The diagnostics of control rod vibrational anomalies using the measurable power spectral densities of in-core neutron detectors are studied analytically. The frequency-dependent Langevin equations are derived in a general way, based on the two-group diffusion theory for a reflected reactor. The adjoint function technique is applied to obtain the fluctuation of the flux. Following this, frequency responses to the control rod vibration are derived and are checked conceptually. A typical pressurized water reactor is used for the numerical investigations. The main computations are associated with the frequency responses to the vibration of a control rod. It is shown that the frequency responses have a plateau region for frequency roughly between 0.5 and 50 rad/s. The frequency of 5 Hz within the plateau region may be chosen for further calculations. The global and local components of the neutron noise are investigated and discussed, following the original global/local concept, which is useful for interpretation. The frequency responses as functions of the position of a detector for a given equilibrium position of an absorber rod are studied in some detail. The noise phenomena as seen from the numerical investigations are explained. The frequency responses as functions of the position of an absorber rod for a given position of a detector, related to the problem of localization of a vibrating control rod, are also investigated. From the signals obtained from in-core neutron detectors and the corresponding frequency responses, one can localize the vibrating rod or even possibly estimate the root-mean-squared amplitude of the vibration using a “contour” method. The result is expected to be that an appropriate action may be taken in time to prevent further deterioration of the vibratory phenomenon.
