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 16
... calculations we claim that the dimensions of both sides of this equation agree. Checking formulas by dimensional analysis is an important general procedure you should follow. Frequently this type of check will detect embarrassing ...
... calculations we claim that the dimensions of both sides of this equation agree. Checking formulas by dimensional analysis is an important general procedure you should follow. Frequently this type of check will detect embarrassing ...
Page 17
... calculation to eliminate possible confusion only one unit of length should be used. In this text we will use metric units in the m—k—s system, i.e., meters for length, kilogram for mass, and seconds for time. However, as an aid in ...
... calculation to eliminate possible confusion only one unit of length should be used. In this text we will use metric units in the m—k—s system, i.e., meters for length, kilogram for mass, and seconds for time. However, as an aid in ...
Page 37
... calculation has been simplified using the rough numerical approximation 1:2 k 10, since 7:2 = 9.8696 . . . . EXERCISES 12.1. If friction is sufficiently small (02 < 4mk), it has been shown that x = Ae“'/2"' sin (tot + (150), where k 02 ...
... calculation has been simplified using the rough numerical approximation 1:2 k 10, since 7:2 = 9.8696 . . . . EXERCISES 12.1. If friction is sufficiently small (02 < 4mk), it has been shown that x = Ae“'/2"' sin (tot + (150), where k 02 ...
Page 39
... Calculate the position and velocity of the mass at t : At. Assume that the force is large for the short length of time At, i.e., as At ——> 0, [0 —-> 00. Further assume that as At -—> O, foAt = I (called the impulse). Calculate the limit ...
... Calculate the position and velocity of the mass at t : At. Assume that the force is large for the short length of time At, i.e., as At ——> 0, [0 —-> 00. Further assume that as At -—> O, foAt = I (called the impulse). Calculate the limit ...
Page 45
... calculate the acceleration vector d zx/dtz, the velocity vector dx/dt must first be calculated: Q:L'i+€l:f dr dr dr ' Since L is a constant for a pendulum, dL/dt : 0, and hence a = a. dr dt From equation 14.3a, df _d@ a - r it _ ...
... calculate the acceleration vector d zx/dtz, the velocity vector dx/dt must first be calculated: Q:L'i+€l:f dr dr dr ' Since L is a constant for a pendulum, dL/dt : 0, and hence a = a. dr dt From equation 14.3a, df _d@ a - r it _ ...
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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