Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 62
... dx / dt , dx d2x m dt dt2 di di2 = f ( x ) dx . The left - hand side is an exact derivative , since Thus , After multiplying by dt , 1 / dx \ 2 dx d2x d [ 1⁄2 ( dr ) 2 ] = di di 2 dt dt dt2 d [ 1 ( dx ) ' ] = − f ( x ) dx . m dt 2 dt ...
... dx / dt , dx d2x m dt dt2 di di2 = f ( x ) dx . The left - hand side is an exact derivative , since Thus , After multiplying by dt , 1 / dx \ 2 dx d2x d [ 1⁄2 ( dr ) 2 ] = di di 2 dt dt dt2 d [ 1 ( dx ) ' ] = − f ( x ) dx . m dt 2 dt ...
Page 68
... dx / dt vs. x space , for each value of E : dx dt X Figure 20-1 Typical energy curve . For each time , the solution x ( t ) corresponds to one point on this curve since if x ( t ) is known so is dx / dt . As time changes , the point ...
... dx / dt vs. x space , for each value of E : dx dt X Figure 20-1 Typical energy curve . For each time , the solution x ( t ) corresponds to one point on this curve since if x ( t ) is known so is dx / dt . As time changes , the point ...
Page 351
... dx , dt = P2u ( P2 ) - P1u ( p1 ) . — P2- P1 ( 77.4 ) At points of discontinuity this shock condition replaces the use of the partial differential equation which is valid elsewhere . However , we have not yet explained where shocks ...
... dx , dt = P2u ( P2 ) - P1u ( p1 ) . — P2- P1 ( 77.4 ) At points of discontinuity this shock condition replaces the use of the partial differential equation which is valid elsewhere . However , we have not yet explained where shocks ...
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