Rayleigh’s quotient–based damage detection algorithm: Theoretical concepts, computational techniques, and field implementation strategies

This article proposes a Rayleigh’s quotient–based damage detection algorithm. It aims at efficiently revealing nascent structural changes on a given structure with the capability to differentiate between an actual damage and a change in operational conditions. The first three damage detection levels are targeted: existence, location, and severity. The proposed algorithm is analytically developed from the dynamics theory and the virtual energy principle. Some computational techniques are proposed for carrying out computations, including discretization, integration, derivation, and suitable optimization methods. Field implementation strategies are also considered for the purpose of online damage monitoring. In order to prove the efficiency of this strategy, one experimental and three numerical case studies were conducted. The proposed algorithm successfully detected the damage in all simulated cases and estimated the damage severity with acceptable accuracy. The conclusion is that the proposed algorithm was able to efficiently detect damage appearance in a range of structures for various damage levels and locations, and under different operational conditions.

Rayleigh's quotient for multiple cracked beam and application

Rayleigh's quotient for Euler-Bernoulli multiple cracked beam with different boundary conditions has been derived from the governed equation of free vibration. An appropriate choosing of approximate shape function in terms of mode shape of uncracked beam and specific functions satisfying conditions at cracks and boundaries leads to an explicit expression of natural frequencies through crack parameters that can simplify not only the analysis of natural frequencies of cracked beam but also the crack detection problem. Numerical analysis of natural frequencies of the cracked beam by using the obtained expression in comparison with the well-known methods such as the characteristic equation and finite element method shows their good agreement. The analytical expression of natural frequencies applied to the crack detection problem allows the result of detection to be improved.

Rayleigh’s Quotient of Linear Piezoelectricity and Its Use in the Approximate Calculation of the Natural Frequencies of Piezoelectric Continua

The paper discusses an extension to the linear theory of piezoelectricity of the Rayleigh quotient used in the analysis of the properties and the approximate calculation of the natural frequencies of elastic continua. It is shown that for a piezoelectric continuum an infinite number of equivalent expressions can be obtained which generalize the classical Rayleigh quotient. The stationarity conditions of any of these quotients under additional constraints imposed by the Gauss equation of electrostatics and the prescribed natural electrical boundary condition are shown to result in the complete set of the governing equations of the free vibration problem of a piezoelectric continuum. The general results discussed in the paper are illustrated by the approximate calculation of the natural frequencies of a piezoelectric rod by the Rayleigh–Ritz method. Unlike in the case of elastic structures, no monotonic convergence of the approximate frequencies is guaranteed for a piezoelectric continuum, the property which can be explained using the introduced Rayleigh quotients.