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. |
From inside the book
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Page 8
... spring force. Thus F=—M pm is a good approximation for the spring-force as long as the mass is not very far from its ... system. EXERCISES 3.1. Newton's first law states that with no external forces a mass will move along a straight line ...
... spring force. Thus F=—M pm is a good approximation for the spring-force as long as the mass is not very far from its ... system. EXERCISES 3.1. Newton's first law states that with no external forces a mass will move along a straight line ...
Page 9
... spring is permanently deformed due to a large force, then do our assumptions fail? 4. Gravity In Sec. 3, we showed that the difi'erential equation 2 m 'Z'Tf = —kx (4.1) describes the motion of a spring ... spring-mass system. 9 Sec. 4 ...
... spring is permanently deformed due to a large force, then do our assumptions fail? 4. Gravity In Sec. 3, we showed that the difi'erential equation 2 m 'Z'Tf = —kx (4.1) describes the motion of a spring ... spring-mass system. 9 Sec. 4 ...
Page 10
... spring-mass system does not apply to the vertical system. There is another force—gravity. We approximate the gravitational force as a constant* —mg, the mass m times the acceleration due to gravity —g. The two forces add vectorially and ...
... spring-mass system does not apply to the vertical system. There is another force—gravity. We approximate the gravitational force as a constant* —mg, the mass m times the acceleration due to gravity —g. The two forces add vectorially and ...
Page 11
... spring. The spring sags downwards a distance mg/k when the mass is added, a result that should not be surprising. For a larger mass, the spring sags more. The stiffer the spring (k ... Spring-Mass System We now proceed to. II Sec.4 Gravity.
... spring. The spring sags downwards a distance mg/k when the mass is added, a result that should not be surprising. For a larger mass, the spring sags more. The stiffer the spring (k ... Spring-Mass System We now proceed to. II Sec.4 Gravity.
Page 12
... Spring-Mass System We now proceed to analyze the differential equation describing a springmass system, 2 m '57; = —kx. (5.1) The restoring force is proportional to the stretching of the spring. Although this equation has been derived ...
... Spring-Mass System We now proceed to analyze the differential equation describing a springmass system, 2 m '57; = —kx. (5.1) The restoring force is proportional to the stretching of the spring. Although this equation has been derived ...
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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