Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 7
... spring . To develop an appropriate model of the spring force , one should study the motions of spring - mass systems under different circumstances . Let us suppose a series of experiments were run in an attempt to measure the spring ...
... spring . To develop an appropriate model of the spring force , one should study the motions of spring - mass systems under different circumstances . Let us suppose a series of experiments were run in an attempt to measure the spring ...
Page 24
... spring may be stretched or X2 X1 compressed . Certainly , for example , we may impose initial conditions such that ... spring . Each force is an application of Hooke's law ; the force is proportional to the stretching of the spring ( it ...
... spring may be stretched or X2 X1 compressed . Certainly , for example , we may impose initial conditions such that ... spring . Each force is an application of Hooke's law ; the force is proportional to the stretching of the spring ( it ...
Page 26
... spring , that is one with spring constant k but fixed at the other end . The mass m necessary for this analogy is such that 1 m - 1 + 1 M2 ( 9.7 ) This mass m is less than either m1 or m2 ( since 1 / m > 1 / m , and 1 / m > 1 / m2 ...
... spring , that is one with spring constant k but fixed at the other end . The mass m necessary for this analogy is such that 1 m - 1 + 1 M2 ( 9.7 ) This mass m is less than either m1 or m2 ( since 1 / m > 1 / m , and 1 / m > 1 / m2 ...
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 |
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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