Spatial flexure mechanism analysiseEnergy-based geometric stiffness modeling

Abstract The influence of tread designs on tire performance is well known. The tire industry spends significant effort in the development process to create and refine tread patterns. Creating an aesthetic yet functional design requires characterization of the tread design using many engineering parameters such as stiffness, moments of inertia, principal angles, etc. The tread element stiffness is of particular interest because of its use to objectively determine differences between tread patterns as the designer refines the design to provide optimum levels of performance. The tread designer monitors the change in stiffness as the design evolves. Changes to the geometry involve many attributes including the number of sipes, sipe depth, sipe location, block element edge taper, nonskid depth, area net-to-gross, and so forth. In this paper, two different formulations for calculating tread element or block stiffness are reviewed and are compared to finite element results in a few cases. A few simple examples are shown demonstrating the basic functionality that is possible with a numerical method.

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Test bench for experimental study of the geometric stiffness in vibration of beams as support of rotating machines

10.26678/abcm.cobem2017.cob17-2821 ◽

2017 ◽

Author(s):

Alexandre de Macêdo Wahrhaftig ◽

Reyolando Brasil ◽

Gabriela Silva Correia Cordeiro

Keyword(s):

Experimental Study ◽

Test Bench ◽

Rotating Machines ◽

Geometric Stiffness

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Consistent co-rotational framework for Euler-Bernoulli and Timoshenko beam-column elements under distributed member loads

Advances in Structural Engineering ◽

10.1177/1369433220986632 ◽

2021 ◽

pp. 136943322098663

Author(s):

Yi-Qun Tang ◽

Wen-Feng Chen ◽

Yao-Peng Liu ◽

Siu-Lai Chan

Keyword(s):

Coordinate System ◽

Stiffness Matrix ◽

Traditional Method ◽

Local Coordinate System ◽

Frame Structures ◽

Global Coordinate System ◽

Single Element ◽

Geometrically Nonlinear ◽

Geometrically Nonlinear Analysis ◽

Geometric Stiffness

Conventional co-rotational formulations for geometrically nonlinear analysis are based on the assumption that the finite element is only subjected to nodal loads and as a result, they are not accurate for the elements under distributed member loads. The magnitude and direction of member loads are treated as constant in the global coordinate system, but they are essentially varying in the local coordinate system for the element undergoing a large rigid body rotation, leading to the change of nodal moments at element ends. Thus, there is a need to improve the co-rotational formulations to allow for the effect. This paper proposes a new consistent co-rotational formulation for both Euler-Bernoulli and Timoshenko two-dimensional beam-column elements subjected to distributed member loads. It is found that the equivalent nodal moments are affected by the element geometric change and consequently contribute to a part of geometric stiffness matrix. From this study, the results of both eigenvalue buckling and second-order direct analyses will be significantly improved. Several examples are used to verify the proposed formulation with comparison of the traditional method, which demonstrate the accuracy and reliability of the proposed method in buckling analysis of frame structures under distributed member loads using a single element per member.

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Stiffness modeling of an industrial robot with a gravity compensator considering link weights

Mechanism and Machine Theory ◽

10.1016/j.mechmachtheory.2021.104331 ◽

2021 ◽

Vol 161 ◽

pp. 104331

Author(s):

Peng Xu ◽

Xiling Yao ◽

Shibo Liu ◽

Hao Wang ◽

Kui Liu ◽

...

Keyword(s):

Industrial Robot ◽

Stiffness Modeling ◽

Gravity Compensator

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Stiffness Modeling of Robotic-Manipulators Under Auxiliary Loadings

Volume 4: 36th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2012-70457 ◽

2012 ◽

Cited By ~ 1

Author(s):

Alexandr Klimchik ◽

Anatol Pashkevich ◽

Stéphane Caro ◽

Damien Chablat

Keyword(s):

Robotic Manipulators ◽

Computationally Efficient ◽

End Effector ◽

Efficient Procedure ◽

Stiffness Modeling ◽

Cartesian Stiffness Matrix ◽

Different Types ◽

Mapping Equation ◽

Linear Force ◽

Cartesian Stiffness

The paper focuses on the extension of the virtual-joint-based stiffness modeling technique for the case of different types of loadings applied both to the robot end-effector and to manipulator intermediate points (auxiliary loading). It is assumed that the manipulator can be presented as a set of compliant links separated by passive or active joints. It proposes a computationally efficient procedure that is able to obtain a non-linear force-deflection relation taking into account the internal and external loadings. It also produces the Cartesian stiffness matrix. This allows to extend the classical stiffness mapping equation for the case of manipulators with auxiliary loading. The results are illustrated by numerical examples.

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

Volume 2: 30th Annual Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2006-99752 ◽

2006 ◽

Cited By ~ 5

Author(s):

Shorya Awtar ◽

Edip Sevincer

Keyword(s):

Mechanism Design ◽

Parametric Model ◽

Nonlinear Load ◽

Geometric Imperfections ◽

Geometric Imperfection ◽

Flexure Mechanism ◽

Important Concern ◽

Proposed Model ◽

Geometric Arrangements

Over-constraint is an important concern in mechanism design because it can lead to a loss in desired mobility. In distributed-compliance flexure mechanisms, this problem is alleviated due to the phenomenon of elastic averaging, thus enabling performance-enhancing geometric arrangements that are otherwise unrealizable. The principle of elastic averaging is illustrated in this paper by means of a multi-beam parallelogram flexure mechanism. In a lumped-compliance configuration, this mechanism is prone to over-constraint 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 load-stiffening and elasto-kinematic effects in the constituent beams have an important role to play in the over-constraint and elastic averaging characteristics of this mechanism. Therefore, a parametric model that incorporates these nonlinearities is utilized in predicting the influence of a representative geometric imperfection on the primary motion stiffness of the mechanism. The proposed model utilizes a beam generalization so that varying degrees of distributed compliance are captured using a single geometric parameter.

