class11 - example: The equation of motion for a certain...

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Unformatted text preview: example: The equation of motion for a certain driven damped oscillator is x + 3 x + 2 x = 10 cos t (83) and initially the particle is at rest at the origin. Find the subsequent motion. solution: Compare this equation of motion to equation 73, we see A = 10 , = 1 , 3 = 2 , 2 = 2 (84) First, we can find the particular solution. Assuming our form above in equation 78: x i ( t ) = D cos( t- ) (85) we can solve for the amplitude D = A ( 2 o- 2 ) 2 + 4 2 2 = 10 1 + 9 = 10 10 = 10 (86) and the phase lag = arctan 2 3 / 2 2- 1 = arctan(3) (87) so x i ( t ) = 10 cos( t- arctan(3)) (88) or, recalling that cos( x- y ) = cos x cos y + sin x sin y x i ( t ) = 10[cos t cos(arctan 3) + sin t sin(arctan 3)] (89) noting cos(arctan 3) = 1 / 10 and sin(arctan 3) = 3 / 10 we can also write this as x i ( t ) = cos t + 3 sin t (90) Next, lets find the complementary function. Noting that 2 = 9 / 4 and 2 = 2, we see that 2 > 2 . For the homogeneous equation, this is the case for....
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class11 - example: The equation of motion for a certain...

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