THE DESIGN OF LINEAR SERVO-MECHANISMS HAVING PRESCRIBED TRANSIENT RESPONSES WITH AN EXPERIMENTAL INVESTIGATION INTO TILE PREDICTION OF TRANSIENT RESPONSE FROM FREQUENCY RESPONSE. PART I . REVIEW OF BASIC THEORY. Page No. Chapter 1. Introduction. 1 2. Transient Analysis of Linear Systems. 7 3. Frequency Analysis of Linear Systems. 25 4. Stability Criteria. 38 5. Criteria of Relative Stability. Design Procedures. 57 PART I I . THE DESIGN OF LINEAR SERVO-MECHANISMS HAVING PRESCRIBED TRANSIEN T RESPONSES. Introduction. 79 A Chapter 6. The Design of ^?(p) Transfer Functions. 80 7. Numerical Work Illustrating the Use of the (A) Design Charts. 107 6i 8. The Design of g(p) Transfer Functions. 122 9. Use of th e 7>tp) Design Charts. Comparison of Qo(p) and* . (P) Methods. 133 &l er ProQuest Number: 13838566 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 13838566 Published by ProQuest LLC(2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 PART I I I . EXPERIMENTAL INVESTIGATION OF TRANSIENT AND FREQUENCY RESPONSES OF METADYNE SERVO-MECHANISM. Page No. In tro d u ctio n . 145 Chapter 10. Description of Apparatus and Methods of T esting. 148 11. Experimental Step and FrequencyResponses. Step Responses calculated from Frequency Responses. 167 12. Discussion of Results. 179 13. Conclusion. Further Work. 202 BIBLIOGRAPHY. Appendix I. The Exponential Fourier Series for a Periodic Function. Appendix II, Tables to Assist in the Calculation of Nyquist Diagrams a t Real and Complex Frequencies. 1 1 Appendix III. Tables of “ sin no and " cos no1 . Appendix IV. Harmonic A nalysis Method o f obtaining Step Response from Frequency Response. The Design Charts pertaining to Part II of the Thesis are separately bound. THE DESIGN OF LINEAR SERVO-MECHANISMS HAVING PRESCRIBED TRANSIENT RESPONSES, WITH AN EXPERIMENTAL INVESTIGATION INTO THE PREDICTION OF TRANSIENT RESPONSE FROM FREQUENCY RESPONSE. Introductory Note. Towards the end of 1946, considerable variety existed in the technical literature on the subject of servo-mechanisms. With the intention of reviewing this literature and thus obtaining guidance for future work, the author, in October 1946, began a programne of study and research in the Electrical Engineering Department. Shortly after this, the volume of published work on servo-mechanisms released at Conferences in the post-war years, and appearing also in text-book form, increased to an extent which made it quite clear that the review originally planned could not be carried out in a work of this nature. The author, therefore, turned to the more restricted problems of the design of servo-mechanisms with specified transient responses and the correlation of frequency and transient response in an experimental systan. At the same time, it was thought that these might be prefaced by a review of the basic theory of the subject, and more particularly of the relationship of relative stability to the transient response. The Thesis, therefore, is divided into three Parts. Part I contains a review of the basic theory. Chapters 4 and 5 of this Part give a full discussion of stability and relative stability not to be found elsewhere in the literature of the subject, and some points, which it is believed, are not so well-known. Part II deals with the specific problem of designing for a prescribed overshoot in the step-function response. The whole of this Part is original and follows up the idea, due to Campbell, of the principal mode representation of the error quantity. Part III gives the description and results of an experimental investigation into the transient and frequency responses of a metadyne-controlled servo-mechanism. The work was carried out during the period October 1946 to October 1951 in the Electrical Engineering Laboratory. The author wishes to thank Professor Bernard Hague, D.Sc., for his permission to use the equipment contained therein and for his encouragement to the author while in pursuance of the work. The author also wishes to acknowledge the benefit derived from a short visit made in 1948 to the Electrical Engineering Department, University of Birmingham and to thank Professor A. Tustin for kindly permitting this. PART I . REVIEW OF BASIC THEORY. 