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
Results 1-5 of 68
Page 6
... depends on the position along the bar, then what is the total mass m? (b) Using the result of exercise 2.2, where is the center of mass 52",? [Hintz Divide the bar up into N equal pieces and take the limit as N —-> 00.] (c) If the total ...
... depends on the position along the bar, then what is the total mass m? (b) Using the result of exercise 2.2, where is the center of mass 52",? [Hintz Divide the bar up into N equal pieces and take the limit as N —-> 00.] (c) If the total ...
Page 7
... depends on the amount of stretching of the spring; the force does not depend on any other quantities. Thus, for example, the force is assumed to be the same no matter what speed the mass is moving at. *Througho ut this text, we assume ...
... depends on the amount of stretching of the spring; the force does not depend on any other quantities. Thus, for example, the force is assumed to be the same no matter what speed the mass is moving at. *Througho ut this text, we assume ...
Page 8
... depends, in a complex manner, on the stretching. However, for stretching of the spring which is not too large (corresponding to at most a moderate force), Fig. 3-3 shows that this curve can be approximated by a straight line: Figure 3-3 ...
... depends, in a complex manner, on the stretching. However, for stretching of the spring which is not too large (corresponding to at most a moderate force), Fig. 3-3 shows that this curve can be approximated by a straight line: Figure 3-3 ...
Page 16
... depend on c1 and c2? 6. Dimensions and Units In the previous section, the formula for the circular frequency of a simple spring-mass system was derived, a, : x/£. m As a check on our calculations we claim that the dimensions of both ...
... depend on c1 and c2? 6. Dimensions and Units In the previous section, the formula for the circular frequency of a simple spring-mass system was derived, a, : x/£. m As a check on our calculations we claim that the dimensions of both ...
Page 22
... depends in a reasonable way on 110, k, and m. 8.2. Suppose that a mass is intially at x = x0 with an initial velocity 1:0. Show that the resulting motion is the sum of two oscillations, one corresponding to the mass initially at rest at ...
... depends in a reasonable way on 110, k, and m. 8.2. Suppose that a mass is intially at x = x0 with an initial velocity 1:0. Show that the resulting motion is the sum of two oscillations, one corresponding to the mass initially at rest at ...
Other editions - View all
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