Vibrations of flexible pylons with time‐dependent mass attachments under ground induced motions

Abstract

This work examines how mass attachments that vary with time influence the motion of flexible pylons undergoing ground-induced vibrations. Specifically, an analytical solution is derived for a time-dependent, lumped mass attachment placed at the top of a cantilevered pylon undergoing time-harmonic longitudinal vibrations. At first, the solution to the governing equation of motion entails the recovery of the eigenvalue problem in the presence of a fixed mass. This is then followed by a generalization of this solution to allow for mass variability with time. It is noted that modal analysis no longer yields an uncoupled system of equations for the generalized coordinates, this necessitating a second step modal analysis, which is complicated by the fact that the system matrices are now time dependent. Results are presented in terms of time histories and frequency plots for the pylon displacement amplitude at the top, resulting from harmonic base motion of unit amplitude. The aim is to identify the frequency range over which the presence of a time-varying mass is either beneficial or detrimental in minimizing the pylon's kinematic response. Since the rate of change of the mass can be either positive or negative, the special cases of constant mass and of no mass can be recovered from the solution. Finally, this methodology can be generalized to include transverse and rotational vibrations of flexible pylons.

Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

“…Another noteworthy problem in practical engineering is a cantilever with a lumped mass and rotary inertia at the free end. The Euler-Bernoulli beam, even in the non-homogeneous case, has been applied in many works, 2,[16][17][18][19] whereas Timoshenko beams were considered in other studies. [20][21][22] Manolis and Dadoulis 17 proposed a study on the dynamics of a pylon with an attached mass using an Euler-Bernoulli beam with axial deformability.…”

“…The Euler-Bernoulli beam, even in the non-homogeneous case, has been applied in many works, 2,[16][17][18][19] whereas Timoshenko beams were considered in other studies. [20][21][22] Manolis and Dadoulis 17 proposed a study on the dynamics of a pylon with an attached mass using an Euler-Bernoulli beam with axial deformability. Additionally, they proposed an alternative method to consider the lumped mass directly in the equations of motion, where the Dirac delta function was used instead of applying the mass contribution in the boundary conditions.…”

“…Instead of being considered in the boundary conditions, the lumped mass is included in the equations using the Dirac delta function, as was performed for the Euler-Bernoulli beam by Manolis and Dadoulis. 17 The proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self-adjoint, and the solution in the case of forced response using modal superposition. In addition, the closed-form expression of modal shapes for uniform beam properties is derived, and a Galerkin approach is presented for non-uniform beam properties.…”

“…Finally, in the case of seismic motion at the base, considering Equation (17), Equation (62) assumes the following form:…”

“…After fully non‐dimensional equations are derived, the approach presented herein transforms the second‐order coupled system into a first‐order system, which can be solved more easily using matrix algebra and Laplace transform. Instead of being considered in the boundary conditions, the lumped mass is included in the equations using the Dirac delta function, as was performed for the Euler‐Bernoulli beam by Manolis and Dadoulis 17 . The proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self‐adjoint, and the solution in the case of forced response using modal superposition.…”