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.

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.

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.

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.

Application of Regional Strain Energy in Topology Optimization With Inertia Relief Analysis

2013 ◽  
Author(s):  
Wei Song ◽  
Hae Chang Gea ◽  
Ren-Jye Yang ◽  
Ching-Hung Chuang
Keyword(s):  
Finite Element ◽  
Topology Optimization ◽  
Strain Energy ◽  
Computational Cost ◽  
Structure Design ◽  
Protective Structure ◽  
Element Analysis ◽  
Regional Strain ◽  
Unbalanced Loads ◽  
Energy Formulation

In finite element analysis, inertia relief solves the response of an unconstrained structure subject to constant or slowly varying external loads with static analysis computational cost. It is very attractive to utilize it in topology optimization to design structures under unbalanced loads, such as in impact and drop phenomena. In this paper, regional strain energy formulation and inertia relief is integrated into topology optimization to design protective structure under unbalanced loads. For background, the equations of inertia relief are introduced and a commonly used solving method is revisited. Then the regional strain energy formulation for topology optimization with inertia relief is proposed and its sensitivity is derived from the adjoint method. Based on the solving method, the sensitivity is evaluated term by term to simplify the results. The simplified sensitivity can be calculated easily using the output of commercial finite element packages. Finally, the effectiveness of this formulation is shown in the first example and the proposed regional strain energy formulation for topology optimization with inertia relief are presented and discussed in the protective structure design examples.

scholarly journals Effects of boundary conditions on bistable behaviour in axisymmetrical shallow shells

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170230 ◽  
Author(s):  
P. M. Sobota ◽  
K. A. Seffen
Keyword(s):  
Boundary Conditions ◽  
Strain Energy ◽  
Stable State ◽  
Spring Stiffness ◽  
Element Analysis ◽  
Single Degree Of Freedom ◽  
Shallow Shells ◽  
Single Degree ◽  
Thin Walled Structures ◽  
Energy Formulation

Multistable shells are thin-walled structures that have more than one stable state of self-stress. We consider isotropic axisymmetrical shallow shells of arbitrary polynomial shapes using a Föppl–von Kármán analytical model. By employing a Rayleigh–Ritz approach, we identify stable shapes from local minima in the strain energy formulation, and we formally characterize the level of influence of the boundary conditions on the critical geometry for achieving bistable inversion—an effect not directly answered in the literature. Systematic insight is afforded by connecting the boundary to ground through sets of extensional and rotational linear springs. For typical cap-like shells, it is shown that bistability is generally enhanced when the extensional spring stiffness increases and when the rotational spring stiffness decreases, i.e. when boundary movements in-plane are resisted but when their rotations are not; however, for certain other shapes and large in-plane stiffness values, bistability can be enhanced by resisting but not entirely preventing edge rotations. Our predictions are furnished as detailed regime maps of the critical geometry, which are accurately correlated against finite-element analysis. Furthermore, the suitabilities of single degree-of-freedom models, for which solutions are achieved in closed form, are evaluated and compared to our more accurate predictions.

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.

Stiffness characteristics of inner–outer ring flexure pivots applied to the ultra-precision instruments

2017 ◽  
Vol 232 (13) ◽  
pp. 2441-2457
Author(s):  
Lang Liu ◽  
Shusheng Bi ◽  
Qizi Yang
Keyword(s):  
Analytical Model ◽  
Strain Energy ◽  
Compliant Mechanisms ◽  
Outer Ring ◽  
Element Analysis ◽  
Qualitative Comparison ◽  
Ultra Precision ◽  
Stiffness Characteristics ◽  
Work First ◽  
Energy Formulation

The inner-outer ring flexure pivot (IORFP), composed of three straight springs that cross each other in space, is studied in this work. First, to emphasize the study value of IORFP, qualitative comparison is applied to IORFP and some of most commonly used flexure pivots. Then an analytical model for the rotational stiffness of IORFP is developed based on the strain energy formulation of a beam flexure, and model applicability is provided as well. Analysis of stiffness, buckling load, and the nonlinear of moment–rotation relation is then carried out. Subsequently, the analytical model is verified by finite element analysis. After that, seven prototypes of IORFP are manufactured, and their rotational stiffnesses are tested. The results show that the analytical model can be used for analysis and designing of compliant mechanisms that contain IORFP. Finally, the study quantitatively compares stiffness characteristics and axis drift of IORFP and the generalized cross-spring pivot, indicating that the former significantly outperforms the latter. IORFP possesses excellent performances and can be widely used to supplant generalized cross-spring pivot in compliant mechanisms and ultra-precision instruments.

Vibration-Based Crack Detection in L-Frames

2014 ◽  
Vol 658 ◽  
pp. 261-268
Author(s):  
Jean Louis Ntakpe ◽  
Gilbert Rainer Gillich ◽  
Florian Muntean ◽  
Zeno Iosif Praisach ◽  
Peter Lorenz
Keyword(s):  
Strain Energy ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Vibration Modes ◽  
Structural Elements ◽  
Element Analysis ◽  
Accurate Localization ◽  
Mathematical Expressions ◽  
Measurement Results ◽  
Non Destructive

This paper presents a novel non-destructive method to locate and size damages in frame structures, performed by examining and interpreting changes in measured vibration response. The method bases on a relation, prior contrived by the authors, between the strain energy distribution in the structure for the transversal vibration modes and the modal changes (in terms of natural frequencies) due to damage. Using this relation a damage location indicator DLI was derived, which permits to locate cracks in spatial structures. In this paper an L-frame is considered for proving the applicability of this method. First the mathematical expressions for the modes shapes and their derivatives were determined and simulation result compared with that obtained by finite element analysis. Afterwards patterns characterizing damage locations were derived and compared with measurement results on the real structure; the DLI permitted accurate localization of any crack placed in the two structural elements.

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