achapt 1 - COMPLEX DYNAMICS SIMPLIFIED 1 COMPLEX DYNAMICS...

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COMPLEX DYNAMICS SIMPLIFIED 1 COMPLEX DYNAMICS SIMPLIFIED To develop a control system for a dynamical system one must first understand precisely how the system behaves. One can arrive at this understanding using mathematics by performing a dynamic analysis. Dynamic analysis is performed in two steps, often called the formulation or the modeling step and the second simply called the solution step. In the formulation step the equations that describe the system are developed. In the solution step, the equations are solved. One often distinguishes between solutions that are obtained analytically and those that are obtained numerically. The equations, when solving them analytically, are often approximated by constant- coefficient linear differential equations, because they can be solved analytically with relative ease. When solving the equations analytically, quantities like displacement, velocity, and force are expressed as algebraic functions of time. Displacements, velocities and forces are called time responses. The other way to solve the original equations, the numerical approach, produces time responses in the form of computer graphs. MAE 461: DYNAMICS AND CONTROLS
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COMPLEX DYNAMICS SIMPLIFIED Sometimes the dynamical behavior of a system is relatively complicated in which case the motion is divided into regions of behavior called stability regions. When developing a control system for such a system, the control system designer usually focuses on one stability region at-a-time. If necessary, the control systems for the different stability regions are patched together to produce a control system for the different regions. In this section, we first discuss the development of the equations that describe the motion of systems. We’ll restrict our attention to systems that have one independent degree-of-freedom and we’ll confine the motion to a plane. We’ll find the stability regions of these planar single degree-of- freedom systems. 1. Equations Let’s first review how to find the equation that describes the motion of a planar single degree-of- freedom system. As an example of such a system, we consider a pendulum. The pendulum is composed of a light rod pinned at one end (point O ) and free at the other end (point A ). At point A , the rod is attached to a small but relatively heavy sphere that has mass M . The mass of the rod is neglected. Figure 1 – 1: Pendulum and its Free-Body Diagram MAE 461: DYNAMICS AND CONTROLS
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COMPLEX DYNAMICS SIMPLIFIED The development of the equation that describes the pendulum’s motion, like the equation that describes the motion of other planar bodies, begins by looking at the equations that govern the three degrees-of-freedom of a planar body, namely, the three equations associated with a body’s rotational motion and its two translational motions. The equation that describes its rotational motion is usually found either by summing moments about a fixed point O , if one exists, or by summing moments about the system’s mass center
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achapt 1 - COMPLEX DYNAMICS SIMPLIFIED 1 COMPLEX DYNAMICS...

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