have been working to develop design tools that yield distributed compliant devices from a continuum design domain. However, it has been demonstrated in the literature that much of this work has resulted in mechanisms that localize compliance rather than distribute it as desired. Inaccurate representation of the stiffness or strain energy due to the existence of point flexures in the mechanism was identified as the cause of this behavior by early researchers. To eliminate this cause, several approaches have been tried to improve the design of distributed mechanisms, for example additional constraints on the optimization process, alternate parameterization techniques that avoid point flexures and additional objective functions evaluated as Pareto sets. In this paper, the authors further investigate the fundamental reasons for the prevalence of lumped designs. Representative simple compliant mechanisms are investigated analytically and numerically and the influence of various additional objectives on the final design is evaluated. To extrapolate these results to more complex mechanisms, examples are constructed that show evidence that a preference remains for lumped compliance, despite the countermeasures that have been applied. Pareto compatibility analysis developed by the authors is used to analyze the influence of various objectives on the distributive nature of the final design. These conditions that influence the distribution of compliance fall into two basic categories: those specific to the numerical methods applied and those of purely mechanical (i.e. fundamental) nature. This work will examine conditions of the latter type and will demonstrate that such a preference for lumped compliance exists. This preference is shown to be contained in the classic objectives; flexibility and stiffness. Based on these results, greater insight into the optimization process is gained and applied to improve the search for distributed compliant mechanisms.

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