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RESEARCH PAPERS

J. Basic Eng. 1962;84(1):1-11. doi:10.1115/1.3657253.

A large class of optimum control problems can be formulated as the variational problem of minimizing a known functional subject to isoperimetric and nonholonomic constraints. The (vector) Euler equation for this problem leads directly to the structure of the optimum controller, which turns out to comprise a dynamic portion which is the adjoint of the plant to be controlled and instantaneous nonlinear elements determined by the performance functional and input constraints. Continuous measurement of the state of the plant results in the elimination of the dynamic portion, and the entire optimum controller is instantaneous. An example is given which illustrates the complete design of regulators for a simple plant with constraints on either amplitude or energy of the actuating signal which minimize response time or integrated square error.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):13-20. doi:10.1115/1.3657236.

The problem of existence of various types of optimum controls for controlling processes which are described by ordinary differential equation models is considered. The results presented enable one to test if there does exist an optimum control in the class of controls under consideration before proceeding to the construction of an optimal control.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):23-29. doi:10.1115/1.3657260.
Abstract
Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):30-32. doi:10.1115/1.3657261.

A rule for computing the initial relay position for time-optimal regulation corresponding to any set of initial conditions without solving the transcendental switching equations is presented. The discussion is restricted to systems represented by ordinary differential equations with constant coefficients with real characteristic roots. The calculation requires the evaluation of an (n − 1)-order surface integral and cannot be considered a practical solution of the problem.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):33-37. doi:10.1115/1.3657263.

A class of problems that has received considerable attention in recent years from both control theorists and engineers is the following:

Given  = Fx + du,  x(0) = cDetermine |u(t)| ≤ 1  such that x(T) = 0  and x(t) ≠ 0for 0 ≤ t < T and  where T is a minimum    (P-1)
A related and perhaps more practical class of problems can be stated as
Given  = Fx + du,  x(0) = cDetermine |u(t)| ≤ 1   such that ‖x(T)‖2P is  a minimum for given T   (P-2)
Although a considerable amount of effort has been expended on (P-1), and to a lesser extent on (P-2), yet computational techniques which enable one to solve numerically the above problems are still lacking except in restricted cases [7, 8]. This paper presents such a technique which completely solves this problem by successive approximation. The convergence of this solution is proved, and it is shown to satisfy all known properties of the problems.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):41-53. doi:10.1115/1.3657268.

The problem of controlling the attitude of a space vehicle is unusual in several respects. While the required precision may be extreme—less than 0.1 second of arc in certain cases—the required response time may often be very slow, measured in minutes or hours. Vehicles weighing tons may have to be controlled by inch-ounces of torque, and control energy is at an extreme premium. The present paper discusses the effects, on performance, of inter-axis coupling due to internal spinning parts. A de-coupling computer to nullify gyroscopic torque is described, and its utility is evaluated. The computer is found to improve precision, but to reduce energy consumption only in certain cases. It is shown that by postulating such a computer the performance of a given system may be accurately evaluated on the basis of much simpler single-axis relations, even though strong coupling is present. Specifically: (1) It is shown that the best available performance is established by postulating de-coupling control; and (2) a method is given for determining the amount by which a conventional system will fail to achieve that performance.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):54-60. doi:10.1115/1.3657269.

A gyro-free nonlinear attitude control system for a spacecraft is analyzed. On-off jet actuators are used. Hysteresis and a dead zone are intentionally put into the system. Under certain conditions the feedback signal in the control system is proportional to an angular velocity increment of the system. This is called the derived-rate increment feedback signal. The analysis for a single axis of the attitude control system is given in two parts. One part is concerned with the performance of the system in a limit cycle. The other part discusses the convergence to a limit cycle after a disturbance has occurred. Experimental results verify the results of the analysis. Typical results show the performance of the system during convergence to and operation in a limit cycle. Although the technique is described for use in an attitude control system, it can be successfully employed in other applications.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):61-69. doi:10.1115/1.3657270.

A stability criterion for certain types of nonlinear feedback systems in the presence of Gaussian noise is established here. This criterion may be considered as a natural extension of the describing function method. It is assumed that the lowest frequency component in the power spectral density of the noise is at least ten times higher than the highest significant frequency of the system. The method developed here is applicable to feedback systems with just one instantaneous, nonmemory type nonlinearity in the loop. The results mentioned in this paper have been experimentally verified on an analog computer. The theory explained here may be used by the designer to predict the manner in which noise will affect the performance of a system.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):71-81. doi:10.1115/1.3657273.

A general procedure for obtaining information on the periodic modes of oscillation in PWM and nonlinear sampled-data feedback systems is considered in this paper. Based on the equivalence of PWM in the state of limit cycles to the finite pulsed systems with the periodically varying sampling pattern, the methods of analysis applied to the latter are extended to obtain these limit cycles. In particular, the final value theorem is applied to obtain the fundamental response equation which gives rise to the limit cycles for the various specified modes. The theory is applied to systems with and without integrator and the results are checked by the phase-plane approach. Two kinds of nonlinearities, namely, pulse-width modulation and saturating gain, are discussed among the various nonlinearities, and examples are presented for each of these cases. Furthermore, both self-excited and forced oscillations are examined as well as the possible existence of limit cycles for certain specified modes. This approach to examining the periodic modes is not restricted to the type of non-linearity or the order of the system and thus can be applied to various forms of nonlinear discrete systems. However, it is based on the assumption that the mode of the limit cycle is specified, as can be done in certain cases, and thus the method of this paper permits the study of the conditions that sustain those oscillations.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):85-90. doi:10.1115/1.3657276.

