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 6
... Figure 2-4 shows a rigid bar of length L: [:1 <_—L——-——> Figure 2-4. a.) If the mass density p(x) (mass per unit length) depends on the position along the bar, then what is the total mass m? (b) Using the result of exercise 2.2, where ...
... Figure 2-4 shows a rigid bar of length L: [:1 <_—L——-——> Figure 2-4. a.) If the mass density p(x) (mass per unit length) depends on the position along the bar, then what is the total mass m? (b) Using the result of exercise 2.2, where ...
Page 7
... Figure 3-1 Equilibrium: no force exerted by the spring. X = O x-—» The distance x is then referred to as the displacement from equilibrium or the amount of stretching of the spring. If we stretch the spring (that is let x > 0), then the ...
... Figure 3-1 Equilibrium: no force exerted by the spring. X = O x-—» The distance x is then referred to as the displacement from equilibrium or the amount of stretching of the spring. If we stretch the spring (that is let x > 0), then the ...
Page 9
... 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.
... 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 23
... Figure 9-1. We wish to know the manner in which the two masses, ml and m2, move. To analyze that question, we must return to fundamental principles; we must formulate Newton's law of motion for each mass. The force on each mass equals ...
... Figure 9-1. We wish to know the manner in which the two masses, ml and m2, move. To analyze that question, we must return to fundamental principles; we must formulate Newton's law of motion for each mass. The force on each mass equals ...
Page 27
... Figure 9-3. Suppose that a mass m were attached between two walls a distance d apart (refer to Figures 9-3 and 9-4): k2 m. d. n. Figure 9-4. (a) Briefly explain why it is not necessary for d = II + 12. (b) What position of the mass would ...
... Figure 9-3. Suppose that a mass m were attached between two walls a distance d apart (refer to Figures 9-3 and 9-4): k2 m. d. n. Figure 9-4. (a) Briefly explain why it is not necessary for d = II + 12. (b) What position of the mass would ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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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