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 50
... at the equator, approximately calculate your present acceleration. InAwhat direction is it? From equation 14.3, show that f is perpendicular to 0. 14.12. Consider any nonconstant vector f of constant length. Show. 50 Mechanical Vibrations.
... at the equator, approximately calculate your present acceleration. InAwhat direction is it? From equation 14.3, show that f is perpendicular to 0. 14.12. Consider any nonconstant vector f of constant length. Show. 50 Mechanical Vibrations.
Page 53
... calculation, 00, 9°, and g/L. To determine all solutions to the differential equation, it appears we must vary these three parameters. However, if time is measured in a certain way, then we will show that only two parameters are ...
... calculation, 00, 9°, and g/L. To determine all solutions to the differential equation, it appears we must vary these three parameters. However, if time is measured in a certain way, then we will show that only two parameters are ...
Page 58
... calculated. As an example, let us investigate the linearized stability of the equilibrium positions of a nonlinear pendulum: 1129 L? : —~gsin0. The equilibrium positions are 0E : 0 and QE : n. f (0) = g sin 0, and therefore f '(0) = g ...
... calculated. As an example, let us investigate the linearized stability of the equilibrium positions of a nonlinear pendulum: 1129 L? : —~gsin0. The equilibrium positions are 0E : 0 and QE : n. f (0) = g sin 0, and therefore f '(0) = g ...
Page 60
... Calculate the equilibrium position of the moonship. (0) Is the equilibrium position stable? Is your conclusion reasonable? (d) Compare this problem to exercise 18.7. Consider an isolated positive electrically charged particle of charge ...
... Calculate the equilibrium position of the moonship. (0) Is the equilibrium position stable? Is your conclusion reasonable? (d) Compare this problem to exercise 18.7. Consider an isolated positive electrically charged particle of charge ...
Page 61
... Calculate the tension Tin the string. (The tension is the force exerted by the string.) Assume that the tension remains the same when the mass is displaced vertically a small distance y. Calculate the period of oscillation of the mass ...
... Calculate the tension Tin the string. (The tension is the force exerted by the string.) Assume that the tension remains the same when the mass is displaced vertically a small distance y. Calculate the period of oscillation of the mass ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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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