## 1 Introduction  
The presence of mass attachments in structural elements such as beams and columns is quite common in civil engineering infrastructure, as is the case with pylons supporting cables, wind turbines, antennas, lights, etc., [1]. In certain cases, these mass attachments may vary with time on a scale comparable to that of the duration of the external dynamic loads. This may occur under certain conditions, as for example if the mass attachment plays the role of a damping device [2]. Specifically, pylons are infrastructure components serving a variety of functions and can be efficiently modeled as waveguides [3]. The presence of attachments on pylons leads to the concept of secondary systems, which may serve as passive or semi-active structural control devices, a typical example being liquid column dampers [4]. Furthermore, the attached mass may in turn be connected to a nonlinear spring which acts as a vibration absorber [5]. In general, it is possible to view the pylon with an attached time-variable mass as a system comprising a primary linear oscillator and a nonlinear secondary system. Much work has been done in this field from a mechanical engineering viewpoint, where the secondary system is often viewed as a Vibro-impact nonlinear energy sink [6]. As a consequence, new methods of solving the problem must be sought, such as multiple-scale expansions, which are possible for low mass ratios of the combined primary-secondary system. For heavily attached masses that are also time-dependent, recourse must be made to alternative techniques [7]. Here, we solve for the time-dependent, heavy top mass on a beam element by developing a technique based on a method originally outlined in [8] and further improved in [9], whereby the free vibration boundary-value problem (BVP) of a beam with point masses is reduced to an eigenvalue problem through the separation of variables, but the characteristic equation involving transcendental functions is bypassed. Instead, the solution is expanded in series involving the eigenfunctions of the beam without any attachments, plus a new set of generalized coordinates which can be recovered from the solution of single-degree-of-freedom (SDOF) equations with generalized forces at the right-hand side (RHS) that correspond to the point masses. An additional complication arises when ground vibrations are considered because the motion of an attached mass on the pylon cannot be uncoupled by modal analysis, since the absolute accelerations experienced by the mass require contributions from all modes at the same instant [10]. This necessitates a second-stage modal analysis, and if the attached mass is time-dependent, then the new eigen properties of the combined system are time-dependent since they are influenced by the mass rate of change [7]. Based on this solution methodology, the transient pylon vibrations can be Fourier transformed to extract the time dependence of the eigenfrequencies of the combined pylon-mass system. The usual model for pylons is based on the Bernoulli-Euler fourth-order differential equation for bending, or second order differential equations for axial and torsional responses [3]. For an effective modelling of the lumped mass, it is necessary to introduce generalized functions [11], also known as Schwartz distributions, making it possible to differentiate functions whose derivatives do not exist in the classical sense. In this work, the inertia term that develops at the top of the beam is a temporal discontinuity, and the Dirac delta function appears as a force term on the RHS of the differential equation under study. It is noted that the Dirac delta function can also be used to model the presence of a crack in a structural member, which reduces stiffness and alters the dynamic response [12]. From the above discussion, it follows that a computationally demanding problem arises with the presence of point mass attachments in flexible structures.