Numerical Integration of Dynamic Models for the Protection of the Millefiori Cup with Error Bounding Aspects

The present doctoral thesis originates within the PhD Program in Information Technologies for Engineering, XXXVIII cycle, curriculum Energy and Environment. Its aim is to provide a scientific contribution to engineering problems sufficiently centered on themes belonging to Civil Engineering, while at the same time directing research efforts toward concrete and relevant applications for the protection of the built environment and cultural heritage. These two dimensions of protection are today jointly recognized in Article 9 of the Italian Constitution, as recently updated in the Official Gazette of 22/02/2022, within the broader and unified principle of the joint necessity to safeguard cultural and environmental heritage. This principle encompasses eight constitutionally recognized dimensions: protection of the landscape, of the historical and artistic heritage of the Nation, of the environment, biodiversity, ecosystems, the interests of future generations, and animals. These elements complement the opening part of the article, which establishes the principle of promoting culture and scientific and technical research. The foundation of this study therefore lies in the need to identify a research path in which Civil Engineering can employ its classical analytical tools — mechanical analysis and structural dynamics — to provide a meaningful contribution to the protection of cultural and environmental heritage. From this perspective, the study of the dynamic behavior of archaeological artifacts displayed in museums, characterized by different shapes and dimensions and subjected to environmental excitations of various kinds, falls fully within the domain of Civil Engineering. Indeed, although these objects are typically small compared with the dimensions of civil engineering structures, their mechanical behavior under dynamic excitations is essentially not dissimilar, from a mechanical standpoint, to that of large-scale artifacts such as buildings and structural elements traditionally analyzed and designed within this discipline. The focus can therefore legitimately be extended to artifacts belonging to museum collections, whose mechanical behavior can be investigated using the same theoretical and methodological tools adopted for the analysis of both small and large civil structures. During the first year of the PhD program, within the framework of the bibliographic research, a particularly significant moment was represented by the critical review of a work published in 1963 by George W. Housner. This contribution, widely cited both in the earthquake engineering literature and in numerous studies concerning the mechanical vulnerability of archaeological artifacts and museum objects, represented an important turning point in the development of the present research and led to the formulation of many of the questions that guided the investigation in the subsequent years of the doctoral program. The model proposed by Housner (frequently referred to in the scientific literature as the “rocking block model”) considers a rigid planar rectangular body whose kinematics admit only rotations about the base vertices (“no bouncing”, or the assumption of absence of rebounds) and impacts between the base of the rectangle and the horizontal support. In this scheme, the impulsive contact of all the points of the base of the rectangle is idealized as a single simultaneous event. This model can essentially be interpreted as a combination of the classical models of Rational Mechanics of the physical pendulum and impact between planar rigid bodies (already widely treated in classical Italian textbooks such as the second 1927 edition of the famous lectures by Tullio Levi-Civita and Ugo Amaldi). It can also be regarded as a variant of a model proposed before World War II by Kirkpatrick, also in 1927, to measure seismic excitation through the overturning of column-shaped objects. The 1963 model, however, differs from these earlier studies by introducing several particular assumptions. Among them stands out the unusual hypothesis of conservation of angular momentum across the instant of impact, evaluated with respect to the pole located at the base vertex opposite to the vertex about which rotation occurs before impact. From this assumption, somewhat surprisingly, a coefficient of restitution emerges that depends exclusively on the ratio between the base and height of the rectangle and is entirely independent of other constitutive or empirical factors. This assumption, although unusual, apparently resolves the indeterminacy in the mathematical formulation of the impact model (that is, the mismatch between the number of equations and the number of unknowns) introduced by the adopted scheme of simultaneous impulsive contact of all the points of the rectangle’s base. A research effort aimed at developing sufficiently predictive analytical models for the protection of simply supported museum artifacts subjected to environmental excitations — such as anthropogenic vibrations produced in urban contexts and seismic actions — must naturally examine the modeling assumptions proposed by Housner from both a physical–mathematical and an engineering perspective. The critical review of several modeling choices proposed in Housner’s study therefore naturally led to the formulation of the following questions:  Does the rocking model proposed by Housner, which considers exclusively alternating rotations about the two base vertices of the rectangle, actually represent the only kinematic mechanism (or even the primary mechanism) that deserves consideration for the protection of museum artifacts?  