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# 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 , ω = 1 , 3 = 2 γ, 2 = ω 2 (84) First, we can find the particular solution. Assuming our form above in equation 78: x i ( t ) = D cos( ωt- φ ) (85) we can solve for the amplitude D = A ( ω 2 o- ω 2 ) 2 + 4 γ 2 ω 2 = 10 √ 1 + 9 = 10 √ 10 = √ 10 (86) and the phase lag φ = 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, let’s find the complementary function. Noting that γ 2 = 9 / 4 and ω 2 = 2, we see that γ 2 > ω 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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