The great evolution of the data-processing tools during the last years allowed for the development of the computer aided design in the field of mechanical structures. Controlling the clearance in joints between parts, is one of the required objectives to provide accurate relative movements and to minimize geometrical errors. For that purpose, a new method of static study allowing for the computation of the equilibrium positions of various elements in spatial mechanisms constituted by parallel joints and subjected to mechanical loadings is proposed. The isostatic study takes into account the presence of the clearance in the mechanism joints. The method is based to the minimization of the potential energy by means of some algorithms of optimization. The results obtained show the effectiveness of the method.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The method is demonstrated to analyze an isotropic beam undergoing bending when subjected to a distributed load.
A computational approach for evaluating helical compression springseSAT Journals
Abstract Helical compression springs are generally synthesized and evaluated by determining the maximum torsional stress, fatigue life, natural frequency, and/or load loss due to stress relaxation. To this end, researchers have developed finite element analysis (FEA) modeling methods to simulate the design performance of helical compression springs. The intent of this paper was to make a useful contribution to the published works for evaluating round wire helical compression springs. Specifically, commercially available FEA software was used to construct a structural model of a helical compression spring to simulate its full range of compression. The proposed FEA modeling methodology considers coil-to-coil contact and the end coils were modeled as rigid surfaces. With 9 mm of compression, the predicted spring rate correlated with the analytically calculated value to within 5%. Keywords: Helical compression spring, machine component design, spring FEA
Modeling and simulation of four bar planar mechanisms using adamsIAEME Publication
This document discusses the modeling and simulation of four-bar planar mechanisms using ADAMS software. It begins with an abstract describing four-bar linkages and modeling them in ADAMS. It then reviews relevant literature on modeling multibody systems and closed kinematic chains. Mathematical models of displacement, velocity and acceleration are developed. The modeling process in ADAMS is described, including defining geometry, joints and motions. Simulation results like angular position, speed and acceleration graphs are presented. It concludes ADAMS provides reliable results by considering link properties and eliminates errors from other methods.
Finite element analysis in orthodontics/ /certified fixed orthodontic courses...Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Isoparametric formulation allows for more accurate modeling of curved boundaries by mapping regular element shapes from a natural coordinate system to the actual curved geometric shapes in the global coordinate system. This mapping technique revolutionized finite element analysis by reducing unnecessary stress concentrations compared to using only straight-edged elements. The document goes on to define isoparametric, superparametric, and subparametric elements, and explains the basic theorems and uniqueness conditions for valid mappings between coordinate systems. It also describes Gaussian numerical integration for assembling stiffness matrices in isoparametric finite element analysis and provides some illustrative numerical examples.
This study compares experimental and finite element analysis results for stress analysis. Strain gauges were placed on test materials (a beam and contacting blocks) at locations corresponding to finite element mesh nodes. Testing involved applying loads and measuring strain. Results showed good agreement between experimental and numerical analyses for the linear beam problem. For the nonlinear contacting blocks problem, close placement of strain gauges was important due to high stress gradients at contact points. Small gauge placement errors could cause up to 10% difference in strain measurements. The approach demonstrated that matching strain gauge locations to finite element meshes facilitated accurate validation of numerical models.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document summarizes a finite element analysis of a torpedo battery tray conducted to evaluate its performance under severe vibration. The battery tray was modeled and meshed in ANSYS. Static, modal, harmonic and shock analyses were performed by applying loads in different axes. Results from the ANSYS simulation like deformation, stresses, natural frequencies and frequency response were compared to experimental test data. The maximum errors between simulation and experimental results for deformation and stresses were within 10%.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The method is demonstrated to analyze an isotropic beam undergoing bending when subjected to a distributed load.
A computational approach for evaluating helical compression springseSAT Journals
Abstract Helical compression springs are generally synthesized and evaluated by determining the maximum torsional stress, fatigue life, natural frequency, and/or load loss due to stress relaxation. To this end, researchers have developed finite element analysis (FEA) modeling methods to simulate the design performance of helical compression springs. The intent of this paper was to make a useful contribution to the published works for evaluating round wire helical compression springs. Specifically, commercially available FEA software was used to construct a structural model of a helical compression spring to simulate its full range of compression. The proposed FEA modeling methodology considers coil-to-coil contact and the end coils were modeled as rigid surfaces. With 9 mm of compression, the predicted spring rate correlated with the analytically calculated value to within 5%. Keywords: Helical compression spring, machine component design, spring FEA
Modeling and simulation of four bar planar mechanisms using adamsIAEME Publication
This document discusses the modeling and simulation of four-bar planar mechanisms using ADAMS software. It begins with an abstract describing four-bar linkages and modeling them in ADAMS. It then reviews relevant literature on modeling multibody systems and closed kinematic chains. Mathematical models of displacement, velocity and acceleration are developed. The modeling process in ADAMS is described, including defining geometry, joints and motions. Simulation results like angular position, speed and acceleration graphs are presented. It concludes ADAMS provides reliable results by considering link properties and eliminates errors from other methods.
Finite element analysis in orthodontics/ /certified fixed orthodontic courses...Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Isoparametric formulation allows for more accurate modeling of curved boundaries by mapping regular element shapes from a natural coordinate system to the actual curved geometric shapes in the global coordinate system. This mapping technique revolutionized finite element analysis by reducing unnecessary stress concentrations compared to using only straight-edged elements. The document goes on to define isoparametric, superparametric, and subparametric elements, and explains the basic theorems and uniqueness conditions for valid mappings between coordinate systems. It also describes Gaussian numerical integration for assembling stiffness matrices in isoparametric finite element analysis and provides some illustrative numerical examples.
This study compares experimental and finite element analysis results for stress analysis. Strain gauges were placed on test materials (a beam and contacting blocks) at locations corresponding to finite element mesh nodes. Testing involved applying loads and measuring strain. Results showed good agreement between experimental and numerical analyses for the linear beam problem. For the nonlinear contacting blocks problem, close placement of strain gauges was important due to high stress gradients at contact points. Small gauge placement errors could cause up to 10% difference in strain measurements. The approach demonstrated that matching strain gauge locations to finite element meshes facilitated accurate validation of numerical models.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document summarizes a finite element analysis of a torpedo battery tray conducted to evaluate its performance under severe vibration. The battery tray was modeled and meshed in ANSYS. Static, modal, harmonic and shock analyses were performed by applying loads in different axes. Results from the ANSYS simulation like deformation, stresses, natural frequencies and frequency response were compared to experimental test data. The maximum errors between simulation and experimental results for deformation and stresses were within 10%.
Fem /certified fixed orthodontic courses by Indian dental academy Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Indian dental academy provides dental crown & Bridge,rotary endodontics,fixed orthodontics,
Dental implants courses.for details pls visit www.indiandentalacademy.com ,or call
0091-9248678078
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
Simplified approach to consider cracking effect on the behavior of laterally ...Ahmed Ebid
DOI: 10.15680/IJIRSET.2015.0410015
Laterally loaded pile is a famous case of soil-structure interaction problem which was intensively studied by many researchers before. The techniques used to predict the behavior of laterally loaded piles were developed with increasing of the available computational capabilities from closed mathematical formulas to finite differences technique and finally linear finite elements technique. Recently, very sophisticated 3D elasto-plastic non-linear finite element models were used to accurately predict that behavior. Unfortunately, those sophisticated models are too complicated to be used in practical design. Hence, the aim of this research is to introduce a much simpler and practical approach to predict the behavior of the laterally loaded concrete piles considering the nonlinear effect of concrete cracking. Special calculating tool based on finite elements is developed to carry out a parametric study of the behavior of a set of 24 piles with different aspect ratios, reinforcement ratios, relative stiffness and head constrains. The validity of the calculating tool is checked against case history field tests. The results of the parametric study show three different failure modes according to the flexibility of the pile. Comparing the results with the formulas of ECP (202/4) shows the matching in the ultimate lateral capacity, while the ultimate lateral deformations are about (127 to 132%) of the code prediction.
