Kinetic field theory: Non-linear cosmic power spectra in the mean-field approximation

We use the recently developed Kinetic Field Theory (KFT) for cosmic structure formation to show how non-linear power spectra for cosmic density fluctuations can be calculated in a mean-field approximation to the particle interactions. Our main result is a simple, closed and analytic, approximate expression for this power spectrum. This expression has two parameters characterising non-linear structure growth which can be calibrated within KFT itself. Using this self-calibration, the non-linear power spectrum agrees with results obtained from numerical simulations to within typically \lesssim10\,\%≲10% up to wave numbers k\lesssim10\,h\,\mathrm{Mpc}^{-1}k≲10hMpc−1 at redshift z = 0z=0. Adjusting the two parameters to optimise agreement with numerical simulations, the relative difference to numerical results shrinks to typically \lesssim 5\,\%≲5%. As part of the derivation of our mean-field approximation, we show that the effective interaction potential between dark-matter particles relative to Zel’dovich trajectories is sourced by non-linear cosmic density fluctuations only, and is approximately of Yukawa rather than Newtonian shape.

Determination of natural frequencies and mode shapes of a wind turbine rotor blade using Timoshenko beam elements

When simulating a wind turbine, the lowest eigenmodes of the rotor blades are usually used to describe their elastic deformation in the frame of a multi-body system. In this paper, a finite element beam model for the rotor blades is proposed which is based on the transfer matrix method. Both static and kinetic field matrices for the 3-D Timoshenko beam element are derived by the numerical integration of the differential equations of motion using a Runge–Kutta fourth-order procedure. In the general case, the beam reference axis is at an arbitrary location in the cross section. The inertia term in the motion differential equation is expressed using appropriate shape functions for the Timoshenko beam. The kinetic field matrix is built by numerical integration applied on the approximated inertia term. The beam element stiffness and mass matrices are calculated by simple matrix operations from both field matrices. The system stiffness and mass matrices of the rotor blade model are assembled in the usual finite element manner in a global coordinate system accounting for the structural twist angle and possible pre-bending. The program developed for the above-mentioned calculations and the final solution of the eigenvalue problem is accomplished using MuPAD, a symbolic math toolbox in MATLAB®. The natural frequencies calculated using generic rotor blade data are compared with the results proposed from the FAST and ADAMS software.

Determination of Natural Frequencies and Mode Shapes of a Wind Turbine Rotor Blades using Timoshenko Beam Elements

In the simulation of a wind turbine, the lowest eigenmodes of the rotor blades are usually used to describe their elastic deformation in the frame of a multibody system. In this paper, a finite element beam model for the rotor blades based on the transfer matrix method is proposed. Both static and kinetic field matrices for the 3D Timoshenko beam element are derived by numerical integration of the differential equations of motion using RUNGE KUTTA 4th order procedure. The beam reference axis in the general case is at an arbitrary location in the cross section. The inertia term in the motion differential equation is expressed using appropriate shape functions for the Timoshenko beam. The kinetic field matrix is built by numerical integration applied on the approximated inertia term. The beam element stiffness and mass matrices are calculated by simple matrix operations from both field matrices. The system stiffness and mass matrices of the rotor blade model are assembled in the usual finite element manner in a global coordinate system with the accounting for structural twist angle and possibly pre-bending. The program developed for the above calculations and the final solution of the eigenvalue problem is accomplished using MuPAD, a symbolic math toolbox of MATLAB®. The calculated natural frequencies using generic rotor blade data are compared with the results proposed from FAST and ADAMS software.