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/termsChapter 5LUMPED-PARAMETERELECTROMECHANICAL DYNAMICS5.0 INTRODUCTIONThe representation of lumped-parameter electromechanical systems bymeans of mathematical models has been the subject of the preceding chapters.Our objective in this chapter is to study their dynamical behavior. Mathe-matically, we are interested in the solution of differential equations ofmotion for given initial conditions and with given driving sources. Physically,we are interested in important phenomena that occur in electromechanicalsystems.It is clear from previous examples that the differential equations thatdescribe electromechanical systems are in most cases nonlinear. Consequently,it is impossible to develop a concise and complete mathematical theory, as isdone for linear circuit theory. We shall find many systems for which we canassume "small-signal" behavior and linearize the differential equations.In these cases we have available to us the complete mathematicalanalysis developed for linear systems. If exact solutions are required fornonlinear differential equations, each situation must be considered separately.Machine computation is often the only efficient way of obtaining theoreticalpredictions. Some simple cases however, are amenable to direct integration.The physical aspects of a given problem often motivate simplifications of themathematical model and lead to meaningful but tractable descriptions.Hence in this chapter we are as much concerned with illustrating approxima-tions that have been found useful as with reviewing and expanding funda-mental analytical techniques.Lumped-parameter systems are described by ordinary differential equations.The partial differential equations of continuous or distributed systems areoften 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 DynamicsSimilarly, the physical behavior of a distributed system is sometimes mosteasily understood in terms of lumped parameter concepts. Examples discussedin this chapter are in many cases motivated by the physical background thatthey 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 electro-mechanical systems by considering the several types of system for which alinear model provides an adequate description. We shall then consider thetypes of system that are basically nonlinear and for which the differentialequations can be integrated directly.5.1 LINEAR SYSTEMSWe have stated that electromechanical systems are not usually described bylinear differential equations. Many devices, however, called incremental-motion transducers, are designed to operate approximatelyas linear systems.Moreover, meaningful descriptions of the basic properties of nonlinearsystems 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 EquationsFirst, we should recall the definition of a linear ordinary differentialequation.* An nth-order equation has the formd"r dn-xx+ AI(t) + ... + A,(t)x = f(t), (5.1.1)dt" dt"-1where the order is determined by the highest derivative. Note that thecoefficients 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. Thegeneral solution to (5.1.1) is a linear combination of these homogeneoussolutions, 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, DifferentialEquations, 2nd Ed., McGraw-Hill, New York, 1955._ independent variable and this can cause complications; for example, iff(t) is a steady-state sinusoid of a given frequency, the solution may containall harmonics of the driving frequency. Alternatively, if f(t) is an impulse,the response varies with the time at which the impulse is applied. Thesecomplications 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) becomesd__(t) dn-xZ(t)dnX + a + " ' -+ anx(t) = f(t). (5.1.3)dt" dtn- 1The solution to equations having this form is the central theme of circuittheory.* The solutions xs,(t) to the homogeneous equation, when the co-efficients are constant, are exponentials e*t, where s can in general be complex;that is, if we letX(t) = ce"t (5.1.4)and substitute it in the homogeneous equation, we obtain(s," + alsr-' + " + a.) ce"'t = 0 (5.1.5)i=1and (5.1.4) is a solution, provided that the complex frequencies satisfythe conditionsin + als'- 1 + " + a7 = 0. (5.1.6)Here we have an nth-order polynomial in s, hence a condition that definesthe n possible values of s required in (5.1.4). The frequencies s, that satisfy(5.1.6) are called the natural frequencies of the system and (5.1.6) is sometimescalled the characteristic equation.tMany commonly used devices are driven in the sinusoidal steady state.In this case the driving function f(t) has the formf(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 Physical Systems, Wiley, New York,1963 (especially Chapter 7).t If the
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