Finite element analysis in orthodontics /certified fixed orthodontic courses ...Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Indian dental academy provides dental crown & Bridge,rotary endodontics,fixed orthodontics,
Dental implants courses.for details pls visit www.indiandentalacademy.com ,or call
0091-9248678078
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document discusses modal analysis of rotating structures using active magnetic bearings. It begins by introducing modal analysis and its importance for analyzing high-speed rotating machinery. It then describes the experimental setup which uses active magnetic bearings to both levitate and excite a test rotor. Natural frequencies and mode shapes are identified for the free-free rotor both with and without bearing stiffness. Introducing bearing stiffness is found to increase natural frequencies and introduce additional modes. The document concludes by explaining how the active magnetic bearings can be used to artificially excite the rotor to measure its frequency response functions and identify its dynamic characteristics.
Fea 1 /certified fixed orthodontic courses by Indian dental academy Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Indian dental academy provides dental crown & Bridge,rotary endodontics,fixed orthodontics,
Dental implants courses.for details pls visit www.indiandentalacademy.com ,or call
0091-9248678078
Contact is the principal method of applying loads between deformable solids,and therefore is present in a wide variety of mechanical components. In addition,contacts usually act as stress concentrations,and are thus probable locations for mechanical failure. Some of the most typical mechanical failures involving contact include:fretting,fretting fatigue,wear,fret ting wear and false brinelling. The contact of UIC - 60 and IRS - T12 rail - wheel has been analyzed by hand calculations to determine contact stresses. A three dimen sional finite element model of both type of rail - wheel contact is developed in ANSYS to compare to the typical hand - calculated s tresses. A vertical force is applied railway wheel & simulates the effects frictional surfaces. The results of the finite element model analysis contained herein are compared to hand calculations. Based on these results,a finite element analysis should be used if a greater level of detail is required for the analysis of the rail - wheel contact.
1) Finite element analysis is a numerical method used to solve engineering problems by breaking structures down into small discrete elements. It involves modeling structures as assemblies of simple geometric shapes called finite elements.
2) The key steps in finite element analysis include discretizing the structure into elements, selecting element types, defining displacement and strain/stress relationships within each element, deriving the element stiffness matrix, and assembling individual element equations into a system of equations for the overall structure.
3) Common approaches include the displacement method, which uses nodal displacements as unknowns, and the force method, which uses internal forces. The displacement method is typically more suitable for computational analysis.
Attitude Control of Quadrotor Using PD Plus Feedforward Controller on SO(3)IJECEIAES
This paper proposes a simple scheme of Proportional-Derivative (PD) plus Feedforward controller on SO(3) to control the attitude of a quadrotor. This controller only needs the measurement of angular velocity to calculate the exponential coordinates of the rotation matrix. With rotation matrix as an error variable of the controller, the simulation shows that the controller is able to drive the attitude of the quadrotor from hovering condition to desired attitude and from an attitude condition goes to the hovering condition, despite the system is disturbed. When the system is convergent, the rotation error matrix will be a 3 3 identity matrix.
This document discusses using genetic algorithms to optimize shape control of a cantilever beam using laminated piezoelectric actuators (LPAs). The modeling is based on Timoshenko beam theory and piezoelectricity. Variables like number of actuators, their size, location, and control voltage are optimized to minimize error between the achieved beam shape and desired shapes, which include higher order polynomial curves. Genetic algorithms as implemented in Matlab are used to calculate the optimal variable values for shape control.
Influence of joint clearance on kinematic and dynamic parameters of mechanismIOSR Journals
Kinematic joints for dynamic analysis of multi-body mechanical systems assumed ideal or perfect.
However, in a real mechanical kinematical joint clearance is always present. Such clearance is necessary for
the component assemblage and to allow the relative motion between the connected bodies. This clearance is
inevitable due to the manufacturing tolerances, material deformations, wear, and imperfections. The presence of
such joint clearance degrades the performance of mechanical systems in virtue of the contact and impact forces
that take place at the joint. Contact analysis is a computational bottleneck in mechanical design where the
contact changes. Manual analysis is time-consuming and prone to error. To address these problems, a
geometrical contact analysis method based on kinematic simulation, using CAD software is developed. An
equivalent kinematic linkage mechanism is constructed according to contact position of pin and hole assembly.
Results of kinematic and dynamic analysis of a four bar linkage with joint clearance shows that the contribution
of joint forces at slower input speed also degrades the performance of mechanism
Structural engineering r 16 regulationspavani reddy
The document outlines the academic regulations and course structure for the Structural Engineering program at Jawaharlal Nehru Technological University for batches admitted from 2016-2017. It details the subjects covered in each semester across the 4 semesters of the program. In the first year, the first semester covers topics such as advanced mathematics, theory of elasticity, matrix analysis of structures, and structural dynamics. The second semester covers finite element method, earthquake resistant design, theory of plates and shells, and CAD laboratory. The third semester involves a comprehensive viva-voce and seminar, as well as project work. The final semester focuses on the completion of the project work and a second seminar.
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANKASHOK KUMAR RAJENDRAN
This document contains a collection of practice problems related to finite element analysis of two-dimensional vector variable problems, including axisymmetric problems. The problems cover derivation of element stiffness matrices and strain-displacement matrices for various element types under different conditions, calculation of element stresses and displacements, modeling of cylinders under pressure, and determination of global stiffness matrices for structures. The elements and conditions include constant strain triangles, linear strain triangles, axisymmetric triangles, plane stress, plane strain, and shells.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Analysis of Four-Bar Linkages Model using Regressioninventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
IRJET- Kinematic Analysis of Planar and Spatial Mechanisms using MatpackIRJET Journal
This document discusses kinematic analysis of planar and spatial mechanisms using computational methods in MATLAB. It develops a MATLAB package called MATPACK for numerical analysis of planar and spatial mechanisms. It uses vector notation to analyze planar mechanisms and Denavit-Hartenberg parameters to analyze spatial mechanisms. Results for velocities and accelerations are obtained from MATPACK and compared to theoretical results. The objective is to introduce existing notation and methods to analyze spatial mechanisms using computational tools like MATPACK, AutoCAD, and CATIA.
Modeling, simulation & dynamic analysis of four bar planarIAEME Publication
This document discusses modeling, simulating, and analyzing the dynamic forces of a four-bar planar mechanism using CATIA V5 software. The document begins with an introduction to four-bar mechanisms and their importance. It then describes the mathematical modeling of displacement, velocity, and acceleration analysis of four-bar linkages. Next, it explains how to model a four-bar mechanism using different CATIA tools. The document presents results of the simulation in CATIA including graphs of link angle, speed, and acceleration over time. It concludes that CATIA allows simulation of link motion at different positions and validation of analytical equations, providing a valuable tool for mechanism analysis and design optimization.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The method is demonstrated to analyze an isotropic beam undergoing bending when subjected to a distributed load.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The analysis shows this finite element approach can accurately model the bending behavior of beams under distributed loads.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The method is demonstrated to analyze an isotropic beam undergoing bending when subjected to a distributed load.
Fem /certified fixed orthodontic courses by Indian dental academy Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Indian dental academy provides dental crown & Bridge,rotary endodontics,fixed orthodontics,
Dental implants courses.for details pls visit www.indiandentalacademy.com ,or call
0091-9248678078
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
Simplified approach to consider cracking effect on the behavior of laterally ...Ahmed Ebid
DOI: 10.15680/IJIRSET.2015.0410015
Laterally loaded pile is a famous case of soil-structure interaction problem which was intensively studied by many researchers before. The techniques used to predict the behavior of laterally loaded piles were developed with increasing of the available computational capabilities from closed mathematical formulas to finite differences technique and finally linear finite elements technique. Recently, very sophisticated 3D elasto-plastic non-linear finite element models were used to accurately predict that behavior. Unfortunately, those sophisticated models are too complicated to be used in practical design. Hence, the aim of this research is to introduce a much simpler and practical approach to predict the behavior of the laterally loaded concrete piles considering the nonlinear effect of concrete cracking. Special calculating tool based on finite elements is developed to carry out a parametric study of the behavior of a set of 24 piles with different aspect ratios, reinforcement ratios, relative stiffness and head constrains. The validity of the calculating tool is checked against case history field tests. The results of the parametric study show three different failure modes according to the flexibility of the pile. Comparing the results with the formulas of ECP (202/4) shows the matching in the ultimate lateral capacity, while the ultimate lateral deformations are about (127 to 132%) of the code prediction.
