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
... maximum' displacement (x largest), again returns to the same position T units of time later. The entire oscillation repeats itself every T units of time, called the period of oscillation. Mathematically a function f (t) is said to be ...
... maximum' displacement (x largest), again returns to the same position T units of time later. The entire oscillation repeats itself every T units of time, called the period of oscillation. Mathematically a function f (t) is said to be ...
Page 23
... maximum and minimum displacements occur halfway between times at which the mass passes its equilibrium position. 9. A Two-Mass Oscillator In the previous sections we carefully developed a mathematical model of a spring-mass system. We ...
... maximum and minimum displacements occur halfway between times at which the mass passes its equilibrium position. 9. A Two-Mass Oscillator In the previous sections we carefully developed a mathematical model of a spring-mass system. We ...
Page 35
... maximum value of the sinusoidal function, sin (out + $0) Figure 12-1. x equals the exponential alone, while at the minimum value of the sinusoidal function, x equals minus the exponential. Thus we first sketch the exponential Ae_“/2 ...
... maximum value of the sinusoidal function, sin (out + $0) Figure 12-1. x equals the exponential alone, while at the minimum value of the sinusoidal function, x equals minus the exponential. Thus we first sketch the exponential Ae_“/2 ...
Page 38
... maximum displacements is a constant. If c < 0, the force is called a negative friction force. (a) In this case, show that x —> 00 as t —> 00. (b) If in addition 02 < 4mk, then roughly sketch the solution. Determine the motion of an ...
... maximum displacements is a constant. If c < 0, the force is called a negative friction force. (a) In this case, show that x —> 00 as t —> 00. (b) If in addition 02 < 4mk, then roughly sketch the solution. Determine the motion of an ...
Page 41
... maximum amplitude? 13.4. Assume that friction is extremely large (c2 > 4mk). (a) If the mass is initially at x = x0 with velocity v0, then approximate the solution. [Hint: Use the result of exercise 13.3b.] (b) Solve the differential ...
... maximum amplitude? 13.4. Assume that friction is extremely large (c2 > 4mk). (a) If the mass is initially at x = x0 with velocity v0, then approximate the solution. [Hint: Use the result of exercise 13.3b.] (b) Solve the differential ...
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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