Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 56
... Solution The concept of the stability of solutions is considered to be one of the fundamental aspects of applied mathematics . Now we are pursuing this subject with respect to the stability of the equilibrium positions of a pen- dulum ...
... Solution The concept of the stability of solutions is considered to be one of the fundamental aspects of applied mathematics . Now we are pursuing this subject with respect to the stability of the equilibrium positions of a pen- dulum ...
Page 95
... solution in the phase plane . Letting v = dx / dt and using the chain rule ( d2x / dt2 ) = v ( dv / dx ) yields the ... solution by first drawing the isoclines , curves along which the slope of the solution is constant . As above along v ...
... solution in the phase plane . Letting v = dx / dt and using the chain rule ( d2x / dt2 ) = v ( dv / dx ) yields the ... solution by first drawing the isoclines , curves along which the slope of the solution is constant . As above along v ...
Page 98
... solution ? In this case show that the isocline itself is a solution curve . 26.2 . Consider a linear oscillator with linear friction : m d2x dt2 dx + cdt + kx = 0 . ( a ) Show that E = m / 2 ( dx / dt ) 2 + ( k / 2 ) x2 is a decreasing ...
... solution ? In this case show that the isocline itself is a solution curve . 26.2 . Consider a linear oscillator with linear friction : m d2x dt2 dx + cdt + kx = 0 . ( a ) Show that E = m / 2 ( dx / dt ) 2 + ( k / 2 ) x2 is a decreasing ...
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