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Practice Prelim 2 Solution 2

# Practice Prelim 2 Solution 2 - 2 The position x(t of a mass...

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2. The position x ( t ) of a mass on a spring experiencing periodic forcing obeys the equation: 2 x 0 +4 x 0 +52 x = 2 sin( ωt ) a) There is a transient solution of x tr ( t )= 3 e 2 t sin(4 3 t ) e 2 t cos(4 3 t ) Express this transient solution in the form x tr ( t )= Ce ρt cos( ω 1 t α ) . b) Find the value of ω that gives practical resonance for this system. Solution: x tr ( t )= 3 e 2 t sin(4 3 t ) e 2 t cos(4 3 t ) =2 e 2 t 1 2 cos(4 3 t )+ 3 2 sin(4 3 t ) ! =2 e 2 t cos 4 3 t π + tan 1 3 2 1 2 ! =2 e 2 t cos ± 4 3 t ± π tan 1 ± 3 ² =2 e 2 t cos ± 4 3 t ± π π 3 ² =2 e 2 t cos ³ 4 3 t 2 π 3 ´ Let x ( t )= A sin( ωt )+ B cos( ωt ) be a steady state solution. Then x 0 ( t )= cos( ωt ) sin( ωt ) x 0 ( t )= 2 sin( ωt ) 2 cos( ωt ) 2 ( 2 sin( ωt ) 2 cos( ωt ) ) +4( cos( ωt ) sin( ωt )) + 52( A sin( ωt )+ B cos( ωt )) = 2 sin( ωt ) 1

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2 2 4 +52 A =2 2 2 +4 +52 B =0 2 2 +26 A =1 2 +2 +26 B =0 2 2 +26 A =1 A 2 ω ω 2 26 = B 2 2 A 2 ω ω 2 26 ω +26 A =1 A 2 ω ω 2 26 = B A ± ω 2 2 2 ω 2 ω 2 26 +26 ² =1 A 2 ω ω 2 26 = B A 4 ω
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Practice Prelim 2 Solution 2 - 2 The position x(t of a mass...

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