Studies on dynamic of rigidly/flexibly coupled pneumatic free-piston machines

STUDIES ON DYNAMICS OF RIGIDLY/FLEXIBLY COUPLED PNEUMAIOND FRSTON MACHINES

by

BAKSHI LAL SACHDEVA

A THESIS SUBMITTED

IN FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

1989

To my mother and wife

CERTIFICATE

This is to certify that the thesis entitled "Studies on Dynamics of Rigidly/Flexibly Coupled Pneumatic Free- Piston Machines" being submitted by Mr. Bakshi Lal Sachdeva to the Indian Institute of Technology, 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 our guidance and supervision and has fulfilled the requirements for the submission of this thesis.

The results contained in this Thesis have not been submitted in part or in full to any other University or Institute for the award of any degree or diploma.

(K ilash Nath Gupta) Professor

Department of Mechanical Engineering Indian Institute of Technology, Delhi New Delhi-110016.

ACKNOWLEDGEMENTS

It is with great pleasure that I wish to express my gratefulness to Dr. K.N. Gupta, Professor of Mechanical Engineering Department for supervising the work of this thesis. His inspiring guidance, valuable discussions and constant encouragement helped me to complete this work.

I also wish to thank Prof. J.S. Rao, who provided supervision at the inception stage of the project and Dr. E.

Raghavacharyulu, who continued to provide guidance and inspiration till he left for abroad in October, 1987.

I sincerely express my sense of gratitude to the Director, the Head of Mech. Engg. Deptt. and the Head of Computer Service Center for the permission and for the use of various facilities made available to me for completing the Ph.D work as a part-time Research Scholar.

I also acknowledge the receipt of whole-hearted co-operation from Mr. B.D. Sharma, Foreman, Mr. H.L. Sharma S.T.A. and Mr. K.N. Madhu, S.L.A. of the Departmental Workshop and Vibration and Instrumentation Research Laboratory, respectively, at various stages of the rig fabrication and experimental work. I thank all my friends who helped me at different stages of my research work.

Dated: July 21, 1989 (B.L. Sachdeva) Place: New Delhi

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ABSTRACT

The present dissertation reports the theoretical and experimental investigations carried out for the dynamic analysis of a free-piston pneumatic compressor driven by a Stirling engine or a cam mechanism. The drive system is considered as if imparting a sinusoidal or harmonic motion to the compressor cylinder. The cyclic variation of gas pressure in cylinders gives rise to non-linear differential gas force characteristic and hysteresis damping in the system, and thus it has led to the formulation of non-linear equations of motion. Method of Krylov and Bogoliubov or Ritz averaging method with the approximation of only first harmonic terms has been used to find out the steady-state solution to these equations.

For the dynamic analysis two versions of the system have been considered. In the first version the free-piston pneumatic machine is considered to be coupled to the drive system rigidly, while in the second version it is connected flexibly. The former version has resulted into a single- degree-of-freedom system, while the latter to a two-degree- of-freedom system, but both having non-linearity due to the presence of hysteresis damping and non-linear gas force characteristic.

A detailed theoretical study has been carried out on the first version of the system to obtain steady-state solutions by using method of Krylov and Bogoliubov, Ritz averaging method, equivalent linearization technique and

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numerical integration technique. With the help of first two methods expression for response of the system has been obtained in dimensionless form. Effect of variation of different parameters on the response has been studied and discussed. The numerical integration technique has shown that the response of the system is fairly harmonic and thus it has supported the approximation of first harmonic terms in the Ritz averaging method.

For validation of the results, an experimental rig has been designed, fabricated and developed. Experimental results have been obtained for different values of parameters like pressure ratio, inertial mass, spring stiffness and cam rise. These results are then compared and discussed in the light of theoretical results obtained.

In the second version of the system two variations have been considered. In the variation-I, cylinder motion of the pneumatic machine is unrestrained, while in the variation-II its motion is restrained by a spring. Steady-state solutions have been obtained in both cases by using the method of Krylov and Bogoliubov. Results are presented in dimensionless form. Effect of coupling stiffness and cylinder mass on response of the system are then studied and discussed.

General conclusions drawn from the whole study are listed at the end. Suggestions for future work are also indicated.

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CONTENTS

PAGE NO.

CERTIFICATE

ACKNOWLEDGEMENTS ii

ABSTRACT iii

NOMENCLATURE viii

LIST OF FIGURES

LIST OF PLATES xiii

CHAPTER-1 INTRODUCTION 1 CHAPTER-2 LITERATURE SURVEY 5 CHAPTER-3 MODELLING OF RIGIDLY COUPLED

FREE-PISTON PNEUMATIC MACHINE (SINGLE-DEGREE-OF-FREEDOM

SYSTEM MODEL) 22

3.1 INTRODUCTION 22

3.2 SYSTEM DESCRIPTION AND WORKING 23 3.3 MATHEMATICAL FORMULATION 27 3.4 METHOD OF SOLUTION 28 3.4.1. Krylov and Bogoliubov Method 28 3.4.2. Ritz Averaging Method 35 3.4.3. Method of Equivalent

Linearization 38 3.4.4. Time Response by Numerical

Integration 45

3.5 RESULTS AND DISCUSSION 46

3.6 CONCLUSIONS 65

CHAPTER-4 RIG DEVELOPMENT AND EXPERIMENTAL WORK 67

4.1 INTRODUCTION 67

4.2 RIG DESIGN 67

4.2.1. Conceptual Design 67 4.2.2. Selection of Material 68 4.2.3. Detailed Design, Selection and

Fabrication of Components. 70 4.3 RIG ASSEMBLY AND COMISSIONING 89 4.3.1. Foundation 89 4.3.2. Rig Assembly 90 4.3.3. Commissioning of Rig 93 4.4. EXPERIMENTAL PROCEDURE AND MEASUREMENT 95.

4.4.1. Instrumentation 95 4.4.2 Callibration of Instrumentation 100 4.4.3. Error Analysis 103 4.4.4. Experimental Procedure 104 4.5. RESULTS AND DISCUSSIONS 106

4.6. CONCLUSION 125

CHAPTER-5 MODELLING OF FLEXIBLY COUPLED FREE-PISTON PNEUMATIC MACHINE

(Two-Degree-of-Freedom System Model) 127 5.1. INTRODUCTION 127 5.2. MODEL OF VARIATION-I 127 5.2.1. System Description 127 5.2.2. Mathematical Formulation 129 5.2.3. Method of Solution 130

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5.3. MODEL OF VARIATION-II 139 5.3.1. System Description 139 5.3.2. Mathematical Formulation 139 5.3.3. Method of Solution 141 5.4. RESULTS AND DISCUSSION 149

5.4.1. Model of Variation-I 149 5.4.2. Model of Variation-II 158

5.5. CONCLUSIONS 166

CHAPTER-6 GENERAL CONCLUSIONS AND SUGGESTIONS

FOR FUTURE WORK 167

6.1

REFERENCES 171

APPENDIX - A 178

APPENDIX - B 191

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