Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 82
... Pendulum Stop ? - -л The phase plane , Fig . 22-10 , for the limiting energy curve E = 2g shows that the pendulum tends towards the inverted position ( either = — ог = π ) . For example , E = 2g corresponds to initially starting a pendulum ...
... Pendulum Stop ? - -л The phase plane , Fig . 22-10 , for the limiting energy curve E = 2g shows that the pendulum tends towards the inverted position ( either = — ог = π ) . For example , E = 2g corresponds to initially starting a pendulum ...
Page 84
... pendulum never pass its equilib- rium position again ? 23.2 . If a pendulum is initially at its unstable equilibrium position , then how large an initial angular velocity is necessary for the pendulum to go completely around ? 23.3 ...
... pendulum never pass its equilib- rium position again ? 23.2 . If a pendulum is initially at its unstable equilibrium position , then how large an initial angular velocity is necessary for the pendulum to go completely around ? 23.3 ...
Page 111
... pendulum diminishes after each oscillation due to the small friction ( as illustrated in Fig . 28-10 ) . V = d0 dt Figure 28-10 Phase plane if k2 < 4Lg : sketch illustrating ... pendulum 111 Sec . 28 Nonlinear Pendulum with Damping.
... pendulum diminishes after each oscillation due to the small friction ( as illustrated in Fig . 28-10 ) . V = d0 dt Figure 28-10 Phase plane if k2 < 4Lg : sketch illustrating ... pendulum 111 Sec . 28 Nonlinear Pendulum with Damping.
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