Design of Foundations for Offshore Wind Turbines

Appendix ANatural Frequency of a Cantilever Beam with Variable Cross Section

The aim of the appendix is to provide the theoretical basis of the natural frequency estimation of monopile‐supported wind turbine. Please also refer to Chapter 5(see Sections 5.4.1 and 5.8) and publications by Arany et al. (2015a,b, 2016, 2017).

The motion of the cantilever beam can be described as a single degree of freedom mass‐spring system. The free vibration of this system is given by

A.1

Assuming harmonic vibration, the following equation can be obtained:

A.2

and from the circular frequency the Hertz frequency is easily obtained:

A.3

The flexibility of the substructure expresses the dependence of the natural frequency on the water depth i.e. the flexibility of the monopile above the mudline and that of the transition piece. For the sake of simplicity, the model used in Chapter 5assumes that the monopile's bending stiffness continues throughout the water depth and up to the root of the tower. For clarity, see Figure A.1.

A.4

A.5

Figure A.1 Cantilever beam with variable cross section.

Castigliano's ...