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. |
From inside the book
Results 6-10 of 72
Page 25
... velocity (determined from initial conditions). If this system of two masses attached to a spring is viewed as a single entity, then since there are no external forces on it, the system obeys Newton's first law and will not accelerate ...
... velocity (determined from initial conditions). If this system of two masses attached to a spring is viewed as a single entity, then since there are no external forces on it, the system obeys Newton's first law and will not accelerate ...
Page 30
... velocity: (10.1) where c is a positive constant (c > 0) referred to as the friction coefficient. This force-velocity relationship is called a linear damping force; a damped oscillation meaning the same as an oscillation which decays ...
... velocity: (10.1) where c is a positive constant (c > 0) referred to as the friction coefficient. This force-velocity relationship is called a linear damping force; a damped oscillation meaning the same as an oscillation which decays ...
Page 32
... velocity. In certain physical situations the damping force is proportional to the velocity squared, known as Newtonian damping. In this case show that dzx dx dxl mdt—z=—kx—06E a?' where 06 > 0. If gravity is approximated by a constant ...
... velocity. In certain physical situations the damping force is proportional to the velocity squared, known as Newtonian damping. In this case show that dzx dx dxl mdt—z=—kx—06E a?' where 06 > 0. If gravity is approximated by a constant ...
Page 39
... velocity by I/m. This explains a method by which a nonzero initial velocity occurs. Reconsider exercise 12.8 for an alternate derivation. (a) Show that for 0 g t g At, (d) I mj—f+cx—cxo+kf xdt'=f(,t. O (b) If At is small, show that d m ...
... velocity by I/m. This explains a method by which a nonzero initial velocity occurs. Reconsider exercise 12.8 for an alternate derivation. (a) Show that for 0 g t g At, (d) I mj—f+cx—cxo+kf xdt'=f(,t. O (b) If At is small, show that d m ...
Page 41
... velocity is sufficiently negative. What is the value of this critical velocity ? Does it depend in a reasonable way on x0, 0, m, and k? 13.2. Do exercise 13.1 for the critically damped case 02 = 4mk. 13.3. Assume that friction is ...
... velocity is sufficiently negative. What is the value of this critical velocity ? Does it depend in a reasonable way on x0, 0, m, and k? 13.2. Do exercise 13.1 for the critically damped case 02 = 4mk. 13.3. Assume that friction is ...
Other editions - View all
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