A novel method of distributed dynamic load identification for aircraft structure considering multi-source uncertainties

Abstract

A series of work for distributed dynamic load identification is investigated in this paper considering unknown-but-bounded uncertainties in the aircraft structure. To facilitate the analysis, the complicated rudder structure is simplified to a plate structure based on the robust equivalence principle of mechanical property under multi-cases of flight environments. Aiming at the plate structure, a time domain–based model for distributed dynamic load identification is established through the acceleration response measured by sensors. Among them, the spatial distributed load is approximated by Chebyshev orthogonal polynomials at each sampling time, and load boundaries can be calculated by the Taylor-expansion-based uncertain propagation analysis. As keys to improve the reliability of recognition results, the optimization process for sensor placement is constructed by the particle swarm optimization algorithm, taking the robustness evaluation index and sensor distribution index into consideration. The validity and the feasibility of the proposed methodology are demonstrated by several numerical examples, and the results reveal that designer can make a rational tradeoff choice among the cost of sensor placement and the performance of load identification in a systematic framework.

Appendices

Appendix 1

In view of the nominal value of the honeycomb equivalent model ${E}_x^c=3.2\mathrm{MPa}$, ${E}_y^c=3.2\mathrm{MPa}$, ${E}_z^c=6043.3\mathrm{MPa}$, ${\mu}_{xy}^c=0.9986$, ${\mu}_{yz}^c=1.2427\times {10}^{-4}$, ${\mu}_{xz}^c=1.2421\times {10}^{-4}$, ${G}_{xy}^c=1.9\mathrm{MPa}$, ${G}_{yz}^c=1654.2\mathrm{MPa}$, ${G}_{xz}^c=827.1\mathrm{MPa}$, and ρ^c = 0.2438g/cm³, the displacement vector ${\mathbf{u}}_i^c$ can be obtained by CFD calculation for the i-th load condition. To get the boundaries of the displacements, the sensitivity analysis is introduced, in which the perturbation of each parameter will be operate. Taking the parameter E_x as an example, the displacement vector ${\left.{\mathbf{u}}_i\right|}_{E_x={E}_x^c+\delta {E}_x}$ can also be calculated when the E_x has the small perturbation δE_x and other parameters remain unchanged. Employing the difference technique, the sensitivity of displacement vector u_i with respect to parameter E_x can be expressed as $\frac{\partial {\mathrm{u}}_i}{\partial {E}_x}=\frac{{\mathrm{u}}_i\left|{}_{E_x={E}_x^c+\delta {E}_x}-{\mathrm{u}}_i^c\right.}{\delta {E}_x}$. As shown in Fig. 18 , the upper bound ${\overline{\mathbf{u}}}_i$ and lower bound ${\underset{\_}{\mathbf{u}}}_i$ of displacement vector u_i can be obtained through the approximate linearization which is often used in engineering. Therefore, the boundaries of displacement vector u_i considering the uncertainties of each parameter can be eventually given by

$$ {\displaystyle \begin{array}{l}{\overline{\mathbf{u}}}_i={\mathbf{u}}_i^c+\left|\frac{{\left.{\mathbf{u}}_i\right|}_{E_x={E}_x^c+\delta {E}_x}-\kern0.33em {\mathbf{u}}_i^c}{2\delta {E}_x}\right|\left({\overline{E}}_x-{\underset{\_}{E}}_x\right)+\cdots +\left|\frac{{\left.{\mathbf{u}}_i\right|}_{\mu_{xy}={\mu}_{xy}^c+\delta {\mu}_{xy}}-\kern0.33em {\mathbf{u}}_i^c}{2\delta {\mu}_{xy}}\right|\left({\overline{\mu}}_{xy}-{\underset{\_}{\mu}}_{xy}\right)\\ {}\kern1.98em +\cdots +\left|\frac{{\left.{\mathbf{u}}_i\right|}_{G_{xy}={G}_{xy}^c+\delta {G}_{xy}}-\kern0.33em {\mathbf{u}}_i^c}{2\delta {G}_{xy}}\right|\left({\overline{G}}_{xy}-{\underset{\_}{G}}_{xy}\right)+\cdots +\left|\frac{{\left.{\mathbf{u}}_i\right|}_{\rho ={\rho}^c+\delta \rho}-\kern0.33em {\mathbf{u}}_i^c}{2\delta \rho}\right|\left(\overline{\rho}-\underset{\_}{\rho}\right)\end{array}} $$

