Detection of Structural Damage in Rotating Beams Using Modal Sensitivity Analysis and Sparse Regularization

Rotating beams are often encountered in the wind turbines and the rotors, and detection of the damages in rotating beams as earlier as possible is central to ensuring the safety and serviceability of practical structures. To this end, a modal sensitivity approach in conjunction with the sparse regularization is proposed in this paper. First, the eigen equations for the flap-wise and chord-wise vibrations of a rotating beam are established upon Hamilton’s principle. Then, damage detection is formulated as a nonlinear least-squares problem that finds the damage coefficients to minimize the error between the measured and calculated data. To solve the nonlinear least-squares problem, the sensitivity method that requires the modal sensitivity analysis is developed. In real applications, damage detection is usually an ill-posed problem and to circumvent the ill-posedness, the sparse regularization is introduced due to the fact that the numbers of actual damage locations are often scarce. Numerical examples are studied and results show that the proposed approach is more accurate than the enhanced sensitivity approach and the flap-wise modal data outperforms the chord-wise modal data in damage detection of rotating beams.

Design of distributed piezoelectric modal sensor for a rotating beam with elastic boundary restraints

As an advanced sensing technique, modal sensors have been attracting a lot of research interest in modal filtering and active control fields. Most of the existing investigations are mainly focused on the static structure. In contrast, there is little effort made for its rotating counterpart, which is frequently encountered in various power machineries. Motivated by such limitation, a unified framework for the distributed piezoelectric modal sensor design of rotating beams with elastic boundary restraints is proposed using polyvinylidene fluoride piezoelectric integral equation and the second-order structural modal functions. A boundary smoothed Fourier series is used to obtain the modal information of rotating beams by solving the differential governing equation and elastic boundary conditions, simultaneously. Modal sensor shape of rotating beams can be determined for any boundary condition by simply setting the elastic restraining coefficients accordingly, instead of reformulating the equation or rewriting the codes like other approach usually does. Numerical examples are presented to demonstrate the correctness and effectiveness of the proposed framework. Modal sensitivity coefficient and charge output frequency response under external excitation are calculated to demonstrate the performance of the designed piezoelectric modal sensors. Influence of rotation speed and boundary restraining stiffness on the modal sensing accuracy of the shaped polyvinylidene fluoride sensor is analyzed and addressed. To our best knowledge, this work represents the first time that an analytical solution for the distributed piezoelectric modal sensor design of a rotating beam with general boundary conditions is derived, which can shed some new lights on further design and implementation of polyvinylidene fluoride modal sensing technique for rotating structures.

Analysis of Complex Modal Characteristics of Fractional Derivative Viscoelastic Rotating Beams

For the transverse vibration problem of a fractional derivative viscoelastic rotating beam, the differential equation of the system is obtained based on the Euler–Bernoulli beam theory and Hamilton principle. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained by the differential quadrature method. The effects of the slenderness ratio, the viscoelastic ratio, the hub radius-beam length ratio, and dimensionless hub speed and fractional order on the vibration characteristics of fractional derivative viscoelastic rotating beams are discussed by numerical examples. Numerical calculations show that when the dimensionless hub speed is constant, the real part of complex frequency increases with the increase of the fractional order, and the higher-order growth trend is more obvious. Through the study of displacement response at different points on the beam, it can be seen that the closer to the free end, the larger the response amplitude. And, the amplitude of response has been attenuated, which is also consistent with the vibration law of free vibration considering damping.