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 6
... Suppose that an external force a, is applied to m], and 62 to m2. By applying Newton's second law to each mass, show the law can be applied to the rigid body consisting of both masses, if 55 is replaced by the center of mass 9?", [i.e. ...
... Suppose that an external force a, is applied to m], and 62 to m2. By applying Newton's second law to each mass, show the law can be applied to the rigid body consisting of both masses, if 55 is replaced by the center of mass 9?", [i.e. ...
Page 7
... suppose a series of experiments were run in an attempt to measure the spring force. At some position the mass could be placed and it would not move; there the spring exerts no force on the mass. This place at which we center our ...
... suppose a series of experiments were run in an attempt to measure the spring force. At some position the mass could be placed and it would not move; there the spring exerts no force on the mass. This place at which we center our ...
Page 18
... Suppose a quantity y having dimensions of time is only a function of k and m. (a) Give an example of a possible dependence of y on k and m. (b) Can you describe the most general dependence that y can have on k and m ? 7. Qualitative and ...
... Suppose a quantity y having dimensions of time is only a function of k and m. (a) Give an example of a possible dependence of y on k and m. (b) Can you describe the most general dependence that y can have on k and m ? 7. Qualitative and ...
Page 19
... Suppose that we use a firmer spring, that is one whose spring constant k is larger, with the same mass. Without relying on the mathematical formula, what differences in the motion should occur? Let us compare two different springs ...
... Suppose that we use a firmer spring, that is one whose spring constant k is larger, with the same mass. Without relying on the mathematical formula, what differences in the motion should occur? Let us compare two different springs ...
Page 22
... Suppose that a mass is initially at the equilibrium position of a spring, but is initially moving with velocity 00. What is the amplitude of oscillation? Show that the amplitude depends in a reasonable way on 110, k, and m. 8.2. Suppose ...
... Suppose that a mass is initially at the equilibrium position of a spring, but is initially moving with velocity 00. What is the amplitude of oscillation? Show that the amplitude depends in a reasonable way on 110, k, and m. 8.2. Suppose ...
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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