A natural response

Abstract

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 e st , where s is possibly 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 second-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 5