Vibration Mitigation of Wind Turbine Towers with Tuned Mass Dampers - Wind Turbine Control and Monitoring - page 341

Fig. 12.8 A horizontally excited structure with a tuned liquid column damper

The equation of motion and the fundamental circular frequency of a TLCD are
given in Eq. 12.8 - 12.9 .

jj u þ x D u ¼ c 1 ð x þ x g Þ

u þ d P ð 12 : 8 Þ

2g
L 1 sin a

r

x D ¼

ð 12 : 9 Þ

Hereby, the motion of the liquid column is defined by u(t) and the motion of the
structure, which is caused by a dynamic force and base excitation equal to
x(t) + x g (t). The coefficient d P specifies the pressure loss, which is caused by
turbulence and friction effects induced by changes in the stream direction and
sectional area of the tank. The geometric factor c 1 (Eq. 12.10 ) scales the inter-
action force between structure and TLCD depending on the geometry of the tank.
The fundamental circular frequency x D of TLCD depends on the so-called
effective length L 1 (Eq. 12.11 ) of the liquid column, inclination of the vertical tank
parts a, and the acceleration g due to gravity. The geometric factor c 1 and the
effective length L 1 depend on the angle a, vertical length V and the horizontal
length H of the liquid stream, and sectional areas A V , A H .

c 1 ¼ H þ 2V cos a
L 1

ð 12 : 10 Þ

L 1 ¼ 2V þ A V

A H H

ð 12 : 11 Þ

The damping forces resulting from the impulse of the liquid mass are given in
the following equation for a structure, which is idealized as an SDOF oscillator.
Hereby, D H and x H are the damping ratio and fundamental circular frequency of
the structure, l the mass ratio of liquid to modal mass of the structure, x g the base
motion, f(t) an excitation force, and c 2 a further geometric factor of TLCD.

x þ 2D H x H x þ x H x ¼ x g þ f ð t Þ l ð x þ x g þ c 2 u Þ

ð 12 : 12 Þ

|{z}
Damping forces