An Algorithm to Study the Elastodynamics of Parallel Kinematic Machines With Lower Kinematic Pairs

2012 ◽  
Vol 5 (1) ◽  
Author(s):  
Alessandro Cammarata ◽  
Davide Condorelli ◽  
Rosario Sinatra
Keyword(s):  
Stiffness Matrix ◽  
Rigid Bodies ◽  
Parallel Kinematic Machine ◽  
Element Analysis ◽  
Stiffness Matrices ◽  
The Matrix ◽  
Parallel Kinematic ◽  
Global Stiffness Matrix ◽  
Local Stiffness

In this paper, an algorithm to help designers to integrate the elastodynamics analysis along with the inverse positioning and orienting problems of a parallel kinematic machine (PKM) into a single package is conceived. The proposed algorithm applies concepts from the matrix structural analysis (MSA) and finite element analysis (FEA) to determine the generalized stiffness matrix and the linearized elastodynamics equations of a PKM with only lower kinematic pairs. A PKM is modeled as a system of flexible links and rigid bodies connected by means of joints. Three cases are analyzed to consider the combinations between flexible and rigid bodies in order to find the local stiffness matrices. The latter are combined to obtain the limb matrices and, then, the global stiffness matrix of the whole robotic system. The same nodes coming from the links discretization are considered to include joint masses/inertias into the model. Finally, a case study is proposed to show some feasible applications and to compare results to commercial software for validation.

A modified elasto-dynamic model based static stiffness evaluation for a 3-PRS PKM

2015 ◽  
Vol 230 (3) ◽  
pp. 353-366 ◽  
Author(s):  
Yan-Qin Zhao ◽  
Jun Zhang ◽  
Ling-Yan Ruan ◽  
Hai-Wei Luo ◽  
Xiao-Liu Yu
Keyword(s):  
Dynamic Model ◽  
Stiffness Matrix ◽  
Performance Enhancement ◽  
Element Formulation ◽  
Parallel Kinematic Machine ◽  
Design Variables ◽  
Parallel Kinematic ◽  
And Performance ◽  
Global Stiffness Matrix ◽  
Spring Constants

This paper proposes a modified elasto-dynamic model for a three-prismatic revolute spherical parallel kinematic machine, in which the flexibility of the prismatic revolute spherical limb structures are accounted in and modeled as a hollowed spatial beam with nonuniform cross section. The governing equations are derived through substructure synthesis and finite element formulation. The stiffness matrix of the platform is then extracted from global stiffness matrix and its characteristics at typical configurations are calculated to reveal complicated coupling effects of diagonal and nondiagonal elements of the stiffness matrix. The concept of principle stiffness and coupled stiffness are proposed and their distributions over the workspace are predicted with numerical simulations in a quick manner. Then the stiffness of the platform is physically interpreted as a kinematically unconstrained rigid body suspended by six screw springs with equivalent spring constants and pitches through eigenscrew decomposition. The distributions of screw spring constants over the workspace are then plotted to demonstrate a duality property. At last, the effects of some design variables such as structural and dimensional parameters on system rigidity performance are investigated with the purpose of providing useful information for the structural design and performance enhancement of the parallel kinematic machine.

Decoupling of the Cartesian Stiffness Matrix: A Case Study on Accelerometer Design

2011 ◽  
Author(s):  
Ting Zou ◽  
Jorge Angeles
Keyword(s):  
Mechanical Properties ◽  
Finite Element Analysis ◽  
Stiffness Matrix ◽  
Screw Theory ◽  
Rotational Stiffness ◽  
Element Analysis ◽  
Cartesian Stiffness Matrix ◽  
Stiffness Matrices ◽  
Cartesian Stiffness

The 6 × 6 Cartesian stiffness matrix obtained through finite element analysis for structures designed with material and geometric symmetries may lead to unexpected coupling that stems from discretization error. Hence, decoupling of the Cartesian stiffness matrix becomes essential for design and analysis. This paper reports a numerical method for decoupling the Cartesian stiffness matrix, based on screw theory. With the aid of this method, the translational and rotational stiffness matrices can be analyzed independently. The mechanical properties of the decoupled stiffness submatrices are investigated via their associated eigenvalue analyses. The decoupling technique is applied to the design of two accelerometer layouts, uniaxial and biaxial, with what the authors term simplicial architectures. The decoupled stiffness matrices reveal acceptable compliance along the sensitive axes and high off-axis stiffness.

scholarly journals Rapid Evaluation for Position-Dependent Dynamics of a 3-DOF PKM Module

