84 P. van der Male and E. Lourens structures would provide us with knowledge of the actual fatigue damage accumulation, enabling an estimation of the remaining service life-time. One approach could be to continually monitor the strain at a number of critical locations. Knowing these locations beforehand, however, could be problematic – consider for instance a lattice structure. Moreover, the failure of a single sensor could frustrate the monitoring process considerably. A need thus exists for a robust integrated fatigue monitoring strategy capable of decreasing the current uncertainty regarding damage development in support structures offshore. With the exception of [12], very little has been done with respect to integrated structural health monitoring of offshore wind turbines. Their focus, however, was on damage detection, sensor-fault detection and load identification. In this contribution, the strain response in a lattice support structure – allowing for the estimation of the accumulated fatigue damage – is estimated on the basis of operational vibrations using a joint input-state estimation algorithm proposed by Lourens et al. [13], in which it is assumed that no prior knowledge of the dynamic characteristics of the input forces is available. Particular attention is paid to the placement of the sensors, which should be within reach for maintenance. The choice for a lattice support structure follows from the expectation that in moderate water depths its application will be more common. Previous studies to the dynamics of lattice support structures can be found in [14] and [15]. To generate artificial measurement data, a reference finite element model, consisting of a simplified wind turbine on a lattice foundation, is constructed. The response data results from combined aerodynamic and hydrodynamic loading. After inclusion of measurement noise, the generated data and an erroneous design model are used to estimate the input forces, the states, and subsequently the strains required to predict fatigue. The erroneous model deviates significantly from the true finite element model, illustrating that despite a relatively weak model representation, the accumulated fatigue damage can be estimated accurately. Apart from the deliberate inclusion of modelling errors, Papadimitriou et al. [16] presented a similar, successful application of the joint input-state estimator for fatigue prediction. 9.2 Method 9.2.1 Joint Input-State Estimation The joint input-state estimator used for the current analysis was presented by Lourens et al. [13]. The algorithm allows for the estimation of states, in terms of displacements and velocities, and the input forces on the basis of a limited number of measurement signals, displacements and accelerations, given that the location of the input forces is known. The starting point of the estimation algorithm is the modally reduced formulation of the system under consideration: Rq.t/ C Pq.t/ C 2q.t/ Dˆ TSpp.t/ (9.1) Here, q.t/ 2Rnm represents the vector of generalized coordinates and p.t/ 2Rnp the input force vector, withnm the number of modes and np the number of input forces. The matrix 2 Rnm nm is the modal damping matrix and 2 Rnm nm a diagonal matrix, containing the natural frequencies related to the nm modes on its diagonal. The corresponding mass normalized mode shapes are collected in the matrix ˆ 2 Rndof nm, with ndof the number of degrees of freedom of the unreduced space-discretized model, and the mode vectors ®j, for j D1; : : : ;nm as its columns. The force selection matrix Sp 2Rndof np specifies the force locations. A dot indicates a derivative with respect to time and the superscript T implies a transpose. The measured quantities are combined in the output vector d.t/ 2Rnd , withnd the number of measured locations: d.t/ DSaˆRq.t/ CSvˆPq.t/ CSdˆq.t/; (9.2) The selection matrices Sa, Sv andSd 2Rndof nd specify the locations of the acceleration, velocity and displacement/strain measurements, respectively. After adopting the state-space formulation for both Eqs. (9.1) and (9.2), where x.t/ D q.t/ Pq.t/ T , and discretizing the continuous-time components, the system can be rewritten in terms of the following discrete-time combined deterministic-stochastic state-space model [13]: xkC1 DAxk CBpk Cwk (9.3) dk DGxk CJpk Cvk (9.4)
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