Finite element analysis in orthodontics /certified fixed orthodontic courses ...Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Indian dental academy provides dental crown & Bridge,rotary endodontics,fixed orthodontics,
Dental implants courses.for details pls visit www.indiandentalacademy.com ,or call
0091-9248678078
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document discusses modal analysis of rotating structures using active magnetic bearings. It begins by introducing modal analysis and its importance for analyzing high-speed rotating machinery. It then describes the experimental setup which uses active magnetic bearings to both levitate and excite a test rotor. Natural frequencies and mode shapes are identified for the free-free rotor both with and without bearing stiffness. Introducing bearing stiffness is found to increase natural frequencies and introduce additional modes. The document concludes by explaining how the active magnetic bearings can be used to artificially excite the rotor to measure its frequency response functions and identify its dynamic characteristics.
Fea 1 /certified fixed orthodontic courses by Indian dental academy Indian dental academy
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and offering a wide range of dental certified courses in different formats.
Indian dental academy provides dental crown & Bridge,rotary endodontics,fixed orthodontics,
Dental implants courses.for details pls visit www.indiandentalacademy.com ,or call
0091-9248678078
Contact is the principal method of applying loads between deformable solids,and therefore is present in a wide variety of mechanical components. In addition,contacts usually act as stress concentrations,and are thus probable locations for mechanical failure. Some of the most typical mechanical failures involving contact include:fretting,fretting fatigue,wear,fret ting wear and false brinelling. The contact of UIC - 60 and IRS - T12 rail - wheel has been analyzed by hand calculations to determine contact stresses. A three dimen sional finite element model of both type of rail - wheel contact is developed in ANSYS to compare to the typical hand - calculated s tresses. A vertical force is applied railway wheel & simulates the effects frictional surfaces. The results of the finite element model analysis contained herein are compared to hand calculations. Based on these results,a finite element analysis should be used if a greater level of detail is required for the analysis of the rail - wheel contact.
1) Finite element analysis is a numerical method used to solve engineering problems by breaking structures down into small discrete elements. It involves modeling structures as assemblies of simple geometric shapes called finite elements.
2) The key steps in finite element analysis include discretizing the structure into elements, selecting element types, defining displacement and strain/stress relationships within each element, deriving the element stiffness matrix, and assembling individual element equations into a system of equations for the overall structure.
3) Common approaches include the displacement method, which uses nodal displacements as unknowns, and the force method, which uses internal forces. The displacement method is typically more suitable for computational analysis.
Attitude Control of Quadrotor Using PD Plus Feedforward Controller on SO(3)IJECEIAES
This paper proposes a simple scheme of Proportional-Derivative (PD) plus Feedforward controller on SO(3) to control the attitude of a quadrotor. This controller only needs the measurement of angular velocity to calculate the exponential coordinates of the rotation matrix. With rotation matrix as an error variable of the controller, the simulation shows that the controller is able to drive the attitude of the quadrotor from hovering condition to desired attitude and from an attitude condition goes to the hovering condition, despite the system is disturbed. When the system is convergent, the rotation error matrix will be a 3 3 identity matrix.
This document discusses using genetic algorithms to optimize shape control of a cantilever beam using laminated piezoelectric actuators (LPAs). The modeling is based on Timoshenko beam theory and piezoelectricity. Variables like number of actuators, their size, location, and control voltage are optimized to minimize error between the achieved beam shape and desired shapes, which include higher order polynomial curves. Genetic algorithms as implemented in Matlab are used to calculate the optimal variable values for shape control.
Influence of joint clearance on kinematic and dynamic parameters of mechanismIOSR Journals
Kinematic joints for dynamic analysis of multi-body mechanical systems assumed ideal or perfect.
However, in a real mechanical kinematical joint clearance is always present. Such clearance is necessary for
the component assemblage and to allow the relative motion between the connected bodies. This clearance is
inevitable due to the manufacturing tolerances, material deformations, wear, and imperfections. The presence of
such joint clearance degrades the performance of mechanical systems in virtue of the contact and impact forces
that take place at the joint. Contact analysis is a computational bottleneck in mechanical design where the
contact changes. Manual analysis is time-consuming and prone to error. To address these problems, a
geometrical contact analysis method based on kinematic simulation, using CAD software is developed. An
equivalent kinematic linkage mechanism is constructed according to contact position of pin and hole assembly.
Results of kinematic and dynamic analysis of a four bar linkage with joint clearance shows that the contribution
of joint forces at slower input speed also degrades the performance of mechanism
Structural engineering r 16 regulationspavani reddy
The document outlines the academic regulations and course structure for the Structural Engineering program at Jawaharlal Nehru Technological University for batches admitted from 2016-2017. It details the subjects covered in each semester across the 4 semesters of the program. In the first year, the first semester covers topics such as advanced mathematics, theory of elasticity, matrix analysis of structures, and structural dynamics. The second semester covers finite element method, earthquake resistant design, theory of plates and shells, and CAD laboratory. The third semester involves a comprehensive viva-voce and seminar, as well as project work. The final semester focuses on the completion of the project work and a second seminar.
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANKASHOK KUMAR RAJENDRAN
This document contains a collection of practice problems related to finite element analysis of two-dimensional vector variable problems, including axisymmetric problems. The problems cover derivation of element stiffness matrices and strain-displacement matrices for various element types under different conditions, calculation of element stresses and displacements, modeling of cylinders under pressure, and determination of global stiffness matrices for structures. The elements and conditions include constant strain triangles, linear strain triangles, axisymmetric triangles, plane stress, plane strain, and shells.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Analysis of Four-Bar Linkages Model using Regressioninventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
IRJET- Kinematic Analysis of Planar and Spatial Mechanisms using MatpackIRJET Journal
This document discusses kinematic analysis of planar and spatial mechanisms using computational methods in MATLAB. It develops a MATLAB package called MATPACK for numerical analysis of planar and spatial mechanisms. It uses vector notation to analyze planar mechanisms and Denavit-Hartenberg parameters to analyze spatial mechanisms. Results for velocities and accelerations are obtained from MATPACK and compared to theoretical results. The objective is to introduce existing notation and methods to analyze spatial mechanisms using computational tools like MATPACK, AutoCAD, and CATIA.
Modeling, simulation & dynamic analysis of four bar planarIAEME Publication
This document discusses modeling, simulating, and analyzing the dynamic forces of a four-bar planar mechanism using CATIA V5 software. The document begins with an introduction to four-bar mechanisms and their importance. It then describes the mathematical modeling of displacement, velocity, and acceleration analysis of four-bar linkages. Next, it explains how to model a four-bar mechanism using different CATIA tools. The document presents results of the simulation in CATIA including graphs of link angle, speed, and acceleration over time. It concludes that CATIA allows simulation of link motion at different positions and validation of analytical equations, providing a valuable tool for mechanism analysis and design optimization.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The method is demonstrated to analyze an isotropic beam undergoing bending when subjected to a distributed load.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The analysis shows this finite element approach can accurately model the bending behavior of beams under distributed loads.
This document summarizes research on using finite element analysis in time to model the nonlinear dynamic response of beams. It describes discretizing a beam into finite elements, with each node having two degrees of freedom for displacement and slope. Hermite shape functions are used to interpolate displacement within each element. The total potential energy of the system is derived and used to form the element stiffness matrix. The method is demonstrated to analyze an isotropic beam undergoing bending when subjected to a distributed load.
NON LINEAR DYNAMIC AND STABILITY ANALYSIS OF BEAM USING FINITE ELEMENT IN TIME IAEME Publication
In this article the main focus is to predict nonlinear dynamic response of a beam using finite element in time under given condition. To do the nonlinear dynamic analysis of a beam, a distributed load is being applied and the beam is experiencing bending. The given beam is homogeneous in composition and isotropic in nature. Here, considering the stiffness of the beam and its effect on the deflection, under the distributed load conditions
Dynamics Behaviour of Multi Storeys Framed Structures by of Iterative Method AM Publications
Dynamics refers to the branch of mechanics that deals with the movement of objects and the forces that drive that movement. Structural analysis which covers the behaviour of structures subjected to dynamic (actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, and the frequency content of the load. One analysis technique for calculating the linear response of structures to dynamic loading is a modal analysis. In modal analysis, we decompose the response of the structure into several vibration modes. A mode is defined by its frequency and shape. Structural engineers call the mode with the shortest frequency (the longest period) the fundamental mode. This paper presents a study on mode shape, inertia force, spring force and deflection of multi storied framed structures by comparison of stodola’s and Holzer method. This study involves in examination of theoretical investigations of multi storied framed structures. Overall four storey multi storied framed structures and two methods were analysed & comparison of all the mode shape, inertia force, spring force and deflection at the critical cross-section with same configuration loading by keeping all other parameters constant. The theoretical data are calculated using code IS 1893, IS 4326, IS 13920. The all storey mass and stiffens are analysed under the cantilever condition. The research project aims to provide which method is most accuracy to find the mode shape, spring force deflection and inertia force. The studies reveal that the theoretical investigations Stodola’s method is most accuracy compare to the Holzer method. The maximum mode shape, spring force, spring deflection and inertia force is 87.29%, 80 %, 89% and 72% is higher the Stodola’s method compare than Holzer method in same configuration.
