A Closed-Form Nonlinear Model for the Constraint Characteristics of Symmetric Spatial Beams

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Shiladitya Sen ◽  
Shorya Awtar
Keyword(s):  
Closed Form ◽  
Virtual Work ◽  
Practical Interest ◽  
Element Analysis ◽  
Nonlinear Load ◽  
Flexure Mechanism ◽  
Spatial Beams ◽  
Spatial Beam ◽  
Load Displacement ◽  
Constraint Model

The constraint-based design of flexure mechanisms requires a qualitative and quantitative understanding of the constraint characteristics of flexure elements that serve as constraints. This paper presents the constraint characterization of a uniform and symmetric cross-section, slender, spatial beam—a basic flexure element commonly used in three-dimensional flexure mechanisms. The constraint characteristics of interest, namely stiffness and error motions, are determined from the nonlinear load–displacement relations at the beam end. Appropriate assumptions are made while formulating the strain and strain energy expressions for the spatial beam to retain relevant geometric nonlinearities. Using the principle of virtual work, nonlinear beam governing equations are derived and subsequently solved for general end loads. The resulting nonlinear load–displacement relations capture the constraint characteristics of the spatial beam in a compact, closed-form, and parametric manner. This constraint model is shown to be accurate using nonlinear finite element analysis, within a load and displacement range of practical interest. The utility of this model lies in the physical and analytical insight that it offers into the constraint behavior of a spatial beam flexure, its use in design and optimization of 3D flexure mechanism geometries, and its elucidation of fundamental performance tradeoffs in flexure mechanism design.

Nonlinear Constraint Model for Symmetric Three-Dimensional Beams

2010 ◽  
Author(s):  
Shiladitya Sen ◽  
Shorya Awtar
Keyword(s):  
Three Dimensional ◽  
Practical Interest ◽  
Nonlinear Constraint ◽  
Element Analysis ◽  
Flexure Mechanism ◽  
Spatial Beam ◽  
Non Linear ◽  
Simplifying Assumptions ◽  
Non Linear Load ◽  
Constraint Model

The constraint-based design of flexure mechanisms requires a qualitative and quantitative understanding of the constraint characteristics of flexure elements that serve as constraints. This paper presents the constraint characterization of a slender, uniform and symmetric cross-section, spatial beam, which is one of the most basic flexure elements used in three-dimensional flexure mechanisms. The constraint characteristics of interest, namely stiffness and error motions, are determined from the non-linear load-displacement relations of the beam. Appropriate simplifying assumptions are made in deriving these relations so that relevant non-linear effects (load-stiffening, kinematic, and elastokinematic) are captured in a compact, closed-form, and parametric manner. The resulting spatial beam constraint model is shown to be accurate, using non-linear finite element analysis, within a load and displacement range of practical interest. The utility of this model lies in the physical and analytical insight that it offers into the constraint behavior of a spatial beam flexure, its use in 3D flexure mechanism geometries, and fundamental performance tradeoffs in flexure mechanism design.

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.

A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Shorya Awtar ◽  
Shiladitya Sen
Keyword(s):  
Strain Energy ◽  
Virtual Work ◽  
Geometric Constraint ◽  
Element Analysis ◽  
Beam Shape ◽  
Nonlinear Strain ◽  
Energy Expression ◽  
Load Displacement ◽  
Constraint Model ◽  
Energy Formulation

The beam constraint model (BCM), presented previously, captures pertinent nonlinearities to predict the constraint characteristics of a generalized beam flexure in terms of its stiffness and error motions. In this paper, a nonlinear strain energy formulation for the beam flexure, consistent with the transverse-direction load-displacement and axial-direction geometric constraint relations in the BCM, is presented. An explicit strain energy expression, in terms of beam end displacements, that accommodates generalized loading conditions, boundary conditions, initial curvature, and beam shape, is derived. Using energy-based arguments, new insight into the BCM is elucidated by fundamental relations among its stiffness, constraint, and energy coefficients. The presence of axial load in the geometric constraint and strain energy expressions—a unique attribute of distributed compliance flexures that leads to the elastokinematic effect—is highlighted. Using the principle of virtual work, this strain energy expression for a generalized beam is employed in determining the load-displacement relations, and therefore constraint characteristics, of a flexure mechanism comprising multiple beams. The benefit of this approach is evident in its mathematical efficiency and succinctness, which is to be expected with the use of energy methods. All analytical results are validated to a high degree of accuracy via nonlinear finite element analysis.

