During the damage detection process, cause of system boundary condition and numerous DOFs, often the load which act on the substructure is unknown. Also there are many concerns about system excitation. Although diagonal mass matrix which is used in recent literatures have simplified the simulation and decreased the computational effort, but model accuracy can be highly influenced in these systems. Therefore in this subchapter parameter estimation with complete mass matrix and unknown forces is investigated.
The force estimation process is performed based on concept of least square method, although special consideration for the convergence is required. The Kalman estimator inherently use a gain value (Kalman gain) for improvement of the estimation in each global iteration, which is in correlation with initial covariance matrix of the initial guess in each stage of global iteration. The initial guess vector and its correlated initial covariance matrix are updating in start of each global iteration, which affecting all internal iterations of Kalman estimator belong to that specific global loop.
Obtaining converged results in reasonable time is only possible with help global weighting factor, which is multiplied to initial covariance matrix in start of each global iteration. This weighting naturally can make Kalman oscillations in calculation stages, leading to divergence of whole estimation. In other hand without such weighting, results are diverged after numerous numbers of estimation with inappropriate initial guess. As initial guess need to remain unconditional and free of assumption for a robust method, use of weighting is normally required when unknown forces are involved. In Figure 4-9 a beam is schematically connected to an unknown structure.
Figure 4-9. Substructured beam under unknown forces.
In estimation with unknown forces, when the load is not good estimation of the actual force value through simulation steps, this oscillation will easily lead to divergence estimated parameters. This diverged estimated parameter will again make an improper guess for the estimated force in following global iteration, as the force estimation loop is one loop delayed from parameter estimation. In this way, whole estimator is diverged without possible successful results.
The solution is possible by help of variable weighting of the Kalman. In variable weighting, early global loops will improve the force estimation. The small weighting factor keep the initial guess of system parameters without change, meanwhile the force estimation process forms a good estimation through time steps. It is noteworthy to mention minimum size of unknown variable force is as large as number of time steps but not same as integration time steps. After primitive estimation of the force, the weighting factor increased in several stages to speed up the divergence process.
The most important aspect in start of the estimation is proper detection of the force, in Figure 4-10 first estimation of the load is shown. In this stage, the system parameters are not close to real values. To show robustness of proposed method, for first guess is chosen with a large difference with real value, these parameter are presented in Table 4-2.
Figure 4-10. Convergence of the estimated force in after first iteration.
Table 4-2. Initial guess for concerned beam substructure.
Parameter Initial guess Real value Stiffness correlation factor of k1 25000 50000 Stiffness correlation factor of k2 100000 50000 Stiffness correlation factor of k3 58000 50000
After several initial iteration, force estimation reach the close to real value, in Figure 4-11 this trend is shown. It is noteworthy to notice in this stage of estimation, parameters are not good approximation of actual values.
Figure 4-11. Convergence of the estimated force in after 10th iteration.
This trend is continued by several early iterations, till unknown force will not change in comparison of the previous iteration. Here in the experiment the extract acting force which is used for simulation of the substructure is drawn alongside of the estimated force. It is obviously can be observed that when the force estimation is not changing anymore through the loops, the estimated force is closely converged to real excitation force. It shall noticed that the force is only used in the graph presented from but not for estimation iteration, as it is initially considered as unknown. Unknown force in this estimation have 10000 independent unknown variables, same number as time steps, which is needed to be estimated.
Convergence of system parameters are shown in Figure 4-12, the early iterations from 1 to 26 are not embarking a change in initial guessed values. As an example, the k1 correlated value is 27450 (comparing to 25000 in initial guess) and k2 is 90935 which is almost 4 times more than correct stiffness. Obviously main convergence of the parameters is happened after 26th iteration.
Figure 4-12. Convergence of correlated stiffness values through global iterations.
Steps of global iteration, the variable weighing is shown in Table 4-3. As a general rule based on presented result, the weighting need to be delayed till there is no big change in the estimated force, then on next stage weighting can be applied for estimation of the parameters.
Table 4-3. Variable weighting of the global iteration through initial covariance matrix.
Global iteration Global weighting on initial covariance matrix
Loop 1 to 19 No weighting
Loop 20 to 50 100
Loop 50 to 200 1000
Sample result of convergence in substructure dynamics is shown is presented through Figure 4-13 to Figure 4-18. The estimated results are good estimation of the real values. The amount of error is almost negligible and error is mainly is result of numerical inaccuracy which can be improved by smaller time steps.
Figure 4-13. Estimation of 1st DOF velocity in comparison to real value.
Figure 4-14. Estimation of 2nd DOF displacement in comparison to real value.
Figure 4-15. Estimation of 3rd DOF displacement in comparison to real value.
Figure 4-16. Estimation of 4th DOF velocity in comparison to real value.
Figure 4-17. Estimation of 5th DOF displacement in comparison to real value.
Figure 4-18. Estimation of 4th DOF accelration in comparison to real value.
In Table 4-4 final result of the model estimation is presented, 30th iteration as sample is compared to final stage of the estimation to investigate the convergence behaviour.
Table 4-4. Final results and relative error after convergence.
Parameter Real
Highest amount of error belongs to first correlated value, but as seen in previous estimations, this error can be reduced by finer time step. This estimation carried out with 0.02 millisecond step size, in presence 5 percent noise and for 10000 time steps. Meanwhile different step size is considered for Newmark integration, so the simulation had 4 times smaller time step.
5 APPLICATION OF THE METHOD FOR WIND TURBINE BLADE
Damage detection techniques based on substructuring can be applied in parallel or independent from non-destructive test of mechanical systems. Once measurement is done with help of accelerometer sensors, gathered data is used to reveal possible fault considering its location and intensity. This chapter explain application of the presented method on a wind turbine blade as test bench.