MIT OpenCourseWare http ocw mit edu Electromechanical Dynamics For any use or distribution of this textbook please cite as follows Woodson Herbert H and James R Melcher Electromechanical Dynamics 3 vols Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms Chapter 5 LUMPED PARAMETER ELECTROMECHANICAL DYNAMICS 5 0 INTRODUCTION The representation of lumped parameter electromechanical systems by means of mathematical models has been the subject of the preceding chapters Our objective in this chapter is to study their dynamical behavior Mathematically we are interested in the solution of differential equations of motion for given initial conditions and with given driving sources Physically we are interested in important phenomena that occur in electromechanical systems It is clear from previous examples that the differential equations that describe electromechanical systems are in most cases nonlinear Consequently it is impossible to develop a concise and complete mathematical theory as is done for linear circuit theory We shall find many systems for which we can assume small signal behavior and linearize the differential equations In these cases we have available to us the complete mathematical analysis developed for linear systems If exact solutions are required for nonlinear differential equations each situation must be considered separately Machine computation is often the only efficient way of obtaining theoretical predictions Some simple cases however are amenable to direct integration The physical aspects of a given problem often motivate simplifications of the mathematical model and lead to meaningful but tractable descriptions Hence in this chapter we are as much concerned with illustrating approximations that have been found useful as with reviewing and expanding fundamental analytical techniques Lumped parameter systems are described by ordinary differential equations The partial differential equations of continuous or distributed systems are often solved by a reduction to one or more ordinary differential equations Hence many concepts used here will prove useful in the chapters that follow Lumped Parameter Electromechnical Dynamics Similarly the physical behavior of a distributed system is sometimes most easily understood in terms of lumped parameter concepts Examples discussed in this chapter are in many cases motivated by the physical background that they provide for more complicated interactions to be considered later Because the mathematics of linear systems is comparatively simple we begin our study of the dynamic behavior of lumped parameter electromechanical systems by considering the several types of system for which a linear model provides an adequate description We shall then consider the types of system that are basically nonlinear and for which the differential equations can be integrated directly 5 1 LINEAR SYSTEMS We have stated that electromechanical systems are not usually described by linear differential equations Many devices however called incrementalmotion transducers are designed to operate approximatelyas linear systems Moreover meaningful descriptions of the basic properties of nonlinear systems can often be obtained by making small signal linear analyses In the following sections we develop and illustrate linearization techniques linearized models and the dynamical behavior of typical systems 5 1 1 Linear Differential Equations First we should recall the definition of a linear ordinary differential equation An nth order equation has the form d r dt AI t dn xx dt 1 A t x f t 5 1 1 where the order is determined by the highest derivative Note that the coefficients A t can in general be functions of the independent variable t If however the coefficients were functions of the dependent unknown variable z t the equation would be nonlinear The driving function f t is a known function of time The homogeneous form of 5 1 1 is provided by making f t 0 There are n independent solutions x t to the homogeneous equation The general solution to 5 1 1 is a linear combination of these homogeneous solutions plus a particular solution x t to the complete equation X t c1lx t cXn t t 5 1 2 Although 5 1 1 is linear it has coefficients that are functions of the A review of differential equations can be found in such texts as L R Ford Differential Equations 2nd Ed McGraw Hill New York 1955 5 1 1 Linear Systems independent variable and this can cause complications for example if f t is a steady state sinusoid of a given frequency the solution may contain all harmonics of the driving frequency Alternatively if f t is an impulse the response varies with the time at which the impulse is applied These complications are necessary in some cases most of our linear systems however are described by differential equations with constant coefficients For now we limit ourselves to the case in which the coefficients A a constant and 5 1 1 becomes d t dn xZ t dnX a anx t f t dt dt n 1 5 1 3 The solution to equations having this form is the central theme of circuit theory The solutions xs t to the homogeneous equation when the coefficients are constant are exponentials e t where s can in general be complex that is if we let X t ce t 5 1 4 and substitute it in the homogeneous equation we obtain s alsr a i 1 ce t 0 5 1 5 and 5 1 4 is a solution provided that the complex frequencies satisfy the condition sin als 1 a7 0 5 1 6 Here we have an nth order polynomial in s hence a condition that defines the n possible values of s required in 5 1 4 The frequencies s that satisfy 5 1 6 are called the naturalfrequencies of the system and 5 1 6 is sometimes called the characteristicequation t Many commonly used devices are driven in the sinusoidal steady state In this case the driving function f t has the form f t Re Pee 5 1 7 Here P is in general complex and determines the phase of the driving signal for example if P 1 f t cos cot but if P j f t sin cot To find See for example E A Guillemin Theory of Linear PhysicalSystems Wiley New York 1963 especially Chapter 7 t If the characteristic equation has repeated roots the solution must be modified slightly see for example M F Gardner and J L Barnes Transients in Linear Systems Wiley New York 1942 pp 159 163 I Lumped Parameter Electromechanical Dynamics
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