Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 63
... ( hence the term kinetic ) . [ * f ( x ) dx is the work * necessary to raise the 1 X1 mass from x1 to x . The force necessary to raise the mass is minus the external force [ for example , with gravity —mgj , the force necessary to raise a ...
... ( hence the term kinetic ) . [ * f ( x ) dx is the work * necessary to raise the 1 X1 mass from x1 to x . The force necessary to raise the mass is minus the external force [ for example , with gravity —mgj , the force necessary to raise a ...
Page 325
... Hence the characteristic which emanates from x = x 。( xo > 0 ) at t O is given by = x = Umaxt + xo ( xo > 0 ) . = Various of these characteristics are sketched in Fig . 72-3 . The first charac- teristic in this region starts at x O and ...
... Hence the characteristic which emanates from x = x 。( xo > 0 ) at t O is given by = x = Umaxt + xo ( xo > 0 ) . = Various of these characteristics are sketched in Fig . 72-3 . The first charac- teristic in this region starts at x O and ...
Page 392
... hence moves more slowly until t = tg ( x1 = 0 ) = t ,. After leaving the entrance region , the velocity of the characteristic is dx dt = Umax ( 1 2p / Pmax ) , but p is the constant Bot . Consequently , x = Umax ( 12ẞote / Pmax ) ( tte ) ...
... hence moves more slowly until t = tg ( x1 = 0 ) = t ,. After leaving the entrance region , the velocity of the characteristic is dx dt = Umax ( 1 2p / Pmax ) , but p is the constant Bot . Consequently , x = Umax ( 12ẞote / Pmax ) ( tte ) ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
AS APPLIED TO A SPRINGMASS SYSTEM | 6 |
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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