Chapter 1 Natural Response A significant portion of these notes are concerned with the study of finite-dimensional, linear time-invariant (LTI) systems. We will define this term with more care in section 1.3.2. Such systems can be described by finite-order linear constant coefficient differential equations. Such models are widely applicable to physical systems. In this chapter, we will be primarily concerned with the natural response of such models, which is defined as the response which occurs solely from initial conditions with no other inputs. The natural response is also known as the unforced response or characteristic response. The model differential equation for such a system is homogeneous, in that there is no forcing term. There is a beautiful property of LTI systems: the homogeneous or natural response can be very simply found. It is composed of weighted sums of functions est, where s is poss ibly complex (or most generally such functions multiplied by polynomials in the time variable t). This is a statement about the solution of differential equations. However, it is a remarkable empirical result that such differential equations well-describe many physical systems. Said another way, the types of natural responses discussed below can be easily observed in an experimental context, and in observations of many physical phenomena. The natural response ties things together. A further surprising result is that real-world systems are frequently able to be represented in terms of very simple models of first- or second-order. When higher-order models are required, these systems have responses com-posed of sums of first- and second-order responses. So it’s very worthwhile to understand the building-block first- and se cond-order responses in depth. This chapter is organized as follows: We present first-order systems, and their natural response, starting with a mechanical example. The charac-56 CHAPTER 1. NATURAL RESPONSE teristics of such first-order responses in time are discussed in detail. These responses involve only real functions and thus use only real mathematics. Next we present the similar first-order responses encountered in electrical, thermal, and fluidic systems. Second-order systems in general have complex-valued natural responses. Thus the section on second-order systems starts with a review of complex numbers. The natural responses for a second-order mechanical system are presented, with individual attention to the overdamped, critically-damped, and underdamped cases. Section 1.2 presents second-order system natural responses. Analogous electrical, thermal, and fluidic second-order systems are discussed next. Finally, the chapter concludes with a discussion of the natural response of higher-order systems, and a discussion of linearity. 1.1 First-order systems The canonical1 homogeneous first-order differential e quation is dy(t)τ + y(t) = 0, (1.1)dt where we assume τ =! 0. The variable τ is the system time constant and has units of seconds. Here we have explicitly shown the time dependence of y(t). It is also acceptable and more compact to use the form τ dy + y = 0. (1.2)dt The response of a such an unforced first-order system is always of the form y(t) = cest . This is a simple and beautiful result, easy to rememb e r, and extends to higher-order systems in a natural way. The variable s has units of frequency (sec−1). The differential equation (1.1) will only allow one value of s = λ1. We call λ1 the characteristic frequency or equivalently the eigenvalue of the system (1.1). In this first-order case, λ1 is a real number, but in higher-order cases the eigenvalues are more generally complex-valued. The constant c is a real number with the same units as y; it is used to set the value of the function at some point in time, typically t = 0. The value at t = 0 is called the initial condition of the homogeneous response. You can find the homogeneous solution as follows: First, substitute the assumed form y(t) = cest into the differential equation. The deriviative 1prototypical7 1.1. FIRST-ORDER SYSTEMS operation just brings down a multiplicative term s, and so you have τsce st + ce st = 0. (1.3) This can be factored as (τ s + 1)ce st = 0. (1.4) Setting the initial condition c = 0 satisfies this equation but is not very interesting, since this give s y = 0 for all time. The function est is nonzero for finite s and t, and thus can be divided out to give the characteristic equation (τ s + 1) = 0. This has the solution s = λ1 = −1/τ , which is the one and only characteristic frequency (eigenvalue)2 associated with this first-order system. Thus we have arrived at the homogeneous solution t y(t) = ce− τ . (1.5) The response decays to z ero with increasing time if τ > 0; if the natural response of a system always decays to zero with increasing time for any initial conditions, we say that the system is stable. If the res ponse goes off to infinity with increasing time for some initial conditions, the system is unstable. The response (1.5) has the initial value y(0) = c. The graph of this response is shown in Figure 1.1 for an initial condition c = 1 with the four values τ = 2, 1, 0.5, 0.1. As you can see, τ represents the characteristic time for the response to decay toward zero; smaller values of τ correspond with faster responses. Because such a response is widely applicable in real engineering systems, we will take a bit of time to understand it in more depth. Your efforts here to internalize an understanding of this response and its characteristics will pay dividends throughout your engineering studies and practice. Specifically, in an interval of one time c onstant, the response shown decays to a value of 0.37 times the value at the start of the interval. This is so because e−1 = 0.3679 ≈ 0.37. Since this response has an initial value of 1, the response decays to a value of 0.37 in one time constant, 0.372 in two time constants, and a value of 0.37n in n time constants. You should verify this result to graphical accuracy for all four of the time constant values; they pass through the dashed line y = 0.37 in an interval equal to τ. And, in 3 seconds, the response for τ = 1 sec passes through a