Research Article Oscillations Control of Rocking-Block ... · solution of the free oscillation...

Research ArticleOscillations Control of Rocking-Block-Type Buildings bythe Addition of a Tuned Pendulum

Luca Collini,1 Rinaldo Garziera,1 Kseniia Riabova,1

Mariya Munitsyna,2 and Alessandro Tasora1

1Department of Industrial Engineering, University of Parma, 43100 Parma, Italy2M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia

Correspondence should be addressed to Luca Collini; [email protected]

Received 31 May 2015; Accepted 20 September 2015

Academic Editor: Chao Tao

Copyright © 2016 Luca Collini 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.

This study deals with the dynamical evolutions exhibited by a simple mechanical model of building, comprising a parallelepipedstanding on a horizontal plane.Themain goal is the introduction of a pendulum in order to reduce oscillations.The theoretical partof 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 thedesign of absorbers in practical cases.

1. Statement of the Problem

This work deals with the dynamical evolutions played bya simple mechanical model. The model comprises a paral-lelepiped standing on a horizontal plane. As the plane per-forms vibrational displacements, the parallelepiped beginsto evolve its position and these “parallelepiped movements”constitute the subject of our concern. It is worth noting thatthis study has to be ascribed to the “applied mathematicalsphere of searches.” Nevertheless, its motivations firmlybelong to the “technical sphere of investigation.” Hence, inthis introduction, a brief description of the underlying tech-nical problem is presented.

The technical scenarios that provoke these studies orig-inate from earthquakes. Interactions between buildings andearthquakes have been studied for a long time and there ismuch literature about them. In Figure 1, examples of collapsesdue to earthquakes are reported. These collapses evidence arocking-block behavior of the structures, namely, rigidmove-ment of the block with respect to the ground, as illustratedin the scheme of Figure 2. When vibrations of buildings maybe assumed (and proved) to be elastic and linear, one maythink to reduce them by applying some “appropriate mass”[1]. It is in fact possible to absorb energy at certain frequencies

(or ranges) by coupling an oscillator with the main structure.This oscillator is often coupled by a spring-damper system,in order to sink energy from the system. However, whenconsidering masonry buildings or no-tension structures ingeneral, the situation changes considerably. Consider, forexample, a slender tower, such as the masonry chimneyillustrated in Figure 3.When the tower oscillates, for example,under the forcing of an earthquake, a plausible model for itsmotion is that of a rigid parallelepiped tilting back and forthon a horizontal plane. In this situation, oscillations are nolonger linear.

Following the seminal work by Housner in 1963 [2], theproblem of the motion of an oscillating parallelepiped-shaped rigid body, the so-called “rocking-block” model, hasbeen widely investigated. The main issue investigated in thisand other earlier studies [3–11] was the performance andthe overturning of a rigid body against the base motiondynamics, for example, an earthquake, assuming the block tobe a buildingmodel.Thework byHousner was devoted to theexplanation of the collapse of many bulky structures duringthe catastrophic Chilean earthquake of May 1960. From thesolution of the free oscillation problem of a rigid rockingblock bouncing on its vertices, it was demonstrated that tallslender structures show a better stability than expected.

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 8570538, 11 pageshttp://dx.doi.org/10.1155/2016/8570538

2 Shock and Vibration

(a) (b)

Figure 1: Catastrophic collapses due to earthquakes.

Block movement

Base excitation

Figure 2: Scheme of a rocking block under excitation of the base.

In subsequent years, the works by Hogan [12–15] refinedthe pioneering study byHousner, assuming a similar rocking-block model and analyzing in depth its dynamics underharmonic forcing. Here, the simple block model is shownto possess extremely complicated dynamics, including chaos.The existence and the form of subharmonic and asymmetricresponses are explained and validated by experimental worksby Wong and Tso [16]. A rocking-block model has alsobeen employed to study the response of structures, such ascolumns or monumentary walls subjected to frictionlessmultiple impacts [17], showing that one may have singleand/or simultaneousmultiple collisions at the surface contactpoints. Furthermore, recently, the dynamic response of therocking block subjected to base excitation has been revisitedto offer new closed-form solutions and original similaritylaws that shed light on the fundamental aspects of the originalmodel [18].

Finally, similar to the concern of the present work, theproblem of vibration control is investigated in [19, 20].

Figure 3: An old masonry chimney.