Estimating damping in structure made of different m aterials (steel,brass,aluminum) and processes sti ll remains as one of the biggest challengers. All mate rials posses certain amount of internal damping,wh ich manifested as dissipation of energy from the system . This energy in a vibratory system is either dissipated into heat or radiated away from the syst em. Material damping or internal damping contribute s to about 10-15% of total system damping. Cantilever beams of required size & shape are prepared for experimental purpose & damping ratio is investigate d. Damping ratio is determined by half-power bandwidth method. It is observed that damping ratio is higher for steel than brass than aluminum.
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This document describes the design and analysis of a planar positioning stage based on a redundantly actuated parallel linkage. The positioning stage has three degrees of freedom (x-y-θ) and uses a 3-PRPR linkage with six actuators. The document analyzes the kinematics and workspace of the linkage. It describes the procedure for static analysis to calculate the actuated joint torques. The positioning stage could enable high-precision motion with a compact design and improved stiffness compared to serial linkages. It is proposed for potential use in micro-scale applications.
This document describes the design and analysis of a planar positioning stage based on a redundantly actuated parallel linkage with six degrees of freedom (three translations and three rotations). The kinematics and workspace analysis of the linkage are presented. A static analysis method to calculate the actuator torques required for a given end-effector force and trajectory is also described. MATLAB programs were developed to analyze the workspace and perform the static analysis. The results show that the redundant actuation can help improve the workspace characteristics and prevent singular configurations compared to non-redundant parallel manipulators. The stage design has potential applications in micro-positioning.
An innovative fea methodology for modeling fastenerseSAT Journals
This document presents an innovative finite element analysis (FEA) methodology for modeling fasteners in bolted joints. The methodology models the actual geometry of fasteners as force-controlled rigid surfaces, which is computationally efficient while considering contact between mating parts. The methodology was demonstrated on a clevis bracket bolted to a rigid wall. Results from the FEA model correlated well with analytical calculations, with surface pressure predictions within 13.5% and bending stress predictions within 5.2-8.8%. This innovative methodology provides an efficient and accurate way to model bolted joints in FEA.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document discusses a method for simulating interactions between objects in mechanisms using small-scale interference detection. It represents the contour shapes of objects using molecular models, where molecules prevent overlap between driver and driven objects. It can analyze and synthesize planar kinematic pairs. Examples show it can simulate gears, cams, ratchets, and generateva mechanisms. The method is generalized and automatically optimizes shapes based on functionality plots and molecular time plots. Local shape changes are shown to affect the simulation results.
A two-stage closed-form planetary gear train dynamic model is presented with 3 sentences:
The model considers translational and rotational displacements of all components in both stages of the gear train. Natural frequencies and vibration modes are determined by formulating the dynamic equations into a matrix form and calculating the eigenvalues and eigenvectors. Parameter effects like planet number and coupling stiffness on natural modes are analyzed to better understand the vibration characteristics.
This document provides a critical review of research on structural dynamic modification of beam structures. It discusses the origins and developments in structural dynamic modification techniques from the late 1970s through the 2000s. Key developments include the use of finite element models, estimation of rotational degrees of freedom, consideration of damping effects, optimization approaches, sensitivity analysis, and both direct and inverse modification problems. The review covers applications of structural dynamic modification to real structures like engines and machine tools to improve dynamic behavior.
The performance evaluation is one of most important issues in the analysis and design of parallel manipulators.
Characteristics such as manipulability and minimum singular value are used to determine the performance of the manipulators. The performance indices are used to eliminate the singularity and it’s near configurations. In this paper 6-UPS spatial parallel manipulator is considered and its performance indices such as condition number, manipulability and minimum singular value are determined for different structures.
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THREE-DIMENSIONAL STUDY FOR THE RELATIVE POSITIONING OF MECHANICAL ELEMENTS IN MECHANISMS CONSTITUTED BY PARALLEL JOINTS WITH CLEARANCES
1. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
1
THREE-DIMENSIONAL STUDY FOR THE RELATIVE
POSITIONING OF MECHANICAL ELEMENTS IN
MECHANISMS CONSTITUTED BY PARALLEL JOINTS
WITH CLEARANCES
Mohamad Younes1
, Philippe Dal Santo2
, Alain. Potiron2
1
Lebanese University, University Institute of Technology, Saida, Lebanon
2
Ecole Nationale Supérieure d'Arts et Métiers, Laboratoire LAMPA Arts et Métiers Paris
Tech Angers, 2 boulevard du Ronceray, BP 3525, 49035 Angers Cedex, France
ABSTRACT
The great evolution of the data-processing tools during the last years allowed for the development of the
computer aided design in the field of mechanical structures. Controlling the clearance in joints between
parts, is one of the required objectives to provide accurate relative movements and to minimize geometrical
errors. For that purpose, a new method of static study allowing for the computation of the equilibrium
positions of various elements in spatial mechanisms constituted by parallel joints and subjected to
mechanical loadings is proposed. The isostatic study takes into account the presence of the clearance in the
mechanism joints. The method is based to the minimization of the potential energy by means of some
algorithms of optimization. The results obtained show the effectiveness of the method.
KEYWORDS
Mechanism Analysis, Joint Clearance, Optimization
1. INTRODUCTION
Between bodies assembly constituting a machine, clearance in the joints are necessary to ensure
the relative movements of links. Unfortunately, it is that presence of clearance which causes
mechanical vibrations, noise and inaccuracy in the relative movement of multi-links mechanism.
Some numerical methods were developed in the C.A.D field in order to control the joints
clearance and to minimize geometrical errors of position in the mechanism.
Nevertheless, it appears that a rather small number of researches were carried out in this field.
The main part of the works concerning the computation of the relative positions of mechanism
elements linked by joints with clearance, are carried out from a strict geometrical point of view.
Potiron et al. [1] proposed a new method of static analysis in order to determine the arrangement
of the various components of planar mechanisms subjected to mechanical loadings. The study
takes into account the presence of linkage clearance and allows for the computation of the small
variations of the parts position compared to the large amplitude of the movements useful for the
power transmission.
2. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
2
It appears that a rather small number of research tasks were carried out in this particular field.
Funabashi et al. [2] tackled the problem by carrying out a dynamic, theoretical and experimental
study of some simple mechanisms. In order to specify the influence of the clearance in the links
on machine operations, they derived the equations of the movement of links including parts
stiffnesses, viscous friction and Coulomb's friction in joints. The results are interesting for the
specific models suggested but they don’t lead to a general usable method suited for the study of
complex mechanisms.
A model of mechanism with joint's clearance was defined by Giordano et al. [3] when researching
the dimensional and geometrical tolerances associated with machine elements. The method is
based on the definition of small rigid-body displacements and the use of closed loops equations
for the associated kinematic chains.
With Giordano and Duret [4], they developed a mathematical model and give a geometrical
interpretation of the gaps by defining a «hyperspace of gaps». By means of this «gap-space», it is
possible to define the compatibility of the tolerances assigned to the manufactured parts.
However, the study didn't take into account neither the geometrical defects of the parts nor the
deformations and feature variations resulting from the applied loads.
To improve the quality of manufactured products and reduce their total cost, Gao et al. [5] and
Chase et al. [6] have developed a method for the tolerance analysis of two and three-dimensional
mechanical assemblies. This method is carried out by a direct linearization of a geometrical non-
linear problem. It was implemented in a commercial C.A.D. code, in order to extract from the
results, acceptable tolerances and the dimensions of the related parts.
