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 7
... distance x is then referred to as the displacement from equilibrium or the amount of stretching of the spring. If we stretch the spring (that is let x > 0), then the spring exerts a force pulling the mass back towards the equilibrium ...
... distance x is then referred to as the displacement from equilibrium or the amount of stretching of the spring. If we stretch the spring (that is let x > 0), then the spring exerts a force pulling the mass back towards the equilibrium ...
Page 10
... distance between them, r, Gmlmz rZ ' 1F|= the so-called inverse-square law, where G is a universal constant determined experimentally. If the earth is spherically symmetric, then the force due to the earth's mass acting on any point ...
... distance between them, r, Gmlmz rZ ' 1F|= the so-called inverse-square law, where G is a universal constant determined experimentally. If the earth is spherically symmetric, then the force due to the earth's mass acting on any point ...
Page 11
... distance mg/k when the mass is added, a result that should not be surprising. For a larger mass, the spring sags more. The stiffer the spring (k larger), the smaller the sag of the spring (also quite reasonable). It is frequently ...
... distance mg/k when the mass is added, a result that should not be surprising. For a larger mass, the spring sags more. The stiffer the spring (k larger), the smaller the sag of the spring (also quite reasonable). It is frequently ...
Page 17
... distance across the Charles River in Boston on the frequently walked Harvard Bridge. Local folklore says that this unit was the length of a slightly inebriated student as he was rolled across the bridge by some “friends”. as it is in ...
... distance across the Charles River in Boston on the frequently walked Harvard Bridge. Local folklore says that this unit was the length of a slightly inebriated student as he was rolled across the bridge by some “friends”. as it is in ...
Page 23
... 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 * Figure 9-2. Although the unstretched length of the spring is I, it. 23 Sec. 9 A Two-Mass Oscillator.
... 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 * Figure 9-2. Although the unstretched length of the spring is I, it. 23 Sec. 9 A Two-Mass Oscillator.
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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