On what mathematical, physical, and/or engineering bases can Housner’s assumption be justified that angular momentum, evaluated with respect to the pole located at the base vertex opposite the pre-impact rotation vertex, is conserved across the impact? What physical implications does this hypothesis entail, and what consequences does it have in terms of the predictive capability of the model for vulnerability analyses of museum artifacts?  To what extent does the simplification introduced by Housner — based on the aforementioned postulate of conservation of angular momentum across the impact — address the problem of indeterminacy in the rocking impact model, and what are the limits of validity and applicability of this idealization?  On the basis of which physical and engineering criteria is it possible to assume that the impact is perfectly inelastic, thus postulating the hypothesis of “no bouncing”? Can this assumption be considered admissible or applicable in engineering problems aimed at the protection of museum artifacts?  In order to obtain a sufficiently predictive model, can the idealization proposed by Housner also be applied to objects that are not particularly slender, such as many museum artifacts?  What theoretical and experimental bases support, for predictive purposes (or engineering usefulness in safety assessments), the assumption of “sufficiently high friction” at the interface between the block and the rigid horizontal support proposed by Housner in the 1963 study?  Is it admissible that the coefficient of restitution emerges as a function exclusively of geometry, as proposed by a large number of authors starting from Housner (1963) and also recently (Destro Bisol, DeJong et al., 2023), assuming that this coefficient does not depend on the properties of the materials in contact, as suggested by most classical textbooks of physics and rational mechanics (and as one might expect from experience)? Within what approximations can a coefficient of restitution calibrated solely on geometric parameters — without accounting for the nature of the bodies in contact — be considered engineeringly valid for the protection of museum artifacts? Although many studies following Housner have raised questions similar to those listed above and have proposed variants and reinterpretations of the rocking block model—including some of the works already mentioned, and in particular several studies by Sinopoli and Augusti published between the 1980s and 1990s on the dynamic modeling of block structures and impacts in the presence of unilateral constraints — several of the questions raised do not appear to have received entirely satisfactory answers. A considerable portion of the studies that reworked Housner’s model (for example, among others, those by Ishiyama (1984), Psycharis (1990), Lipscombe and Pellegrino (1993), and the more recent work by Destro Bisol, DeJong and co-authors (2024)) does not seem to have provided significant insights into the limits of physical admissibility of the model based on the simultaneous impact of all points of the base. Instead, they often introduce conjectured restitution factors or coefficients that still depend exclusively on the geometry of the block, without addressing the underlying physics of the impact phenomenon (such as the materials involved, the nature of the contact, specific application conditions, surface deformability, etc.), perhaps partly disregarding recommendations clearly stated in many celebrated treatises of Rational Mechanics. In the lectures by Levi-Civita and Amaldi (second edition, 1927), in Chapter XXII on impact, one can read, for example: “the coefficient of restitution depends exclusively on the physical constitution of the bodies that collide.” Although the physical constitution of bodies certainly includes geometric ratios, it also undoubtedly includes among the constitutive factors the mechanical characteristics of the materials, as well as additional technical and empirical factors that only experimentation can provide. Even if attention is restricted solely to the theoretical and modeling issues necessary to address the questions raised above, it is important to observe that the analysis of the impact problem, as posed by the abstraction used in Housner’s 1963 model, confronts several singularities, such as the mismatch between the number of equations and unknowns and the temporal discontinuity in the position of the instantaneous center of rotation. Structural Mechanics frequently deals with problems affected by singularities, since even the simplest static problems may involve statically indeterminate structures. These problems teach that statically indeterminate systems can be effectively analyzed by first considering statically determinate systems free from singularities, such as equivalent isostatic systems. Based on this key reflection, the principal methodological choice that guided the research path described in this dissertation was the restoration of a hierarchical and progressive approach to mechanical modeling. Care was taken to ensure that the analysis of more complex models — or those affected by singularities — was always founded on the preliminary analysis, integration, and validation of simpler and more elementary models. Within this framework, it was considered that the analysis of a model affected by singularities or characterized by a mismatch between equations and unknowns could only be effectively addressed starting from basic models free from singularities