Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 56
... distinguish between stable and unstable equilibrium solutions . If a mass m is acted upon by a force f ( x ) , then m d2x dt2 = -f ( x ) . x 56 Mechanical Vibrations LINEARIZED STABILITY ANALYSIS OF AN EQUILIBRIUM SOLUTION.
... distinguish between stable and unstable equilibrium solutions . If a mass m is acted upon by a force f ( x ) , then m d2x dt2 = -f ( x ) . x 56 Mechanical Vibrations LINEARIZED STABILITY ANALYSIS OF AN EQUILIBRIUM SOLUTION.
Page 57
... is determined by the time dependence of the displacement from equilibrium . A simple analysis of equation 18.3 , known as a linearized stability analysis , shows that : 57 Sec . 18 Linearized Stability Analysis of an Equilibrium Solution.
... is determined by the time dependence of the displacement from equilibrium . A simple analysis of equation 18.3 , known as a linearized stability analysis , shows that : 57 Sec . 18 Linearized Stability Analysis of an Equilibrium Solution.
Page 75
... solution oscillates periodically . The amplitude of oscillation is obtained from the initial conditions . Thus the ... equilibrium positions ? ( c ) Derive an expression for conservation of energy . ( d ) Using a phase plane analysis , show ...
... solution oscillates periodically . The amplitude of oscillation is obtained from the initial conditions . Thus the ... equilibrium positions ? ( c ) Derive an expression for conservation of energy . ( d ) Using a phase plane analysis , show ...
Contents
NEWTONS | 6 |
OSCILLATION OF A SPRINGMASS SYSTEM | 12 |
QUALITATIVE AND QUANTITATIVE BEHAVIOR | 18 |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
Common terms and phrases
amplitude of oscillation analyze approximation Assume birth c₁ calculation characteristics Consider constant coefficient corresponding d2x dt2 damping de/dt decreases delay depend derived determine difference equation discussed dx dt dx/dt energy integral equilibrium population equilibrium solution equivalent example exercise exponential Figure formula function growth rate Hint increases initial conditions initial value problem isoclines linearized stability analysis logistic equation mass mathematical model maximum method of characteristics motion moving N₁ Newton's nonlinear pendulum number of cars obtained occur ordinary differential equations oscillation P₁ partial differential equation period phase plane Pmax population growth potential energy r₁ result sharks shock Show shown in Fig simple harmonic motion Sketch the solution sketched in Fig slope solution curves solve species spring spring-mass system stable straight line Suppose Taylor series tion traffic flow trajectories Umax unstable equilibrium position variables vector x₁ yields zero