The impact–echo method is a superior method for detecting the health of concrete structures, but it has the disadvantage of significant errors when identifying structural boundaries. In order to overcome this limitation, this paper proposes a calculation method using a cross-sectional vibration mode in combination with the impact–echo method to detect concrete columns. The variation of the predominant frequency in the mid-column region and the critical boundary is studied. The influence of the edge on the detection results is analyzed. The differences in eigenvalues under different cross-section vibration modes are revealed. A quantitative method for evaluating concrete health using the impact–echo method is further established. Through field tests and finite element simulation calculations, it was verified that the eigenvalues in the fixed mode are very consistent with the predominant frequency measured near the edge region. This makes up for the defect of inaccurate measurements when the impact–echo method is used to detect the edge area of concrete members. The impact–echo method can be better used for the non-destructive testing of concrete members.

The seismic response characteristics of a lead-rubber-bearing(LRB) base-isolated structure under rare and very rare earthquakes were investigated. The acceleration, ductility coefficient, and shear strain of the LRB increase significantly under very rare earthquakes in comparison to rare earthquakes; in particular, the shear strain of the LRB may exceed the ultimate shear strain and cause damage to the base-isolated structure. The criterion selected for the optimum tuned inerter damper (TID) of the TID–LRB hybrid control system is the minimization of the mean value of the maximum shear strain of the LRB. For each inertance mass ratio of the TID, there exists an optimum tuning frequency ratio and damping ratio of the TID to minimize the shear strain of the LRB, and the effectiveness is increased with a higher inertance mass ratio. By equipping the TID with appropriate parameters, the safety of the LRB during rare and very rare earthquakes can be ensured. Finally, the pounding response of the base-isolated structure collision with the moat wall under very rare earthquakes was analyzed. It was observed that under very rare earthquakes, the ductility coefficients of the superstructure by equipping with the suitable TID were improved, and the shear strain of the LRB was reduced. In addition, equipping the TID can reduce the required width of the isolation joint to avoid collision between the isolation layer and the moat wall, and with an increase in the inertance mass ratio, the required width of the isolation joint is smaller.

The Hilbert transform (HT) is an important method for signal demodulation and instantaneous frequency (IF) estimation. The modulus of the analytic signal constructed by the HT is regarded as the amplitude, and the derivative of the instantaneous phase of the extracted pure frequency modulation signal is the IF. When the spectrums of the amplitude function and the oscillation term overlap so as not to satisfy the Bedrosian condition, the instantaneous amplitude (IA) and frequency calculated by the Hilbert transform will contain errors. The recursive Hilbert transform (RHT) is an effective method to overcome this problem. The RHT regards the pure frequency modulation signal obtained by the previous HT as a new signal and recursively computes its Hilbert transform until convergence. The final pure frequency modulation signal of the recursive procedure has the same zero-crossing points as the original signal, and its corresponding quadrature error signal vanishes. We emphasize the convergence analysis of the algorithm and study the convergent tendency of the quadrature error signal in each recursive step. The key to the proof is that the discrete Fourier transform value of the quadrature error signal is regarded as a vector, and the length/norm of the vector decreases with the recursion process. Finally, three examples are used to demonstrate the effective application of this method in signal demodulation, IF identification and damped vibrating signal analysis, which indicate the application potential of the RHT method in mono-component signal processing.