A tuned mass damper in a 120 m tall wind turbine tower consists of a massive metal block suspended inside the nacelle at the top of the tower to dampen oscillations caused by high winds and wave action.
The system is modeled as the metal block held between two horizontal springs inside the turbine's nacelle structure, which oscillates in simple harmonic motion.
In a severe wind storm, the natural frequency of the oscillations of the tower is 0.35 Hz and the maximum acceleration of the damper block is 0.15 m s-2.
Calculate the maximum displacement of the damper block during this storm.
Explain why the natural frequency of the damper system must be about 0.35 Hz.
The acceleration a a\,a of the block is given by the equation:
a=−(km)xa = -\left(\frac{k}{m}\right)xa=−(mk)x
where k k\,k is the force constant of the spring combination, x x\,x is the displacement of the block, and m m\,m is the mass of the block.
The mass of the block is 1.5 × 104 kg and its natural frequency of oscillation is 0.35 Hz.
Show that the force constant k k\,k of the spring combination is about 7.3 × 104 N m-1.
A sudden wind gust causes a rapid movement of the nacelle, displacing the block by 0.45 m from its equilibrium position relative to the nacelle. Use the force constant from (i) to calculate the energy transferred to the springs of the damper system.