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Luke Anderson

Differential Equations

Rick ReevesApril 20, 2006

Wing Resonance

1. A)

900y '' ( t) C 8100y ( t) = 1800sin(4 t) y ''C 9 y = 2 sin(4 t)

r2C 9 = 0 r=G 3yh = C1cos(3 t) C C2sin(3 t)

yp = A sin(4 t) C B cos(4 t)

y 'p

= 4 Acos(4 t) K 4 Bsin (4 t)

y ' 'p = K 16 Asin (4 t)K 16 Bcos (4 t)

K 16Asin(4 t) K 16Bcos(4 t) C 9 Asin(4 t)C 9 Bcos(4 t) = 2 sin(4 t)

K 7 Asin(4 t) K 7 Bcos(4 t) = 2 sin(4 t)

sin :K 7 A = 2, A =K 7

2

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cosine:K 7 B = 0, B = 0

yp =K 7

2 sin(4 t)

y ( t) =K 2

7 sin(4 t) C C1 cos (3 t) C C2 sin(3 t)

y ( 0) = 0, 0 = C1

y ' ( t) =K 8

7 cos (4 t) K 3 C1 sin (3 t) C 3 C1 cos(3 t)

y ' (0 ) = 0, 0 =K 8

7C 3 C2 , C2 =

8

21

y ( t) = 8sin(3 t)

21K 2

sin(4 t)

7

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f( t) = 1800sin(3 t)

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The main difference between the two forcing functions is that the first one has a shorter

period along with a higher frequency.

B)

900y ''C 8100y = 1800sin(3 t) y ''C 9 y = 2 sin(3 t)

from part A : yh = C1cos(3 t) C C2sin(3 t)

yp = A tsin (3 t) C B tcos(3 t)

y 'p = A sin(3 t) C 3 A tcos(3 t) C B cos(3 t) K 3 tBsin(3 t)

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y ' 'p = 3 A cos(3 t)

C 3 A cos(3 t) K 9 A t sin(3 t) K 3 B sin(3 t) K 3B sin(3 t) K 9 t B cos(3 t)

3 A cos(3 t) C 3 A cos(3 t) K 9 A t sin(3 t) K 3B sin(3 t) K 3 B sin(3 t) K 9 t B cos(3 t)C 9 A t sin(3 t) C 9 B t cos(3 t) = 2 sin (3 t)

6 A cos(3 t) K 6 B sin(3 t) = 3 sin(3 t)

sine : K 6 B = 2, B = K 3

cosine: 6 A = 0, A = 0

yp = K 3 t cos(3 t)

y ( t) = K 3 t cos(3 t) C C1cos(3 t) C C2sin(3 t)

y (0 ) = C1, C1 = 0

y ' ( t) = K 3 cos(3 t) C 9 tsin(3 t) K 3 C1sin(3 t)

C 3 C2cos(3 t)

y (0) = K 3C 3 C2 , C2 = 1

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y ( t) = K 3 tcos(3 t) C sin(3 t)

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As t grows large, there are subsequent larger displacements in the sinusoidal movement

(the amplitude steadily increases). The wings would snap once they are pushed beyond

their maximum tolerance for flexibility.

3. There were certain forces that we did not include in our model. We did not account for

friction from air, or within the wing itself (no damping). We did not include the force

due to lift, drag, or the weight of the plane itself.

An engineer would need to analyze the density and composition of the wings in use.

Analyze. He would need to take into account the thermal conditions of the material or

the environment in which the resonance occurs. Along the same lines, he would also

want to consider the altitude at which this occurs along with the air pressure and how

changes will affect the wing resonance. Also, it would be appropriate to analyze how

much displacement the wing is capable of. Finally, the speeds at which resonance occurs

should be analyzed.