In the same topic, Chase and Parkinson [7] presented an outline on recent research in the analysis
of mechanical tolerances, from which it is possible to have an idea of how to handle the study of
the joints' clearance in mechanisms.
Flores and Ambrosio [8] presented a computational methodology for dynamic analysis of
multibody mechanical systems with joint clearance. The model for planar revolute joints is based
on a thorough geometric description of contact conditions and on a continuous contact force
model, which represents the impact forces. It is shown that the model proposed lead to realistic
contact forces. These forces correlate well with the joint reaction forces of an ideal revolute joint,
which correspond to a joint with zero clearance.
Turner et al. [9] proposed a methodology for constructing spherical four-bar mechanisms with an
emphasis on utilizing simpler machining processes and part geometries. By building each link out
of easily created pieces instead of a single complex shape, a mechanism can be quickly
prototyped and tested.
Liu et al. [10] presented a method concerned with the determination of the normal force-
displacement relation for the contact problem of cylindrical joints with clearance. A simple
formulation for this contact problem is developed by modeling the pin as a rigid wedge and the
elastic plate as a simple Winkler elastic foundation.
Thomas et al. [11] proposed a new notation for kinematic structures which allows a unified
description of serial, parallel, and hybrid robots or articulated machine tools. They tried to fill this
gap by presenting a new notation, which is based on the graph representation known from gear
trains.
3. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
3
In the works of Fischer [12], the ball joint often referred to as a spherical or ‘S’ joint is modeled
using dual-number coordinate-transformation matrices. The joint consists of concave and convex
spherical surfaces engaged to prevent translations but allowing three degrees of freedom, all of
which are rotations.
Hsieh [13] has proposed a method allowing for the kinematic description of mechanisms
containing prismatic, revolute, helical and cylindrical joints to be explicitly defined, it cannot be
directly applied to mechanical systems containing spherical pairs. Accordingly, he proposed an
extended D-H notation which allows the independent parameters of any spatial mechanism,
including one with spherical pairs, to be derived for analysis and synthesis purposes. The validity
of the proposed notation is demonstrated via its application to the analysis of mechanisms
containing revolute, spherical, cylindrical and prismatic joints.
In the study of Erkaya and Uzmay [14] a dynamic response of mechanism having revolute joints
with clearance is investigated. A four-bar mechanism having two joints with clearance is
considered as a model of mechanism. A neural network was used to model several characteristics
of joint clearance. Kinematic and dynamic analyses were achieved using continuous contact mode
between journal and bearing. A genetic algorithm was also used to determine the appropriate
values of design variables for reducing the additional vibration effect due primarily to the joint
clearance.
In this work, a general quasi-static method is proposed to determine the relative positions of
mechanisms links with joints clearance.
Being given a geometrical position resulting from the great amplitude of movements in the
parallel mechanism, it will be possible to compute the equilibrium positions of the various parts
by taking into account the clearance in the linkages. The main idea is to define and minimize an
objective function which, in the present case, is the potential energy of the mechanism. This is
carried out by taking account of the geometrical constraints imposed by the clearance on
infinitely small displacements in the joints.
During these studies, several simplifying assumptions will be formulated:
- the joints in the mechanism are carried out with clearances,
- the solids are not deformable,
- the solids are geometrically perfect, i.e. the shape defects due to machining tolerances are
ignored,
- the gravity force is neglected compared to others applied forces.
2. MECHANISM MODELING ASSUMPTIONS
2.1. Examples of real mechanisms
In various mechanical devices such as pumps, hydraulic motors, hydraulic tubes, etc. each
component is linked to others by means of different joints (spherical, cylindrical, prismatic, etc.).
In each of them there exists some clearance which can perturb the movements or (and) reduces
the relative position accuracy between the links.
Some mechanisms are represented in the following figures which exhibit several components
linked together with some common joints.
4. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
4
Pump housing (0), plunger (1), shaft (2), cylinder barrel (3), hydraulic distributor (4)
Figure 1. Two mechanical devices with common joints
2.2. Clearance space modeling
In Figure 1 above, it is supposed that some clearance with values exists in the different joints.
In the study, the real parts are associated with a joint space. Each part is replaced by lines without
transverse dimensions and the joint is replaced by a geometrical space with dimensions equal to
clearance. The distances between joints on the same part are conserved. An example is given in
Figure 2 below.
Figure 2. Clearance modeling
The sketch in Figure 2 and the others presented in the following expand the joint clearance
compared to the other dimensions of the mechanism and they do not take into account the scale of
the figure.
3. BASIC THEORY FOR THE MECHANISM WITH PARALLEL
JOINTS
3.1. Rigid-body small displacements
The mechanism is composed of two parts S1 and So linked together by two joints l1 and l2. The
points A1 and A2 appearing in equation (1), lie on the center line of the link. They coincide
respectively with the two points O1 and O2 "centers" of the joints l1 and l2 in the initial position.
Hydraulic tube
Hydraulic
pump
Spherical
clearance
Cylindrical
clearance
5. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
5
OXYZ is the global reference frame with its origin O. O1X1Y1Z1 and O2X2Y2Z2 are the local
reference frames of the joints with origins O1 and O2. O1 is currently chosen to coincide with O.
In the absence of great amplitude movements, the relative motion of a body S1 compared to a
body S0 in the center Ai (i=1,2) of the joint li is represented by the following kinematic vectorial-
set { })S/S( 01iτ :
{ } { } ii A01A01i01i )S/S(),S/S()S/S( VΩ=τ (1)
)S/S( 01iΩ and )S/S( 01Ai
V are respectively the rotation vector and the displacement vector
of S1 at point Ai of S1 relative to the origin Oi of the joint li. In the case of three-dimensional
study, )/( 01 SSiΩ and )/( 01 SSiAV are written in the form of:
γ
β
α
=
i
i
i
01i )S/S(Ω (2.a) and
=
i
i
i
01A
w
v
u
)S/S(i
V (2.b)
The unknowns parameters of the problem are the components ui, vi, wi, αi, βi, et γi, of the two
vectors )S/S( 01iΩ and )S/S( 01Ai
V .
Remark
In the following, for infinitely small rotation θi, sinθi ≈ tanθi ≈ θi. and cosθi ≈ 1.
3.2. Potential energy
The theorem of potential energy, as stated by Germain and Muller in [15], reads:
For a given problem, among all cinematically admissible-fields (C’) the real displacement field is
the one that minimizes the potential energy of the system.
The potential energy V(C’) of a body, resulting from a cinematically admissible displacement-
field is defined by:
∫∫∫ ∫∫−−= dsU.TdvU.f)'C(W)'C(V '
kk
'
kk (3)
W(C') is the strain energy of the structure, fk is the component of the external volume force, Tk is
the component of the surface force being exerted on the system and )M(U'
k is the component of
the admissible displacement of any point M of the system.
It is this property of the real displacement field which will be exploited in the proposed
optimization method.
3.3. Relationships between the design variables
The kinematic composition law for the relative motions allowed by each of the joints implies that:
6. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
6
{ } { })S/S()S/S( 012011 ττ = (4)
In equality (4), each kinematic vectorial-set should be reduced to the same point.
The small displacements { } 1
)/( 011 A
SS and { } 2
)/( 011 A
SS are respectively written as:
{ }
=
γ
β
α
τ
1
1
1
1w
1v
1u
A011 1
)S/S( (a) and { }
=
γ
β
α
τ
2
2
2
2w
2v
2u
A011 2
)S/S( (b) (5)
The development of (4) gives:
)S/S()S/S( 012011 ΩΩ = (6)
and
)S/S()S/S()S/S( 01101A01A i2
ΩΛ+= 12OOVV (7)
In the initial position of mechanism, the points A1 and A2 coincide respectively with O1 and O2.
The two equalities (6) and (7) give six linear equations linking the design variables.
4. OPTIMIZATION METHOD FOR THE RESEARCH OF RELATIVE
PARTS' ARRANGEMENTS
The proposed method computes the values of the small displacements, characterizing the
movements authorized by the presence of clearance in the joints, and determines their
components (displacements along the three axes and rotations with respect to these axes).
