Resonant vibration control of rotating beams

Rotating structures, like e.g. wind turbine blades, may be prone to vibrations associated with particular modes of vibration. It is demonstrated, how this type of vibrations can be reduced by using a collocated sensor-actuator system, governed by a resonant controller. The theory is here demonstrated by an active strut, connecting two cross-sections of a rotating beam. The structure is modeled by beam elements in a rotating frame of reference following the beam. The geometric stiffness is derived in a compact form from an initial stress formulation in terms of section forces and moments. The stiffness, and thereby the natural frequencies, of the beam depend on the rotation speed and the controller is tuned to current rotation speed to match the resonance frequency of the selected mode. It is demonstrated that resonant control leads to introduction of the intended level of damping in the selected mode and, with good modal connectivity, only very limited modal spill-over is generated. The controller acts by resonance and therefore has only a moderate energy consumption, and successfully reduces modal vibrations at the resonance frequency.     

Identification of Nonlinear Oscillator with Parametric White Noise Excitation

An identification technique is proposed for a nonlinear oscillator excited by response-dependent white noise. Stiffness, damping and excitation are estimated from records of the stationary stochastic response. The estimation of the stiffness is based on a nonparametric procedure in which the potential energy at the displacement extremes is obtained from the kinetic energy at the previous mean-level-crossing. A nonparametric estimate is obtained by an iterative averaging, in which the increased knowledge of the potential energy in each step is used to avoid bias. The second step in the procedure is to estimate the stationary probability potential in a nonparametric form from a histogram of the kinetic energy at mean-level-crossings. The damping is also estimated in a nonparametric way from approximate expressions of the covariance functions of a set of modified phase plane variables at a given energy level. Finally, the excitation is estimated from a relation between the stationary probability potential, the damping and the excitation. The separation of damping and excitation requires a parametric representation. The system identification technique is investigated by application to response records obtained by stochastic simulation. The stiffness estimation generally gives excellent results, while the damping and excitation estimation tend to be slightly biased for systems with strongly nonlinear stiffness.     

Some integral relations of Hankel transform type and applications to elasticity theory

The dominant part of an integral equation arising in connection with boundary value problems for the circular disc is evaluated in terms of orthogonal polynomials. This relation leads to an efficient method for numerical solution of the complete integral equation even in the presence of a complicated bounded kernel. The static problem of a circular crack in an infinite elastic body under general loads is used to illustrate vector boundary conditions leading to two coupled integral equations, while the problem of a vibrating flexible circular plate in frictionless contact with an elastic half space is solved by use of the associated numerical method.     

Stochastic Analysis of Self-Induced Vibrations

Vortex-induced vibrations of a structural element are modelled as a non-linear stochastic single-degree-of-freedom system. The deterministic part of the governing equation represents laminar flow conditions with a stationary non-zero solution corresponding to lock-in. Across-wind turbulence is included as an additive excitation and along-wind turbulence is introduced as a parametric excitation term, both assumed to be white noise processes. An approximate closed-form solution to the corresponding Fokker–Planck equation in terms of the stationary probability density of the energy is obtained. The auto spectral density of the position at a particular energy-level is approximated by the spectral density of a linear system with energy dependent damping. The spectral density is then obtained by integration of the energy conditional spectral density over all energies weighted by the probability density. The approximate theoretical expressions for the probability density of the energy and the auto spectral density of the position compare favourably with results obtained by numerical simulation.     

Whirling motion of a shallow cable with viscous dampers

The paper deals with the non-linear dynamic analysis of cables with a pair of viscous dampers close to one support. Such cables are characterized by a sag-to-chord-length ratio below 0.02, for which natural frequencies for the vertical and the horizontal vibrations are pair-wise close. Under resonance the non-linear coupling of pairs of modes may cause whirling harmonic motions around the chord line. Whirling motion may occur after bifurcation from single-mode response for harmonic loads in either vertical or horizontal direction. The non-linear features are included in the two coupled modes, while all other modes are treated as linear. The motion is discretized by expansion in terms of the damped complex eigenfunctions. The applied base functions fulfil the transition condition at the damper, leading to fast convergence of the expansion. It is demonstrated that the behaviour of the whirling motion is controlled primarily by the damper acting in the direction of the unloaded mode, whereas the magnitude of the damper in the loaded mode is less important. If the dampers in the vertical and horizontal direction are close to the optimal value of the corresponding taut cable case, substantial reduction of the vibration level of the whirling mode as well as the frequency interval of its occurrence is attained.     

Adaptive tuning of elasto-plastic damper

Hysteretic dampers are frequency independent, and thereby potentially effective for several structural vibration modes, provided that the inherent amplitude dependence can be controlled. An adaptive tuning procedure is proposed, aiming at elimination of the amplitude dependence by adjusting the damper parameter(s) with respect to the magnitude of the damper motion. The procedure is demonstrated in terms of the bilinear elasto-plastic damper model, and optimality corresponds to maximum modal damping. A parametric solution for the damping ratio is obtained by a two-component system reduction technique, and maximization leads to an amplitude dependent expression for the optimal yield level. The amplitude is predicted from the most recent extremum of the damper response, and simultaneously used to adjust the yield level. Numerical examples demonstrate that the adaptive tuning procedure succeeds in controlling the amplitude dependence, resulting in equal damping for the first vibration modes.     

On the elastic strip with an internal crack

The paper presents a method to deal with an inclined crack in an elastic strip. No assumptions of symmetry are made. The method involves the solutions for a cracked plane and an uncracked strip and results in two coupled singular integral equations with finite interval of integration. A crack in a half-plane arises as a limiting case. For internal cracks the integral equations are of a standard type and do not present any numerical difficulties. Results are presented for loads according to the technical beam theory.     

Stress concentration around holes in anisotropic sheets

The formulation of stress concentration problems of plane anisotropic elasticity in terms of integral equations is discussed. First the singular solutions of a concentrated force and a dislocation are formulated so that they remain valid in the case of double roots. The distribution of singularities on smooth curves is then treated, and general formulae for the stress discontinuities are given. Finally, the integral equations arising from stress boundary conditions are discussed, and the characteristics and numerical solution of one of the types are treated in detail.     

The stress distribution in an infinite anisotropic plate with co-linear cracks

A general solution of the plane problem of a finite number of co-linear cracks in an anisotropic material is presented. The solution is obtained by reducing the problem to four very simple Riemann-Hilbert problems. From the solution it is concluded that if the loads acting on the cracks have the resultant zero for each of the cracks, then the normal and shear stresses created on the line of the cracks are independent of the elastic constants. Expressions for the stress intensity factors are derived, and some examples are presented.