In these works, a systematic theoretical investigation ofcontrol/anticontrol of the nonlinear dynamics of a rockingblock has been made through the analysis of two curves:the heteroclinic bifurcation and the immediate overturningthresholds, characterizing the system response in excitationparameters space in terms of overturning behavior.

Now, in line with previous investigations, the preliminaryquestions addressed by the present work are as follows. (i) Is itpossible to give some parameterization of these oscillations?(ii) Is it possible to tune a pendulum with this tower so thatoscillations are reduced and controlled? (iii) Is it possible togive some sort of frequency response (with all the limitationsimposed by nonlinearity)? The simple model studied in thefollowing sections partially answers these questions. Themodel is somewhat simplified. Plane movement is assumedand, therefore, we name it a parallelepiped and the structureis actually schematized as a “physical rectangle.” From thissimple model and its elaboration through adding a planarphysical pendulum and some damping between the paral-lelepiped and the ground surface, some interesting resultshave been achieved.

Shock and Vibration 3

l

2b

𝜓

m

M

2a𝜃0𝜑

x(t)

(a) 𝜑 > 0

l

2b 𝜓 m

M

2a 𝜃0𝜑

x(t)

(b) 𝜑 < 0

Figure 4: The rocking block with the added pendulum, scheme of notations.

2. Equations of Motion

The model of block with the pendulum we take into accountfor the study is shown in Figure 4. With the nomenclaturereported in Figure 4, we can write the following geometricalrelations of a block ofmass𝑀 rocking on its corners, towhicha pendulum of mass𝑚 is added:

𝑟2

0= 𝑎2+ 𝑏2,

𝑟2

1= 𝑎2+ 4𝑏2,

sin 𝜃0 =𝑏

𝑟0

,

cos 𝜃0 =𝑎

𝑟0

,

sin 𝜃1 =2𝑏

𝑟1

,

cos 𝜃1 =𝑎

𝑟1

.

(1)

Note that we have two different situations depending on thesign of 𝜑 angle (𝜑 > 0, 𝜑 < 0). We assume that the surfaceon which the block is oscillating moves in accordance with asinusoidal law:

𝑥 = 𝛼 sin (𝜔𝑡) , (2)

hence forcing the block oscillations with amplitude 𝛼 andfrequency 𝜔. The kinetic and potential energies of the blockand of the pendulum can be written as

𝑇𝑀 =𝑀

2{[�̇� − 𝑟0�̇� sin (±𝜑 + 𝜃0)]

2

+ [𝑟0�̇� cos (±𝜑 + 𝜃0)]2} +

𝐽

2�̇�2,

𝑇𝑚 =𝑚

2{[�̇� − 𝑟1�̇� sin (±𝜑 + 𝜃1) + 𝑙�̇� cos𝜓]

2

+ [𝑟1�̇� cos (±𝜑 + 𝜃1) + 𝑙�̇� sin𝜓]2} ,

𝑉𝑀 = 𝑀𝑔 [𝑎 sin (±𝜑) + 𝑏 cos𝜑] ,

𝑉𝑚 = 𝑚𝑔 [𝑎 sin (±𝜑) + 2𝑏 cos𝜑 − 𝑙 cos𝜓] ,(3)

in which 𝐽 is the moment of inertia around the axis ofrotation. The signs ± are valid for both conditions of oscil-lation. After Lagrangian derivation, we obtain the equationsof motion:

(𝑀𝑟2

0+ 𝑚𝑟2

1+ 𝐽) �̈�

− 𝑚𝑙 [±𝑎 sin (𝜑 − 𝜓) + 2𝑏 cos (𝜑 − 𝜓)] �̈�

+ 𝑚𝑙 [±𝑎 cos (𝜑 − 𝜓) − 2𝑏 sin (𝜑 − 𝜓)]

⋅ �̇�2+𝛼𝜔2 [± (𝑀 + 𝑚) 𝑎 sin𝜑 + (𝑀 + 2𝑚) 𝑏 cos𝜑]

⋅ sin (𝜔𝑡)

+ 𝑔 [± (𝑀 + 𝑚) 𝑎 cos𝜑 − (𝑀 + 2𝑚) 𝑏 sin𝜑] = 0

− [±𝑎 sin (𝜑 − 𝜓) + 2𝑏 cos (𝜑 − 𝜓)] �̈� + 𝑙�̈�

− [±𝑎 cos (𝜑 − 𝜓) − 2𝑏 sin (𝜑 − 𝜓)] �̇�2 − 𝛼𝜔2

⋅ sin (𝜔𝑡) cos𝜓 + 𝑔 sin𝜓 = 0.

(4)

Equations (4) can be solved numerically to obtain angles𝜑 and 𝜓 defining the oscillations of the block and of thependulum in time.


3. Oscillations without/with Damping

3.1. No Damping Problem. In order to study our block-pendulum system, it is interesting at first to consider themotion of the parallelepiped after the end of short exci-tation given by the base movement (nonforced linearizedundamped case) and during excitation of the base and steadymotion (forced linearized damped case). Viscous dampingis introduced in order to take into account generic dampingeffects, for example, impact dissipation, internal damping ofthe material, and air drag. We underline that impact dissipa-tion, which is sometimes considered via the introduction of acoefficient of restitution, has been here considered embeddedinto viscous damping. A sort of correspondence betweenthe two, that is, damping and coefficient of restitution, canbe heuristically carried out from real observations of theoscillation decay.

The calculation of this section gives us results that are inaccordance with [2] and validate our nondimensional model.We underline that we obtained these results following anapproach that is quite different from [2].

When, after short forcing excitation at the base, the fol-lowing conditions are satisfied (𝑇 is the period of oscillation):

𝛼𝜔2>

𝑔𝑎

𝑏

𝑇 >1

𝜔arcsin

𝑔𝑎

𝑏𝛼𝜔2,

(5)

the parallelepiped will have nonzero velocity and nonhar-monic oscillations will start.The equation ofmotion takes theform

�̈� − 𝜔2

0sin𝜑 = ∓𝜔2

0] cos𝜑, (6)

where𝜔0 = √𝑏𝑔/(𝑎2 + 𝑏2 + 𝑐2) is the “natural” frequency of aparallelepiped pinned at the center of the base, 𝑐 is the radiusof inertia of the parallelepiped (𝐽/𝑀), and ] = 𝑎/𝑏 indicatesthe dimensionless “width” of the parallelepiped.

From a simple geometrical consideration, the boundaryof nonoverturning of the block is defined by the expression𝜑 ≤ arctan(]). Our study considers a slender parallelepiped(𝑎 ≪ 𝑏) and very small angles of motion (𝜑 ≪ 1).

If it is rewritten in dimensionless time 𝜏 = 𝑡𝜔0, thenlinearized equation (6) takes the form

�̈� − 𝜑 = ∓]. (7)

Let us now introduce the quantityΦ indicating the amplitudeof the natural oscillations of the block and Ω0 its frequency.Also, consider that, at time 𝑡 = 0, the following conditions aresatisfied:

𝜑 (0) = Φ > 0

�̇� (0) = 0.

(8)

Then, at time interval 𝑡 ∈ [0, 𝑡1], where 𝑡1 = 𝜋/(2Ω0) is aquarter of the oscillation period, themotion can be describedby the equation

�̈� − 𝜑 = −]. (9)

� = 1� = 0.5� = 0.2

0.2

0.5

1

1 2 3 4 50Ω0

Φ

Figure 5: Amplitude of oscillation versus frequency of oscillation atvarious ] = 𝑎/𝑏 ratios.

The solution of (9), with the initial conditions (8), becomes

𝜑 (𝑡) = ] + (Φ − ]) cosh (𝑡) , (10)

which also satisfies the condition 𝜑(𝑡1) = 0, from which thefollowing dependence of the amplitude of oscillation from itsfrequency can be derived:

Φ = ](1 − cosh−1 𝜋2Ω0

) . (11)

This function is plotted in Figure 5 for different values of theparameter ] (solid lines). Points in the plot of Figure 5 reportthe results of the numerical integration of the equations ofmotion (4). For large values of ], that is, for wide blocks, theplot of (11) and the results of the numerical integration are inagreement only in the field of small amplitudes.

These results are somewhat in accordance with thosereported in previous works [2, 14].

Let us notice that, from (10), it is possible to find theangular velocity �̇�1 = �̇�(𝑡1) of the block in correspondencewith the sign change in the equation of oscillations (7) and toproceed with the second quarter of the period of oscillation,integrating the equation �̈� − 𝜑 = +] with the new entrycondition 𝜑(𝑡1) = 0, �̇�(𝑡1) = �̇�1.

3.2. Steady-State Problem. Now, let us consider the problemof the steady motion of the parallelepiped on a vibratingsurface. Such a regime of motion originates in the real worldduring earthquakes and, therefore, we introduce dissipationinto the model, as described in the previous section, in order


to make the model more realistic. For example, dissipationcan be considered in the following equation of motion:

(𝑎2+ 𝑏2+ 𝑐2) �̈� + 𝛼𝜔

2[±𝑎 sin𝜑 + 𝑏 cos𝜑] sin (𝜔𝑡)

+ 𝑔 (±𝑎 cos𝜑 − 𝑏 sin𝜑) = 0,(12)

by adding the term 𝑘 sgn(�̇�), with 𝑘 = 𝜅/𝜔20, where 𝜅 is the

constant of “dry friction.” In dimensionless time, the equationof motion of the parallelepiped results as

�̈� + 𝑘 sgn (�̇�) − sin𝜑 − 𝑓] sin𝜑 cos 𝜉𝑡

= ∓] cos𝜑 + 𝑓 cos𝜑 cos 𝜉𝑡,(13)

where 𝑓 = 𝛼𝜔2/𝑔 is the dimensionless amplitude of theexciting force and 𝜉 = 𝜔/𝜔0 is the dimensionless frequencyof oscillation.

If parameter ] is small enough, and angle 𝜑 is conse-quently small as well, the last term in the left part of (13),namely, ] sin(𝜑), is of the second order of infinitesimal and,hence, negligible. The linearized equation (13) becomes

�̈� + 𝑘 sgn (�̇�) − 𝜑 = ∓] + 𝑓 cos 𝜉𝑡. (14)

As the aim of the present work is to investigate the worstcases of forcing conditions, such as the source of resonanceprovided by earthquakes, let us assume that the frequency ofthe block oscillations Ω coincides with the frequency of theexciting force; that is,Ω = 𝜉. Let the initial conditions at 𝑡 = 0be

𝜑 (0) = Φ > 0

�̇� (0) = 0;

(15)

then, in the time interval 𝑡 ∈ [0, 𝑡1], where 𝑡1 = 𝜋/(2Ω0), theequation of motion becomes

�̈� − 𝜑 = −] + 𝑘 + 𝑓 cos (𝜉𝑡 − 𝛾) (16)

and in the time interval 𝑡 ∈ [𝑡1, 2𝑡1]

�̈� − 𝜑 = ] + 𝑘 + 𝑓 cos (𝜉𝑡 − 𝛾) . (17)

Here, 𝛾 denotes the phase difference between the oscillationof the parallelepiped and the motion of the base.

The equation of motion (16) with the initial conditionsexpressed by (15) can be written as

𝜑 (𝑡) = ] − 𝑘 + (Φ − ] + 𝑘 +𝑓 cos 𝛾1 + 𝜉2

) cosh 𝑡

+𝜉𝑓 sin 𝛾1 + 𝜉2

sinh 𝑡 −𝑓

1 + 𝜇2cos (𝜉𝑡 − 𝛾) ,

(18)

where the dimensionless parameter 𝜇 = 𝑚/𝑀 is introduced.From this law of motion, it is possible to find out theangular velocity �̇�1 = �̇�(𝑡1) at which the block changes insign. Integrating (17) with the initial conditions 𝜑(𝑡1) = 0and �̇�(𝑡1) = �̇�1, we will determine the motion of theparallelepiped during the second interval 𝑡 ∈ [𝑡1, 2𝑡1],

depending on the amplitudeΦ and on the phase 𝛾 of steady-state oscillations.

Additionally, because in the steady-state motion theequalities 𝜑(2𝑡1) = −Φ and �̇�(2𝑡1) = 0 hold, the amplitudeand the phase of oscillations can be determined, dependingon the excitation parameters, as follows:

sin 𝛾 =𝑘 (1 + 𝜉

2) sinh (𝜋/2Ω)

𝑓 cosh (𝜋/2Ω)

Φ = ] − 𝑘 −𝑓 cos 𝛾1 + 𝜉2

+

−] + 𝑘 + (𝑓 sin 𝛾/ (1 + 𝜉2)) (1 − 𝜉 sinh (𝜋/2Ω))cosh (𝜋/2Ω)

.

(19)

Note that (19) tends to (11) when 𝑘 = 𝜉 = 𝑓 = 0. From(19), using as nondimensional parameters ] = 0.2, 𝑘 = 0.05,and 𝑓 = 0.1, we can obtain the curves displayed inFigures 6(a) and 6(b). In Φ − 𝜉 graph of Figure 6(a), thebold solid curves in blue represent the amplitude-frequencyand the phase-frequency characteristics corresponding to theresonance conditionΩ = 𝜉; the same curves extrapolated forvery small 𝜉 are drawn in grey color. In the same plots, thethin curves represent the same characteristics correspondingto the frequency of excitation Ω = 𝜉/3, obtained for a biggerinitial tilt angle. The dashed line is the free vibration curve,equation of motion (11), included as a reference. In the samefigure, points designate the results obtained via the numericalintegration of the equations of motion, validating (19). Also,illustrated in Figure 6(a) is 𝛾 − 𝜉 graph, which represents thephase characteristics. Figure 6(b) encloses the behavior justdescribed above, for initial conditions leading toΩ = 𝜉/2 andΩ = 𝜉/4.

It is clear that formula (19) greatly supports the designof a block, by giving a clear insight of the amplitude ofmotion when in the presence of oscillations of the base.The identification of the most dangerous components inearthquakesmight not be a straightforward task; however, therange of frequencies never goes out of the range of validity ofthis formula.

3.3. Resolution by Bubnov-Galerkin’s Method. Now, wepresent a short development of the previously treated prob-lem via Bubnov-Galerkin’s method. The quantities ] and 𝜑no longer need to be very small. In order to simplify thecalculations, we will use an unknown phase of oscillation 𝛾and we will write (13) in the form 𝐹(𝜑) = 0, where

𝐹 (𝜑) = 𝜑+ 𝑘 sgn (𝜑) − 𝜑 ± 𝑓]𝜑 cos (𝜉𝜏 + 𝛾) ± ]

− 𝑓 cos (𝜉𝜏 + 𝛾) .(20)

Using one term of the Fourier series, we will determine themotion in the form

𝜑 = Φ cos (𝜉𝜏) . (21)


0

Ω = 𝜉/3Ω = 𝜉

𝜋

2

−𝜋

2

𝛾

0.1

0.2

0.3

1 2 3 4 50𝜉

1 2 3 4 50𝜉

Φ

(a)

0

−𝜋

2

𝛾

−𝜋

0.1

0.2

0.3

1 2 3 4 50𝜉

1 2 3 4 50𝜉

Ω = 𝜉/2Ω = 𝜉/4

Φ

(b)Figure 6: Amplitude- and phase-frequency characteristics of the system at different ratios Ω/𝜉.

Applying the procedure of orthogonalization of basic func-tions, that is, from the conditions

∫

2𝜋/𝜔

0

𝐹 (𝜑) sin (𝜔𝜏) 𝑑𝜏 = 0,

∫

2𝜋/𝜔

0

𝐹 (𝜑) cos (𝜔𝜏) 𝑑𝜏 = 0,

(22)

and taking into account (21), it is possible to obtain theequations of the phase 𝛾 of the system in the form

cos 𝛾 =4] − 𝜋Φ(1 + 𝜉2)𝑓 (𝜋 + (8/3) ]Φ)

,

sin 𝛾 = 4𝑘𝑓 (𝜋 + (4/3) ]Φ)

.

(23)

Here, conditions (22) have been used with the followingequalities:

∫

2𝜋/𝜔

0

sgn [sin (𝜔𝜏)] cos (𝜔𝜏) 𝑑𝜏 = 0,

∫

2𝜋/𝜔

0

sgn [cos (𝜔𝜏)] sin (𝜔𝜏) 𝑑𝜏 = 0,

∫

2𝜋/𝜔

0

sgn [sin (𝜔𝜏)] sin (𝜔𝜏) 𝑑𝜏 = −4,

∫

2𝜋/𝜔

0

sgn [cos (𝜔𝜏)] cos (𝜔𝜏) 𝑑𝜏 = 4,

∫

2𝜋/𝜔

0

sgn [cos (𝜔𝜏)] cos3 (𝜔𝜏) 𝑑𝜏 = 83,

∫

2𝜋/𝜔

0

sgn [cos (𝜔𝜏)] sin3 (𝜔𝜏) 𝑑𝜏 = 0,

∫

2𝜋/𝜔

0

sgn [cos (𝜔𝜏)] sin (𝜔𝜏) cos2 (𝜔𝜏) 𝑑𝜏 = 0,

∫

2𝜋/𝜔

0

sgn [cos (𝜔𝜏)] sin2 (𝜔𝜏) 𝑑𝜏 = 43.

(24)

The characteristics of amplitude-frequency and of phase-frequency are shown in Figures 7(a) and 7(b), respectively,for the parameters ] = 0.2 and 𝑓 = 0.1 and variousfriction coefficients 𝑘. From Figure 7, it can be seen that,


0.1

0.2

0.3

0.4

1 2 3 4 50𝜉

k = 0.01k = 0.05

k = 0.075k = 0.079

Φ

(a)

0

𝜋

2

−𝜋

2 1 2 3 4 50𝜉

k = 0.01k = 0.05

k = 0.075k = 0.079

𝛾

(b)

Figure 7: (a) Amplitude-frequency and (b) phase-frequency characteristics of the system at various friction coefficients 𝑘, for ] = 0.2 and𝑓 = 0.1.

with the increasing of the dry friction coefficient 𝑘, a branchof the amplitude-frequency characteristic appears, reflectingin the phase-frequency characteristic around the value 𝛾 =𝜋/2. On the other hand, when the friction coefficient ishigh enough, the oscillations are absent. Figure 7 synthetizes,with a different degree of approximation, the results alreadyplotted in Figure 6.

4. Rocking Block with the Pendulum

Themain object of this study is to investigate some device orsystem suitable for reducing oscillations during earthquakes.Hence, using most of the results calculated in the previoussections, we now consider a system, which includes a pen-dulum of mass 𝑚. In the following approach, we will useBubnov-Galerkin’s method and, at this stage, we will notconsider friction.

The linearized equations of oscillation of such a system indimensionless time are

[1 + 𝜇𝜌 (4 + ]2)] 𝜑 − 2𝜇𝜌𝜆𝜓 − (1 + 2𝜇) 𝜑

± (1 + 𝜇) ]

+ 𝑓 [± (1 + 𝜇) ]𝜌 + (1 + 2𝜇)] sin (𝜉𝜏) = 0

− 2𝜌�̈� + 𝜆𝜌�̈� + 𝑓 sin (𝜉𝜏) + 𝜓 = 0,

(25)

where the following dimensionless parameters are intro-duced: 𝜆 = 𝑙/𝑏 and 𝜌 = 𝑏2/𝐽.

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

𝜑 = Φ cos (𝜉𝜏) ,

𝜓 = Ψ cos (𝜉𝜏) .(26)

0.1

0.2

0.3

0.4

1 2 3 4 50𝜉

Φ

Figure 8: Amplitude of free oscillation of the block (bold red line),of the pendulum (thin grey line), and of the block without thependulum (dashed line) for ] = 0.2, 𝜇 = 0.01, and 𝜆 = 0.1.

The set of equations describing the oscillation amplitude Φof the block and the oscillation amplitudeΨ of the pendulumare, respectively,

[1 + 2𝜇 + (1 + 𝜇𝜌) 𝜉2+

8

3]𝑓 (1 + 𝜇)]Φ + 2𝜆𝜌𝜇𝜉2Ψ

= 𝑓 (1 + 𝜇) −4

𝜋] (1 + 𝜇)

− 2𝜌𝜉2Φ + (1 − 𝜆𝜌𝜉

2)Ψ = −𝑓.

(27)

In the case of 𝑓 = 0, (27) describes the dependence ofthe natural oscillation amplitude of the rocking block withpendulum versus frequency. This dependence is plotted inFigure 8 for the parameters ] = 0.2, 𝜇 = 0.01, and 𝜆 = 1.


0.1

0.2

0.3

0.4

1 2 3 4 50𝜉

Φ

(a)

0.1

0.2

0.3

0.4

1 2 3 4 50𝜉

Φ

(b)Figure 9: Amplitude of forced oscillation of the block for different slenderness: (a) ] = 0.15, 𝜇 = 0.01, 𝜆 = 0.1, and 𝑓 = 0.1; (b) ] = 0.40,𝜇 = 0.01, 𝜆 = 0.1, and 𝑓 = 0.1.

The bold solid curve in Figure 8 represents the amplitudeof the oscillation of the block, while the solid thin curve isthe amplitude of the pendulum oscillations.The dotted curveshows the dependence of the block amplitude from frequencyin the case of no pendulum, as already shown in Figure 6.Hence, it is shown that the presence of the pendulum candrastically reduce the amplitude of the oscillation of theparallelepiped when its frequency of oscillations does not liewithin the pendulum resonance area.

Similar curves can be plotted for the case of forced oscilla-tions; for example, for𝑓 = 0.1, the systembehaves as depictedin Figures 9(a) and 9(b). Here, the dotted curves representthe two branches of the amplitude-frequency characteristicsof the parallelepiped without the addition of a pendulum (seeFigure 7(a)). From these plots, it is deduced that the presenceof the pendulum can reduce the amplitude of the oscillationof the block, even under forced oscillations at the base, whenthe pendulum’s natural frequency of oscillation is marginallyexcited.

In order to better support the theory contained in thiswork, we have carried out a realistic simulation via a researchlevel multibody dynamic simulator specifically developed byone of the authors, and this simulation has been addressedto a real rocking-block case that is reported in the nextsection.

5. A Case Study

In this section, we present numerical simulations of rockingblocks equipped with pendulums and we compare the out-come of the simulations with the prediction of the analyticalapproach discussed in this paper.

From a numerical point of view, a model of a rockingblock connected with a pendulum leads to a multibody prob-lem involving both bilateral constraints (the hinge betweenthe pendulum and the block) and unilateral contacts thatmight experience impacts and stick-slip phenomena and,thus, requires numerical schemes for nonsmooth dynamicproblems. A conventional strategy for the solution of such

a class of problems is based on the regularization of dis-continuous terms, which are approximated by Lipschitz-continuous mollifiers. This casts the original problem intoconventional Ordinary Differential Equations (ODEs) orDifferential Algebraic Equations (DAEs) that can be solvedby well-known numerical integrators. However, a drawbackof such regularization approaches is that regularization couldlead to extremely stiff functions that hinder the efficiencyof ODE/DAE solvers; consequently, very short time stepsor sophisticated implicit integrators are required. Therefore,as an alternative to regularization, we use a more advancedmathematical framework that deals directly with the dis-continuous nature of friction and contacts, expressing themultibody problem with the tools of Differential VariationalInequalities (DVIs) [21]. In our Chrono::Engine multibodysimulation software, we endorse the DVI formulation and,thus, obtain high computational efficiency, good robust-ness, and numerical stability when simulating problems ofmultibody frictional contacts. The DVI problem is solvedby means of a time-stepping scheme, which requires thesolution of a convex second-order Cone ComplementarityProblem (CCP) at each time step. In general, CCP problemsinclude the more popular Linear Complementarity Problems(LCPs) as subcases; as for LCPs, there are theoretical resultsfor the existence and uniqueness of the solution under mildassumptions [22]. We solve the CCPs using either a fixed-point iteration [23] or a Spectral Projected Gradient (SPG)Barzilai-Borwein method.

Impacts between rigid shapes can be handled via theintroduction of restitution coefficients, but, if preferred, oursoftware can also handle the case of nonrigid frictionalcontacts, which fit in the broad context of DVIs. More detailson this can be found, for instance, in [24].

The three-dimensional simulation of the rocking blockhas been performed by introducing three rigid bodies,namely, the moving floor, the block (with dimensions 0.1m,0.2m, and 0.4m), and the pendulum, this being connectedby a spherical joint placed at the top of the block. In addition,two box collision shapes have been assigned to the block


NPl/2b = 0.4

l/2b = 0.1l/2b = 0.6

(a)

62 3 4 51 70

t (s)

−0.24

−0.23

−0.22

−0.21

−0.20

−0.19

−0.18

−0.17

−0.16

z(m

)

C1

No penduluml/2b = 0.1

l/2b = 0.4

l/2b = 0.6

(b)

62 3 4 51 70

t (s)

−0.50

−0.45

−0.40

−0.35

−0.30

−0.25

−0.20

−0.15

z(m

)

C2


l/2b = 0.4

l/2b = 0.6

(c)

62 3 4 51 70

t (s)

−0.24

−0.22

−0.20

−0.18

−0.16

−0.14

−0.12z

(m)

C3


l/2b = 0.4

l/2b = 0.6

(d)

62 3 4 51 70

t (s)

−0.23

−0.22

−0.21

−0.20

−0.19

−0.18

−0.17

−0.16

z(m

)

C4


l/2b = 0.4

l/2b = 0.6

(e)

62 3 4 51 70

t (s)

−0.23

−0.22

−0.21

−0.20

−0.19

−0.18

−0.17

z(m

)

C5


l/2b = 0.4

l/2b = 0.6

(f)

Figure 10: Multibody model and base-block relative motion amplitude (𝑧) under forced oscillations for cases C1, . . . ,C5 of Table 1.


Table 1: Cases studied in the multibody simulations.

Case Mass ratio 𝜇 = 𝑚/𝑀 (%) Length ratio 𝜆 = 𝑙/2𝑏C1 5 0.1–0.4–0.6C2 10 0.1–0.4–0.6C3 20 0.1–0.4–0.6C4 30 0.1–0.4–0.6C5 40 0.1–0.4–0.6

and to the fixed floor. A collision detection algorithm findsthe contact points at each time step and feeds them into theCCP solver, for advancing the DVI integration. A frictionmodel of Amontons-Coulomb type is associated with eachcontact point, thus automatically taking into account thestick-slip effects. In the presented simulations, we used staticand dynamic friction coefficients 𝜇𝑠 = 𝜇𝑑 = 0.6.

The density of the simulated blocks is 2028 kg/m3, and themotion of the floor is defined via a rheonomic constraint thatimposes a harmonic horizontal motion along the horizontal𝑧-axis.

In this case, we used a cosine wave with frequency 𝑓𝑧 =2.5Hz and amplitude 𝐴𝑧 = 0.015m. Various ratios of pen-dulum lengths, masses, and horizontal frequency have beensimulated, obtaining results that, although not exhibiting aperfectly steady-state periodic pattern in all cases becauseof the numerical nature of the simulation, can confirm theprediction of the analytical model regarding the beneficialeffect of the pendulum. This accordance emphasizes theimportance of preliminary analysis of real situations using theresults of our analytical approach.

Table 1 summarizes this set of cases, and Figure 10 illus-trates the main results, that is, the block-table relative motionduring the simulation. It is shown that the two heaviestpendulums (mass over 20% of the block mass) are able toprevent the rocking of the block, whereas in the absenceof the pendulum the block would oscillate noticeably. It isa matter of fact that heavy pendulums dramatically reduceoscillations; we point out that these simulations, done takinginto account damping and slender structures, are quite newin the rocking-block literature.

Moreover, the simulations show that for a low masspendulum (i.e., up to 20% of the block mass) no oscillationsreduction occurs despite the pendulum length. It is worthnoticing that the effect of length of the pendulum is nearlynegligible, with the pendulummass being themost importantparameter for this frequency of “table” oscillation. Note that,due to friction sensitivity of the system, case C2 presents theoverturning of the block.

We finally remark that the numerical method is able tosimulate transient phenomena that are not considered in theanalytical model and that, optionally, accelerograms can beassigned to all three directions of the floor, thereby simulatinga real earthquake.

6. Conclusions

The rocking-block problem has been investigated underdifferent points of view, as discussed in Section 1.

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

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

Analyzing the problemof the forced oscillations of a rock-ing block connected with a pendulum, Bubnov-Galerkin’smethod was applied and the analytical results are exposed interms of gain-frequency and phase-frequency characteristics.It is of great interest to notice that the presence of thependulum greatly reduces the amplitude of vibrations inthose cases, when the frequency of excitation is not withinthe resonance area of pendulum.This means that it would bepossible to study some passive tuned pendulum to be addedto real “rocking blocks” like ancient towers, and so forth, inorder to reduce oscillations induced by wind, earthquakes,and so on.

Adding damping, we investigated two situations, bothwith no pendulum. At first, the case of the free oscillationsof a block with dry friction is analytically solved. Thissolution has been carried out considering small tilt angles.The main results, amply reported in this work, are presentedin graphs where the amplitude of the free oscillations isplotted versus frequency. It has to be remarked that, forblocks presenting low base/highness ratios, the theoreticaland numerical results are in good accordance. When forcingis added, given by the movement of the base, we still applyBubnov-Galerkin’s method. Starting from certain amplitude,we can define a pseudo resonant frequency andwe investigatethe block motion in terms of gain-frequency and phase-frequency characteristics, as reported in Figures 8 and 9,which give a good scenario for the influence of an addedpendulum on the frequency response of the system. Also inthis case, there is good accordance between the theoreticaland numerical results.

Finally, a multibody dynamical simulation has been com-pared against theoretical numerical results on a real case, giv-ing a very satisfactory reduction of oscillations.

Conflict of Interests

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

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