The idea of direct optimization is briefly presented and an introduction given to the divider-optimizer. By investigation of the operating principles of the divider-optimizer, it is shown that the divider functions can be accomplished in separate and distinct manners. The optimizing behavior of the modified form of the divider-optimizer is investigated in the optimization of two quite different dynamic optimal systems. Experimental results of an analog computer study are presented. It is also shown that the modified form of the optimizer presents the possibility of a more flexible optimizing control using simple, reliable, and relatively inexpensive components.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):91-99. doi:10.1115/1.3657277.

This paper considers the problem of optimal (minimum-time) on-off regulation of a dynamic plant whose state is known only at periodic instants of time. It is shown that on-off control inputs which change value only at the sampling instants inherently lack the capability of providing accurate (dead-beat) control. If, however, the on-off controller is modified to include pulse width control, it not only has the capability for accurate control, but also has the capability for optimal control comparable to that of a saturating amplifier controller. In addition, the optimal pulse width control inputs can be specified as a function of the state of the plant at each sampling instant. The application of this method is discussed in detail for the cases of pure inertial and second-order underdamped plants.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):101-109. doi:10.1115/1.3657227.

This paper treats the minimal time control problem for two second order pulse-width-modulated sampled-data systems, one with a double integrator type plant and one with a plant described by an integral and a time constant. Such plants are encountered in systems with hydraulic components. It is shown rigorously that for minimal time control the phase plane can be divided into two regions: a striplike region around the optimal switching trajectory for a continuous relay system with the same plants, in which the pulse width must be adjusted for optimal action; and the rest of the phase plane in which an optimal p.w.m. system of the type described behaves like a continuous optimal relay system, the pulse duration being equal to the sampling period. A brief description of an electromechanical computer capable of implementing minimal time control for these systems is also given.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):110-118. doi:10.1115/1.3657228.

A theoretical derivation is given showing that whenever the center of curvature of a bearing does not coincide with its center of rotation, synchronous counterrotation will produce a load capacity. Test results are given for a number of bearing designs that operated successfully under synchronous counterrotation with load capacities beyond 1500 psi. The only bearing that scored under counterrotation was a perfectly circular ungrooved bearing, as predicted by theory.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):119-122. doi:10.1115/1.3657229.

This paper presents theoretical studies of the method of optimization of the stiffness of externally pressurized bearings. When an externally pressurized bearing is designed to operate at any given film thickness, it is found that the maximum bearing stiffness can be obtained by proper selection of the value of the ratio of recess to supply pressure, Pr /Ps . While various values of Pr /Ps can be attained by either varying the restrictor constant for a given film thickness or varying the film thickness for a given restrictor constant, the important quantity to vary in determining the optimum condition is the restrictor constant rather than the film thickness. For an incompressible fluid, the Pr /Ps value for which the bearing stiffness is optimized depends only on the type of compensation used, while for a compressible fluid it is slightly affected by exhaust pressure.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):123-131. doi:10.1115/1.3657231.

As a theoretical study of the hydrodynamic, gas-lubricated journal bearings, the paper presents approximate solutions with fair accuracy for high values of eccentricity ratio. In addition, an approximate solution available for clarifying the characteristics of journal bearing operating at low revolution speeds is reported. Comparison with experimental data reveals the fact that actual bearings operate under an intermediate condition between isothermal and adiabatic when running at high revolution speeds, while under the isothermal condition at low speeds.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):132-138. doi:10.1115/1.3657234.

The nature of whirl of a rigid rotor in externally pressurized air-lubricated journal bearings, including self-acting bearings as a special case, is discussed and an experimental investigation described. Rotor unbalance leads to synchronous whirl which can persist through one or more critical speeds. Self-excited whirl is likely when the rotational frequency is about twice the lowest critical frequency. We discuss the effects of supply pressure, number and location of sources, rotational speed, mass, unbalance, and load upon the onset of self-excited whirl, and describe first-order prediction techniques.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):139-151. doi:10.1115/1.3657235.

Proceeding from the results obtained previously [5] this paper analyzes theoretically the three-dimensional motion in the lubricant layer by using Prandtl’s mixing length theory. Formulas and diagram are presented for calculating journal and thrust bearings subjected to turbulent lubrication.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):152-158. doi:10.1115/1.3657237.

