engleski

Bearing stiffness optimization with respect to vibration response

This paper presents a novel technique for structural optimization with respect to the vibration response. The vibration system to be optimized is described by the linear second-order time-invariant system. Optimization criterion is derived from the system H-infinity norm, or in other words, system worst-case vibration response of the system for all steady-state sinusoidal excitations over all frequencies. Optimization variables are stiffness parameters of the structure, and it is assumed that the system stiffness matrix is affine function of the optimization variables. The optimization algorithm is derived from the H-infinity optimization criterion, expressed in terms of bilinear matrix inequality. Such approach adds additional optimization variables to the problem, and this is a consequence of the introduction of so-called Lyapunov matrix with dimensions equal to the number of system state variables. Obviously, this increases the computational cost significantly, limiting the proposed optimization procedure to small-scale problems. To overcome this, special structure of the system matrices is utilized to eliminate some of the optimization variables. Furthermore, a reduced-order optimization procedure is proposed, as follows. For some fixed optimization variables (e.q. their initial values), a matrix containing several system mode shapes is constructed. Such mode shapes are chosen such that they represent the critical structure vibration modes. Then, parametrized system matrices, i.e. matrices which are functions of optimization variables, are projected to the subspace spanned by such mode shape matrices. This results in small reduced-order parametrized system, which is used in the above described optimization procedure. This reduced-order optimization procedure relies on the assumption of small sensitivity of the system mode shapes with respect to optimization variables. In other words, the system mode shapes matrix, which normally depends on the system parameters (optimization variables), is kept constant throughout the optimization procedure. The validity of this assumption for each problem can easily be checked a-posteriori. Finally, a numerical example which clearly illustrates the applicability and efficiency of the proposed procedure is presented. For such purpose, a finite element model of a real-world power plant comprising of a turbine and a generator connected by a shaft is constructed. The shaft is placed at two bearing blocks, and its vibrations are excited by two harmonic forces acting perpendicularly to the shaft at the turbine and the generator. Radial stiffness parameters of the bearing blocks are optimized to attenuate the system steady-state vibrations, which are defined as the radial vibration displacements of the turbine and the generator. By applying the above described optimization method, both turbine and generator steady-state vibration response is attenuated significantly.

vibration control; bearing stiffness

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