Structural Dynamic Behavior of Wind Turbines - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 612

Equilibrium Equations

The equations of equilibrium are derived by summing forces and moments in the three
coordinate-axis directions and equating the sums to zero. Referring to Figure 11-2,
summing of forces and moments gives the following six equations:

V x , p + p x , p = 0 (11-4a)

V y , p + p y , p = 0 (11-4b)

T + p z , p = 0

(11-4c)

M x , p - V y , p + Tv + q x , p = 0

(11-5a)

M y , p + V x , p + q y , p = 0 (11-5b)

M �
z , p

− V x , p v � + q z , p = 0

(11-5c)

where
p x ,, p , p y, p , p z , p = applied (external) force loading per unit length (N/m)
q x, p , q y, p , q z ,, p = applied (external) moment loading per unit length (N-m/m)
M x, p , M y, p , M z , p = internal bending moment loads (N-m)

V x, p , V y, p , T = internal force loads (N)

The terms load and loading are often used interchangeably in the literature, so some
clarification of their use in this chapter (and generally in this topic) is in order. We shall
use the term loading to describe an external action applied onto the structure. Thus, p and
q are the sum of all aerodynamic pressures and inertial body forces and are here called
loadings. The internal moment and force responses of the structure to these loadings -- M ,
V , and T -- are termed loads. It has become the general practice in the structural analysis
of wind turbines that if the term “load” stands alone it refers to internal moment and force
responses.
Differentiation twice of Equation (11-3) and once each of Equations (11-5a) and (11-
5b) allows these three equations to be combined with each other and with Equations (11-4a)
and (11-4b) to eliminate M x,p , V x,p , and V y,p . In addition, Equations (11-5b) and (11-5c) can
be combined to eliminate the shear force load V x,p . These eliminations reduce the previous
seven equations to the following four combined moment-curvature-equilibrium equations
in the four unknowns v , M x,p , M y,p , and T:

Flapwise Bending : (- v EI y , p ) + ( T v ) + q x , p + p y , p = 0

(11-6a)

Edgewise Bending : M y , p + q y , p - p x , p = 0

(11-6b)

Pitchwise Torsion : M z , p + M y , p v + q y , p v + q z , p = 0 (11-6c)

Spanwise Tension : T + p z , p = 0

(11-6d)