The optimization problem deals with the minimization of the potential energy of the system and
accounts for the geometrical, mechanical and technological constraints imposed by the design of
the machine. A mathematical optimization-algorithm is used in order to compute the values of the
small displacements in the joints which are the design variables of the problem. Mathematically,
the optimization problem can be formulated in the form of:
Minimize V(x) (8a)
Subjected to the following optimization constraints:
0)(g j ≤x j = 1,..,m (8b)
hk(x) = 0 k = 1,..,n (8c)
x is the vector of the design variables which are the components ui, vi, wi, αi, βi, and γi, of the two
vectors )S/S( 01iΩ and )S/S( 01Ai
V with i = 1, 2.
gj(x) and hk(x) are respectively the inequality and equality constraints of the problem. In this
study, they arise from the geometry of the linkages, the displacement's limits of particular points
and from the closing loops' equations (6) and (7).
7. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
7
5. OPTIMIZATION PROCESS
The optimization process can be sketched in the following diagram:
Figure 3. Optimization Algorithm
The purpose of this study is the computation of the equilibrium positions of various elements in
spatial mechanisms with parallel joints subjected to mechanical loadings. The values of the
design variables being initialized, the initial values of the objective function and the constraints
are calculated.
Concerning the search for the optimal solution of the optimization problem, it will be noticed that
the majority of optimization methods involve mathematical calculation of the gradients of the
objective-function and constraints, by means of the design parameters. This can be achieved by a
sensitivity analysis of the objective function and the constraints when the design parameters vary,
leading to the convergence of the optimization process towards the optimal solution. The
optimization method implemented is the sequential quadratic programming method [16].
The algorithm is based upon an iterative process in which, at each stage, the design parameters
take new values, allowing for the convergence towards the optimal solution. In the case of non-
convergence, new values are assigned to the design variables and a new iteration is carried out.
This process is repeated until convergence is reached.
Calculation of the Potential Energy
and the imposed constraints
Sensitivity Analysis of the Potential
Energy and the constraints
Optimization of design variables by
a Mathematical Optimization Code
yes
No
Stop
New values of
design variables
convergence
?
Initial values of design variables
8. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
8
6. MECHANISMS WITH CLEARANCES IN THE PARALLEL
JOINTS
6.1. Mechanism with a spherical joint and a cylindrical joint
The mechanism is composed of two parts, a shaft S1 and a reference body So linked together by
two joints l1 and l2. l1 is a spherical joint and l2 is a cylindrical joint. L2 is the length of the
cylindrical joint. J1 and J2 are respectively the joint clearances of l1 and l2. A real arrangement of
such mechanism can be viewed in Figure 2.
The points A1 and A2 lie on the center line of the shaft. They coincide respectively with O and O2
in the initial position. OXYZ is the global reference frame with its origin O and O2X2Y2Z2 the
local reference frame of the cylindrical joint of center O2. The scheme of this mechanism is given
by figure 4:
Figure 4. Clearance model of spherical and cylindrical joints
The design variables of the problem are:
- the displacements u1, v1 and w1 of points A1 along X, Y and Z axis
- the rotations α1, β1 and γ1 of the shaft at points A1 around the X, Y and Z axis
- the displacements u2, v2 and w2 of points A2 along X2, Y2 and Z2 axis
the rotations α2, β2 and γ2 of the shaft at points A2 around the X2, Y2 and Z2 axis.
6.1.1. Objective-function
A torque C and a force F applied at point B on the shaft are the components of the mechanical
loading. The potential energy is expressed as:
V(C') = - (F.UB + C.B) (9)
UB and B, respectively the displacement vector and the rotation vector of point B, are calculate
by means of { } iA01i )S/S(τ in a same manner as equation (7).
6.1.2. Limits of design variables
The values of the design variables are limited by the geometry of the joints. They are
characterized by the following inequalities:
Y
X
Z
O
So
S1
y2
x2
23
z2
O
2
9. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
9
a. Spherical joint
2
1J
1u
2
1J
≤≤− (a);
2
1J
1v
2
1J
≤≤− (b);
2
1J
1w
2
1J
≤≤− (c) (10)
∞≤α≤∞− 1 (a); ∞≤β≤∞− 1 (b); ∞≤γ≤∞− 1 (c) (11)
b. Cylindrical joint
∞≤≤∞− 2u (a);
2
2J
2v
2
2J
≤≤− (b);
2
2J
2w
2
2J
≤≤− (c) (12)
∞≤α≤∞− 2 (a);
2L
2J
2
2L
2J
≤β≤− (b);
2L
2J
2
2L
2J
≤γ≤− (c) (13)
6.1.3. Linear geometric constraints
That kind of constraints appears when the contacts between the shaft and the joints are
accounted for and when closing the kinematic vectors chain.
a. Cylindrical joint
For that joint, in the O2x2y2 plane, the displacements along the axis y2 are limited by
upper and lower contacts shaft/cylinder of the points with abscises L2/2 and –L2/2. They
are respectively written in explicit form by:
2
2L
22v γ+ (14a)
2
2L
22v γ− (14b)
These displacements should not exceed the boundaries
−
2
2,
2
2
JJ
, involving the two
following linear inequalities:
2
2J
2
2L
22v
2
2J
≤γ+≤− (15a)
2
2J
2
2L
22v
2
2J
≤γ−≤− (15b)
For the same joint but in the O2x2z2 plane, we obtain:
2
2J
2
2L
22w
2
2J
≤β+≤− (16a)
2
2J
2
2L
22w
2
2J
≤β−≤− (16b)
b. Kinematic equation
Now, for each kind of joint, the explicit form of the kinematic closing loop (6) and (7)
leads to the six linear independent equations:
10. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
10
{}0
2
2
2
1
1
1
=
−
γ
β
α
γ
β
α
(17a)
{}0
1
1
1
1w
1v
1u
2w
2v
2u
=
Λ−
−
γ
β
α
12OO (17b)
6.1.4. Non-Linear geometric constraints
These constraints are due to the limited movement of the shaft in the cylindrical or spherical
clearance space. It leads to the following quadratic inequalities:
2
2
1J2
1w2
1v2
1u0
≤++≤ (18)
2
2
2J
2
2
2L
22w
2
2
2L
22v0
+
≤β−γ+≤ (19)
2
2
2J
2
2
2L
22w
2
2
2L
22v0
+
≤β+γ−≤ (20)
6.1.5. Shaft loading
The shaft can be loaded in several ways, including forces and torques applied at various points.
Three cases of shaft mounting will be studied, each one being subjected two various loading
conditions. The numerical applications will allow to validate the formulation.
6.1.6 Spherical and cylindrical joints arrangement
The spherical joint is placed at the origin O and the clearance value J1 is 0.2mm. The cylindrical
joint at the other end has a length L2=40mm and a clearance value J2 = 0.2mm. The distance
A1A2 = 200mm on X axis. In unloaded position, A1 coincides with O.
6.1.6.1 First loading case
The shaft is loaded at right and out of the cylindrical joint by a force
=
N1
N1
N1
F . The distance A2B
will have no influence on the optimal relative position.
The method allows finding the optimal solution of the displacements and rotations of the shaft at
points A1 and A2. The numerical results obtained at the end of the optimization process, are
reported in the following vectors sets (the first column refers to the displacements (mm) and the
second to the rotations of the shaft (rad)):
11. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
11
{ }
=γ
=β
=α
=
=
=
=
rad104.684034
rad10-4.684034
rad10-1.562943
mm10-3.233806w
mm10-3.233808v
mm8.892973u
)/S(S
4-
1
4-
1
306-
1
2-
1
2-
1
2-
1
A011
10
1
τ
{ }
=γ
=β
=α
=
=
=
=
rad104.684034
rad10-4.684034
rad10-1.524142
mm106.134262w
mm106.134259v
mm8.892973u
)/S(S
4-
2
4-
2
306-
2
2-
2
2-
2
2-
2
A012
10
2
τ
From these values, it follows that the displacements of the center A1 of the spherical joint satisfy
the relation mm1.0
2
1J2
1w2
1v2
1u ==++ . Thus, the final position of point A1 is located on the
sphere with a radius equal to the clearance value.
