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 vii
... Friction Z9 11. Oscillations of a Damped System 33 12. Underdamped Oscillations 34 13. Overdamped and Critically Damped Oscillations .............................................. .. 4O 14. A Pendulum 42 15. How Small is Small? 51 ...
... Friction Z9 11. Oscillations of a Damped System 33 12. Underdamped Oscillations 34 13. Overdamped and Critically Damped Oscillations .............................................. .. 4O 14. A Pendulum 42 15. How Small is Small? 51 ...
Page 3
... frictional forces (Secs. 10—13). A pendulum is then analyzed (Secs. l4~l6) since its properties are similar to those of a spring-mass system. The nonlinear frictionless pendulum and spring-mass systems are briefly studied, stressing the ...
... frictional forces (Secs. 10—13). A pendulum is then analyzed (Secs. l4~l6) since its properties are similar to those of a spring-mass system. The nonlinear frictionless pendulum and spring-mass systems are briefly studied, stressing the ...
Page 4
... friction. 3. nonlinear systems (frictionless). 4. nonlinear systems with friction. 2. Newton's Law To begin our investigations of mathematical models, a problem with which most of you are somewhat familiar will be considered. We will ...
... friction. 3. nonlinear systems (frictionless). 4. nonlinear systems with friction. 2. Newton's Law To begin our investigations of mathematical models, a problem with which most of you are somewhat familiar will be considered. We will ...
Page 28
... m; is suddenly removed (for example, by cutting the string connecting m1 and m2), then what is the period and amplitude of oscillation of m, ? 10. Friction Our mathematical model shows that the displacement of. 28 Mechanical Vibrations.
... m; is suddenly removed (for example, by cutting the string connecting m1 and m2), then what is the period and amplitude of oscillation of m, ? 10. Friction Our mathematical model shows that the displacement of. 28 Mechanical Vibrations.
Page 29
... significantly decay. In this case the mathematical model of a spring-mass system, d 2x is a good approximation for times that are not particularly. m'dv — '~kx, dx dt Figure 10-6 Coulomb friction. x : e—ct/2m(aeimt +. 29 Sec.10 Friction.
... significantly decay. In this case the mathematical model of a spring-mass system, d 2x is a good approximation for times that are not particularly. m'dv — '~kx, dx dt Figure 10-6 Coulomb friction. x : e—ct/2m(aeimt +. 29 Sec.10 Friction.
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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