Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic FlowThe author uses mathematical techniques along with observations and experiments to give an in-depth look at models for mechanical vibrations, population dynamics, and traffic flow. Equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. In the sections on mechanical vibrations and population dynamics, the author emphasizes the nonlinear aspects of ordinary differential equations and develops the concepts of equilibrium solutions and their stability. He introduces phase plane methods for the nonlinear pendulum and for predator-prey and competing species models. Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Fan-shaped characteristics describe the traffic situation that occurs when a traffic light turns green and shock waves describe the effects of a red light or traffic accident. Although it was written over 20 years ago, this book is still relevant. It is intended as an introduction to applied mathematics, but can be used for undergraduate courses in mathematical modeling or nonlinear dynamical systems or to supplement courses in ordinary or partial differential equations. |
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Page 7
... what speed the mass is moving at. *Througho ut this text, we assume that the width of the mass is negligible. A careful examination of the experimental data shows that the. 7 Sec. 3 Newton's Law as Applied to a Spring-Mass System.
... what speed the mass is moving at. *Througho ut this text, we assume that the width of the mass is negligible. A careful examination of the experimental data shows that the. 7 Sec. 3 Newton's Law as Applied to a Spring-Mass System.
Page 16
... moving around a circle, with its position designated by the polar angle 0. Assume its angular velocity dQ/dt is constant, d0/dt = a). Show that the x component of the particle's position executes simple harmonic motion (and also the y ...
... moving around a circle, with its position designated by the polar angle 0. Assume its angular velocity dQ/dt is constant, d0/dt = a). Show that the x component of the particle's position executes simple harmonic motion (and also the y ...
Page 22
... moving with velocity 00. What is the amplitude of oscillation? Show that the amplitude depends in a reasonable way on 110, k, and m. 8.2. Suppose that a mass is intially at x = x0 with an initial velocity 1:0. Show that the resulting ...
... moving with velocity 00. What is the amplitude of oscillation? Show that the amplitude depends in a reasonable way on 110, k, and m. 8.2. Suppose that a mass is intially at x = x0 with an initial velocity 1:0. Show that the resulting ...
Page 30
... moving to the right, then there is a force exerted to the left, as shown in Fig. 10-3, resisting, the motion of the mass. Figure 10-4 shows that when the spring is moving to the left, then there is a force exerted to the right. 7 _> F ...
... moving to the right, then there is a force exerted to the left, as shown in Fig. 10-3, resisting, the motion of the mass. Figure 10-4 shows that when the spring is moving to the left, then there is a force exerted to the right. 7 _> F ...
Page 31
... moving / Figure 10-5. the friction force is resistive but has a magnitude which is approximately constant independent of the velocity. We model this experimental result by stating for d dt < 0 f I { Y x/ (10.3) —y for dx/dt > O, as ...
... moving / Figure 10-5. the friction force is resistive but has a magnitude which is approximately constant independent of the velocity. We model this experimental result by stating for d dt < 0 f I { Y x/ (10.3) —y for dx/dt > O, as ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
Common terms and phrases
amplitude analysis applied approximately Assume birth calculated called cars characteristics Consider constant continuous corresponding curve decreases delay depends derived described determine differential equation discussed distance energy equal equilibrium population equilibrium position equivalent example exercise experiments expression Figure first fish flow force formulate friction function given growth rate hence highway illustrated increases initial initial conditions integral isoclines known length light limit linear manner mass mathematical model maximum measured method motion moving nonlinear number of cars observer obtained occurs oscillation partial differential equation pendulum period phase plane possible probability problem region result roots sharks shock Show shown in Fig simple sketched sketched in Fig solution solve species spring spring-mass system stable straight line Suppose tion traffic density traflic trajectories unstable variables velocity yields zero