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An XYZ Parallel Kinematic Flexure Mechanism With Geometrically Decoupled Degrees of Freedom

Volume 6: 35th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2011-47713 ◽

2011 ◽

Cited By ~ 1

Author(s):

Shorya Awtar ◽

John Ustick ◽

Shiladitya Sen

Keyword(s):

Degrees Of Freedom ◽

Element Analysis ◽

Motion Direction ◽

Form Analysis ◽

Flexure Mechanism ◽

Non Linear ◽

Decoupled Motion ◽

Motion Range ◽

Parallel Kinematic ◽

Cross Axis

We present the constraint-based design of a novel parallel kinematic flexure mechanism that provides highly decoupled motions along the three translational directions (X, Y, and Z) and high stiffness along the three rotational directions (θx, θy, and θz). The geometric decoupling ensures large motion range along each translational direction and enables integration with large-stroke ground-mounted linear actuators or generators, depending on the application. The proposed design, which is based on a systematic arrangement of multiple rigid stages and parallelogram flexure modules, is analyzed via non-linear finite element analysis. A proof-of-concept prototype of the flexure mechanism is fabricated to validate its large range and decoupled motion capability. The analyses as well as the hardware demonstrate an XYZ motion range of 10 mm × 10 mm × 10 mm. Over this motion range, the non-linear FEA predicts a cross-axis error of less than 3%, parasitic rotations less than 2 mrad, less than 4% lost motion, actuator isolation less than 1.5%, and no perceptible motion direction stiffness variation. Ongoing work includes non-linear closed-form analysis and experimental measurement of these error motion and stiffness characteristics.

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Geometrical Stiffness of Thin-Walled I-Beam Element Based on Rigid-Beam Assemblage Concept

Journal of Mechanics ◽

10.1017/jmech.2012.10 ◽

2012 ◽

Vol 28 (1) ◽

pp. 97-106 ◽

Cited By ~ 1

Author(s):

J. D. Yau ◽

S.-R. Kuo

Keyword(s):

Stiffness Matrix ◽

Virtual Work ◽

Bending Moment ◽

Beam Element ◽

Narrow Beam ◽

Cross Sectional ◽

Geometric Stiffness ◽

Buckling Behavior ◽

Post Buckling ◽

Thin Walled

ABSTRACTUsing conventional virtual work method to derive geometric stiffness of a thin-walled beam element, researchers usually have to deal with nonlinear strains with high order terms and the induced moments caused by cross sectional stress results under rotations. To simplify the laborious procedure, this study decomposes an I-beam element into three narrow beam components in conjunction with geometrical hypothesis of rigid cross section. Then let us adopt Yanget al.'s simplified geometric stiffness matrix [kg]12×12of a rigid beam element as the basis of geometric stiffness of a narrow beam element. Finally, we can use rigid beam assemblage and stiffness transformation procedure to derivate the geometric stiffness matrix [kg]14×14of an I-beam element, in which two nodal warping deformations are included. From the derived [kg]14×14matrix, it can take into account the nature of various rotational moments, such as semi-tangential (ST) property for St. Venant torque and quasi-tangential (QT) property for both bending moment and warping torque. The applicability of the proposed [kg]14×14matrix to buckling problem and geometric nonlinear analysis of loaded I-shaped beam structures will be verified and compared with the results presented in existing literatures. Moreover, the post-buckling behavior of a centrally-load web-tapered I-beam with warping restraints will be investigated as well.

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An XYZ Parallel-Kinematic Flexure Mechanism With Geometrically Decoupled Degrees of Freedom

Journal of Mechanisms and Robotics ◽

10.1115/1.4007768 ◽

2012 ◽

Vol 5 (1) ◽

Cited By ~ 25

Author(s):

Shorya Awtar ◽

John Ustick ◽

Shiladitya Sen

Keyword(s):

Degrees Of Freedom ◽

Finite Elements Analysis ◽

Motion Direction ◽

Flexure Mechanism ◽

Nonlinear Finite Elements ◽

Decoupled Motion ◽

Motion Range ◽

Stiffness Variation ◽

Parallel Kinematic ◽

Cross Axis

A novel parallel-kinematic flexure mechanism that provides highly decoupled motions along the three translational directions (X, Y, and Z) and high stiffness along the three rotational directions (θx, θy, and θz) is presented. Geometric decoupling ensures large motion range along each translational direction and enables integration with large-stroke ground-mounted linear actuators or generators, depending on the application. The proposed design, which is based on a systematic arrangement of multiple rigid stages and parallelogram flexure modules, is analyzed via nonlinear finite elements analysis (FEA). A proof-of-concept prototype is fabricated to validate the predicted large range and decoupled motion capabilities. The analysis and the hardware prototype demonstrate an XYZ motion range of 10 mm × 10 mm × 10 mm. Over this motion range, the nonlinear FEA predicts cross-axis errors of less than 7.8%, parasitic rotations less than 10.8 mrad, less than 14.4% lost motion, actuator isolation better than 1.5%, and no perceptible motion direction stiffness variation.

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