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 11
... equal the displacement from this equilibrium position: I _ _.@5 2 m_g. Z J' ( k l y + k Upon this substitution, equation 4.2 becomes dZZ * m d', — —-kZ. This is the same as equation 4.1. Thus the mass will move vertically around the new ...
... equal the displacement from this equilibrium position: I _ _.@5 2 m_g. Z J' ( k l y + k Upon this substitution, equation 4.2 becomes dZZ * m d', — —-kZ. This is the same as equation 4.1. Thus the mass will move vertically around the new ...
Page 21
... equal to zero 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 ...
... equal to zero 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 ...
Page 23
... equals its mass times its acceleration. In order to obtain expressions for the accelerations, we introduce the position of each mass (for example, in Fig. 9-2, x1 and x2 are the distances each mass is from a fixed origin): —.X1—<> X2 ...
... equals its mass times its acceleration. In order to obtain expressions for the accelerations, we introduce the position of each mass (for example, in Fig. 9-2, x1 and x2 are the distances each mass is from a fixed origin): —.X1—<> X2 ...
Page 34
... equal, and complex. EXERCISES 11.1. Show that c2 has the same dimension as mk. 11.2. What dimension should the roots of the characteristic equation have? Verify that the roots have this dimension. 12. Underdamped Oscillations If c2 ...
... equal, and complex. EXERCISES 11.1. Show that c2 has the same dimension as mk. 11.2. What dimension should the roots of the characteristic equation have? Verify that the roots have this dimension. 12. Underdamped Oscillations If c2 ...
Page 35
... equals the exponential alone, while at the minimum value of the sinusoidal function, x equals minus the exponential. Thus we first sketch the exponential Ae_“/2"' and its negative —Ae_"/2"' in dashed lines. Periodically at the “x's” the ...
... equals the exponential alone, while at the minimum value of the sinusoidal function, x equals minus the exponential. Thus we first sketch the exponential Ae_“/2"' and its negative —Ae_"/2"' in dashed lines. Periodically at the “x's” the ...
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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