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
... show the law can be applied to the rigid body consisting of both masses, if 55 is replaced by the center of mass 9?", [i.e., show m(d7-J_c\c,,,/dt 2) = E, where m is the total mass, m : m1 + m;, it”, is the center of mass, in, = (mp-c ...
... show the law can be applied to the rigid body consisting of both masses, if 55 is replaced by the center of mass 9?", [i.e., show m(d7-J_c\c,,,/dt 2) = E, where m is the total mass, m : m1 + m;, it”, is the center of mass, in, = (mp-c ...
Page 16
... Show that elm' = cos on + isin tor, for ca real. (b) Show that e""" = cos cot — isin wt, for (0 real. 5.7. Consider a particle moving around a circle, with its position designated by the polar angle 0. Assume its angular velocity dQ/dt ...
... Show that elm' = cos on + isin tor, for ca real. (b) Show that e""" = cos cot — isin wt, for (0 real. 5.7. Consider a particle moving around a circle, with its position designated by the polar angle 0. Assume its angular velocity dQ/dt ...
Page 18
... Show that 60 has the same dimensions as J You will probably need to note that a radian has no dimension. The formula (d/d6) sin 6 = cos 0 shows this to be true. 6.2 Suppose a quantity y having dimensions of time is only a function of k ...
... Show that 60 has the same dimensions as J You will probably need to note that a radian has no dimension. The formula (d/d6) sin 6 = cos 0 shows this to be true. 6.2 Suppose a quantity y having dimensions of time is only a function of k ...
Page 22
... 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, one corresponding to the ...
... 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, one corresponding to the ...
Page 27
... Show that your formula is in agreement. ' (c) Show that the mass executes simple harmonic motion about its equilibrium position. ((1) What is the period of oscillation? (c) How does the period of oscillation depend on d? Suppose that a ...
... Show that your formula is in agreement. ' (c) Show that the mass executes simple harmonic motion about its equilibrium position. ((1) What is the period of oscillation? (c) How does the period of oscillation depend on d? Suppose that a ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
Common terms and phrases
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