1 - CHAPTER I . INTRODUCTION1. Part I of this thesis provides a review of the basic theory governing the operation and design of servo-mechanisms. Following a general description of the closed-sequence type of control, the development of servo-mechanism theory is outlined. Transient and frequency analyses are dealt with in the succeeding chapters and a discussion of stability is then given. The Part concludes with a consideration of relative stability and design procedure, 1,1, Closed-Sequence Control Systems, A broad division of control systems can be made according to whether the control is of the open-sequence or closed-sequence type, the fundamental difference between the two being the effective cause ■which operates the system. In the first case, the quantity operating the system in no way depends on the value of the controlled quantity, but is fixed in size and variation merely by previous design. In the second case, the effective cause operating the system is the difference between a prescribed quantity and another quantity which is proportional to the actual value of the controlled quantity of the system, with the direction of operation so arranged that this difference always tends to zero. The schematic diagram of a closed-sequence control system is therefore as given by Fig. 1. - 2 - <t>-k(p e,-% Control (If Control a t^L System -%^-f System k<i> & Proportional Proportional Element - k Element ~ / Fig.l. Closed-sequence Fig.2. Angular oosition control control system. s ervo-me cha ni sm. In the diagram,-^ is the controlled quantity, e.g. voltage, temperature, speed, etc. It is measured by some means which produces a proportional q u an tity kf for comparison with <f>, the quantity corresponding to the desired value of the controlled variable. The value of (f> will in general not be % pre-determined. The difference e is now arranged to move V' in such a way that c tends to zero, that is, that 'f' tends to which is the desired value of the output quantity. The sub-division of closed-sequence control sjrstems usually designated by the term servo-mechanisms refers to such systems in which the quantities <j) and ^ are of a mechanical nature and a one-to-one relationship is desired between them. Agreement on this definition is, however, by no means universal. If this statement is adopted, however, the diagram of Fig. 2 results. The prescribed value of the controlled quantity or output quantity 0O, is given by 0• and is designated the input quantity. The result of the ^ I f (p is pre-determined, some type of regulator results. In the case of <f> being constant, it is usually referred to as the datum quantity of the regula tor. 3 - control action is that 60 tends to the value QL under both steady-state and changing conditions. Such a system oocurs, for example, in the following by a gun-mounting of its prescribed aiming angle as provided by the angle of rotation of a light computing shaft. This particular problem of fire-control is one of remote positioning of a large mass. Such angular position controls form the great bulk of servo-mechanism applications, and it has become customary to represent the input and)^ z \ / ^ . * 1 \ quantities, therefore, as the shaft angles and 60 . The adoption of closed-sequence control immediately confers two main advantages compared with the more simple open-sequence type. These are, firstly, the accuracy of correspondence of output and input which is obtained, and secondly, the speed of response of the output when following a changing input. Both these advantages are secured merely by inserting sufficient amplification into the main control sequence from the error quantity to the output quantity. It is, of course, fundamental that power amplification be present in order to move an output having inertia, and possibly resisted by external torques. While, in the open-sequence type of control, aocuracy can only be achieved by a power amplification which is both high and constant, a closed-sequence control system only requires that it should be high. In practice, It is impossible for an open-sequence control to retain its initial calibration in the face of normal temperature and load charges. It is, therefore, essential that fast and accurate control systems be of the closed-sequence type, and it is with the subdivision of servo-mechanisms that this thesis is concerned, although little extension is required to cover the general case. - 4 The advantages mentioned in the preceding paragraph are not, however, obtained without sacrifice in the simplicity and operation of the system. In particular, self-excited oscillations requiring no external input signal, will usually appear unless steps have otherwise been taken to counteract such behaviour. This possibility is, of course, common to all systems having feedback over an energy source and constitutes the main undesirable feature of a closed-sequence control. The design problem is therefore the achievement of accuracy and speed of response, which require high amplification, without such amplification causing undue loss of s ta b i lit y .