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 14
... amplitude of oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + 450, ¢O being the phase at t I 0. In many. 14 Mechanical ...
... amplitude of oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + 450, ¢O being the phase at t I 0. In many. 14 Mechanical ...
Page 15
... drum head! EXERCISES 5.1. Sketch x = 2 sin (3t — tt/2). 5.2. If x = —cos t + 3 sin t, what is the amplitude and phase of the oscillation? Sketch this function. 5.3. If x = —cos t + 3 sin (2'. 15 Sec. 5 Oscillation of a Spring-Mass System.
... drum head! EXERCISES 5.1. Sketch x = 2 sin (3t — tt/2). 5.2. If x = —cos t + 3 sin t, what is the amplitude and phase of the oscillation? Sketch this function. 5.3. If x = —cos t + 3 sin (2'. 15 Sec. 5 Oscillation of a Spring-Mass System.
Page 16
... amplitude of the oscillation? 5.4. If x =, —sin 2t, what is the frequency, circular frequency, period, and amplitude of the oscillation? Sketch this function. 5.5. It was shown that x = ae'“" + be"“" is equivalent to x = c1 cos cor + c ...
... amplitude of the oscillation? 5.4. If x =, —sin 2t, what is the frequency, circular frequency, period, and amplitude of the oscillation? Sketch this function. 5.5. It was shown that x = ae'“" + be"“" is equivalent to x = c1 cos cor + c ...
Page 22
... amplitude of oscillation? Show that the amplitude depends in a reasonable way on 110, k, and m. 8.2. Suppose that a mass is intially at x = x0 with an initial velocity 1:0. Show that the resulting motion is the sum of two oscillations ...
... amplitude of oscillation? Show that the amplitude depends in a reasonable way on 110, k, and m. 8.2. Suppose that a mass is intially at x = x0 with an initial velocity 1:0. Show that the resulting motion is the sum of two oscillations ...
Page 28
... m; is suddenly removed (for example, by cutting the string connecting m1 and m2), then what is the period and amplitude of oscillation of m, ? 10. Friction Our mathematical model shows that the displacement of. 28 Mechanical Vibrations.
... m; is suddenly removed (for example, by cutting the string connecting m1 and m2), then what is the period and amplitude of oscillation of m, ? 10. Friction Our mathematical model shows that the displacement of. 28 Mechanical Vibrations.
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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