Mathematical Problems in Engineering

Volume 2019, Article ID 4843738, 14 pages

https://doi.org/10.1155/2019/4843738

## The Use of a Pendulum Dynamic Mass Absorber to Protect a Trilithic Symmetric System from the Overturning

^{1}Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale, University of L’Aquila, Italy^{2}Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, USA

Correspondence should be addressed to Angelo Di Egidio; ti.qavinu@oidigeid.olegna

Received 16 October 2018; Revised 10 December 2018; Accepted 18 December 2018; Published 14 January 2019

Academic Editor: Francesco Aymerich

Copyright © 2019 Angelo Di Egidio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The trilith consists of two vertical elements (columns) supporting a horizontal element (lintel). The understanding of the dynamic behaviour of triliths is an important step towards their preservation and starts with the knowledge of the dynamics of rigid blocks. A passive method based on a dynamic mass absorber is used to protect a trilith from overturning. The protection system is modelled as a pendulum, hinged on the lintel, with the mass lumped at the end. The equations of rocking motion, uplift and the impact conditions are obtained for the coupled system trilith-mass absorber. An extensive parametric analysis is performed with the aim to compare the behaviour of the system with and without the pendulum, under impulsive one-sine (or one-cosine) base excitations. In order to point out the effectiveness of the protection system, overturning spectra, providing the amplitude of the excitation versus its frequency, are obtained. The pendulum mass absorber results effective in avoiding overturning in specific ranges of the frequency of the excitation. However, outside these ranges the mass absorber never compromises the safety of the trilith.

#### 1. Introduction

The dynamical behaviour of block-like structures has focused the interest of many researchers in the last fifty years. The reason is that several elements such as hospital equipment, storage boxes, and in some cases historical monument and art objects, show a rocking motion typical of block-like structures when subjected to earthquakes. Therefore, the initial studies on this kind of structures used two-dimensional models to analyze the dynamics of symmetric blocks subject to earthquakes excitations [1, 2].

Even though the response of block-like structures to earthquakes remains a central topic [3, 4], other kinds of ground excitation have been considered in subsequent studies. Random and harmonic excitations were discussed in [5–9], respectively. Several papers enriched the original two-dimensional symmetric model of rigid block. Non-symmetric rigid blocks are modelled in [10, 11]. In [12, 13], instead, the possibility for combined slide-rocking motions is contemplated. The transitions among the different phases of motions are analyzed in details in [14–16]. An alternative three-dimensional formulation for the rigid block motion that considers the rocking and the spinning over one vertex of the block-like structure is presented in [17–19].

In the last years, a topic of broad interest is the protection of block-like structures. Several papers investigated different methodologies and devices. The simplest method is the anchorage of the block-like structures, as in [20, 21] where a semi-active control of the rocking motion is also considered. Instead, an active control method is used in [22]. The most investigated protection methodology is the use of base isolation [10, 23]. In the majority of the paper on base isolation, the element is constrained to remain on the base since the sliding is usually prevented. In [24], instead, the block-like element is allowed to slide or rock partially outside the oscillating base. The research presented in [25] investigates the effectiveness of base isolation for block-like elements at different levels of a multi-story frame. Another protection methodology for block-like elements is the use of Tuned Mass Dampers (TMD). The effectiveness of a TMD constituted by a single degree of freedom oscillating mass connected to the block with a linear visco-elastic device is analyzed in [26–28]. The use of a different kind of TMD is proposed in [29] where the motion is controlled using a sloshing water damper. Some recent studies proved, with both deterministic [30, 31] and probabilistic [32] approaches, the effectiveness of pendulum damping systems to control the rocking motion.

Some papers considered multiple blocks, either stacked or in the configuration of a trilith. A contact model for rigid blocks is proposed in [33]. In [4, 34], the rocking of two stacked rigid block is analyzed considering different possible patterns and deriving the equations of motion for each pattern. The identification of the minimum amplitude that leads to the overturning of multiple block systems is studied in [35, 36]. A few papers are focused on the dynamics of triliths. The main difference among them is in the modeling of the impact between two elements of the trilith and with the ground. The studies presented in [37, 38] make use of specific formulations that are discussed in [33, 39], respectively. In [40], instead, the loss of energy associated with the impact is used to define the initial conditions for the post-impact motion. The angular momentum-impulse theorem applied on one column of the frame before and after the impact is used in [41] to define the maximum coefficient of restitution of the rocking frame. In particular, in the same paper, an equivalent single block model is used to analyze the motion of a multicolumn rocking frame. Using the same model, in [42], it is demonstrated that the stability of the rigid frame increase with the weight of the lintel.

In this paper a passive method of motion control, based on the use of dynamic mass absorber is used to protect from overturning a trilithic structure with equal columns and generic shaped and positioned lintel. The only restriction considered on the geometry of the lintel is that the contact zones with the top of the columns is horizontal. In trilithic structures with equal columns, the columns rotate of the same angle and the lintel undergoes only to translational motion. The adopted protection system is constituted by a mass absorber, modelled as a pendulum hinged on the lintel with the mass lumped at the end. The equations of motion of the trilith coupled with the mass absorber are obtained; the uplift and the impact conditions are derived in a rigorous way, contrarily to what done in [31].

An extensive parametric analysis is performed with the aim to compare the behaviour of the system with and without pendulum, under impulsive external excitations. Specifically, in order to point out the efficiency of the pendulum mass absorber, a one-sine (or one-cosine) base excitations is considered. Overturning spectra, providing the amplitude of the excitation versus its frequency, are obtained for a wide class of triliths characterized by different geometrical properties, with and without pendulum mass absorbers. The results show the effectiveness of this kind of protection in avoiding overturning.

#### 2. Mechanical Model

The mechanical system is constituted by two equal columns and a generic shaped and positioned lintel. The only restriction considered on the geometry of the lintel is that the contact zones with the top of the columns is horizontal. The pendulum mass absorber is hinged to the lintel in a generic point and it has a lumped mass connected at its end point. The columns and the lintel are assumed to be made of the same material and of unitary depth (the dimension orthogonal the plane containing the trilith). The three bodies of the trilith are modelled as rigid blocks. A sufficiently larger friction coefficient is assumed to prevent the slipping between columns and ground and between columns and lintel as in [41]. So, the trilith can undergo only rocking motion. The two equal columns of the trilith rotate of the same angle, and the lintel undergoes only a translation (it does not rotate, see [41]). Figure 1 shows the geometrical parameters characterizing the mechanical system; , are the mass centers of the three rigid blocks constituting the trilith, and is the position of the mass of the pendulum. The pendulum is hinged on the lintel at point .