Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 37
... Taylor series expansion of the period . * In this manner we can roughly estimate the amplitude of oscillation after one period , Ae - ct / 2m≈ Ae ̄2x [ c / 2m√ ( k / m ) ] = Ae - 2x ( c2 / 4mk ) 1/2 Since e- * approximately equals 1 ...
... Taylor series expansion of the period . * In this manner we can roughly estimate the amplitude of oscillation after one period , Ae - ct / 2m≈ Ae ̄2x [ c / 2m√ ( k / m ) ] = Ae - 2x ( c2 / 4mk ) 1/2 Since e- * approximately equals 1 ...
Page 51
... Taylor series of sin 0 , sin = Ꮎ - 07 + 03 05 + 3 ! 5 ! 7 ! - is valid for all 0. What error is introduced by neglecting all the nonlinear terms ? An application of an extension of the mean value theorem ( more easily remembered as the ...
... Taylor series of sin 0 , sin = Ꮎ - 07 + 03 05 + 3 ! 5 ! 7 ! - is valid for all 0. What error is introduced by neglecting all the nonlinear terms ? An application of an extension of the mean value theorem ( more easily remembered as the ...
Page 89
... Taylor expansion of the integrand ( around E = 0 ) will yield a good approximation - E ( 1 2g - u ) 7-1 / 2 E = 1+ ... series . It is valid for all n ( including negative and noninteger n ) as long as a < 1. Many approximations requiring a ...
... Taylor expansion of the integrand ( around E = 0 ) will yield a good approximation - E ( 1 2g - u ) 7-1 / 2 E = 1+ ... series . It is valid for all n ( including negative and noninteger n ) as long as a < 1. Many approximations requiring a ...
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