Analytical Elastostatic Stiffness Modeling of Overconstrained Parallel Manipulators Using Geometric Algebra and Strain Energy

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Qinchuan Li ◽  
Lingmin Xu ◽  
Qiaohong Chen ◽  
Xinxue Chai
Keyword(s):  
Geometric Algebra ◽  
Strain Energy ◽  
Linear Equations ◽  
Parallel Manipulators ◽  
Equilibrium Equations ◽  
Element Analysis ◽  
Outer Product ◽  
Stiffness Analysis ◽  
Stiffness Modeling ◽  
Stiffness Matrices

A general method for the analytical elastostatic stiffness modeling of overconstrained parallel manipulators (PMs) using geometric algebra and strain energy is proposed. First, an analytical solution of the constraint and actuation wrenches exerted on the moving platform is obtained using the outer product and dual operation of geometric algebra, which avoids solving complex symbolic linear equations. Second, considering the compliances of the limbs, an analytical elastostatic model is established using the strain energy to obtain the stiffness matrices of the limbs. Finally, the deformation compatibility equations are added into equilibrium equations to obtain the overall stiffness matrix of the PM, which has a concise expression and a clear physical meaning. The proposed method is applied to the Tex3 overconstrained PM and the Tex4 overconstrained PM with redundant actuation to prove its validity. Comparable results between the theoretical analysis and the finite-element analysis (FEA) show that the former could be used as an effective alternative to the FEA method in the predesign stage. This new approach is universally applicable to the elastostatic stiffness analysis of overconstrained PMs.

scholarly journals Elastostatic Stiffness Analysis for the US/UPS Parallel Manipulators

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Weizhong Zhang ◽  
Wei Ye ◽  
Chao Yang
Keyword(s):  
Strain Energy ◽  
Screw Theory ◽  
Virtual Work ◽  
Element Model ◽  
Parallel Manipulators ◽  
Modeling Method ◽  
Stiffness Analysis ◽  
Design And Optimization ◽  
Stiffness Matrices ◽  
The Us

The virtual joint method (VJM) cannot calculate the strain energy stored in each rod. In order to solve the problem, a modeling method of the elastostatic stiffness was investigated for the UP/UPS parallel manipulators (PMs), taking the example of the 6-SPS PM. The modeling method was based on screw theory, Castigliano’s theorem, and strain energy (where U, P, and S, respectively, denote universal, prismatic, and spherical joints). First, the actuator and constraint wrenches of the mechanism were obtained by screw theory. Second, compact limb stiffness matrices were obtained in terms of strain energy and Castigliano’s second theorem. Finally, analytic expressions for the overall stiffness matrix of the mechanism and the amplitudes of the actuator force were obtained by adopting the virtual work principle and the balance equation for the mobile platform. All relative errors between the results of the analytical model and the finite element model are below 2%, which validates the effectiveness of the elastostatic stiffness model. The virtual work index was adopted to evaluate the stiffness performance of the mechanism, and the results show that the stiffness is not only related to position and orientation but also closely related to the directions of external loads. It is also demonstrated that the method has general adaptability for the stiffness analysis for the US/UPS PMs, laying the foundation for further reasonable dynamic design and optimization of such manipulators.

A Method for Compliance Modeling of Five Degree-of-Freedom Overconstrained Parallel Robotic Mechanisms With 3T2R Output Motion

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
Wen-ao Cao ◽  
Huafeng Ding ◽  
Donghao Yang
Keyword(s):  
Screw Theory ◽  
Parallel Manipulators ◽  
Static Equilibrium ◽  
Equilibrium Equations ◽  
Element Analysis ◽  
Compliance Matrix ◽  
Fea Model ◽  
Stiffness Model ◽  
Compliance Modeling ◽  
The One

This paper presents an approach to compliance modeling of three-translation and two-rotation (3T2R) overconstrained parallel manipulators, especially for those with multilink and multijoint limbs. The expressions of applied wrenches (forces/torques) exerted on joints are solved with few static equilibrium equations based on screw theory. A systematic method is proposed for deriving the stiffness model of a limb with considering the couplings between the stiffness along the constrained wrench and the one along the actuated wrench based on strain energy analysis. The compliance model of a 3T2R overconstrained parallel mechanism is established based on stiffness models of limbs and the static equilibrium equation of the moving platform. Comparisons show that the compliance matrix obtained from the method is close to the one obtained from a finite-element analysis (FEA) model. The proposed method has the characteristics of involving low computational efforts and considering stiffness couplings of each limb.

Stiffness Analysis of a Spatial Parallel Mechanism With Flexible Moving Platform

2010 ◽  
Author(s):  
Amir Rezaei ◽  
Alireza Akbarzadeh ◽  
Javad Enferadi
Keyword(s):  
Inverse Kinematics ◽  
Strain Energy ◽  
Parallel Manipulators ◽  
Physical Structure ◽  
Fem Model ◽  
Lumped Model ◽  
Stiffness Analysis ◽  
Spatial Parallel Mechanism ◽  
Inverse Kinematics Problem ◽  
Moving Platform

In this paper, stiffness analysis of a 3-DOF spatial, 3-PSP type, parallel manipulator is investigated. Most previous stiffness analysis studies of parallel manipulators are performed using lumped model as well as assuming a rigid moving platform. In this paper, unlike traditional stiffness analysis, the moving platform is assumed to be flexible. Additionally, a continuous method is used for obtaining mathematical model of the manipulator stiffness matrix. This method is based on strain energy and Castigliano’s theorem [1]. For this purpose, first we solve inverse kinematics problem then We must find relationship between the applied external torques on the moving platform and the resultant joints forces. Next, strain energy moving platform is calculated. Strain energy of this element is calculated using force analysis and inverse kinematics problem. Finally, a FEM model is generated and used to simulate the physical structure. Simulation results are compared with the analytical model.

