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 4
... shown in Fig. 2-1: Figure 2-1 Spring-mass system. Observations of this kind of apparatus show that the mass, once set in motion, moves back and forth (oscillates). Although few people today have any intrinsic interest in such a spring ...
... shown in Fig. 2-1: Figure 2-1 Spring-mass system. Observations of this kind of apparatus show that the mass, once set in motion, moves back and forth (oscillates). Although few people today have any intrinsic interest in such a spring ...
Page 5
... shown to be invalid as the velocities involved approach the speed of light. However, as long as the velocity of a ... Fig. 2-2, which shows two masses (m1 and m2) attached to the opposite ends of a rigid (and massless) bar: Figure '2-2 ...
... shown to be invalid as the velocities involved approach the speed of light. However, as long as the velocity of a ... Fig. 2-2, which shows two masses (m1 and m2) attached to the opposite ends of a rigid (and massless) bar: Figure '2-2 ...
Page 7
... shown in Fig. 3-2, where a curve is smoothly drawn connecting the experimental data points marked with an a n, X . F Figure 3-2 Experimental spring force. We have assumed that the force only depends on the amount of stretching of the ...
... shown in Fig. 3-2, where a curve is smoothly drawn connecting the experimental data points marked with an a n, X . F Figure 3-2 Experimental spring force. We have assumed that the force only depends on the amount of stretching of the ...
Page 9
... shown in Fig. 4-1: 7 Figure 4-1. It may seem more reasonable to consider a vertical spring-mass system as illustrated in Fig. 4-2: Figure 4-2. The derivation of the equation governing a horizontal spring-mass system. 9 Sec. 4 Gravity.
... shown in Fig. 4-1: 7 Figure 4-1. It may seem more reasonable to consider a vertical spring-mass system as illustrated in Fig. 4-2: Figure 4-2. The derivation of the equation governing a horizontal spring-mass system. 9 Sec. 4 Gravity.
Page 14
... shown by noting sin (cot + (#0) : sin (or cos ¢,, + cos cot sin ¢,, in which case 0, I A sin 45,, c2 : A cos ¢,. If you are given ... Fig. 5-1: |<—T—>l \/ \/ t Figure 5-1 Period and amplitude of oscillation. A is called the amplitude of the ...
... shown by noting sin (cot + (#0) : sin (or cos ¢,, + cos cot sin ¢,, in which case 0, I A sin 45,, c2 : A cos ¢,. If you are given ... Fig. 5-1: |<—T—>l \/ \/ t Figure 5-1 Period and amplitude of oscillation. A is called the amplitude of 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