Fourth-order Perturbed Eigenvalue Equation for Stepwise Damage Detection of Aeroplane Wing

. Perturbed eigenvalue equations up to fourth-order are established to detect structural damage in aeroplane wing. Complete set of perturbation terms including orthogonal and non-orthogonal coefficients are computed using perturbed eigenvalue and orthonormal equations. Then the perturbed eigenparameters are optimized using BFGS approach. Finite element model with small to large stepwise damage is used to represent actual aeroplane wing. In small damaged level, termination number is the same for both approaches, while rms errors and termination d-norms are very close. For medium damaged level, termination number is larger for third-order perturbation with lower d-norm and smaller rms error. In large damaged level, termination number is much larger for third-order perturbation with same d-norm and larger rms error. These trends are more significant as the damaged level increases. As the stepwise damage effect increases with damage level, the increase in stepwise effect leads to the increase in model order. Hence, fourth-order perturbation is more accurate to estimate the model solution.


Introduction
To solve physical problem without exact solution, perturbation theory is established to obtain their approximate solution. A general approach is carried out using various perturbation order expansions on its equation. Adopting the advanced modeling method of element stiffness matrix modification, the order of the structure stiffness matrix was kept invariable in establishing the model of intact and damaged structures by Yu et al. [1]. First-order eigenvalue perturbation theory is introduced to obtain the eigenvalues and eigenvectors of the damaged structure. In the first-order perturbed eigenvalue equation of Wang et al. [2], modal strain energy was used as damage indicator in the damage detection of simply supported beam. Second-order perturbation technique was developed by Chen et al. [3] for damage detection of simply supported beam. When the extent of damage was small, first-order perturbation equation was adopted. Meanwhile the extent was large, second-order perturbation equation was employed to improve the identification precision.
A general order perturbation method involving multiple perturbation parameters was developed by Wong et al. [4] for eigenvalue problems with changes in the stiffness parameters. Symmetric perturbation solutions and eigenparameter sensitivities of different orders were derived. The perturbed eigenvalue problem was established from the perturbations of stiffness matrix, eigenvector, and eigenvalue. Then stiffness parameters of fixed-fixed modular beam was estimated from this equation using the DFP approach [5]. The perturbed orthonormal equation was generated from the perturbation of the eigenvectors and eigenvalues to obtain the k-th skew-symmetric coefficients [6].
In this effort, third and fourth-order perturbation equations are established. Complete set of perturbation terms including orthogonal and non-orthogonal coefficients are computed using perturbed eigenvalue and orthonormal equations. Then the perturbed eigenparameters are optimized using BFGS approach. Limited number of eigenparameters are used to detect different level stepwise damages within aeroplane wing. Finite element model with variational elastic moduli is used to represent the aeroplane wing.

Derivation of fourth-order perturbation equation
The stiffness parameters of the healthy structure prior to any damage are denoted by hi G ( 1, 2,..., i m ), where m is the number of the stiffness parameters. Structural damage is characterized by reductions in the stiffness parameters. The estimated stiffness parameters of the damaged structure before each iteration are denoted by where M is the constant mass matrix, and , By Eq.

Geometric dimension of aeroplane wing
XX-XXX series aeroplane belongs to air-jet type passenger aeroplane. It is a short distance twin turboengine aeroplane. It meets the requirements of middle and short distance flights, reliable, simple and fast, with economical operational and maintenance characteristics. . Figure 1. Span-chord dimensions of aeroplane wing.
Another geometric dimension is the span-chord aspect ratio. Indicated as sc A , its calculation formula is sc s m A W b . Simultaneously, aspect ratio can be represented as the ratio between square of wing span and area of wing surface. When sc A increases, wing lift force coefficient increases, and its drag force increases simultaneously. Therefore, small sc A is utilized in high speed aeroplane wing.   Wc . Then they converge smoothly to the correct values in 52 th iteration with rms error of 0.265 %. In the third order perturbation, termination number is larger at 76. It's rms error is larger at 0.465 %, while it's termination d-norm is exactly the same.

Summary
Finite element model of aeroplane wing is constructed. Third and fourth-order perturbation equations with orthogonal and non-orthogonal coefficients are computed using perturbed eigenvalue and orthonormal equations. In small stepwise damaged level, termination number is the same for both approaches, while rms error and termination d-norm are very close. For medium stepwise damaged level, termination number is larger for thirdorder perturbation with lower d-norm and smaller rms error. In large damaged level, termination number is much larger for third-order perturbation with same dnorm and larger rms error. This trend is more significant as the damaged level increases. This indicates that the increase in stepwise effect leads to the increase in model order. Hence, fourth-order perturbation is more accurate to approximate the model solution.