KINETO-ELASTODYNAMIC ANALYSIS OF SPATIAL MECHANISMS
A Thesis submitted in partial fulfilment of the requirements for the award of the Degree of
DOCTOR OF PHILOSOPHY
V. K. MANNA
DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY
NEW DELHI-110016 August 1990
CERTIFICATE
This is to certify that the thesis entitled "KINETO- ELASTODYNAMIC ANALYSIS OF SPATIAL MECHANISMS", being submitted by V.K.Mahna to the Indian Institute of Technology, New Delhi, for the award of the degree of Doctor of Philosophy in Mechanical Engineering is a record of bonafide research work carried out by him. He has worked under my guidance and supervision and has fulfilled the requirements for the submission of this thesis, which has attained the standard required for a Ph.D. degree of the Institute.
The results contained in this thesis have not been submitted in part or in full elsewhere for the award of any degree or diploma.
(Dr.N.T.ASNANI), Professor,
Department of Mechanical Engineering, Indian Institute of Technology,
NEW DELHI-110016
C NT W E GEME 1\T S
I gratefully acknowledge the guidance of Dr.N.T.Asnani, Professor in the Department of Mechanical Engineering, Indian Institute of Technolgy, Delhi, the supervisor for this research endeavour, who initiated me to the area of kineto-elastodynamic analysis of mechanisms and directed me with remarkable adroitness and aplomb.
Prof.Asnani's keenness and interest were unflagging and he has all along been a source of great motivation and encouragement.
It gives me great pleasure to record my deep sense of gratitude to Dr.P.C.Dumir, Professor in the Department of Applied Mechanics, Indian Institute of Technology,Delhi, for the discussions I was privileged to have had with him on a number of occasions without count.
I further acknowledge the contribution of Dr.Y.V.S.R. Sastry, Professor and Head of the Department of Mechanical Engineering, Delhi College of Engineering, Delhi, who was ever forthcoming to share his profound knowledge and provided me with excellent facilities in the Computation Laboratory during the preparation of the manuscript.
I appreciatively acknowledge the efforts of Mr.H.L.Sharma in the Vibrations Laboratory and Mr.B.D. Sharma in the Workshop, Department of Mechanical Engineering, Indian Institute of Technology, Delhi during the experimental work.
Finally,I wish to express my gratitude to the Principal, Delhi College of Engineering, Delhi for having sponsored me under the Quality Improvement Programme of the Ministry of Human Resource Develop ment (Government of India) for this academic pursuit.
I
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(VINOD K. MAHNA)
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The objective of this research work entitled
"KINETO-ELASTODYNAMIC ANALYSIS OF SPATIAL MECHANISMS" is to develop a general procedure for the kineto-elastodynamic analysis of spatial mechanisms using the displacement finite element method. In the first place, rigid-body kinematic and force analysis of spatial mechanisms has been accomplished in an iteration-free procedure using the 4x4 transformation matrix approach. The method obviates the need of making any initial estimate of the unknown variables-an arduous ordeal for a 3-dimensional motion. The closed-form relations obtained, are made use of to provide the requisite data for the subsequent kineto-elastodynamic analysis.
Using the displacement finite element method (the stiffness technique of structural analysis) the kineto- elastodynamic formulations are accomplished based on Lagrange's equation. A multi-element model consisting of space-frame elements with six degrees of freedom at each nodal point, has been employed to idealize each flexible link. The effect of axial forces on the transverse vibrations of the link as well as the coupling terms (between rigid-body and elastic motion) have been duly reckoned with and subsumed into the system equations. Using an efficient assembly scheme, the elemental matrices are appropriately hooked up to the system matrices and the rigid-body degrees of freedom are quarantined
using the matrix decomposition method to preclude the complications arising out of the singularity of stiffness matrix.
A closed-form numerical algorithm has been used to an advantage to solve the kineto-elastodynamic equations using the modal analysis. Numerical integration techniques have also been employed to obtain solution of the kineto- elastodynamic equations. In-depth kineto-elastodynamic analysis in respect of skew slider-crank, skew 4-bar, 5-link (RCPCR) and 7-link (4R3P) mechanisms as the case-study
problems have been conducted to demonstrate the general nature and efficacy of the procedure. An experimental set-up for the skew slider-crank mechanism has been fabricated and experimented upon to establish the veracity of the analytical results. A conformity to a fairly good degree, between the analytical and experimental results has been achieved.
Abstract
List of Important Symbols List of Figures
CONTENTS
1.INTRODUCTION 1
1.1 Genesis of Kineto-Elastodynamic Analysis. 1 1.2 Review of the Past work. 4 1.3 Objectives of the Present work. 13 1.4 Organization of the Text. 14
2.DISPLACEMENT ANALYSIS OF SPATIAL MECHANISMS. 16
2.1 Introduction 16
2.2 Mobility Equation for Spatial Mechanisms. 19 2.3 4x4 Transformation Matrices. 20 2.4 General Formulations of Displacement Equations.24 2.5 Displacement Analysis of Basic Spatial 27
Mechanisms.
