Forced Vibrations Method of Undetermined Coefficients Method of Undetermined

Forced vibrations method of undetermined coefficients

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Forced Vibrations Method of Undetermined Coefficients Method of Undetermined Coefficients: Consider the problem ay 00 + by 0 + cy = g ( t ) First solve the homogeneous equation , which must have constant coefficients The nonhomogeneous function , g ( t ), must be in the class of functions with polynomials, exponentials, sines, cosines, and products of these functions g ( t ) = g 1 ( t ) + ... + g n ( t ) is a sum the type of functions listed above Find particular solutions , y ip ( t ), for each g i ( t ) General solution combines the homogeneous solution with all the particular solutions The arbitrary constants with the homogeneous solution are found to satisfy initial conditions for unique solution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (23/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Method of Undetermined Coefficients Summary Table for Method of Undetermined Coefficients The table below shows how to choose a particular solution Particular solution for ay 00 + by 0 + cy = g ( t ) g ( t ) y p ( t ) P n ( t ) = a n t n + ... + a 1 t + a 0 t s ( A n t n + ... + A 1 t + A 0 ) P n ( t ) e αt t s ( A n t n + ... + A 1 t + A 0 ) e αt P n ( t ) e αt sin( βt ) cos( βt ) t s ( A n t n + ... + A 1 t + A 0 ) e αt cos( βt ) + ( B n t n + ... + B 1 t + B 0 ) e αt sin( βt ) Note: The s is the smallest integer ( s = 0 , 1 , 2) that ensures no term in y p ( t ) is a solution of the homogeneous equation Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (24/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Forced Vibrations Forced Vibrations: The damped spring-mass system with an external force satisfies the equation: my 00 + γy 0 + ky = F ( t ) Example 1 Assume a 2 kg mass and that a 4 N force is required to maintain the spring stretched 0.2 m Suppose that there is a damping coefficient of γ = 4 kg/sec Assume that an external force, F ( t ) = 0 . 5 sin(4 t ) is applied to this spring-mass system The mass begins at rest, so y (0) = y 0 (0) = 0 Set up and solve this system Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (25/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Example 1 1 Example 1: The first condition allows computation of the spring constant, k Since a 4 N force is required to maintain the spring stretched 0.2 m, k (0 . 2) = 4 or k = 20 It follows that the damped spring-mass system described in this problem satisfies: 2 y 00 + 4 y 0 + 20 y = 0 . 5 sin(4 t ) or equivalently y 00 + 2 y 0 + 10 y = 0 . 25 sin(4 t ) , with y (0) = y 0 (0) = 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (26/32)
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Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Homogeneous Equations Method of Undetermined Coefficients Forced Vibrations Example 1 2 Solution: Apply the Method of Undetermined Coefficients to y 00 + 2 y 0 + 10 y = 0 . 25 sin(4 t ) The Homogeneous Solution: The characteristic equation is λ 2 + 2 λ + 10 = 0, which has solution λ = - 1 ± 3 i , so the homogeneous solution is
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