Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 336
... Umax - ( 73.7 ) ( 73.8 ) From equation 73.6 the car starts moving with zero initial velocity ; it slowly accelerates . Its velocity is always less than umax . For very large t , the car approaches maximum velocity ; dx / dt → Umax as ...
... Umax - ( 73.7 ) ( 73.8 ) From equation 73.6 the car starts moving with zero initial velocity ; it slowly accelerates . Its velocity is always less than umax . For very large t , the car approaches maximum velocity ; dx / dt → Umax as ...
Page 374
... Umax 2x ( 1 - 20 ) . Dmax ) Furthermore the general expression for the shock velocity may be simplified as follows : dxs dt = Umax P2 - [ q ] 92-91 == P2 P1 P2 Pmax P2P1 – P1 ( 1 - P1 Pmax / or equivalently P2 dx , = Umax dt - P1 + - P ...
... Umax 2x ( 1 - 20 ) . Dmax ) Furthermore the general expression for the shock velocity may be simplified as follows : dxs dt = Umax P2 - [ q ] 92-91 == P2 P1 P2 Pmax P2P1 – P1 ( 1 - P1 Pmax / or equivalently P2 dx , = Umax dt - P1 + - P ...
Page 375
... Umax ( tu — T ) T ) = u ( po ) tu ( following from Fig . 82-5 ) or thus when tu = - umaxt Umax u ( po ) Umaxt Pmaxt , Umax Umax ( 1 Po Po ( 82.5 ) Pmax a very long time for light traffic , a shorter time for heavier traffic . t , can ...
... Umax ( tu — T ) T ) = u ( po ) tu ( following from Fig . 82-5 ) or thus when tu = - umaxt Umax u ( po ) Umaxt Pmaxt , Umax Umax ( 1 Po Po ( 82.5 ) Pmax a very long time for light traffic , a shorter time for heavier traffic . t , can ...
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