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.

scholarly journals Axisymmetric Large Deflection Elastic Analysis of Hollow Annular Membranes under Transverse Uniform Loading

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1770
Author(s):  
Jun-Yi Sun ◽  
Qi Zhang ◽  
Xue Li ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Circular Plate ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Circular Ring ◽  
Form Solution ◽  
Elastic Response ◽  
Geometrically Nonlinear ◽  
Closed Form Solutions ◽  
Annular Membrane

The anticipated use of a hollow linearly elastic annular membrane for designing elastic shells has provided an impetus for this paper to investigate the large deflection geometrically nonlinear phenomena of such a hollow linearly elastic annular membrane under transverse uniform loads. The so-called hollow annular membranes differ from the traditional annular membranes available in the literature only in that the former has the inner edge attached to a movable but weightless rigid concentric circular ring while the latter has the inner edge attached to a movable but weightless rigid concentric circular plate. The hollow annular membranes remove the transverse uniform loads distributed on “circular plate” due to the use of “circular ring” and result in a reduction in elastic response. In this paper, the large deflection geometrically nonlinear problem of an initially flat, peripherally fixed, linearly elastic, transversely uniformly loaded hollow annular membrane is formulated, the problem formulated is solved by using power series method, and its closed-form solution is presented for the first time. The convergence and effectiveness of the closed-form solution presented are investigated numerically. A comparison between closed-form solutions for hollow and traditional annular membranes under the same conditions is conducted, to reveal the difference in elastic response, as well as the influence of different closed-form solutions on the anticipated use for designing elastic shells.

scholarly journals Self-moments stiffening effect and buckling strength of periodic Vierendeel beams

2020 ◽  
Author(s):  
Francesco Penta
Keyword(s):  
Closed Form ◽  
Large Deflection ◽  
Continuous System ◽  
Equivalent Model ◽  
Geometrically Nonlinear ◽  
Closed Form Solutions ◽  
Expansion Technique ◽  
Polar Model ◽  
The Galerkin Method ◽  
Stiffening Effect

AbstractThis paper deals with the buckling phenomenon of periodic Vierendeel beams. Closed-form solutions for critical loads and deformed shapes are presented. They are built by exploiting several auxiliary solutions obtained for the discrete periodic girder and for a geometrically nonlinear micro-polar equivalent model. In particular, the girder when subjected to sinusoidal self-equilibrated systems of inner bending moments (self-moments) is analysed. The corresponding results are used for solving the large-deflection equilibrium problem of the continuous equivalent model by means of the eigenfunction expansion technique. Girder buckling conditions are then defined in terms of kinematics of the micro-polar model: more precisely, they are attained when special distributions of self-moments, able to bend the continuous system without violating compatibility of shear strains, act in the girder. It is shown that these systems, neglected in the theories presented so far, have a significant stiffening effect on the buckling girder behaviour. Moreover, they are governed by the continuity equation for micro-rotations that is solved in closed form by the Galerkin method, with the micro-polar model eigenfunctions as basis functions. The accuracy of the proposed solutions is verified by comparing them with those achieved by a series of finite element girder models.

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.

Closed Form Solutions of Joint Water-Filling for Coordinated Transmission

2010 ◽  
Vol E93-B (12) ◽  
pp. 3461-3468 ◽  
Author(s):  
Bing LUO ◽  
Qimei CUI ◽  
Hui WANG ◽  
Xiaofeng TAO ◽  
Ping ZHANG
Keyword(s):  
Closed Form ◽  
Closed Form Solutions ◽  
Water Filling
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