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 15
... determine the periogL T, we recall that the trigonometric functions are periodic with period 21f." {Thus for a complete oscillation, as 1 increases to t + T, from equation 5.5 wt + (#0 must change by 21:: a>(t+T)+¢0——wt—¢o—_~21z ...
... determine the periogL T, we recall that the trigonometric functions are periodic with period 21f." {Thus for a complete oscillation, as 1 increases to t + T, from equation 5.5 wt + (#0 must change by 21:: a>(t+T)+¢0——wt—¢o—_~21z ...
Page 17
... determine the dimensions of an acceleration: dzx _[x]_L_ WJ—[tTZ—F dzx L2 [W] i 'F This is obvious from a physical point of view. However, this result is shown below (since sometimes the dimension of a quantity might not be as obvious ...
... determine the dimensions of an acceleration: dzx _[x]_L_ WJ—[tTZ—F dzx L2 [W] i 'F This is obvious from a physical point of view. However, this result is shown below (since sometimes the dimension of a quantity might not be as obvious ...
Page 20
... determined from the initial conditions of the spring-mass system. One way to initiate motion in a spring-mass system ... determine the solution of equation 8.2 which satisfies the initial conditions that the mass is at x0 at t = 0, x(0) ...
... determined from the initial conditions of the spring-mass system. One way to initiate motion in a spring-mass system ... determine the solution of equation 8.2 which satisfies the initial conditions that the mass is at x0 at t = 0, x(0) ...
Page 24
... determine the two forces, F1 and F2. The only force on each mass is due to the spring. Each force is an application of Hooke's law; the force is proportional to the stretching of the spring (it is not proportional to the length of the ...
... determine the two forces, F1 and F2. The only force on each mass is due to the spring. Each force is an application of Hooke's law; the force is proportional to the stretching of the spring (it is not proportional to the length of the ...
Page 36
... determine the amplitude of oscillation after one period. The period of oscillation follows from equations 12.1 and 12.2 and is _ 2n = 2n ' _ x/k/m - 62/4m2 (/k/m[1 _ (cZ/4mk)] However if c2 <<4mk, we can approximate the period by its ...
... determine the amplitude of oscillation after one period. The period of oscillation follows from equations 12.1 and 12.2 and is _ 2n = 2n ' _ x/k/m - 62/4m2 (/k/m[1 _ (cZ/4mk)] However if c2 <<4mk, we can approximate the period by its ...
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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