Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 7
... spring . To develop an appropriate model of the spring force , one should study the motions of spring - mass systems under different circumstances . Let us suppose a series of experiments were run in an attempt to measure the spring ...
... spring . To develop an appropriate model of the spring force , one should study the motions of spring - mass systems under different circumstances . Let us suppose a series of experiments were run in an attempt to measure the spring ...
Page 24
... spring may be stretched or X2 X1 compressed . Certainly , for example , we may impose initial conditions such that ... spring . Each force is an application of Hooke's law ; the force is proportional to the stretching of the spring ( it ...
... spring may be stretched or X2 X1 compressed . Certainly , for example , we may impose initial conditions such that ... spring . Each force is an application of Hooke's law ; the force is proportional to the stretching of the spring ( it ...
Page 26
... spring , that is one with spring constant k but fixed at the other end . The mass m necessary for this analogy is such that 1 m - 1 + 1 M2 ( 9.7 ) This mass m is less than either m1 or m2 ( since 1 / m > 1 / m , and 1 / m > 1 / m2 ...
... spring , that is one with spring constant k but fixed at the other end . The mass m necessary for this analogy is such that 1 m - 1 + 1 M2 ( 9.7 ) This mass m is less than either m1 or m2 ( since 1 / m > 1 / m , and 1 / m > 1 / m2 ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
AS APPLIED TO A SPRINGMASS SYSTEM | 6 |
Copyright | |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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amplitude applied approximation Assume calculated called cars characteristics conservation Consider constant continuous corresponding curve decreases density wave depends derived described determine differential equation discussed distance dx dt dx/dt energy equal equilibrium population equilibrium position equivalent example exercise expression Figure force formula friction function given growth rate hence highway illustrated increases indicate initial conditions integral intersect 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 Pmax possible problem region result road satisfies shock Show shown in Fig simple sketched sketched in Fig slope solution solve species spring spring-mass system stable straight line Suppose tion traffic density traffic flow trajectories Umax unstable valid variables yields zero