Concerning the point A2, it moves along the three axes x2, y2 and z2 with rotations β2 and γ2
between the boundaries - J2/L2 and J2/L2. The rotation α2 is negligible. In addition, the linear
constraints given in the relations (15) and (16) respect their boundaries - J2/2 and J2/2:
mm10 2071066.7
2
2L
22v −=γ+ , mm10 25.197453
2
2L
22v −=γ− ,
mm105.197456 2
2
L
22w −=β+ , mm107.071069 2
2
L
22w −=β− .
Furthermore, since the value of nonlinear constraint (19) is equal to the upper boundary:
2
2
2
2J
2
2
2L
22w
2
2
2L
22v mm01.0=
=
+
β−γ+ , then the contact between the shaft and
the right side of the cylindrical joint is assured. But the nonlinear constraint (20) does not achieve
their upper boundart:
2
2
2J23
2
2
2L
22w
2
2
2L
22v mm10402706.5
=
+
<−β+γ− , so the
left side of the cylindrical joint has no contact with the shaft as indicated in figure 5.
Figure 5. Displacement of the shaft after loading
Now if the force values are different from 1N, it will not be obvious to give an accurate prediction
of the exact position of the parts. A purely geometrical calculation is no more possible for the
Y
X
Z
O
So
S1
y2
x2
23
z2
O
2
=
N
N
N
1
1
1
F
B
A1
A2
12. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
12
computation of the relative displacements values because the components of the force will
influence the result.
In the case where spherical joint has a clearance very large compared to that of cylindrical joint
(J1 = 3 mm, J2 = 0.1 mm and L2 = 40 mm), the optimization algorithm converges to optimal
values of displacements and rotations of shaft in the points A1 and A2 given respectively in
{ } 1A011 )S/S(τ and { } 2A012 )S/S(τ :
{ }
=
rad101.76776694
rad101.76776695-
rad105.25281282-
mm0.35355339-
mm0.35355338-
mm1.41421356
)/S(S
3-
3-
16-
A011 1
τ
{ }
=
rad101.76776694
rad101.76776695-
rad105.25281282-
mm103.20706891
mm103.20688982-
mm1.41421356
)/S(S
3-
3-
16-
10-
10-
A012 2
τ
The nonlinear constraints have the following values:
2
2
1J22
1w2
1v2
1u mm2.250000
==++
2
2
2J23
2
2
2L
22w
2
2
2L
22v mm102.50000
==
+
−β−γ+
2
2
2J23
2
2
2L
22w
2
2
2L
22v mm102.50000
==
+
−β+γ−
Since the nonlinear constraints reach their upper boundaries, the point A1 is located on the sphere
of the spherical joint and there are two contacts between the shaft and the cylindrical joint.
Figure 6. Contact of the shaft with the joints
Y
X
Z
O
So
S1
y2
x2
23
z2
O
2
=
N
N
N
1
1
1
F
B
A1 A2
13. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
13
6.1.6.2 Second loading case
The dimensions and clearances of the first case are unchanged (J1 = 0.2 mm, J2 = 0.2 mm, L2 = 40
mm and A1A2 =200mm) but the shaft is loaded at its middle point B by a force
=
N1
N1
N0
F .
After computations and convergence of the algorithm, the optimum values are reported below:
{ }
=
rad101.65050781
rad101.65058153
rad101.59740463-
mm107.07106781
mm107.07106780
mm9.09985988
)/S(S
13-
13-
274-
2-
2-
259-
A011
10
1
τ
{ }
=
rad101.65051596
rad101.65058234
rad101.59740463-
mm107.07106781
mm107.07106781
mm109.09985988
)/S(S
13-
13-
274-
2-
2-
259-
A012 2
τ
Results show that the shaft moves only in the plane OYZ without any rotation. The schematic
view of the mechanism is presented in figure 7.
Figure 7. Shaft position after loading
The optimal values of design variables show that the shaft moves in a parallel direction to the x
axis without moving along this axis. Since the clearance values of joints are equal, then there is
no rotation of shaft. The final position of point A1 is located on the sphere considering that
2
J
10071.7071.7 122 ==+ . The final position of point A2 lies on the circle with a radius
2
2J
because
2
2J2
2w2
2v2
2u =++ . Since the clearances in two joints are equal and the load is
parallel to the plane OYZ, { } 1A011 )/S(Sτ and { } 2A012 )/S(Sτ are identical.
If the clearances in two joints are different, the two following cases are presented:
J1 > J2 and J1 < J2. The other dimensions and the load case are unchanged.
a- Case 1: J1 > J2
F
Y
X
Z
O
So
S1
y2
x2
23
z2
O
2
A
1
A2
14. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
14
The clearance in spherical joint is modified from 0.2mm to 0.3mm. The algorithm converges to
the following optimum values:
{ }
=
rad101.96418919-
rad101.96418180
rad101.25554283-
mm0.10606598
mm0.10606604
mm105.40044426-
)/S(S
4-
4-
15-9-
A011 1
τ
{ }
=
rad101.96418919-
rad101.96418180
rad101.25554283-
mm106.67823498
mm106.67822643
mm105.40044426-
)/S(S
4-
4-
15-
2-
2-
9-
A012 2
τ
Since J1 > J2, the optimal values of displacements of A1 are greater than those of A2 and there is
only a rotation of the shaft relative to the two axes y2 and z2.
The point A1 is located on the sphere of the spherical joint because
2
1J2
1w2
1v2
1u =++ . In
addition, 2
2
2
2J23
2
2
2L
22w
2
2
2L
22v mm01.0mm10901235,7 =
<=
+
−β−γ+ and
2
2
2
2J
2
2
2L
22w
2
2
2L
22v mm01.0=
=
+
β+γ− . This shows that there is only one
contact between the shaft and the left side of the second joint.
b-Case 2: J1 < J2
In this case, the clearance of the spherical joint is equal to 0.1mm. After computations and
convergence of the algorithm, the numerical values which have been obtained, are reported in the
kinematic vectorial-sets:
{ }
=
rad101.60706086
rad101.60706086-
rad101.67486499-
mm103.53553390
mm103.53553390
mm103.56393699-
)/S(S
4-
4-
17-
2-
2-
14-
A011 1
τ
{ }
=
rad101.60706086
rad101.60706086-
rad101.67486499-
mm106.74965563
mm106.74965563
mm103.56393699-
)/S(S
4-
4-
17-
2-
2-
14-
A012 2
τ
The optimal values of displacements of A1 are smaller than those of A2 and there are only
rotations of the shaft relative to the two axis y2 and z2.
Since 23
2
2
1J2
1w2
1v2
1u mm102.50000 −==++
,
22
2
2
2J
2
2
2L
22w
2
2
2L
22v mm10000.1 −β−γ+ =
=
+
and
15. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
15
2
2
2J23
2
2
2L
22w
2
2
2L
22v mm10264463.8
≤=
+
−β+γ− , the conclusions obtained,
for this case, are:
- The point A1 is located on the sphere of the spherical joint.
- There is only one contact between the shaft and the right side of cylindrical joint.
The effectiveness of the method is thus verified.
6.2. Mechanism with two spherical joints
The mechanism has two parts, a shaft S1 and a reference body So linked together by two spherical
joints. A real arrangement of such mechanism can be viewed in Figure 8:
Figure 8. Rod with two spherical joints
In the model, the points A1 and A2 at the ends of the shaft are the centers of the spheres on the
shaft. They coincide respectively with O and O2 in the initial position. OXYZ is the global
reference frame with its origin O and O2X2Y2Z2 (the local reference frame of the second spherical
joint of center O2). The scheme of this mechanism is given by Figure 9:
Figure 9. Clearance model of mechanism with two spherical joints
The components of the displacements of points A1 and A2 and the rotation of the shaft in the So
frame are the design variables. As in the preceding example, the optimization problem is carried
out in several cases of shaft loading.
Y
X
Z
O
Y2
X2
Z2
So
S1
O2
16. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
16
As in the preceding example, an optimization problem is raised allowing the evaluation of the
design variables. Several cases of loading will be now considered.
6.2.1. First case of two spherical joints
For the numerical study, the clearance values are chosen to be 0,3 mm in the first joint and
0,6 mm in the second joint.
The center O2 of the second sphere has as Cartesian coordinates .
The shaft is loaded at its middle point B by a force
=
N1
N1
N1
F .
