On a Graphical Solution for the Forced Vibration of a System with Non-linear Restoring Force

This paper deals with the forced vibration of a single-degree-of-freedom system under simple harmonic force. The author discussed this problem in two cases separately; namely, the one of symmetric restoring force and the other of unsymmetric one. As the result of studying, the author expounded a graphical solution which is an approximate one but will be very conveniently used to get the steady vibration of this system. In this paper, the author proves by adequate calculations that the graphical solution is practically available for obtaining resonance curves and phase difference curves in the system with any non-linear restoring force. Generally, the equation of motion in this case is expressed as mx⁻⁻+cx⁻+f(x)=P₀sinωt, (1) where m is the vibrating mass, c the viscous damping coefficient, f(x) the restoring force with any non-linear characteristic and P₀sinωt the simple harmonic force acting on the vibrating mass, and x means displacement of the mass, t time, x⁻ and x⁻⁻ velocity and accelaration respectively. Dealing with this problem, it is conventional to discuss the solutions in the two cases ; that is, the case when f(x) is symmetric as to x=O and the other case f(x) is non-symmetric

Exact Periodic Solutions for the Forced Oscillations of a Symmetric Nonlinear System with “Set-up Springs”

This paper presents the exact periodic solutions for the forced oscillation of a symmetric nonlinear system with “set-up springs”. There exist many kinds of solutions, the “harmonic” solution, the subharmonic solution of odd order and the subharmonic solution of even order of equi-interval type and the “harmonic” and the even order subharmonic solution of non-equi-interval type. By “equi-interval” is meant that the time interval in which the solutions take positive or negative values in a cycle is a half of the fundamental period of the solution. The stability of the periodic solutions is investigated exactly with the aid of the Tsypkin's method used for the stability problems of relay control systems

Some Numerical Methods for Analysis of Transient Responses of Nonlinear Control System

This paper deals with some numerical methods for the analysis of transient responses of a nonlinear control system and their computational errors, as shown in Fig. 1. Three types of mothods are proposed, which will be called the “zero-, the “first- and the “second-order data hold” types. They are all based on the same principle of replacing the output of the nonlinear element by another time function convenient to analyze. Excepting the second-order data hold type, each type consists of two alternative methods: one is useful when the indicial response of G(s) is obtained only experimentally ; the other can be used when G(s) is obtained in an analytic form

Multiple-Criteria Optimization for Environment Development System : Constrained Case

This paper deals with the multiple-criteria optimization problem for environment development systems, where the global amounts of each exogenous factor are limited. The model of an environment development system, which was introduced by R. Kulikowski, is described by a system of nonlinear differential equations which include interconnected n endogenous factors and m×n exogenous factors. The problem of multiple-criteria optimization for an environment development system is formulated. A main difficulty of multiple-criteria optimization is that it is no longer clear what one means by an optimal solution. A possible remedy for this situation is to introduce an objective function which is expressed as some function of various criterions. Given the specific objective function, we first optimize the system with respect to another criterion which is a linear combination of the given criteria. For the special case when the systems have similar nonlinearities, the solution of the linear combination problem is obtained in an explicit manner, in terms of the weighting factors in the linear combination functional. A search procedure is then used to determine the optimum values of these weighting factors for the specified objective function

On Adaptive Control Processes with Computing Time Delay

In the theory of R. Bellman's adaptive control processes, computing time to yield optimal control signal from data obtained during normal operation is ideally assumed to be zero. There will be, however, cases where the computing time delay may not be neglected. The subject of this paper is the optimal control policy for adaptive control processes in the case where certain computing time delay is assumed in advance. Functional equations to get such an optimal control policy are described in a fairly general form. These equations are also shown to be applicable to stochastic control processes and deterministic control processes. Two simple examples are presented to illustrate the application of the method

Modeling Support System for Systems Analytic Research

A computer-assisted mathematical modeling system based on an advanced interactive methodology is presented. The system aims at the extraction of a trade-off between human mental models and regression-type models based on the use of numerical data, by using a flexible combination of statistical methods and graph-theoretical analysis. One of the main advantages of using this system is the facilities for the structuring of both mental and mathematical models

The Theory of Creep of Visco-Plasto-Elastomer

Previously the authors proposed the concept of visco-plasto-elastomer, a dynamical model of which is shown in Fig. 1, because industrial materials used in practice usually have visco-plasto-elastic properties in their mechanical behaviors. In this paper various problems are discussed on the creep theory of visco-plasto-elastomer. Creep, recovery and repeated creep phenomena are analytically treated on the dynamical model, and as the result, various interesting phenomena and facts are found, for example, the existence of virgin state, dependence of the creep strain and the recovery strain on stress σ₀, existence of the permanent set, simple relation between the virgin creep strain and non-virgin creep strain, and so on

Accuracy Considerations of the Equivalent Linearization Technique for the Analysis of a Non-Linear Control System with a Gaussian Random Input

In this paper, the accuracy of the results which can be obtained by the linearization technique of a non-linear control system with a Gaussian random input is investigated. Presence of non-linearity in the control system destroys the Gaussian nature of the response signals. In the linearization technique, therefore, assumption of the Gaussian input for the non-linear element is approximate. The accuracy analysis is restricted to consideration of the cumulants of the second and the fourth order of the input to the non-linear element. This results from the consideration that a linear element with a memory tends to make non-Gaussian signals more nearly Gaussian. A simple control system with a Gaussian input is analyzed as an example. It is concluded that the equivalent linearization technique is accurate enough for engineering purposes

Existing Conditions for Self-Oscillation in an Improved On-Off Control System

This paper deals with the self-oscillation occurring in an improved on-off control system, in which the on-off controller is compensated by two first-order linear elements and the controlled system is a first-order linear element with a dead time. The existing conditions for self-oscillation in this system are exactly derived by the periodicity of the time responses of various variables under some assumptions. The present analysis has revealed the followings : a) in respect to the oscillated amplitude of a controlled variable, the performance of a usual on-off control system is improved by introducing compensating elements b) as the value of Kp/Kf is changed monotonously, one mode of self-oscillation changes into another one, that is, the amplitude and period of self-oscillation change discontinuously c) for some values of Kp/Kf, two or more solutions satisfy the existing conditions for self-oscillation d) the self-oscillation in this improved on-off control system never vanishes with an extremely small value of Kp/Kf, but the amplitude and period of the oscillated variables become very small values