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
... called the equilibrium or unstretched position of the spring*: / Figure 3-1 Equilibrium: no force exerted by the spring. X = O x-—» The distance x is then referred to as the displacement from equilibrium or the amount of stretching of ...
... called the equilibrium or unstretched position of the spring*: / Figure 3-1 Equilibrium: no force exerted by the spring. X = O x-—» The distance x is then referred to as the displacement from equilibrium or the amount of stretching of ...
Page 8
... 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 thus known as Hooke's law ...
... 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 thus known as Hooke's law ...
Page 10
... called an equilibrium position? If there is, then it follows that dy/dt : d'y/dtZ = O, and the two forces must balance: 0 : —ky — mg. Thus we see :__fl k8 is the equilibrium position of this spring-mass-gravity system (represented by ...
... called an equilibrium position? If there is, then it follows that dy/dt : d'y/dtZ = O, and the two forces must balance: 0 : —ky — mg. Thus we see :__fl k8 is the equilibrium position of this spring-mass-gravity system (represented by ...
Page 13
... called Euler 's formulas, which when applied to equation 5.3 yield x = (a + b) cos cot + i(a — b) sin cut. The desired result x : c1 cos wt + cz sin cot follows, if the constants c1 and c2 are defined by_ c1 = a + b c2 : i(a — b). The ...
... called Euler 's formulas, which when applied to equation 5.3 yield x = (a + b) cos cot + i(a — b) sin cut. The desired result x : c1 cos wt + cz sin cot follows, if the constants c1 and c2 are defined by_ c1 = a + b c2 : i(a — b). The ...
Page 14
... of oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + 450, ¢O being the phase at t I 0. In many. 14 Mechanical Vibrations.
... of oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + 450, ¢O being the phase at t I 0. In many. 14 Mechanical Vibrations.
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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