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 8
... yields m 2 df=—m, pm Hi— the simplest mathematical model of a spring-mass system. EXERCISES 3.1. Newton's first law states that with no external forces a mass will move along a straight line at constant velocity. (a) If the motion is ...
... yields m 2 df=—m, pm Hi— the simplest mathematical model of a spring-mass system. EXERCISES 3.1. Newton's first law states that with no external forces a mass will move along a straight line at constant velocity. (a) If the motion is ...
Page 14
... yields an expression for tan (#0, and using sinZ on + cos2 (1),, = 1 results in an equation for A2: A I or + cal/2 _ c ¢0:tan1—l6?. The expression, x = A sin (wt + (150), is especially convenient for sketching the displacement as a ...
... yields an expression for tan (#0, and using sinZ on + cos2 (1),, = 1 results in an equation for A2: A I or + cal/2 _ c ¢0:tan1—l6?. The expression, x = A sin (wt + (150), is especially convenient for sketching the displacement as a ...
Page 21
... 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. This ...
... 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. This ...
Page 24
... problem in a different way; one which in fact yields a solution more easily interpreted. By inspection we note that the equations simplify if they are added together. In that manner 2 2 m,% + m, '17'? = 0. 24 Mechanical Vibrations.
... problem in a different way; one which in fact yields a solution more easily interpreted. By inspection we note that the equations simplify if they are added together. In that manner 2 2 m,% + m, '17'? = 0. 24 Mechanical Vibrations.
Page 29
... yields perfect periodic motion ? Absolutely not, for we can at least imagine a spring-mass system that exhibits many oscillations before it finally appears to significantly decay. In this case the mathematical model of a spring-mass ...
... yields perfect periodic motion ? Absolutely not, for we can at least imagine a spring-mass system that exhibits many oscillations before it finally appears to significantly decay. In this case the mathematical model of a spring-mass ...
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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