2014 ◽  
Vol 6 ◽  
pp. 238928 ◽  
Author(s):  
Hai-wei Luo ◽  
Hui Wang ◽  
Jun Zhang ◽  
Qi Li
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
Parallel Kinematic Machine ◽  
Computationally Efficient ◽  
Element Analysis ◽  
Reduction Techniques ◽  
Modal Reduction ◽  
Governing Differential Equation ◽  
Parallel Kinematic ◽  
Analytical System

Based on the substructure synthesis and modal reduction technique, a computationally efficient elastodynamic model for a fully flexible 3-RPS parallel kinematic machine (PKM) tool is proposed, in which the frequency response function (FRF) at the end of the tool can be obtained at any given position throughout its workspace. In the proposed elastodynamic model, the whole system is divided into a moving platform subsystem and three identical RPS limb subsystems, in which all joint compliances are included. The spherical joint and the revolute joint are treated as lumped virtual springs with equal stiffness; the platform is treated as a rigid body and the RPS limbs are modelled with modal reduction techniques. With the compatibility conditions at interfaces between the limbs and the platform, an analytical system governing differential equation is derived. Based on the derived model, the position-dependent dynamic characteristics such as natural frequencies, mode shapes, and FRFs of the 3-RPS PKM are simulated. The simulation results indicate that the distributions of natural frequencies throughout the workspace are strongly dependant on mechanism's configurations and demonstrate an axial-symmetric tendency. The following finite element analysis and modal tests both validate the analytical results of natural frequencies, mode shapes, and the FRFs.

Elastoplastic Analysis of Layered Metal Matrix Composite Cylinders—Part I: Theory

1996 ◽  
Vol 118 (1) ◽  
pp. 13-20 ◽  
Author(s):  
R. S. Salzar ◽  
M.-J. Pindera ◽  
F. W. Barton
Keyword(s):  
Metal Matrix Composite ◽  
Matrix Composite ◽  
Stiffness Matrix ◽  
Metal Matrix ◽  
Plastic Response ◽  
Solution Strategy ◽  
Composite Tubes ◽  
Elastic Plastic ◽  
Stiffness Matrices ◽  
Local Stiffness

An exact elastic-plastic analytical solution for an arbitrarily laminated metal matrix composite tube subjected to axisymmetric thermo-mechanical and torsional loading is presented. First, exact solutions for transversely isotropic and monoclinic (off-axis) elastoplastic cylindrical shells are developed which are then reformulated in terms of the interfacial displacements as the fundamental unknowns by constructing a local stiffness matrix for the shell. Assembly of the local stiffness matrices into a global stiffness matrix in a particular manner ensures satisfaction of interfacial traction and displacement continuity conditions, as well as the external boundary conditions. Due to the lack of a general macroscopic constitutive theory for the elastic-plastic response of unidirectional metal matrix composites, the micromechanics method of cells model is employed to calculate the effective elastic-plastic properties of the individual layers used in determining the elements of the local and thus global stiffness matrices. The resulting system of equations is then solved using Mendelson’s iterative method of successive elastic solutions developed for elastoplastic boundary-value problems. Part I of the paper outlines the aforementioned solution strategy. In Part II (Salzar et al., 1996) this solution strategy is first validated by comparison with available closed-form solutions as well as with results obtained using the finite-element approach. Subsequently, examples are presented that illustrate the utility of the developed solution methodology in predicting the elastic-plastic response of arbitrarily laminated metal matrix composite tubes. In particular, optimization of the response of composite tubes under internal pressure is considered through the use of functionally graded architectures.

Finite Element Analysis of Coupled Lateral and Torsional Vibrations of a Rotor With Multiple Cracks

2005 ◽  
Author(s):  
Xi Wu ◽  
Jerzy T. Sawicki ◽  
Michael I. Friswell ◽  
George Y. Baaklini
Keyword(s):  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Accurate Determination ◽  
Crack Model ◽  
Multiple Cracks ◽  
Torsional Vibrations ◽  
Element Analysis ◽  
Cosine Series ◽  
Six Degrees Of Freedom ◽  
Stiffness Matrices

The coupling between lateral and torsional vibrations has been investigated for a rotor dynamic system with breathing crack model. The stiffness matrix has been developed for the shaft element which accounts for the effect of the crack and all six degrees of freedom per node. Since the off-diagonal terms of the stiffness matrix represent the coupling of the respective modes, the special attention has been paid on accurate determination of their values. Based on the concepts of fracture mechanics, the variation of the stiffness matrix over the full shaft revolution is represented by the truncated cosine series where the fitting coefficient matrices are extracted from the stiffness matrices of the cracked shaft for a number of its different angular positions. The variation of the system eigenfrequencies and dynamic response of the rotor with two cracks have been studied for various shaft geometries, crack axial locations, and relative phase of cracks.

