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
... position the mass could be placed and it would not move; there the spring exerts no force on the mass. This place at which we center our coordinate axis, as we see in Fig. 3-1, x = 0, is called the equilibrium or unstretched position of ...
... position the mass could be placed and it would not move; there the spring exerts no force on the mass. This place at which we center our coordinate axis, as we see in Fig. 3-1, x = 0, is called the equilibrium or unstretched position of ...
Page 8
... equilibrium position. k is called the spring constant. It depends on the elasticity of the spring. This linear relationship between the force and the position of the mass was discovered by the seventeenth century physicist Hooke and is ...
... equilibrium position. k is called the spring constant. It depends on the elasticity of the spring. This linear relationship between the force and the position of the mass was discovered by the seventeenth century physicist Hooke and is ...
Page 10
... position at which the spring exerts no force. Is there a position at which we could place the mass and it would not move, what we have called an equilibrium position? If there is, then it follows that dy/dt : d'y/dtZ = O, and the two ...
... position at which the spring exerts no force. Is there a position at which we could place the mass and it would not move, what we have called an equilibrium position? If there is, then it follows that dy/dt : d'y/dtZ = O, and the two ...
Page 11
... equilibrium position with the mass). Let Z 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 ...
... equilibrium position with the mass). Let Z 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 ...
Page 15
... equilibrium position x : O. The solution is periodic in time. As illustrated in Fig. 5-1, the mass after reaching its maximum' displacement (x largest), again returns to the same position T units of time later. The entire oscillation ...
... equilibrium position x : O. The solution is periodic in time. As illustrated in Fig. 5-1, the mass after reaching its maximum' displacement (x largest), again returns to the same position T units of time later. The entire oscillation ...
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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