Stiffness Estimation and Experiments for the Exechon Parallel Self-Reconfiguring Fixture Mechanism

2012 ◽  
Author(s):  
Xiong Li ◽  
Dimiter Zlatanov ◽  
Matteo Zoppi ◽  
Rezia Molfino
Keyword(s):  
Screw Theory ◽  
Virtual Work ◽  
Element Analysis ◽  
Stiffness Analysis ◽  
Stiffness Modeling ◽  
Successful Series ◽  
Stiffness Model ◽  
Stiffness Map ◽  
Fixture System ◽  
Typical Configuration

The Exechon X150, a new smaller member of a successful series of parallel kinematic machines, has been recently developed as a component of a mobile self-reconfigurable fixture system within an inter-European project. This paper is the first to address the stiffness analysis of the parallel mechanism on which the design is based. The stiffness modeling method uses reciprocal screw theory as well as the virtual work principle, resulting in a simpler formulation and more convenient than ones obtained with traditional stiffness-modeling methods. Based on this model, the stiffness map within the workspace is obtained. The stiffness of the mechanism at a typical configuration is carried out. The complete finite element analysis and simulation used to verify the effectiveness of the stiffness model. Using geometric spatial decomposition, numerical examples of the mechanism at three typical configurations are presented.

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.

Fixture Clamping Force Analysis during Milling Process

2013 ◽  
Vol 278-280 ◽  
pp. 385-388 ◽  
Author(s):  
Shao Gang Liu ◽  
Qiu Jin
Keyword(s):  
Analytical Method ◽  
Linear Equations ◽  
Milling Process ◽  
Contact Forces ◽  
Clamping Force ◽  
Contact Deformation ◽  
Force Analysis ◽  
Equilibrium Equations ◽  
Tangential Contact ◽  
Fixture Element

This paper presents a analytical method to calculate the minimum clamping force to prevent slippage between the workpiece and spherical-tipped fixture elements during milling process. After the contact deformation between the workpiece and spherical-tipped fixture element is determined, the relationships between the workpiece displacement and the contact deformations are obtained. Based on the static equilibrium equations, these equations are combined and linear equations are obtained to calculate the tangential contact forces between the workpiece and spherical-tipped fixture element. According to the maximum tangential contact force, the minimum clamping force to prevent slippage between the workpiece and spherical-tipped fixture elements is calculated. At last, this method is illustrated with a simulation example.

scholarly journals A Kollár and Pluzsik anisotropic composite beam theory for arbitrary multicelled cross sections

2016 ◽  
Vol 35 (23) ◽  
pp. 1696-1711 ◽  
Author(s):  
Danilo S Victorazzo ◽  
Andre De Jesus
Keyword(s):  
Finite Element ◽  
Cross Sections ◽  
Composite Beam ◽  
Linear Equations ◽  
Beam Theory ◽  
Element Formulation ◽  
Compliance Matrix ◽  
Asymmetric System ◽  
Anisotropic Composite ◽  
Stiffness Matrices

In this paper we extend Kollár and Pluzsik’s thin-walled anisotropic composite beam theory to include multiple cells with open branches and booms, and present a finite element formulation utilizing the stiffness matrix obtained from this theory. To recover the 4 × 4 compliance matrix of a beam containing N closed cells, we solve an asymmetric system of 2N + 4 linear equations four times with unitary section loads and extract influence coefficients from the calculated strains. Finally, we compare 4 × 4 stiffness matrices of a multicelled beam using this method against matrices obtained using the finite element method to demonstrate accuracy. Similarly to its originating theory, the effects of shear deformation and restrained warping are assumed negligible.

scholarly journals Effect of chemical reaction and radiation on double diffusive flow of a viscous, dissipative fluid through porous medium in a rectangular cavity with heat sources

2013 ◽  
Vol 18 (4) ◽  
pp. 1115-1150
Author(s):  
T.L. Raju ◽  
P. Muralidhar
Keyword(s):  
Porous Medium ◽  
Stream Function ◽  
Rectangular Cavity ◽  
Heat Sources ◽  
Element Analysis ◽  
Coupled Equations ◽  
Stiffness Matrices ◽  
Diffusive Flow ◽  
Transfer Flow ◽  
And Diffusion

Abstract In this paper, an attempt is made to discuss the combined influence of radiation and dissipation on the convective heat and mass transfer flow of a viscous fluid through a porous medium in a rectangular cavity using the Darcy model. Making use of the incompressibility, the governing non-linear coupled equations for the momentum, energy and diffusion are derived in terms of the non-dimensional stream function, temperature and concentration. The Galerkin finite element analysis with linear triangular elements is used to obtain the global stiffness matrices for the values of stream function, temperature and concentration. These coupled matrices are solved using an iterative procedure and expressions for the stream function, temperature and concentration are obtained as linear combinations of the shape functions. The behavior of temperature, concentration, the Nusselt number and Sherwood number is discussed computationally for different values of the governing parameters, such as the Rayleigh Number (Ra), heat source parameter (α), Eckert number (Ec), Schmidt Number (Sc), Soret parameter (S0), buoyancy ratio (N).

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