scholarly journals Hybrid Modelling; Mixed Rigid and Flexible Systems

1997 ◽  
Author(s):  
Ibrahim Esat ◽  
K. Banisoleiman
Keyword(s):  
Stiffness Matrix ◽  
Finite Size ◽  
Rigid Bodies ◽  
Element Formulation ◽  
Flexible Systems ◽  
Body System ◽  
Oscillatory Behaviour ◽  
Hybrid Modelling ◽  
Global Stiffness Matrix ◽  
Multi Body

The paper presents the Euler Newton formulation of oscillatory behaviour of a multi-body system interconnected by discrete stiffness elements. The dynamical system is treated as geometrically and materially linear, where assembly of the global stiffness matrix may be achieved. The formulation is extended to incorporate flexible shafting systems. It is assumed that flexible shafts are connected to rigid bodies of finite size and the connection is assumed to be built-in or pin jointed. Flexible shafting system is formulated using beam finite element formulation.

High Speed Constructional Method of Stiffness Matrix Based on Object Orientation

1992 ◽  
Author(s):  
H. Takahashi ◽  
S. Nagasawa ◽  
H. Sakuta
Keyword(s):  
Numerical Experiment ◽  
Conventional Method ◽  
Material Properties ◽  
Stiffness Matrix ◽  
High Speed ◽  
Object Oriented ◽  
Object Orientation ◽  
Stiffness Matrices ◽  
Local Coordinates ◽  
Global Stiffness Matrix

Abstract The paper deals with a method of high speed construction of stiffness matrix in FEM. The stiffness matrices of each element becomes to equal in following conditions. 1) Components of the B and D matrices are independent of global coordinate. 2) Elements have the same material properties and the same local coordinates respectively. This equality gives possibility of associable calculation to us. Hence, we can generate the global stiffness matrix at high speed by use of this feature. In this paper, a method of associable calculation by using equality of object ELEMENT based on object oriented concept is proposed. Futhermore, in comparison between conventional method and proposed one, relationship between accelerative rate and ratio of equal object and result of numerical experiment are described.

Dynamic Design for a Novel Parallel Kinematic Machine with Passive Linkage

2007 ◽  
Vol 364-366 ◽  
pp. 1037-1042
Author(s):  
Ying Hu ◽  
Bing Li ◽  
Dai Zhong Su ◽  
Hong Hu
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
Parallel Mechanism ◽  
Parallel Kinematic Machine ◽  
Element Analysis ◽  
Dynamic Design ◽  
Modeling Accuracy ◽  
Axis Parallel ◽  
Parallel Kinematic ◽  
Artificial Neural

Based on the proposed 4PUS-1RPU parallel mechanism a 5-axis Parallel Kinematic Machine (PKM) scheme has been developed and the dynamic characteristics of the PKM have been investigated in detail. To avoid the intensive computation caused by finite element analysis in the research, two typical metamodeling techniques of Response Surface Methodology (RSM) and its artificial neural network enhanced technique were employed as the robust design approaches. Comparing the results obtained from the direct RSM the modeling accuracy by the artificial neural network enhanced RSM can be improved.

ANALYSIS OF SPACE FRAMES WITH GENERALIZED SEMI-RIGID CONNECTIONS

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Shujin Duan ◽  
Zhiyue Li ◽  
Meixiang Liu ◽  
Xiaofeng Xie
Keyword(s):  
Mechanical Model ◽  
Stiffness Matrix ◽  
Element Analysis ◽  
Rigid Frame ◽  
Space Frames ◽  
The Matrix ◽  
Reaction Forces ◽  
Element Number ◽  
Rigid Joints ◽  
Independent Zero

A mechanical model and analytic method are proposed, in which, the axial, the shearing and the bending semi-rigid characteristics of space frames are taken into account. An independent zero-length connection element comprising six translational and rotational springs is used to simulate the beam-to-column connection. The model, namely six-spring mechanical model, has an advantage that the element number of structure does not increase. The matrix displacement method is used to analyze mechanism of the model, including element analysis and structural analysis. The stiffness matrix of the element is derived. Some reaction forces at the end of the element are obtained when it is subjected to two kinds of different loads respectively. The obtained stiffness matrix gets the characteristics of symmetry and singularity and that makes the size of total stiffness matrix for semi-rigid frame the same as that for frame with rigid joints.

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