A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Load-Displacement Formulation

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Shorya Awtar ◽  
Shiladitya Sen
Keyword(s):  
Two Dimensional ◽  
Finite Elements Analysis ◽  
Nonlinear Load ◽  
Beam Shape ◽  
Flexure Mechanism ◽  
Elastic Load ◽  
Nonlinear Finite Elements ◽  
Wide Range ◽  
Load Displacement ◽  
Constraint Model

To utilize beam flexures in constraint-based flexure mechanism design, it is important to develop qualitative and quantitative understanding of their constraint characteristics in terms of stiffness and error motions. This paper provides a highly generalized yet accurate closed-form parametric load-displacement model for two-dimensional beam flexures, taking into account the nonlinearities arising from load equilibrium applied in the deformed configuration. In particular, stiffness and error motions are parametrically quantified in terms of elastic, load-stiffening, kinematic, and elastokinematic effects. The proposed beam constraint model incorporates a wide range of loading conditions, boundary conditions, initial curvature, and beam shape. The accuracy and effectiveness of the proposed beam constraint model is verified by nonlinear finite elements analysis.

Beam Constraint Model: A Non-Linear Strain Energy Formulation for Generalized Two-Dimensional Beam Flexures

2010 ◽  
Author(s):  
Shorya Awtar ◽  
Shiladitya Sen
Keyword(s):  
Strain Energy ◽  
Virtual Work ◽  
Linear Strain ◽  
Element Analysis ◽  
Beam Shape ◽  
Energy Expression ◽  
Non Linear ◽  
Load Displacement ◽  
Constraint Model ◽  
Energy Formulation

In the past, we have introduced the Beam Constraint Model (BCM), which captures pertinent non-linearities to predict the constraint characteristics of a generalized beam flexure in terms of its stiffness and error motions. In this paper, a non-linear strain energy formulation for the beam flexure, consistent with the transverse-direction load-displacement and axial-direction geometric constraint relations in the BCM, is presented. An explicit strain energy expression, in terms of beam end-displacements, that accommodates generalized loading conditions, boundary conditions, initial curvature, and beam shape is derived. Using the Principle of Virtual Work, this strain energy expression for a generalized beam is employed in determining the load-displacement relations, and therefore constraint characteristics, for flexure mechanisms comprising multiple beams. The benefit of this approach is evident in its mathematical efficiency and succinctness, which is to be expected with the use of energy methods. All analytical results are validated to a high degree of accuracy via non-linear Finite Element Analysis. Furthermore, the proposed energy formulation leads to new insights into the nature of the BCM.

Design of Nonlinear Springs for Prescribed Load-Displacement Functions

2008 ◽  
Vol 130 (8) ◽  
Author(s):  
Christine Vehar Jutte ◽  
Sridhar Kota
Keyword(s):  
Displacement Function ◽  
Curved Beams ◽  
Element Analysis ◽  
Nonlinear Load ◽  
Nonlinear Spring ◽  
Nonlinear Springs ◽  
Bending Stresses ◽  
Bending Loads ◽  
Load Displacement ◽  
Synthesis Methodology

A nonlinear spring has a defined nonlinear load-displacement function, which is also equivalent to its strain energy absorption rate. Various applications benefit from nonlinear springs, including prosthetics and microelectromechanical system devices. Since each nonlinear spring application requires a unique load-displacement function, spring configurations must be custom designed, and no generalized design methodology exists. In this paper, we present a generalized nonlinear spring synthesis methodology that (i) synthesizes a spring for any prescribed nonlinear load-displacement function and (ii) generates designs having distributed compliance. We introduce a design parametrization that is conducive to geometric nonlinearities, enabling individual beam segments to vary their effective stiffness as the spring deforms. Key features of our method include (i) a branching network of compliant beams used for topology synthesis rather than a ground structure or a continuum model based design parametrization, (ii) curved beams without sudden changes in cross section, offering a more even stress distribution, and (iii) boundary conditions that impose both axial and bending loads on the compliant members and enable large rotations while minimizing bending stresses. To generate nonlinear spring designs, the design parametrization is implemented into a genetic algorithm, and the objective function evaluates spring designs based on the prescribed load-displacement function. The designs are analyzed using nonlinear finite element analysis. Three nonlinear spring examples are presented. Each has a unique prescribed load-displacement function, including a (i) “J-shaped,” (ii) “S-shaped,” and (iii) constant-force function. A fourth example reveals the methodology’s versatility by generating a large displacement linear spring. The results demonstrate the effectiveness of this generalized synthesis methodology for designing nonlinear springs for any given load-displacement function.