(42)

and

$$ {\displaystyle \begin{array}{l}{\underset{\_}{\mathbf{u}}}_i={\mathbf{u}}_i^c-\left|\frac{{\left.{\mathbf{u}}_i\right|}_{E_x={E}_x^c+\delta {E}_x}-\kern0.33em {\mathbf{u}}_i^c}{2\delta {E}_x}\right|\left({\overline{E}}_x-{\underset{\_}{E}}_x\right)-\cdots -\left|\frac{{\left.{\mathbf{u}}_i\right|}_{\mu_{xy}={\mu}_{xy}^c+\delta {\mu}_{xy}}-\kern0.33em {\mathbf{u}}_i^c}{2\delta {\mu}_{xy}}\right|\left({\overline{\mu}}_{xy}-{\underset{\_}{\mu}}_{xy}\right)\\ {}\kern1.98em -\cdots -\left|\frac{{\left.{\mathbf{u}}_i\right|}_{G_{xy}={G}_{xy}^c+\delta {G}_{xy}}-\kern0.33em {\mathbf{u}}_i^c}{2\delta {G}_{xy}}\right|\left({\overline{G}}_{xy}-{\underset{\_}{G}}_{xy}\right)-\cdots -\left|\frac{{\left.{\mathbf{u}}_i\right|}_{\rho ={\rho}^c+\delta \rho}-\kern0.33em {\mathbf{u}}_i^c}{2\delta \rho}\right|\left(\overline{\rho}-\underset{\_}{\rho}\right)\end{array}} $$

(43)

Fig. 18

The explanation of sensitivity analysis method

Fig. 19

Model equivalence

Fig. 20

Load identification

Appendix 2

According to (15), the recurrence formulas of modal force can be concluded through displacement, velocity, acceleration, or the combination of them. However, the acceleration response is more convenient to acquire through acceleration sensors in comparison with displacement, velocity, strain, or other indexes in practical engineer.

In general, the input of measured acceleration response $\ddot{\mathbf{w}}\left({t}_k,{\varGamma}_{\gamma}\right)$ is depend on the number and position of the sensors, therefore the modal acceleration term ${\ddot{q}}_r\left(\boldsymbol{\upalpha}, {t}_k\right)$ is a variable related to the sensor placement, namely

$$ \ddot{\mathbf{w}}\left({t}_k,{\varGamma}_{\gamma}\right)=\boldsymbol{\Phi} \left(\boldsymbol{\upalpha}, p\right)\ddot{\mathbf{q}}\left(\boldsymbol{\upalpha}, {t}_k,{\varGamma}_{\gamma}\right) $$

(44)

where Γ_γ is the variable of the sensor placement.

Assume that the initial values of modal displacement response q(t₀) and modal velocity response $\dot{\mathbf{q}}\left({t}_0\right)$ are zero. Numerical integration method, such as Newmark-Beta method (Roy and Dash 2002), serve to solve the modal velocity and modal displacement response,

$$ {\dot{\mathbf{q}}}_r\left(\boldsymbol{\upalpha}, {t}_k,{\varGamma}_{\gamma}\right)={\dot{\mathbf{q}}}_r\left(\boldsymbol{\upalpha}, {t}_{k-1},{\varGamma}_{\gamma}\right)+\left[\left(1-\beta \right){\ddot{\mathbf{q}}}_r\Big(\boldsymbol{\upalpha}, {t}_{k-1},{\varGamma}_{\gamma}\left)+\beta {\ddot{\mathbf{q}}}_r\right(\boldsymbol{\upalpha}, {t}_k,{\varGamma}_{\gamma}\Big)\right]\varDelta t $$

(45)

$$ {\mathbf{q}}_r\left(\boldsymbol{\upalpha}, {t}_k,{\varGamma}_{\gamma}\right)={\mathbf{q}}_r\left(\boldsymbol{\upalpha}, {t}_{k-1},{\varGamma}_{\gamma}\right)+{\dot{\mathbf{q}}}_r\left(\boldsymbol{\upalpha}, {t}_{k-1},{\varGamma}_{\gamma}\right)\varDelta t+\left[\left(\frac{1}{2}-\theta \right){\ddot{\mathbf{q}}}_r\Big(\boldsymbol{\upalpha}, {t}_{k-1},{\varGamma}_{\gamma}\left)+\theta {\ddot{\mathbf{q}}}_r\right(\boldsymbol{\upalpha}, {t}_k,{\varGamma}_{\gamma}\Big)\right]\varDelta {t}^2 $$

(46)

where the value of parameters θ and β need to be chosen according to the accuracy and stability of the integration. The parameter θ is generally taken as 0 ∼ 1/4, and β is often taken as 1/2.