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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Results 6-10 of 35
Page 27
... equivalent? Suppose a mass m were attached to two springs in series (refer to Figs. 9-3 and 9-6):. (a). (b). (c). (d). Figure 9-6. 9.6. 9.7. Answer the same questions as in exercise 9.4a-d. 27 See. 9 A Two-Mass Oscillator.
... equivalent? Suppose a mass m were attached to two springs in series (refer to Figs. 9-3 and 9-6):. (a). (b). (c). (d). Figure 9-6. 9.6. 9.7. Answer the same questions as in exercise 9.4a-d. 27 See. 9 A Two-Mass Oscillator.
Page 35
... equivalent to an arbitrary linear combination of cos cot and sin cot, we observe that the motion of a linearly damped spring-mass system in the underdamped case (c2 < 4mk) is described by x -—- e'“/Z'"(c1 cos cot + 02 sin wt) or x : Ae ...
... equivalent to an arbitrary linear combination of cos cot and sin cot, we observe that the motion of a linearly damped spring-mass system in the underdamped case (c2 < 4mk) is described by x -—- e'“/Z'"(c1 cos cot + 02 sin wt) or x : Ae ...
Page 36
... equivalent to say the friction is negligible if after one “period” the amplitude of the oscillation has remained approximately constant. Thus we wish to determine the amplitude of oscillation after one period. The period of oscillation ...
... equivalent to say the friction is negligible if after one “period” the amplitude of the oscillation has remained approximately constant. Thus we wish to determine the amplitude of oscillation after one period. The period of oscillation ...
Page 47
... equivalent to two scalar equations.) T could be obtained from the second equation (if desired, which it frequently isn't), after determining 0 from the first equation. Equation 14.7a implies that the mass times the 0 component of the ...
... equivalent to two scalar equations.) T could be obtained from the second equation (if desired, which it frequently isn't), after determining 0 from the first equation. Equation 14.7a implies that the mass times the 0 component of the ...
Page 55
... equivalent.) 6,, I O is the “natural” position of a pendulum, as shown in Fig. 17-1, while 0,, I a as demonstrated in Fig. 17-2 is the “inverted” position of a pendulum: Figure 17-1 Natural equilibrium position Figure 17-2 Inverted ...
... equivalent.) 6,, I O is the “natural” position of a pendulum, as shown in Fig. 17-1, while 0,, I a as demonstrated in Fig. 17-2 is the “inverted” position of a pendulum: Figure 17-1 Natural equilibrium position Figure 17-2 Inverted ...
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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