The dynamic behavior of rigid-block structures resting on a rigid foundation subjected to horizontal harmonic excitation is examined. For slender structures, the nonlinear equation of motion is approximated by a piecewise linear equation. Using this approximation for an initially quiescent structure, safe or no-toppling and unsafe regions are identified in an excitation amplitude versus excitation frequency plane. Furthermore, several possible modes of steady-state response are detected, and analytical procedures are developed for determining the amplitudes of the predominant modes and for performing stability analyses. It is shown that the produced stability diagrams can be beneficial to assessing the toppling potential of a rigid-block structure under a given amplitude-frequency combination of harmonic excitation; in this manner the integration of the equation of motion is circumvented.

Closure of "Rocking of Rigid Blocks Due to Harmonic Shaking"

Nonlinear vibrations of suspended cables—Part I: Modeling and analysis

In this paper, the dynamic response of the rocking block subjected to base excitation is revisited. The goal is to offer new closed-form solutions and original similarity laws that shed light on the fundamental aspects of the rocking block. The focus is on the transient dynamics of the rocking block under finite-duration excitations. An alternative way to describe the response of the rocking block, informative of the behaviour of rocking structures under excitations of different intensity, is offered. In the process, limitations of standard dimensional analysis, related to the orientations of the involved physical quantities, are revealed. The proposed dimensionless and orientationless groups condense the response and offer a lucid depiction of the rocking phenomenon. When expressed in the appropriate dimensionless–orientationless groups, the rocking response becomes perfectly self-similar for slender blocks (within the small rotations range) and practically self-similar for non-slender blocks (larger rotations). Using this formulation, the nonlinear and non-smooth rocking response to pulse-type ground motion can be directly determined, and need only be scaled by the intensity and frequency of the excitation.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2012.0026

Revisiting the rocking block: closed-form solutions and similarity laws

SUMMARY
This paper examines the rocking response and stability of rigid blocks standing free on an isolated base supported: (a) on linear viscoelastic bearings, (b) on single concave and (c) on double concave spherical sliding bearings. The investigation concludes that seismic isolation is beneficial to improve the stability only of small blocks. This happens because while seismic isolation increase the ‘static’ value of the minimum overturning acceleration, this value remains nearly constant as we move to larger blocks or higher frequency pulses; therefore, seismic isolation removes appreciably from the dynamics of rocking blocks the beneficial property of increasing stability as their size increases or as the excitation pulse period decreases. This remarkable result suggests that free- standing ancient classical columns exhibit superior stability as they are built (standing free on a rigid foundation) rather than if they were seismically isolated even with isolation system with long isolation periods. The study further confirms this finding by examining the seismic response of the columns from the peristyle of two ancient Greek temples when subjected to historic records. Copyright © 2011 John Wiley & Sons, Ltd.

Analysis of the rocking response of rigid blocks standing free on a seismically isolated base

SUMMARY
Predicting the rocking response of structures to ground motion is important for assessment of existing structures, which may be vulnerable to uplift and overturning, as well as for designs which employ rocking as a means of seismic isolation. However, the majority of studies utilize a single rocking block to characterize rocking motion. In this paper, a methodology is proposed to derive equivalence between the single rocking block and various rocking mechanisms, yielding a set of fundamental rocking parameters. Specific structures that have exact dynamic equivalence with a single rocking block, are first reviewed. Subsequently, approximate equivalence between single and multiple block mechanisms is achieved through local linearization of the relevant equations of motion. The approximation error associated with linearization is quantified for three essential mechanisms, providing a measure of the confidence with which the proposed methodology can be applied. Copyright © 2014 John Wiley & Sons, Ltd.

Dynamically equivalent rocking structures

Base isolation of slide-rocking non-symmetric rigid blocks under impulsive and seismic excitations

In this paper the influence of base isolation on the behaviour of a work of art has been analysed. To make things more realistic, the work of art has been modelled with a non-symmetrical rigid body, sitting on a base that is connected to a visco-elastic device, which represents the passive control system. To prevent the breaking of the isolation device, security stops have been introduced to limit the displacement of the oscillating base to a maximum safety value. All analyses have been carried out comparing the behaviour of the non-isolated and the isolated non-symmetric rigid body subject to impulsive and seismic excitations. The analysis is particularly focused on the effects of the eccentricity of the rigid body and on the presence of the security stops. Generally, base isolation improves the behaviour of the system while the presence of an eccentricity makes the performance of the system worse with respect to the symmetric rigid body. Moreover the security stops, although they preserve the isolator devices, cause a worsening in the performance of the systems. Copyright © 2008 John Wiley & Sons, Ltd.

Investigations into the benefits of base isolation for non-symmetric rigid blocks

Seismic response of a non-symmetric rigid block on a constrained oscillating base

Many works have been focused on the use of the base isolation to improve the dynamic response of the rigid blocks, avoiding the overturning of these systems In this paper the effects of a mass damper on the rocking motion of a non-symmetric rigid block, subject to one-sine pulse type excitation, is investigated The damper is modelled as a pendulum, hinged at the top of the block, with the mass lumped at the end The equations of rocking motion, the uplift and the impact conditions are derived and the results are obtained by numerical integration of these equations An extensive parametric analysis is performed, by taking as variable parameters the eccentricity of the centre of mass, the frequency and the amplitude of the excitation and the characteristics of the mass damper Here the geometrical parameters characterizing the block are taken as fixed quantities, since the main objective of the study is understand if it is possible to find the optimal properties of the pendulum, capable to make more difficult the overturning of the body The results show that the presence of the mass damper, if correctly designed, leads to a general improvement of the response of the system, since the overturning of the block occurs for values, of the amplitude of the base excitation, higher than those observed where no mass damper is used Curves providing the optimum value of the characteristics of the mass damper versus the parameters characterizing the excitation, are finally obtained

On the use of a pendulum as mass damper to control the rocking motion of a rigid block with fixed characteristics

A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is considered. A sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space. Because the coalescence of the eigenvalues implies that the Jacobian matrix is defective (or nilpotent ), well-suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities. Different asymptotic methods are used, based on perturbations both of the eigenvalue problem and the characteristic equation. The analysis reveals the existence of a generic (nonsingular ) case and of a nongeneric (singular) case. However, even in the generic case, a codimension-1 subspace exists in the parameter space on which a singularity occurs. By the use of the relevant asymptotic expansions, linear stability diagrams arebuiltup, and different bifurcation mechanisms (divergence‐Hopf, doubledivergence, doubledivergence ‐Hopf, degenerateHopf )arehighlighted. The problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed. It isfound that a second-degreealgebraic equation governsthebehaviorofthecriticaleigenvalues. It also permits clarie cation of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens ‐Bogdanova bifurcation. Finally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.

/pdf/sensitivities-and-linear-stability-analysis-around-a-double-33knj4w3iz.pdf

A. Di Egidio

Papers

Base isolation of slide-rocking non-symmetric rigid blocks under impulsive and seismic excitations

Investigations into the benefits of base isolation for non-symmetric rigid blocks

Seismic response of a non-symmetric rigid block on a constrained oscillating base

On the use of a pendulum as mass damper to control the rocking motion of a rigid block with fixed characteristics

Sensitivities and Linear Stability Analysis Around a Double-Zero Eigenvalue