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 12
... roots are imaginary, r = iico where a) = A/k/m. Thus the general solution is a linear combination of eimt and e—t'art' x : ae'm + be"°", (5.3) where a and b are arbitrary constants. However, the above. 12 Mechanical Vibrations.
... roots are imaginary, r = iico where a) = A/k/m. Thus the general solution is a linear combination of eimt and e—t'art' x : ae'm + be"°", (5.3) where a and b are arbitrary constants. However, the above. 12 Mechanical Vibrations.
Page 34
... roots are respectively real and unequal, real and equal, and complex. EXERCISES 11.1. Show that c2 has the same dimension as mk. 11.2. What dimension should the roots of the characteristic equation have? Verify that the roots have this ...
... roots are respectively real and unequal, real and equal, and complex. EXERCISES 11.1. Show that c2 has the same dimension as mk. 11.2. What dimension should the roots of the characteristic equation have? Verify that the roots have this ...
Page 41
... roots if c2 > 4mk. (c) Based on part (b), sketch the approximate solution. What is the approximate maximum amplitude? 13.4. Assume that friction is extremely large (c2 > 4mk). (a) If the mass is initially at x = x0 with velocity v0 ...
... roots if c2 > 4mk. (c) Based on part (b), sketch the approximate solution. What is the approximate maximum amplitude? 13.4. Assume that friction is extremely large (c2 > 4mk). (a) If the mass is initially at x = x0 with velocity v0 ...
Page 65
... root must be chosen appropriately. It is positive if the velocity is positive and vice versa. Furthermore, this first order differential equation is separable: dx # =d" i-§~;£fla# Integrating from t = to (where x 2 x0), yields an ...
... root must be chosen appropriately. It is positive if the velocity is positive and vice versa. Furthermore, this first order differential equation is separable: dx # =d" i-§~;£fla# Integrating from t = to (where x 2 x0), yields an ...
Page 171
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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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