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 xiii
... known to the average reader. We will not find it necessary to refer to exceedingly technical research results. Furthermore these three topics were chosen for inclusion in this text because each serves as an introduction to more ...
... known to the average reader. We will not find it necessary to refer to exceedingly technical research results. Furthermore these three topics were chosen for inclusion in this text because each serves as an introduction to more ...
Page 7
... known as Newton's first law, is easily verifiable—see exercise 3.1.) Thus the observed variability of the velocity must be due to forces probably exerted by the spring. To develop an appropriate model of the spring force, one should ...
... known as Newton's first law, is easily verifiable—see exercise 3.1.) Thus the observed variability of the velocity must be due to forces probably exerted by the spring. To develop an appropriate model of the spring force, one should ...
Page 8
... known as Hooke's law. Doubling the displacement, doubles the force. Using Hooke's law, Newton's second law of motion yields m 2 df=—m, pm Hi— the simplest mathematical model of a spring-mass system. EXERCISES 3.1. Newton's first law ...
... known as Hooke's law. Doubling the displacement, doubles the force. Using Hooke's law, Newton's second law of motion yields m 2 df=—m, pm Hi— the simplest mathematical model of a spring-mass system. EXERCISES 3.1. Newton's first law ...
Page 14
... oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + 450, ¢O being the phase at t I 0. In many. 14 Mechanical Vibrations.
... oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + 450, ¢O being the phase at t I 0. In many. 14 Mechanical Vibrations.
Page 15
... known as a Hertz). Since a springmass system normally oscillates with frequency l/ZnN/W, this value is referred to as the natural frequency of a spring-mass system of mass m and spring constant k. Other physical systems have natural ...
... known as a Hertz). Since a springmass system normally oscillates with frequency l/ZnN/W, this value is referred to as the natural frequency of a spring-mass system of mass m and spring constant k. Other physical systems have natural ...
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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