## 5 Discussion and Conclusions  
This work investigates the applicability of iterative eigenvalue extraction algorithms for the purpose of structural identification. More specifically, the application example is a flexible, metallic cantilevered pylon with a time-dependent mass attachment. This specific attachment renders the eigenvalue problem time-dependent for the time duration of the action of the applied loads. Once the transient motion at a given station on the pylon is synthesized by modal analysis, the application of the short-time Fourier transform yields a clear picture of the manner in which the eigenvalues of the combined pylon-mass structural system vary during the action of ground-induced motions. Focusing on the eigenvalue analysis of the combined pylon-mass system, at first the coefficients of the polynomial resulting from setting the determinant of the first-order matrix system formulation equal to zero are recovered using the Leverrier-Faddeev algorithm. This is a recursive method for calculating the coefficients of the characteristic polynomial of a square matrix [13]. However, these coefficients show large differences in their absolute values rendering any subsequent method for extracting eigenvalues potentially unstable. Therefore, the eigenvalue extraction is first performed by the Laguerre method, which is the complex number variant of the conventional Newton-Raphson method used for real root extraction [14; 15]. For the Laguerre method, an additional complication is computing the correct initial values at the beginning of each time step, which are the final values of the immediately previous time step. Next, the QR method [16] is implemented in conjunction with the Householder algorithm, which is the best option in terms of both accuracy and efficiency, while all limiting cases are correctly reproduced when the attached mass attains a fixed value. In general, it is estimated that three eigenvalues are necessary  
Figure 7: First three pylon-variable mass structural system eigenvalues: (**a**) negative mass rate \(\mu=-1.0\) (tn/s) and (**b**) positive mass rate \(\mu=1.0\) (tn/s): Top graphs derive from a Runge–Kutta solution of the equations of motion; bottom graphs derive from the iterative methods of solution of the eigenvalue problem. for any subsequent modal analysis when the attached point mass is at a boundary, which in our case is the top of the pylon. If the mass is attached at an intermediate station on the pylon, then two eigenvalues are sufficient, which implies one can use the Laguerre method as well. At first, an analytical solution for the axial motion of the pylon modeled as a distributed mass system is derived, with the formulation augmented to handle an attached, time-dependent mass attachment at the top which is heavy and cannot be viewed as a secondary system. The solution is obtained by modal analysis following the eigenvalue extraction. What complicates the solution is the fact that the mass is time-dependent, i.e., may either increase or decrease during the time interval the external forces are active. This requires a second-tier solution of a time-dependent eigenvalue problem cast as a system of first order differential equations with non-constant coefficients. Next, the eigenvalues are plotted against the time duration of the increasing/decreasing fluid flow from a container at the top of the pylon. The stiffening/softening of the combined fluid mass- pylon system is manifested as a function of fluid decrease/increase in the container. Then, the time histories at any station on the pylon for ground-induced motions can be computed by modal analysis. Finally, if structural identification is the goal, then the aforementioned time histories can be used as raw date, followed by application of the FT with Hanning windows sequentially places across the time axis. In general, the methodology developed herein is applicable to other types of structural systems with transient mass attachments and to general types of dynamic loads, including the case of a moving mass.