47 Identification of Breathing Cracked Shaft Models from Measurements 539 weight dominance is assumed, then the opening and closing of the crack is periodic at the rotor spin speed. This periodic opening and closing may be represented by a general Fourier series, although three specific possibilities will be considered here. The first is that the crack remains open, and hence QKc.t/ D QK1 is constant. In the Mayes model the time dependent stiffness matrix in rotating coordinates (where the choice of time origin is based on the crack orientation) is QKc.t/ DK0 0:5.1 cos t/dQK (47.2) When cos t D 1 the crack is fully closed, the stiffness is the un-cracked rotor stiffness and thus the rotor is axisymmetric. When cos t D 1the crack is fully open, and the rotor is asymmetric. The hinge or Gasch model has an abrupt change in stiffness and is given by QKc.t/ DK0 0:5.1 sgn.cos t//dQK (47.3) where sgn is the signum function that returns the sign of a number. The signum function can be written in terms of a Taylor series so that Eq. (47.3) becomes QKc.t/ DK0 0:5 X 1 nD1 2 n sin n 2 cos.n t/ dQK (47.4) Note that the cos t term of the hinge model is 2/ D0.6366, compared to 0.5 for the Mayes model. The coefficients of a general Fourier series may be used to give a general breathing model of the form QKc.t/ DK0 na0 X 1 nD1 an cos.n t/odQK (47.5) Converting from rotating to fixed co-ordinates gives the stiffness matrix in stationary co-ordinates, Kc(t), which is a periodic function of time only, showing that Eq. (47.1) is a linear parametrically excited equation. Penny and Friswell [12] showed that the Mayes model generates a constant term plus 1X, 2X and 3X rotor angular velocity components in the stiffness matrix. An open crack will generate a constant term plus 1X and 2X components in the stiffness matrix. The hinge model will generate all harmonics. 47.3 Rotor Response to Breathing Crack Models In order to check the frequency content of the machine response a time simulation will be performed on a detailed model of the machine. This will allow realistic features of the real machine to be easily incorporated. To ensure the transient response decays within a reasonable time, damping is added to the bearings and/or disks. The equations of motion are integrated using ode45 in MATLAB. However the number of degrees of freedom of a detailed finite model is likely to be large, requiring a long computational time to simulate the response. Thus the equations of motion in the rotating frame are reduced using the lower mode shapes of the undamped and undamaged machine, neglecting gyroscopic effects. A sufficient number of modes should be included to simulate the range of excitation frequencies, and also any significant combinational frequencies. This reduction has two beneficial effects; not only are the number of degrees of freedom reduced, leading to a lower computational cost per time step, but also the higher frequencies are removed, thus allowing a larger time step. The equations are integrated until a steady state has been established and then the FFT is calculated. The steady state response should only contain the rotor spin frequencies and its harmonics, and therefore the spectrum of the response should only contain discrete frequencies. However, leakage is likely to occur because of the difficulties in choosing a sample period so that every sinusoidal component in the response has an integer number of cycles in the sample. The effect of leakage may be reduced by using time window functions. Furthermore the sample period must be sufficiently long to ensure that the frequency increment is small enough to distinguish the individual frequency components. Friswell et al. [9] gave more detail.
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