Dynamic Strength - Hybrid Anisotropic Materials for Wind Power Turbine Blades - page 130

We.can.represent.the.section.areas.Sx,.S R ,.S θ. as:

. Sx.=.ϒx.X 2 .:.S R .=.ϒ R .R 2 ;.S θ =.ϒ θ .θ 2

We.can.replace.the.angle.deformation.following.as:

R

My
E I

R R

Mz
E I

Mx

z = ; = ; =

R 0

G T

.

1 z 1 R xz x

We. now. substitute. the. angle. deformation. and,. after. differentiation,.
Equation.(5.50).can.be.shown.as:

R

.

2 R

My

R

∂
∂

S

t Mz
E Iz

Mx
G T

M = 2( L - D - W sin )sin θ Ω + +

. .(5.51)

2

E I

t 1 1 R xz x

We. can. substitute. time. t. for. natural. frequency. f. and,. as. a. result,. we. can.
ind.a.material.properties.structure.as.well.as.mass.of.rotor.blade.(m),.critical.
damping.coeficient.(δ),.stiffness.of.rotor.blade.(Q 11 ),.vibration.characteristics,.
forcing.frequency.(Ω),.and.natural.frequency(f).
The.natural.frequency.f.can.be.found.by.solving.the.differential.equation.
for.an.orthotropic.rotor.blade.in.a.polar.coordinate:

2

g
h

4 4 4

=

∂
∂

f ∂

∂

f ∂

∂ ∂

f
x R

∂ f

. + D + 2 D + D 0 . (5.52)

1 3 2

2 2 2 2

θ 2

t η x ∂

Here: . D ij. is.the.stiffness.of.the.rotor.blade.from.the.bending.moment.

. D 1 .=.Q 11 Sx.D 2 .=.Q 22 S R ;. D 3 .=.(Q 12 .+2Q 66 )S θ .. (5.53)

Stiffness.constants.Qij.are.determined.only:..Qij.=.Qji;
Tx.is.a.geometrical.stiffness;
E 1 .is.the.modulus.of.elasticity.in.the.warp-x.direction;
E 2 .is.the.modulus.of.elasticity.in.the.ill-y.direction;
μ 12. μ 21 .are.the.Poisson's.ratio.
h.is.the.height.of.the.rotor.blade;
g.is.the.density.of.the.iber;
η.is.the.acceleration.due.to.gravity.

In.the.case.of.a.free.vibration.rotor.blade,.we.use.the.boundary.conditions:.
If.x.=.0;.x.=.R;.f.=.0;