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 67
... Curves In the previous section a complex expression was derived for the motion of a spring-mass system. A better understanding of the solution can be obtained by analyzing the energy equation 19.2, 1 d 2 " _ _ 1 "° - - +f f(x)d I —2 ...
... Curves In the previous section a complex expression was derived for the motion of a spring-mass system. A better understanding of the solution can be obtained by analyzing the energy equation 19.2, 1 d 2 " _ _ 1 "° - - +f f(x)d I —2 ...
Page 68
... curve in the dx/dt vs. x space, for each value of E: _d_x dt Figure 20-1 Typical energy curve. For each time, the solution x(t) corresponds to one point on this curve since if x(t) is known so is dx/dt. As time changes, the point ...
... curve in the dx/dt vs. x space, for each value of E: _d_x dt Figure 20-1 Typical energy curve. For each time, the solution x(t) corresponds to one point on this curve since if x(t) is known so is dx/dt. As time changes, the point ...
Page 69
... curve is quite significant because we can determine certain qualitative features of the solution directly from it. For example for the curve in Fig. 20-3, since the solution is in the upper half plane, dx/dt > 0, it follows that x ...
... curve is quite significant because we can determine certain qualitative features of the solution directly from it. For example for the curve in Fig. 20-3, since the solution is in the upper half plane, dx/dt > 0, it follows that x ...
Page 70
... curve, defined by equation 21.2 corresponding to one value of E, E : E0, is an ellipse in the phase plane shown in Fig. 21-2 with intercepts at x : ix/ZW and at dx/dt : ix/ZEo/m. / kx2/2 (Potential energy) \ E (Total energy) I I. 70 ...
... curve, defined by equation 21.2 corresponding to one value of E, E : E0, is an ellipse in the phase plane shown in Fig. 21-2 with intercepts at x : ix/ZW and at dx/dt : ix/ZEo/m. / kx2/2 (Potential energy) \ E (Total energy) I I. 70 ...
Page 72
... is the time to go once completely around the closed curve in the phase plane, can be obtained withl \/\/\X/\/\/\/ vv\/\/\/\/r—» Figure 21 -7. out the explicit solution. The velocity v is determined from. 72 Mechanical Vibrations.
... is the time to go once completely around the closed curve in the phase plane, can be obtained withl \/\/\X/\/\/\/ vv\/\/\/\/r—» Figure 21 -7. out the explicit solution. The velocity v is determined from. 72 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