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 20
... constant) compressing it by 2.5 centimeters (one inch). What is the natural frequency of oscillation of this spring ... constants and a) : A/Wl. The constants c1 and c2 will be determined from the initial conditions of the spring-mass ...
... constant) compressing it by 2.5 centimeters (one inch). What is the natural frequency of oscillation of this spring ... constants and a) : A/Wl. The constants c1 and c2 will be determined from the initial conditions of the spring-mass ...
Page 25
... constant velocity (determined from initial conditions). If this system of two masses attached to a spring is viewed as a single entity, then since there are no external forces on it, the system obeys Newton's first law and will not ...
... constant velocity (determined from initial conditions). If this system of two masses attached to a spring is viewed as a single entity, then since there are no external forces on it, the system obeys Newton's first law and will not ...
Page 26
... constant k but fixed at the other end. The mass m necessary for this analogy is such that + ——- (9.7) This mass m is less than either m1 or m2 (since l/m > l/m1 and 1/m > l/mz); it is thus called the reduced mass: 1 _ mlmZ 1 i '— m1 + ...
... constant k but fixed at the other end. The mass m necessary for this analogy is such that + ——- (9.7) This mass m is less than either m1 or m2 (since l/m > l/m1 and 1/m > l/mz); it is thus called the reduced mass: 1 _ mlmZ 1 i '— m1 + ...
Page 28
... constant and unstretched length. (a) Suppose that the left mass is a distance x from the left wall and the right mass a distance y from the right wall. What position of each mass would be called the equilibrium position of the system of ...
... constant and unstretched length. (a) Suppose that the left mass is a distance x from the left wall and the right mass a distance y from the right wall. What position of each mass would be called the equilibrium position of the system of ...
Page 32
... constant. (a) What is the sign of or? (b) What is the dimension of a? (0) Show that the resulting differential equation is nonlinear. From the differential equation for a spring-mass system with linear damping, show that x = 0 is the ...
... constant. (a) What is the sign of or? (b) What is the dimension of a? (0) Show that the resulting differential equation is nonlinear. From the differential equation for a spring-mass system with linear damping, show that x = 0 is the ...
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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