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 |
NEWTONS LAW 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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analyze approximation Assume c₁ c₂ calculated conservation of cars Consider corresponding d2x dt2 de/dt density wave velocity depends derived determine dq/dp dx dt dx/dt equilibrium population equilibrium position equilibrium solution example exercise exponential Figure flow-density force formula friction function growth rate highway increases initial conditions initial density initial traffic density initial value problem integral intersect isoclines logistic equation mass mathematical model maximum method of characteristics motion moving Newton's nonlinear pendulum number of cars observer occurs ordinary differential equation oscillation P/Pmax P₁ partial differential equation period phase plane Pmax potential energy problem qualitative region result sharks shock velocity Show shown in Fig simple harmonic motion Sketch the solution sketched in Fig slope solve species spring spring-mass system stable straight line Suppose Taylor series tion traffic flow traffic light trajectories Umax Umaxt unstable equilibrium variables velocity-density x₁ yields zero др