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 viii
... Curves 67 Phase Plane of a Linear Oscillator 70 Phase Plane of a Nonlinear Pendulum 76 Can a Pendulum Stop? 82 What Happens if a Pendulum is Pushed Too Hard? ....................................... .. 84 Period of a Nonlinear Pendulum ...
... Curves 67 Phase Plane of a Linear Oscillator 70 Phase Plane of a Nonlinear Pendulum 76 Can a Pendulum Stop? 82 What Happens if a Pendulum is Pushed Too Hard? ....................................... .. 84 Period of a Nonlinear Pendulum ...
Page 7
... curve is smoothly drawn connecting the experimental data points marked with an a n, X . F Figure 3-2 Experimental spring force. We have assumed that the force only depends on the amount of stretching of the spring; the force does not ...
... curve is smoothly drawn connecting the experimental data points marked with an a n, X . F Figure 3-2 Experimental spring force. We have assumed that the force only depends on the amount of stretching of the spring; the force does not ...
Page 8
... curve can be approximated by a straight line: Figure 3-3 Hooke's Law: approximation of experimental 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 equilibrium ...
... curve can be approximated by a straight line: Figure 3-3 Hooke's Law: approximation of experimental 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 equilibrium ...
Page 35
... curves drawn in Fig. 12-2 (exactly where depends on the phase (150): 1 -Al// Figure 12-2 underdamped oscillation. ' ' Halfway between the marks, the function is zero. It varies smoothly throughout. Thus in the figure we sketch in a ...
... curves drawn in Fig. 12-2 (exactly where depends on the phase (150): 1 -Al// Figure 12-2 underdamped oscillation. ' ' Halfway between the marks, the function is zero. It varies smoothly throughout. Thus in the figure we sketch in a ...
Page 65
... curve. These facts are noted in Fig. 19-1. . From conservation of energy, equation 19.2, an expression for the velocity, dx/dt, can be obtained, dx __ 2E 2 x _ - E_i-;—;Lnaw The sign of the square root must be chosen appropriately. It ...
... curve. These facts are noted in Fig. 19-1. . From conservation of energy, equation 19.2, an expression for the velocity, dx/dt, can be obtained, dx __ 2E 2 x _ - E_i-;—;Lnaw The sign of the square root must be chosen appropriately. It ...
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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