Elastic Averaging in Flexure Mechanisms: A Three-Beam Parallelogram Flexure Case Study

2010 ◽  
Vol 2 (4) ◽  
Author(s):  
Shorya Awtar ◽  
Kevin Shimotsu ◽  
Shiladitya Sen
Keyword(s):  
Improve Performance ◽  
Mathematical Basis ◽  
Element Analysis ◽  
Redundant Constraints ◽  
Flexure Mechanism ◽  
Design Paradigm ◽  
Constraint Model ◽  
Effective Use ◽  
Exact Constraint

Redundant constraints are generally avoided in mechanism design because they can lead to binding or loss in expected mobility. However, in certain distributed-compliance flexure mechanism geometries, this problem is mitigated by the phenomenon of elastic averaging. Elastic averaging is a design paradigm that, in contrast with exact constraint design principles, makes deliberate and effective use of redundant constraints to improve performance and robustness. The principle of elastic averaging and its advantages are illustrated in this paper by means of a three-beam parallelogram flexure mechanism, which represents an overconstrained geometry. In a lumped-compliance configuration, this mechanism is prone to binding in the presence of nominal manufacturing and assembly errors. However, with an increasing degree of distributed-compliance, the mechanism is shown to become more tolerant to such geometric imperfections. The nonlinear elastokinematic effect in the constituent beams is shown to play an important role in analytically predicting the consequences of overconstraint and provides a mathematical basis for elastic averaging. A generalized beam constraint model is used for these predictions so that varying degrees of distributed compliance are captured using a single geometric parameter. The closed-form analytical results are validated against finite element analysis, as well as experimental measurements.

Nonlinear Strain Energy Formulation to Capture the Constraint Characteristics of a Spatial Symmetric Beam

2011 ◽  
Author(s):  
Shiladitya Sen ◽  
Shorya Awtar
Keyword(s):  
Cross Sections ◽  
Strain Energy ◽  
Virtual Work ◽  
Nonlinear Strain ◽  
The Past ◽  
Transverse Bending ◽  
Spatial Beams ◽  
Cross Axis ◽  
Constraint Model ◽  
Energy Formulation

In the past, a beam constraint model (BCM) that captures pertinent geometric nonlinearities associated with large displacements has been proposed for slender spatial beams with uniform and symmetric cross-sections. By providing closed-form parametric relations between the end-loads and end-displacements of the beam, the BCM quantifies the constraint characteristics of the beam in terms of stiffness variations, parasitic error motions, and the cross-axis coupling. This paper presents a nonlinear strain and strain energy formulation for the spatial symmetric beam, based on assumptions that are consistent with the BCM. This strain energy derivation, employing the Principle of Virtual Work, provides a simpler mathematical approach for the analysis of flexure mechanisms with multiple spatial beams. Using this formulation, we obtain the stiffness relations in the transverse bending directions, the constraint relations in the axial and torsional directions, and the overall strain energy expression in terms of the beam end-loads and end-displacements. These expressions, collectively the BCM, are in form that is suitable for the analysis of multi-beam flexure mechanisms.

scholarly journals An Analytical Formulation for the Lateral Support Stiffness of a Spatial Flexure Strip

2015 ◽  
Author(s):  
Marijn Nijenhuis ◽  
J. P. Meijaard ◽  
Just L. Herder ◽  
Shorya Awtar ◽  
Dannis M. Brouwer
Keyword(s):  
Three Dimensional ◽  
Beam Element ◽  
Element Analysis ◽  
Cross Sectional ◽  
Flexure Mechanism ◽  
Stiffness Properties ◽  
Spatial Beam ◽  
Discrete Nature ◽  
Stiffness Characteristics

A flexure strip has constraint characteristics, such as stiffness properties and error motions, that limit its performance as a basic constituent of flexure mechanisms. This paper presents a framework for modeling the deformation and stiffness characteristics of general three-dimensional flexure strips that exhibit bending, shear and torsion deformation. The formulation is based on a finite strain discrete spatial beam element with refinements to account for plate-like behavior due to constrained cross-sectional warping. This framework is suited for analytical calculations thanks to the accuracy of the beam element, while its discrete nature allows for easy implementation in numeric software to serve as calculation aid. As case study, a closed-form parametric analytical expression is derived for the lateral support stiffness of a parallel flexure mechanism. This captures the deteriorating support stiffness when the mechanism moves in the intended degree of freedom. By incorporating relevant geometric nonlinearities and a warping constraint stiffening factor, an accurate load-displacement and stiffness expression for the lateral support direction is obtained. This result is verified by nonlinear finite element analysis.

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

2015 ◽  
Author(s):  
Guimin Chen ◽  
Ruiyu Bai
Keyword(s):  
Compliant Mechanisms ◽  
Research Community ◽  
Nonlinear Constraint ◽  
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 whether the tip loads are known or the tip deflections are known. The examples show that CSBCM can accurately predict the large spatial deflections of flexible beams, as compared to the available nonlinear 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.

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