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

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Marijn Nijenhuis ◽  
J. P. Meijaard ◽  
Dhanushkodi Mariappan ◽  
Just L. Herder ◽  
Dannis M. Brouwer ◽  
...  
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Flexure Mechanism ◽  
Stiffness Properties ◽  
Spatial Beam ◽  
Bearing Stiffness ◽  
Stiffness Characteristics ◽  
Parametric Expression ◽  
Analytical Result ◽  
Analytical Expressions

A flexure strip has constraint characteristics, such as stiffness properties and error motions, that govern its performance as a basic constituent of flexure mechanisms. This paper presents a new modeling approach for obtaining insight into the deformation and stiffness characteristics of general three-dimensional flexure strips that exhibit bending, shear, and torsion deformation. The approach is based on the use of a discretized version of a finite (i.e., nonlinear) strain spatial beam formulation for extracting analytical expressions that describe deformation and stiffness characteristics of a flexure strip in a parametric format. This particular way of closed-form modeling exploits the inherent finite-element assumptions on interpolation and also lends itself for numeric implementation. As a validating case study, a closed-form parametric expression is derived for the lateral support stiffness of a flexure strip and a parallelogram flexure mechanism. This captures a combined torsion–bending dictated geometrically nonlinear effect that undermines the support bearing stiffness when the mechanism moves in the intended degree of freedom (DoF). The analytical result is verified by simulations and experimental measurements.

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.

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.

The Influence of Internal Friction on the Stability of High Speed Rotors With Anisotropic Supports

1969 ◽  
Vol 91 (4) ◽  
pp. 1105-1113 ◽  
Author(s):  
E. J. Gunter ◽  
P. R. Trumpler
Keyword(s):  
Internal Friction ◽  
High Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Analog Computer ◽  
Three Dimensional Model ◽  
Stiffness Properties ◽  
Stiffness Characteristics ◽  
The Stability ◽  
Internal Rotor

This paper evaluates the stability of the single mass rotor with internal friction on damped, anisotropic supports. The paper shows under what conditions the rotor stability may be improved by an undamped support with anisotropic stiffness properties. A three dimensional model is presented to show the influence of rotor and support stiffness characteristics on stability. Curves are also presented on how support damping may also improve or even reduce rotor stability. An analog computer solution of the governing equations of motion is presented showing the shaft transient motion for various speed ranges, and also plots of the rotor steady state motion are given for various speeds up to and including the stability threshold. The analysis is used to explain many of the experimental observations of B. L. Newkirk concerning stability due to internal rotor friction.

scholarly journals Fabrication, Experiments, and Analysis of an LBM Additive-Manufactured Flexure Parallel Mechanism

Micromachines ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 572 ◽  
Author(s):  
Huaxian Wei ◽  
Li Wang ◽  
Xiaodong Niu ◽  
Jian Zhang ◽  
Alessandro Simeone
Keyword(s):  
Additive Manufacturing ◽  
Parallel Mechanism ◽  
Three Dimensional ◽  
Experimental Tests ◽  
Element Analysis ◽  
Flexure Mechanism ◽  
Laser Beam Melting ◽  
Stiffness Characteristics ◽  
First Time

Additive manufacturing technology has advantages for realizing complex monolithic structures, providing huge potential for developing advanced flexure mechanisms for precision manipulation. However, the characteristics of flexure hinges fabricated by laser beam melting (LBM) additive manufacturing (AM) are currently little known. In this paper, the fabrication and characterization of a flexure parallel mechanism through the LBM process are reported for the first time to demonstrate the development of this technique. The geometrical accuracy of the additive-manufactured flexure mechanism was evaluated by three-dimensional scanning. The stiffness characteristics of the flexure mechanism were investigated through finite element analysis and experimental tests. The effective hinge thickness was determined based on the parameters study of the flexure parallel mechanism. The presented results highlight the promising outlook of LBM flexure parts for developing novel nanomanipulation platforms, while additional attention is required for material properties and manufacturing errors.

A Spatial Parametric Model for the Nonlinear Stiffness Characteristics of Flexure Strips

2018 ◽  
Author(s):  
Marijn Nijenhuis ◽  
J. P. Meijaard ◽  
Dannis M. Brouwer
Keyword(s):  
Closed Form ◽  
Nonlinear Effects ◽  
Geometrically Nonlinear ◽  
Nonlinear Stiffness ◽  
Mechanism Analysis ◽  
Compliance Matrix ◽  
Closed Form Solutions ◽  
Flexure Mechanism ◽  
Stiffness Characteristics ◽  
Torsion Deformation

The flexure strip is commonly used to provide support stiffness in flexure mechanisms for precision applications. While the flexure strip is often treated in a simplified form, e.g. by assuming planar deformation or linearized stiffness, the deformation in practice is spatial and sufficiently large that nonlinear effects due to the geometrical stiffness are significant. This paper presents an understandable analytical model for the nonlinear stiffness characteristics of flexure strips that deform spatially due to a general 3-D loading condition. This model provides closed-form expressions in a mixed stiffness and compliance matrix format that is tailored to flexure mechanism analysis. The effects of bending, elongation, and torsion deformation are taken into account. The geometrically nonlinear effects of the model are verified numerically. The approach for deriving closed-form solutions in a nonlinear context is detailed in this paper. Based on the Hellinger–Reissner variational principle, it can also be extended to the analysis of multi-flexure strip mechanisms. This is demonstrated with the case of a spatially deforming parallelogram flexure mechanism.

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.

On the Stability of Rotation of a Rotor With Rotationally Unsymmetric Inertia and Stiffness Properties

1961 ◽  
Vol 28 (4) ◽  
pp. 567-570 ◽  
Author(s):  
S. H. Crandall ◽  
P. J. Brosens
Keyword(s):  
Three Dimensional ◽  
The Body ◽  
Uniform Rotation ◽  
Stiffness Properties ◽  
Principal Axes ◽  
Rotationally Symmetric ◽  
Speed Range ◽  
Stiffness Characteristics ◽  
The Stability ◽  
Gyroscopic Coupling

The stability of uniform rotation of a rigid body about a principal axis of inertia is analyzed for the case where there is a diametral inertia inequality and there is an elastic restoring mechanism with a diametral stiffness inequality which rotates with the body. This model is an idealization for systems such as a two-bladed propeller rotating on a flexible shaft whose stiffness characteristics are not rotationally symmetric. It is found that many such systems possess unstable speed ranges. The instability may be due to either type of asymmetry alone or due to the interaction of the two. Quantitative analytical results are obtained which relate the unstable speed range to the gyroscopic coupling, the inertia inequality, the stiffness inequality, and the relative orientation of the principal axes of inertia with respect to the principal axes of stiffness. Three-dimensional stability surfaces are plotted to give a qualitative overview of the interplay of the various parameters.

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