At the end of the optimization process, the optimum values of displacements and rotations are
reported below:
{ }
=
rad101.24552477-
rad102.49104927
rad101.24552440-
mm50.10250242
mm108.54799757
mm106.84574801
)/S(S
4-
4-
4-
2-
2-
A011 1
τ
{ }
=
rad101.24552477-
rad102.49104927
rad101.24552440-
mm102.86046329
mm40.13530094
mm70.26774142
)/S(S
4-
4-
4-
3-
A012 2
τ
From these values, it follows that the displacements of each center of the spherical joints satisfy
the relation
2
iJ2
i
w2
i
v2
i
u =++ (i = 1, 2). Thus, the final positions of the points A1 and A2 are
located on the two spheres of clearance as shown in Figure 10:
Figure 10. Displacements of the shaft after loading
6.2.2. Second case of two spherical joints
We suppose in this section that the two joints have the same clearance value equal to 0.3 mm. The
other dimensions and the load case are unchanged.
F =
1
1
1
N
N
N
B
Y
X
Z
O
Y2
X2
Z2
S
o
S
1
A1
A2
O2
A2
17. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
17
{ }
=
rad103.20236638
rad102.13479904
rad101.06758281
mm108.66025403
mm108.66025403
mm108.66025403
)/S(S
15-
15-
15-
2-
2-
2-
A011 1
τ
{ }
=
rad103.20236229
rad102.13481067
rad101.06757756
mm108.66025403
mm108.66025403
mm108.66025403
)/S(S
15-
15-
15-
2-
2-
2-
A012 2
τ
The results show that { } 1A011 )S/S(τ has the same value of { } 2A012 )S/S(τ . In addition, there
are no rotation (numerical values are negligible) of the shaft which is logical.
After computations and convergence of the algorithm, the schematic view of the mechanism is
presented in figure 11.
Figure 11. Shaft position after loading
Since
2
iJ2
i
w2
i
v2
i
u =++ , the final positions of the points A1 and A2 are located on the spheres
defined by the clearance values J1 and J2.
The effectiveness of the method is thus established knowing that the computation of the
optimization algorithms must be reliable.
6.3. Mechanism with two cylindrical joints
The mechanism has two parts, a shaft S1 and a reference body So linked together by two
cylindrical joints l1 and l2.
A real arrangement of such mechanism can be viewed in Figure 12.
S1 BY
X
Z
O
Y2
X2
Z2
So
A1
A2
O2
F =
1
1
1
N
N
N
18. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
18
Figure 12. Hydraulic tube with two cylindrical joint
Ji and Li are respectively the joint clearance and the length of joint li (i = 1, 2). The points A1 and
A2 are the centers of the joints. They coincide respectively with O and O2 in the initial position.
OXYZ is the global reference frame with its origin O and O2X2Y2Z2 (the local reference frame of
the cylindrical joint of center O2). The scheme of this mechanism is given by figure 13:
Figure 13. Clearance model of two cylindrical joints
6.3.1. First case of two cylindrical joints
We take a two identical joints with a clearance value J1 = J2 = 0,2 mm and a length
L1 = L2 = 40 mm. The force is applied in the middle of OO2 with
=
N1
N1
N0
F . The distance
between the origins is OO2 = 100 mm.
After computations and convergence of the algorithm, the numerical values are reported in the
following kinematic vectorial-sets:
{ }
=
rad101.20620232-
rad101.20633278-
rad101.99239567-
mm107.07106797
mm107.07106765
mm104.50527452
)/S(S
10-
10-
306-
2-
2-
306-
A011 1
τ
{ }
=
rad101.20620231-
rad101.20633282-
rad101.99554887-
mm107.07106917
mm107.07106644
mm104.51218348
)/S(S
10-
10-
306-
2-
2-
306-
A012 2
τ
y
2
x2
z2
O
2
Y
X
Z
O
So
S1
19. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
19
According to the optimal values, the nonlinear constraints of problem are limited as follows:
22
2
2
1J22
2
2
1L
11w
2
2
1L
11v0 mm101.000000mm101.000000 −≤−β−γ+≤ =
=
+
22
2
2
1J22
2
2
1L
11w
2
2
1L
11v0 mm101.000000mm101.000000 −≤−β+γ−≤ =
=
+
22
2
2
2J22
2
2
2L
22w
2
2
2L
22v0 mm101.000000mm101.000000 −≤−β−γ+≤ =
=
+
22
2
2
2J22
2
2
2L
22w
2
2
2L
22v0 mm101.000000mm101.000000 −≤−β+γ−≤ =
=
+
Since
2
iJ2
i
w2
i
v =+ , the final positions of the points A1 and A2 are located on the circles
defined by the clearance values J1 and J2. The results show that the shaft moves only along the Y
and Z axis. In addition, there is no rotation of the shaft.
Figure 14. Displacement of the shaft after loading
6.3.2. Second case of two cylindrical joints
A transverse torque Cz = 1 Nm is applied in B to the right of the joint l2, along the X axis. In this
section, the value of the first joint clearance is four times that of the second connection
(J1 = 0,4 mm and J2 = 0,1 mm) but the length of the second joint is greater than that of the first
(L1 = 40 mm and L2 = 60 mm). The other dimensions and load cases are similar to the previous
case.
After computations and convergence of the algorithm, the numerical values are reported in the
kinematic vectorial-sets:
{ }
=
rad101.66666666
rad105.53813565
rad101.82228772
mm105.96041716
mm60.16666666-
mm103.20084144-
)/S(S
3-
8-
308-
5-
308-
A011 1
τ
y2
x
23
z2
O
2
Y
X
Z
O
So
S1
20. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
20
{ }
=
rad101.66666666
rad105.53813565
rad103.13762951
mm105.40660360
mm108.13538007-
mm103.35496565-
)/S(S
3-
8-
308-
5-
19-
308-
A012 2
τ
From the numerical values, the equilibrium position of shaft is given on the following figure 16:
Figure 15. Displacements of the shaft after loading
This equilibrium position of the shaft is normal because
2121
21
2
2
OO2LL
JJ
L
J
++
+
= . This causes
rotations with respect to the Z axis equal to: rad10666.1
OO2LL
JJ
L
J
21
3
2121
21
2
2 −
=
++
+
==γ=γ .
Only in this case, there are three contacts between the shaft and the two mechanism joints.
In the following, consider the case where
2121
21
2
2
OO2LL
JJ
L
J
++
+
< . The clearance J1 is increased to
the value 0.5 mm. Results show that the displacement v1 and the rotations γ1 et γ2 are similar to
the previous case. Other unknowns are also negligible.
The values of nonlinear constraints are similar to those of the previous result. The difference is in
the upper boundaries of the constraints concerning the left joint. They are varied from 4 10-2
mm2
to 6.25 10-2
mm2
. In this case, these constraints are lower than the upper boundaries.
It is found that the shaft is in contact with the two circles of the second joint boundary while there
is no contact with the first joint as presented in the following figure:
Figure 16. Displacements of the shaft after loading
y2
x
23
z2
O
2
Y
X
Z
O
So
S1
Cz
y2
x2
z2
O
2
Y
X
Z
O
So
S1
Cz
21. Mechanical Engineering: An International Journal ( MEIJ), Vol. 1, No. 1, May 2014
21
7. CONCLUSION
The search of maximum accuracy for optimum machine performances requires the control of the
clearance in the mechanical joints between the different components. This could be obtained by
determining the small variations of the parts position subjected to mechanical loadings.
Optimization technique allows the development of new tools for engineers and designers in order
to predict and quantify the effects of the joints' clearance on the geometrical performances of
mechanisms. It has been shown that without a general method, it's difficult, even impossible, to
manually compute the small relative-displacements values induced by the joints' clearance.
This paper has presented a comprehensive method for modeling and analyzing the position of
bodies in three-dimensional mechanical assemblies in the case of mechanism with clearance in
the parallel joints.
Therefore, we propose to study and calculate the relative positions of mechanical elements using
the minimization of the total potential energy in parallel mechanism.
The deformations of the bodies are not considered in the actual formulation, but it can be easily
accounted for, by calculating the strain energy of each structural element.
The solution of this problem must use a mathematical optimization-algorithm computing the
values of the small displacements in the joints, which are the design variables. Several examples
have been presented to evaluate the effectiveness of the formulation. The results obtained from
several simulations show the effectiveness of the method.
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