OPTIMIZATION OF A TRUNCATED CONICAL SHELL WIND TURBINE TOWER

A technique is proposed for the optimization of supports in the form of truncated conical shells with a stepwise change in the wall thickness. The potential energy of deformation, the maximum displacement of the structure and the first frequency of natural vibrations were selected as optimization criteria. The solution is performed using nonlinear optimization methods in combination with the finite element method in the Matlab environment.

Formulation of the problem. This article discusses a technique for optimizing shell structures in the form of a truncated cone with a stepwise varying thickness h.
Such structures act as supports for wind turbines. When solving the optimization problem, the load acting on the tower is simplified by us in the form of the concentrated unit force at the upper end (Fig. 1). This load simulates an upstream wind turbine. The support of the tower at the base is assumed to be hingedly fixed. 3. The first frequency of natural vibrations of the structure ω.
The choice of PED as an objective function is justified by the fact that it is an integral measure that determines the level of the stress-strain state [1]. The lower the PED value is, the better the system resists to external influences [2].
The choice of the maximum displacement of the structure as an optimization criterion does not require additional explanations. The first frequency of natural vibrations was chosen as the objective function, since it shows the ability of a structure to resist the occurrence of the phenomenon of mechanical resonance: the higher it is, the more difficult it is to cause resonance during forced vibrations.
The structure consists of n steps with heights L i and wall thicknesses h i . It is required to find such values of L i and h i , to ensure the optimal value of the objective function. The optimum for the potential energy of deformation and displacement corresponds to the minimum, and for the first frequency to the maximum. In this case, the volume of the construction material must be constant and equal to the specified value. In addition to the restriction on the constancy of the volume, there is one more obvious restriction in this problem: problems. Its solution is carried out in the Matlab software package using the Optimization Toolbox package. The interior point method [3] is used as an optimization method. To calculate the objective functions, a program based on the finite element method was prepared. The structure is modeled by flat triangular shell finite elements with 6 degrees of freedom at the node. It is based on the relations given in [4]. After solving the system of FEM equations, the potential strain energy is determined by the formula:  Table 1.
The results of solving the optimization problem are summarized in Table 2.
When using the criteria of minimum PED and minimum displacement, the same optimal values of varied parameters were obtained. A similar result was obtained earlier in [5,6] when optimizing reinforced concrete slabs. Compared to the structure with constant thickness, the potential energy of deformation and displacement are reduced by about 7%. Also, the first natural frequency is increased by 29%. The variant corresponding to the maximum of the first frequency, from the point of view of the potential energy of deformation and maximum displacement, turned out to be worse than the initial step. The maximum of the first frequency is reached at the lower boundary of the parameter h 2 . Let us consider further the solution for the number of steps n = 3. The number of variable parameters increases from 3 to 5. The lower and upper boundaries for the variables L 1 , L 2 , h 1 , h 2 , h 3 are given in Table 3. Optimal values of parameters L 1 , L 2 , h 1 , h 2 , h 3 found on the basis of three criteria are given in Table 4.
The solutions obtained by the criterion of minimum PED and minimum displacement are slightly different, but provide almost the same values of the functions W and v max . Compared to the constant thickness structure, the potential strain energy and maximum displacement reduced by 8.3% and 8.6%, respectively.