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 31
... approximately constant independent of the velocity. We model this experimental result by stating for d dt < 0 f I { Y x/ (10.3) —y for dx/dt > O, as sketched in Fig. 10-6 (see exercise 10.4). )2 depends on the roughness of the surface ...
... approximately constant independent of the velocity. We model this experimental result by stating for d dt < 0 f I { Y x/ (10.3) —y for dx/dt > O, as sketched in Fig. 10-6 (see exercise 10.4). )2 depends on the roughness of the surface ...
Page 36
... approximately the same as is represented in Fig. 12-3. Figure 12-3 Oscillation with negligible damping. It is equivalent to say the friction is negligible if after one “period” the amplitude of the oscillation has remained approximately ...
... approximately the same as is represented in Fig. 12-3. Figure 12-3 Oscillation with negligible damping. It is equivalent to say the friction is negligible if after one “period” the amplitude of the oscillation has remained approximately ...
Page 37
... approximately equals 1 if x is small, it follows that if 02 << 4mk, then the exponential has not decayed much in one “period.” Using a numerical criteria, the damping might be said to be negligible if after one “period” the mass returns ...
... approximately equals 1 if x is small, it follows that if 02 << 4mk, then the exponential has not decayed much in one “period.” Using a numerical criteria, the damping might be said to be negligible if after one “period” the mass returns ...
Page 38
... approximately obeys the following statements: If the forcing frequency is less than the natural frequency ((00 < x/kl—m), then the mass oscillates “in phase” with the forcing function (i.e., when the forcing function is a maximum, the ...
... approximately obeys the following statements: If the forcing frequency is less than the natural frequency ((00 < x/kl—m), then the mass oscillates “in phase” with the forcing function (i.e., when the forcing function is a maximum, the ...
Page 39
... approximately constant for some short length of time At, and then zero thereafter: 1'., OgtgAt f(t) : {0 t > At. (a) Show that if c2 < 4mk, then C 2m600 L)e'“/2"'(cos wot + sin c001) x = t + (X0 — 1: for 0 g t 3 At. Calculate the ...
... approximately constant for some short length of time At, and then zero thereafter: 1'., OgtgAt f(t) : {0 t > At. (a) Show that if c2 < 4mk, then C 2m600 L)e'“/2"'(cos wot + sin c001) x = t + (X0 — 1: for 0 g t 3 At. Calculate 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