Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 87
... Suppose that a spring - mass system satisfies m ( d2x / dt2 ) = −kx + αx3 , where a > 0. Sketch the solution in the phase plane . Interpret the solution ( see exercise 18.5 ) . 24.6 . Suppose that the potential energy is known ...
... Suppose that a spring - mass system satisfies m ( d2x / dt2 ) = −kx + αx3 , where a > 0. Sketch the solution in the phase plane . Interpret the solution ( see exercise 18.5 ) . 24.6 . Suppose that the potential energy is known ...
Page 264
... Suppose that the car labeled ẞ moves at the velocity 30 + 15ẞ ( dx / dt = 30 + 15ẞ ) and starts at t O at the position ẞL ( x ( 0 ) BL ) . ( a ) Show that the cars ' velocities steadily increase from 30 to 45 as ẞ ranges from 0 to 1 ...
... Suppose that the car labeled ẞ moves at the velocity 30 + 15ẞ ( dx / dt = 30 + 15ẞ ) and starts at t O at the position ẞL ( x ( 0 ) BL ) . ( a ) Show that the cars ' velocities steadily increase from 30 to 45 as ẞ ranges from 0 to 1 ...
Page 389
... Suppose that we approximate the rate of exiting cars as being proportional to the density , i.e. , B -yp . Assume that u = Umax ( 1 - P / Pmax ) . Under what conditions ( if any ) will a traffic shock occur ? [ Hint : See Sec . 80 ...
... Suppose that we approximate the rate of exiting cars as being proportional to the density , i.e. , B -yp . Assume that u = Umax ( 1 - P / Pmax ) . Under what conditions ( if any ) will a traffic shock occur ? [ Hint : See Sec . 80 ...
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