Dynamic Strength - Hybrid Anisotropic Materials for Wind Power Turbine Blades - page 127

displacement.x.will.be:

. x.=.A.cosϕ.sinϖt.+.Asinϕ.cosϖt. (5.31)

or.x.=.Asin(ϖt.+.ϕ)..We.replace.circular.frequency.ω.=.2πf,.where.f.is.a.motion.
frequency,.so:

. x.=.sin(2πf.+.ϕ).. .(5.32)

and:

1 2 /

1
2

Q
m

11

f =

.(5.33)

. .

π

By.differentiating.Equation.(5.33),.we.can.determine.velocity.and.accelera-
tion.of.the.longitudinal.vibration:

. V.=.(2πf.+.ϕ).cos(2πf.+.ϕ). (5.34)

. a.=.-(2πf.+.ϕ) 2 .sin(2πf.+.ϕ). (5.35)

The.equation.for.force.vibration.with.damping.in.the.longitudinal.direc-
tion.becomes:

2

= ∂
∂

∂

= (

)

m + δ + Q x L - D - W sin θ s in Ω t .

. (5.36)

11

2

x x

Here:
m.is.a.mass.of.a.rotor.blade;
δ.is.a.critical.damping.coeficient.

The.particular.solution.that.applies.to.the.steady-state.vibration.of.the.heli-
copter.should.be.a.harmonic.function.of.time,.such.as: 9 . .

. x p. =.Asin(Ωt.-.ϕ). (5.37)

where.A.and.ϕ.are.constant.
Substituting.x p. in.Equation.(5.37),.we.get:

-mΩ 2 Asin(Ωt-ϕ).+.δΩAcos(Ωt.-.ϕ).+.Q 11 Asin(Ωt.-.ϕ)=(T.-.D.-.W.sin.θ).sinΩt
. . .(5.38)

Submitting.two.boundary.conditions.Ωt.-.ϕ.=.0.or.Ωt.-.ϕ.=.π/2.results.in: