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This study deals with the dynamical evolutions exhibited by a simple mechanical model of building, comprising a parallelepiped standing on a horizontal plane. The main goal is the introduction of a pendulum in order to reduce oscillations. The theoretical part of the work consists of a Lagrange formulation and Galerkin approximation method, and dry friction has also been considered. From the analytical/numerical simulations, we derive some important conclusions, providing us with the tools suitable for the design of absorbers in practical cases.

This work deals with the dynamical evolutions played by a simple mechanical model. The model comprises a parallelepiped standing on a horizontal plane. As the plane performs vibrational displacements, the parallelepiped begins to evolve its position and these “parallelepiped movements” constitute the subject of our concern. It is worth noting that this study has to be ascribed to the “applied mathematical sphere of searches.” Nevertheless, its motivations firmly belong to the “technical sphere of investigation.” Hence, in this introduction, a brief description of the underlying technical problem is presented.

The technical scenarios that provoke these studies originate from earthquakes. Interactions between buildings and earthquakes have been studied for a long time and there is much literature about them. In Figure

Catastrophic collapses due to earthquakes.

Scheme of a rocking block under excitation of the base.

An old masonry chimney.

Following the seminal work by Housner in 1963 [

In subsequent years, the works by Hogan [

Finally, similar to the concern of the present work, the problem of vibration control is investigated in [

Now, in line with previous investigations, the preliminary questions addressed by the present work are as follows. (i) Is it possible to give some parameterization of these oscillations? (ii) Is it possible to tune a pendulum with this tower so that oscillations are reduced and controlled? (iii) Is it possible to give some sort of frequency response (with all the limitations imposed by nonlinearity)? The simple model studied in the following sections partially answers these questions. The model is somewhat simplified. Plane movement is assumed and, therefore, we name it a parallelepiped and the structure is actually schematized as a “physical rectangle.” From this simple model and its elaboration through adding a planar physical pendulum and some damping between the parallelepiped and the ground surface, some interesting results have been achieved.

The model of block with the pendulum we take into account for the study is shown in Figure

The rocking block with the added pendulum, scheme of notations.

In order to study our block-pendulum system, it is interesting at first to consider the motion of the parallelepiped after the end of short excitation given by the base movement (nonforced linearized undamped case) and during excitation of the base and steady motion (forced linearized damped case). Viscous damping is introduced in order to take into account generic damping effects, for example, impact dissipation, internal damping of the material, and air drag. We underline that impact dissipation, which is sometimes considered via the introduction of a coefficient of restitution, has been here considered embedded into viscous damping. A sort of correspondence between the two, that is, damping and coefficient of restitution, can be heuristically carried out from real observations of the oscillation decay.

The calculation of this section gives us results that are in accordance with [

When, after short forcing excitation at the base, the following conditions are satisfied (

From a simple geometrical consideration, the boundary of nonoverturning of the block is defined by the expression

If it is rewritten in dimensionless time

Amplitude of oscillation versus frequency of oscillation at various

These results are somewhat in accordance with those reported in previous works [

Let us notice that, from (

Now, let us consider the problem of the steady motion of the parallelepiped on a vibrating surface. Such a regime of motion originates in the real world during earthquakes and, therefore, we introduce dissipation into the model, as described in the previous section, in order to make the model more realistic. For example, dissipation can be considered in the following equation of motion:

If parameter

The equation of motion (

Additionally, because in the steady-state motion the equalities

Amplitude- and phase-frequency characteristics of the system at different ratios

It is clear that formula (

Now, we present a short development of the previously treated problem via Bubnov-Galerkin’s method. The quantities

(a) Amplitude-frequency and (b) phase-frequency characteristics of the system at various friction coefficients

The main object of this study is to investigate some device or system suitable for reducing oscillations during earthquakes. Hence, using most of the results calculated in the previous sections, we now consider a system, which includes a pendulum of mass

The linearized equations of oscillation of such a system in dimensionless time are

Let us apply Bubnov-Galerkin’s method and we will obtain the solution in the form

Amplitude of free oscillation of the block (bold red line), of the pendulum (thin grey line), and of the block without the pendulum (dashed line) for

Similar curves can be plotted for the case of forced oscillations; for example, for

Amplitude of forced oscillation of the block for different slenderness: (a)

In order to better support the theory contained in this work, we have carried out a realistic simulation via a research level multibody dynamic simulator specifically developed by one of the authors, and this simulation has been addressed to a real rocking-block case that is reported in the next section.

In this section, we present numerical simulations of rocking blocks equipped with pendulums and we compare the outcome of the simulations with the prediction of the analytical approach discussed in this paper.

From a numerical point of view, a model of a rocking block connected with a pendulum leads to a multibody problem involving both bilateral constraints (the hinge between the pendulum and the block) and unilateral contacts that might experience impacts and stick-slip phenomena and, thus, requires numerical schemes for nonsmooth dynamic problems. A conventional strategy for the solution of such a class of problems is based on the regularization of discontinuous terms, which are approximated by Lipschitz-continuous mollifiers. This casts the original problem into conventional Ordinary Differential Equations (ODEs) or Differential Algebraic Equations (DAEs) that can be solved by well-known numerical integrators. However, a drawback of such regularization approaches is that regularization could lead to extremely stiff functions that hinder the efficiency of ODE/DAE solvers; consequently, very short time steps or sophisticated implicit integrators are required. Therefore, as an alternative to regularization, we use a more advanced mathematical framework that deals directly with the discontinuous nature of friction and contacts, expressing the multibody problem with the tools of Differential Variational Inequalities (DVIs) [

Impacts between rigid shapes can be handled via the introduction of restitution coefficients, but, if preferred, our software can also handle the case of nonrigid frictional contacts, which fit in the broad context of DVIs. More details on this can be found, for instance, in [

The three-dimensional simulation of the rocking block has been performed by introducing three rigid bodies, namely, the moving floor, the block (with dimensions 0.1 m, 0.2 m, and 0.4 m), and the pendulum, this being connected by a spherical joint placed at the top of the block. In addition, two box collision shapes have been assigned to the block and to the fixed floor. A collision detection algorithm finds the contact points at each time step and feeds them into the CCP solver, for advancing the DVI integration. A friction model of Amontons-Coulomb type is associated with each contact point, thus automatically taking into account the stick-slip effects. In the presented simulations, we used static and dynamic friction coefficients

The density of the simulated blocks is 2028 kg/m^{3}, and the motion of the floor is defined via a rheonomic constraint that imposes a harmonic horizontal motion along the horizontal

In this case, we used a cosine wave with frequency

Table

Cases studied in the multibody simulations.

Case | Mass ratio |
Length ratio |
---|---|---|

C1 | 5 | 0.1–0.4–0.6 |

C2 | 10 | 0.1–0.4–0.6 |

C3 | 20 | 0.1–0.4–0.6 |

C4 | 30 | 0.1–0.4–0.6 |

C5 | 40 | 0.1–0.4–0.6 |

Multibody model and base-block relative motion amplitude (

Moreover, the simulations show that for a low mass pendulum (i.e., up to 20% of the block mass) no oscillations reduction occurs despite the pendulum length. It is worth noticing that the effect of length of the pendulum is nearly negligible, with the pendulum mass being the most important parameter for this frequency of “table” oscillation. Note that, due to friction sensitivity of the system, case

We finally remark that the numerical method is able to simulate transient phenomena that are not considered in the analytical model and that, optionally, accelerograms can be assigned to all three directions of the floor, thereby simulating a real earthquake.

The rocking-block problem has been investigated under different points of view, as discussed in Section

Before plotting a synthesis of the obtained results, it is important to state that this is a first step into a field that is not yet well explored. In fact, while several works deal with the “rocking-block” problem, none of them explores the possibility of adding a pendulum to the rocking block with the aim of controlling the oscillations.

We followed two main steps. First and of great interest in this work are the forced oscillations of a block with a pendulum but without friction. Second, as a first step towards general damping, we studied free and forced oscillations of a block without a pendulum but in the presence of dry friction.

Analyzing the problem of the forced oscillations of a rocking block connected with a pendulum, Bubnov-Galerkin’s method was applied and the analytical results are exposed in terms of gain-frequency and phase-frequency characteristics. It is of great interest to notice that the presence of the pendulum greatly reduces the amplitude of vibrations in those cases, when the frequency of excitation is not within the resonance area of pendulum. This means that it would be possible to study some passive tuned pendulum to be added to real “rocking blocks” like ancient towers, and so forth, in order to reduce oscillations induced by wind, earthquakes, and so on.

Adding damping, we investigated two situations, both with no pendulum. At first, the case of the free oscillations of a block with dry friction is analytically solved. This solution has been carried out considering small tilt angles. The main results, amply reported in this work, are presented in graphs where the amplitude of the free oscillations is plotted versus frequency. It has to be remarked that, for blocks presenting low base/highness ratios, the theoretical and numerical results are in good accordance. When forcing is added, given by the movement of the base, we still apply Bubnov-Galerkin’s method. Starting from certain amplitude, we can define a pseudo resonant frequency and we investigate the block motion in terms of gain-frequency and phase-frequency characteristics, as reported in Figures

Finally, a multibody dynamical simulation has been compared against theoretical numerical results on a real case, giving a very satisfactory reduction of oscillations.

The authors declare that there is no conflict of interests regarding the publication of this paper.