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
... which satisfies the initial conditions that the mass is at x0 at t = 0, x(0) = x0. and at t : O. the velocity of the mass, dx/dt. is zero, dx _ 717(0) if 0. In the above example of an initial value problem the 20 Mechanical Vibrations.
... which satisfies the initial conditions that the mass is at x0 at t = 0, x(0) = x0. and at t : O. the velocity of the mass, dx/dt. is zero, dx _ 717(0) if 0. In the above example of an initial value problem the 20 Mechanical Vibrations.
Page 21
... zero yields 02 = 0. In this manner x = x0 cos cot satisfies this initial value problem. The mass executes simple harmonic motion, as shown in Fig. 8-1. The spring is initially stretched and hence the spring pulls the mass to the left ...
... zero yields 02 = 0. In this manner x = x0 cos cot satisfies this initial value problem. The mass executes simple harmonic motion, as shown in Fig. 8-1. The spring is initially stretched and hence the spring pulls the mass to the left ...
Page 35
... zero. It varies smoothly throughout. Thus in the figure we sketch in a solid line the motion of a spring-mass system in the slightly damped case*. The exponential Ae“'/2"' is called the amplitude of the oscillation. Thus the amplitude ...
... zero. It varies smoothly throughout. Thus in the figure we sketch in a solid line the motion of a spring-mass system in the slightly damped case*. The exponential Ae“'/2"' is called the amplitude of the oscillation. Thus the amplitude ...
Page 38
... zero (i.e., c = 0), then determine the general solution of the differential equation. Show that the solution is oscillatory if 600 i ~/ Show that the solution algebraically grows in time if 600 = ~/k/—m (this is called resonance and ...
... zero (i.e., c = 0), then determine the general solution of the differential equation. Show that the solution is oscillatory if 600 i ~/ Show that the solution algebraically grows in time if 600 = ~/k/—m (this is called resonance and ...
Page 39
... zero thereafter: 1'., OgtgAt f(t) : {0 t > At. (a) Show that if c2 < 4mk, then C 2m600 L)e'“/2"'(cos wot + sin c001) x = t + (X0 — 1: for 0 g t 3 At. Calculate the position and velocity of the mass at t : At. Assume that the force is ...
... zero thereafter: 1'., OgtgAt f(t) : {0 t > At. (a) Show that if c2 < 4mk, then C 2m600 L)e'“/2"'(cos wot + sin c001) x = t + (X0 — 1: for 0 g t 3 At. Calculate the position and velocity of the mass at t : At. Assume that the force is ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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