Mechanical systems are often modeled with the multibody system method or the finite element method and numerically described with systems of differential equations. Increasing demands on detail and the resulting high complexity of these systems make the use of model order reduction inevitable. Frequently, moment matching based on Krylov subspaces is used for the reduction. There, the transfer functions of the full system and of the reduced system are matched at distinct frequency shifts. The selection of these shifts, however, is not trivial. In this contribution we suggest an algorithm that evaluates an increasing number of shifts iteratively until a reduced model that approximates the full model in a subspace with very low approximation error is found. Thereafter, the projection matrix that spans this subspace is decomposed with singular value decomposition and only most important directions are retained. In this way, small reduced models with good approximation properties that do not exceed a predefined error bound can be found or low-error models for a given reduced order can be generated. The evaluation of more shifts than necessary and further reduction by means of singular value decomposition is the novelty of this contribution. In this paper, this novel approach is extensively studied and, furthermore, applied to the numerical example of an industrial helicopter model.
The dynamical behavior of Elastic Multibody Systems (EMBS) is often analyzed using virtual prototypes described by high-dimensional systems of differential equations. Model Order Reduction (MOR) is a key step to permit efficient system evaluations by approximating the full system with a reduced order surrogate model. It is one challenge in MOR of EMBS, to describe the dynamics induced through the coupling of bodies in the reduced system. In this contribution, a workflow for the reduction of EMBS with fast rotating bodies is presented. The rotation causes a change of dynamical behavior due to inertia forces and, therefore, cannot be neglected. In the scope of this work a linear description of rotating bodies with constant angular velocity is given. Different projection-based MOR techniques are compared and applied to an industrial model of a helicopter with rotating rotor. For this purpose, a short introduction on modeling of EMBS and MOR is given. Substructured reduction is then contrasted to the reduction of the coupled system for modal reduction techniques, moment matching based on Krylov subspaces, and Proper Orthogonal Decomposition. The approximation errors of the reduced systems are compared in frequency domain. It is shown that rotation-dependent terms are essential to describe the dynamic behavior of the system correctly. Reduced models with low approximation errors and large speed-up are obtained with substructured Proper Orthogonal Decomposition and outperform the standard techniques modal truncation and Craig-Bampton reduction.
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