Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 57
... displacement from equilibrium y : Using y as the new dependent variable y = x − XE d2y m = -f ' ( x ) y . dt2 ( 18.3 ) The coefficient f ' ( x ) is a constant ; the displacement from equilibrium y approximately satisfies the above ...
... displacement from equilibrium y : Using y as the new dependent variable y = x − XE d2y m = -f ' ( x ) y . dt2 ( 18.3 ) The coefficient f ' ( x ) is a constant ; the displacement from equilibrium y approximately satisfies the above ...
Page 73
... displacement x traverses a complete cycle ( the plus sign in equation 21.4 must be used in equation 21.6 if v is positive and vice versa ) . This calculation is rather awkward . Instead , the time it takes the moving spring - mass ...
... displacement x traverses a complete cycle ( the plus sign in equation 21.4 must be used in equation 21.6 if v is positive and vice versa ) . This calculation is rather awkward . Instead , the time it takes the moving spring - mass ...
Page 169
... displacement is much less than the equilibrium , | Єym ] < a / ß . Thus as a good approximation Ym + 1 - Ym = · αym - 1 · ( 40.12 ) As we could have suspected , small displacements from an equilibrium popula- tion satisfy a linear ...
... displacement is much less than the equilibrium , | Єym ] < a / ß . Thus as a good approximation Ym + 1 - Ym = · αym - 1 · ( 40.12 ) As we could have suspected , small displacements from an equilibrium popula- tion satisfy a linear ...
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