16 Experimental Validation of Modal Parameters in Rotating Machinery 175 Fig. 16.2 The impact hammer is automated by means of a stepping motor that is controlling the hit in order to improve the reproducibility impacthammer shaft stepping motor θ Fig. 16.3 Each element of the finite element model has eight degrees of freedom z y x uyi uxi xi yi uyi+1 uxi+1 xi+1 yi+1 16.4.2 Nonrotating Shaft 16.4.2.1 Modeling, Updating and Reducing of the Model When the shaft is not rotating, there are no speed dependent effects, so a classical mass, spring damper system is assumed. The shaft is modeled with finite elements [18]. Six elements used, with each eight degrees of freedom (Fig. 16.3), so the resulting model has a dimension of 28 28. Only five frequency response functions are measured. The model updating method applied here [19] is based upon the frequency response functions and can deal with incomplete data by using an iterative procedure. The mass and the stiffness matrix of the model are updated and the dimension remains 28 28. Consequently, the (purely imaginary) poles and the corresponding (real) mode shapes of this undamped model are calculated. Because there are only five frequency response functions in the measurement, it is only possible to derive a damping matrix with dimension 5 5. Therefore, the modeled mass and stiffness matrices are reduced to a dimension of 5 5. This is done by truncating both calculated poles and mode shapes to the first five natural frequencies and by deleting the coordinates in the mode shapes that cannot be measured. By doing this the dimensions of the calculated poles and mode shapes correspond to the measurement. The resulting poles and mode shapes are used to reconstruct a new mass and stiffness matrix by transforming the modal matrices back to physical coordinates. 16.4.2.2 Identification of the Damping Matrix The previous description results in updated mass and stiffness matrices with a dimension of 5 5. Next, the measured frequency response functions are used to extract poles and mode shapes with a least squares complex exponential method. The extracted mode shapes are unity modal mass scaled using the reduced mass matrix from the model. The newly scaled mode shapes are used together with the poles to estimate the damping matrixC0.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==