For the theoretical prediction of the dynamical characteristics of a rotor system, it is necessary to have an accurate knowledge of the bearing fluid film forces under dynamical conditions. With a small clearance ratio and at a moderate speed, the motion of the lubricant is governed by the generalized Reynolds equation. If the lubricant is a gaseous medium, the Reynolds equation is complicated by the compressibility effects, which include nonlinearity and time-dependence under dynamic conditions [1]. In the case of a vertical rotor operating in plain cylindrical journal bearings, the steady whirl approximation is appropriate and time-dependence in the Reynolds equation can be removed by a co-ordinate transformation. The form of the transformed equation is identical to the static Reynolds equation except that the compressibility number is modified by a factor which depends on the angular speed of the whirl motion [2, 3]. The altitude angle, in the presence of the whirling motion, is quite different from the static attitude angle. On the other hand, the magnitudes of the forces are not very different. The steady whirl analysis may be used to determine the synchronous whirl motion of an unbalanced rotor. The phase angle between the fluid film force and the maximum film thickness plane is the complement of the attitude angle according to the quasi-static analysis. Experimental data are in excellent agreement with the results of the steady whirl analysis. Also, the modified compressibility number is reduced to zero at half-frequency whirl, and the Reynolds equation, for an isothermal gaseous film with the small eccentricity ratio approximation, becomes identical to that of the liquid film. Since it has been established that the threshold of half-frequency whirl for vertical rotors operating in plain cylindrical journal bearings is at zero speed in [4], the same conclusion applies to the corresponding gas-dynamic bearing.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):159-165. doi:10.1115/1.3657238.

A simplified analysis for calculating the characteristics, such as stiffness, flow, and load of an externally pressurized gas journal bearing, is obtained through a standard one-dimensional flow approach altered to take into account the effect of circumferential pressure variations. From this analysis, it is revealed that bearing stiffness has an optimum which, it is further shown, can be chosen through a proper selection of bearing design parameters, such as radial clearance, upstream resistor, and supply pressure. As the analysis is developed each assumption is experimentally verified. The final predictions of the analysis are also verified. It is felt that the analysis is sufficiently simple to allow comparison of many different bearing configurations, while sufficiently complete to compare very well against experimental measurements in all respects. A comparison is made between the recessed and nonrecessed bearing.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):166-174. doi:10.1115/1.3657240.

This paper which consists of Parts I and II presents a general and practical fluid lubrication theory of roller bearings lubricated by Newtonian and non-Newtonian lubricants with considerations to the effect of sliding of roller and the influence of unsteady load. In Part I, the fundamental theory for the lubrication between two rotating cylinders in contact has been investigated. The load capacity and friction of a non-Newtonian lubricant, supposed to be a Bingham plastic, coincide approximately at high speed with those of a Newtonian lubricant with viscosity equivalent to the plastic viscosity of the non-Newtonian lubricant. Under unsteady loads, the squeeze action works effectively so that the load capacity increases. The amount of friction is 4/3 and the load capacity is 2/3 in the case of two rotating cylinders in contact involving sliding, compared with that involving no sliding.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):175-180. doi:10.1115/1.3657241.

Using the theory for two rotating cylinders in contact obtained in Part I, the fluid lubrication theory of roller bearings for Newtonian and non-Newtonian lubricants was developed in Part II with considerations of the influences of unsteady load and sliding of rollers. It is clarified that the load capacity under unsteady load is generally larger than that under constant basic load and the average friction is nearly equal for both cases. The frictional moment and load capacity for roller bearing including sliding of rollers decrease to 2/3 of the amount of those for roller bearing including no sliding.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):181-189. doi:10.1115/1.3657244.

An understanding of the basic mechanics and statistical variables of the contact roller test as it is applied to surface fatigue phenomena was obtained. One hundred ninety-four contact rollers of AISI 52100 bearing quality steel were tested to establish dispersion limits. No significant fatigue differences were indicated for heats, bars, or test replications among steels of similar quality, composition, and heat-treatment. A statistical evaluation of results points to the contact roller test as a suitable means for studying significant variables that pertain to both the theoretical and material aspects of antifriction bearings. Stresses in the contact surfaces based on the orthogonal shear-stress theories showed good correlation with fatigue life.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):192-195. doi:10.1115/1.3657248.

The relationship between stress and rate of strain for a non-Newtonian fluid with pseudoplastic behavior is assumed to be a cubic power series. Under this assumption a solution of the slider bearing without side leakage can be obtained using a perturbation technique. The solution turns out to be relatively simple in form in contrast to what results when a general power law is used.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1962;84(1):197-202. doi:10.1115/1.3657252.

An analysis is presented of the slider bearing using an electrically conducting lubricant, such as a liquid metal, in the presence of a magnetic field. The solution permits the calculation of the load-carrying capacity of the bearing. A comparison is made with the classical slider bearing solution. It is shown that the load capacity of the bearing depends on the electromagnetic boundary conditions entering through the conductivity of the bearing surfaces. Numerical data are presented for nonconducting surfaces with the emphasis on a comparison between the classical bearing and the magnetohydrodynamic bearing characteristics. It is shown that a significant increase in load capacity is possible with liquid metal lubricants in the presence of a magnetic field.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Basic Eng. 1962;84(1):205-206. doi:10.1115/1.3657256.
Abstract
Commentary by Dr. Valentin Fuster

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