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 |
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