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 13
... expression for e"'°", can be derived from equation 5.4a by replacing to by —co. This results in e"imt = cos cot — i sin cot (5.4b) where the evenness of the cosine function [cos (— y) = cos y] and the oddness of the sine function [sin ...
... expression for e"'°", can be derived from equation 5.4a by replacing to by —co. This results in e"imt = cos cot — i sin cot (5.4b) where the evenness of the cosine function [cos (— y) = cos y] and the oddness of the sine function [sin ...
Page 14
... expression for tan (#0, and using sinZ on + cos2 (1),, = 1 results in an equation for A2: A I or + cal/2 _ c ¢0:tan1—l6?. The expression, x = A sin (wt + (150), is especially convenient for sketching the displacement as a function of ...
... expression for tan (#0, and using sinZ on + cos2 (1),, = 1 results in an equation for A2: A I or + cal/2 _ c ¢0:tan1—l6?. The expression, x = A sin (wt + (150), is especially convenient for sketching the displacement as a function of ...
Page 16
... expression for the general solution is x = Bcos (tot + 90). How do B and 00 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 ...
... expression for the general solution is x = Bcos (tot + 90). How do B and 00 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 ...
Page 23
... expressions for the accelerations, we introduce the position of each mass (for example, in Fig. 9-2, x1 and x2 are the distances each mass is from a fixed origin): —.X1—<> X2 * Figure 9-2. Although the unstretched length of the spring ...
... expressions for the accelerations, we introduce the position of each mass (for example, in Fig. 9-2, x1 and x2 are the distances each mass is from a fixed origin): —.X1—<> X2 * Figure 9-2. Although the unstretched length of the spring ...
Page 25
... expression for the motion of the center of mass is quite interesting, but hardly aids in understanding the possibly complex behavior of each individual mass. Try subtracting equation 9.2 from equation 9.1; you will soon discover that ...
... expression for the motion of the center of mass is quite interesting, but hardly aids in understanding the possibly complex behavior of each individual mass. Try subtracting equation 9.2 from equation 9.1; you will soon discover that ...
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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