2.5.1 Input-Output Equations. 27 2.5.1 (a) Skew 4-Bar Mechanism 29 2.5.1 (b) Skew Slider-Crank Mechanism 31 2.5.2 Displacement Relations for the Coupler 34
of Basic Spatial Mechanisms.
2.6 Displacement Analysis of RCCC Mechanism 37 2.7 Displacement Analysis of 5-Link Mechanisms 41 2.7.1 RCPCR,RCPRC,RRPCC Mechanisms 41 2.7.2 RRCPC,RCRPC,RCCPR,RCRCP,RCCRP Mechanisms 46 2.7.3 RPCCR,RPRCC,RPCRC Mechanisms 51 2.8 Displacement Analysis of 6-Link Mechanisms 55 2.8.1 RPCPRR,RPRPCR,RPRPRC Mechanisms 55 2.8.2 RRRPPC,RCRPPR,RCRPRP,RRCPRP,RRRPCP, 59
RCRRPP Mechanisms
2.8.3 RPRCPR,RPRRPC,RPRCRP,RPCRRP Mechanisms 64 2.8.4 RCPPRR,RRPPRC Mechanisms 66 2.8.5 RPPCRR,RPPRRC Mechanisms 70 2.8.6 RRPRPC Mechanism 73 2.9 Displacement Analysis of 7-Link Mechanisms 74 2.9.1 RPPRRPR,RPPRRRP Mechanisms 74 2.9.2 RPRRRPP,RPRRPRP Mechanisms 78 2.9.3 RPRPRPR,RPRPRRP Mechanisms S1
2.9.4 RRPPPRR Mechanism 84 2.9.5 RPPPRRR Mechanism 86
2.9.6 RRRPPRP Mechanism 88
2.9.7 RRPRPPR Mechanism 90 2.9.8 RRPPRPR Mechanism 91 2.9.9 RPRPPRR Mechanism 95
3.VELOCITY,ACCELERATION AND FORCE ANALYSES OF SPATIAL 97 MECHANISMS
3.1 Introduction 97
3.2 Formulation of Equations 98 3.3 Absolute Linear Velocity and Acceleration 101 3.4 Angular Velocity and Angular Acceleration 104
of a link
3.5 Velocity and Acceleration Analyses of 106 Basic Spatial Mechanisms.
3.5.1 Velocity and Acceleration Analyses of 106 Output Link
3.5.2 Velocity and Accelration Analyses of 107 Coupler for Skew 4-Bar Mechanism
3.5.3 Velocity and Acceleration Analyses of 109 Coupler for Skew Slider-Crank Mechanism
3.6 Force Analysis of Spatial Mechanisms 110
3.6.1 Force Analysis of 5-Link RCPCR Mechanism 113 3.6.2 Force Analysis of 7-Link RRPPRPR 123
Mechanism
3.6.3 Force Analysis of Skew 4-Bar Mechanism 131 3.6.4 Force Analysis of Skew Slider-Crank 135
Mechanism
3.7 Results of Rigid-Body Analyses 138
4.RINETO—ELASTODYNAMIC ANALYSIS OF SPATIAL MECHANISMS 163 --FORMULATIONS
4.1 Introduction 163
4.2 Formulation of Governing Equations of Motion 169 for Kineto-Elastodynamic Analyses of Spatial
Mechanisms
4.3 Geometric Stiffening Effect and Geometric 182 Stiffness Matrix
4.4 Assembly Procedure for Elemental Matrices 185 4.5 Elimination of Rigid-Body Degrees of Freedom 191
4.6 Evaluation of Coupling Terms 196
5.SOLUTION OF KINETO-ELASTODYNAMIC EQUATIONS 203
5.1 Introduction 203
5.2 Imposition of Boundary Conditions and 205 Compatibility Requirements
5.3 Modal Analysis (Mode Superposition Method) 208 5.4 Solution of Uncoupled Second Order 219
Differential Equation Using Closed-Form Algorithm
5.5 Inclusion of Coupling Terms 232 5.6 Solution of Kineto-Elastodynamic Equations 238
Using Numerical Integration Techniques
(a) Newmark's Integration Method 238 (b) Generation of Damping Matrix 241 (c) Wilson-Theta Numerical Integration 242
Method
5.7 Quasi-Static Analysis 244
5.8 Evaluation of Strain-Vector 245
6.KINETO-ELASTODYNAMIC ANALYSIS OF SPATIAL MECHANISMS 247 --ANALYSIS AND EXPERIMENTAL RESULTS
6.1 Introduction 247
6.2 Kineto-Elastodynamic Analysis of Skew Slider 249 -Crank Mechanism
6.3 Experimental Analysis for Skew-Slider 259 Crank Mechanism
6.4 Kineto-Elastodynamic Analysis of Skew 4-Bar 292 Mechanism
6.5 Kineto-Elastodynamic Analysis o1 5-Link RCPCR 306 Mechanism
6.6 Kineto-Elastodynamic Analysis of 7-Link 4R3P 323 Mechanism
6.7 Recapitulation of the Entire Procedure for 338 Kineto-Elastodynamic Analysis
V_CONCLUSIONS 344
Tdentitiable Areas for Further Investigations 346
References 348
Appendices
Brief Bio-Data of the Author