and indeterminacy issues. For this reason, it was deemed necessary to first examine and integrate the rolling model of a convex planar rigid body with a boundary that is continuous and continuously differentiable, before proceeding to the analysis of impact models characterized by irregularities and singularities. This investigative approach redirected the initial attention — originally focused on archaeological artifacts from the Irpinian and Samnite heritage of Campania characterized by more complex geometries (it had initially been considered to center the analysis on the “Red-figure crater with a scene from the myth of Medea,” 5th century BC, from the necropolis of Baronia in Irpinia, preserved at the Archaeological Museum of Carife and Baronia) — toward objects with sufficiently globular and geometrically simple shapes, allowing, within certain limits, a simpler modeling of motion and impact problems. Between the second and third years of the PhD program, in agreement with the supervisors who guided the research, the Millefiori Cup (or more precisely, the Millefiori cups, including the Millefiori Bowl), beautiful artifacts belonging to the Vesuvian collections of the National Archaeological Museum of Naples (MANN), were identified as suitable objects of investigation capable of guiding the theoretical and modeling research. The cup, originally an instrumentum domesticum and instrumentum vitreum (De Caro, 2004), thus also assumed the role of instrumentum inquirendi. The research benefited from scientific collaboration with experts in archaeological research and cultural heritage conservation (Sallicandro, Serpieri, Luciano, 2025), who provided the necessary introduction to the artifact and integrated essential bibliographic information concerning the archaeological and historical context of the Millefiori Cup. This collaboration also enabled effective interaction with the National Archaeological Museum of Naples. Furthermore, the research benefited from the expertise and contribution of the research group “Knowledge, enhancement and digital communication of cultural heritage” of the Department of Architecture and Industrial Design of the University of Campania “Luigi Vanvitelli,” for the design and implementation of a photogrammetric acquisition campaign on the Millefiori Cup. The experimental activities were conducted, with the kind authorization of the direction of MANN, in October 2025 at the Restoration Laboratory of the National Archaeological Museum of Naples. This made it possible for the research activities on the Millefiori artifacts to benefit from fruitful interaction between engineering, scientific, and conservation–archaeological expertise. The structure of this dissertation essentially reflects the hierarchical modeling approach described above. The study therefore proceeds from the elementary model of a planar semicircular disk undergoing pure rolling motion on an oscillating rigid base. For this model, the equations of motion were derived in the form of Euler–Lagrange equations. This development “from scratch” proved necessary because the bibliographic research conducted did not reveal complete treatments or tractable solutions of the equations for this model, thus requiring their analytical derivation and subsequent numerical integration. The semicircular disk model made it possible to address the dynamics of the system in the continuous domain without immediately confronting the indeterminacy of the impact problem that arises in the presence of flat boundary regions (i.e., regions with infinite radius of curvature). Subsequently, the observation of real objects characterized by more complex shapes, such as the Millefiori Bowl and similar artifacts, suggested — within the modeling abstraction that removes boundary regions with infinite radius of curvature — the extension to the more general model of a semi-elliptical body. This apparently simple step required, in a nontrivial way, the ability to treat analytically and numerically the local variability of the radius of curvature along the boundary, thus allowing a more realistic representation of a broader class of planar solids. At the same time, the development of the models highlighted the necessity of addressing the numerical integration of the equations of motion and the associated problem of bounding the numerical error of the algorithms developed. A general review of the main textbooks on computational methods and numerical analysis — including Bachvalov (1980), Demidovich (1981), Hughes (2000), and Quarteroni (2017) — did not allow the identification of explicit error bounds directly applicable to the problem under study. The choice of the time discretization method with midpoint collocation was justified through bounding techniques developed from evaluations of spectral radii and matrix norms, using Leibniz’s theorem, and supported by bounds derived for elementary model equations such as the harmonic equation and the exponential function equation. The determination of the solutions of these equations was carried out using the method of undetermined coefficients, following the classical approach (which, although classical, is not easily found in many textbooks), briefly recalled for example by Piskunov (2004). This phase of the research thus pursued a dual objective: on the one hand, to address the engineering need to understand and model the dynamic behavior of museum objects; on the other hand, to tackle the fundamental problem of bounding the numerical error in the models developed and subsequently implemented in Matlab scripts. In order to approach the behavior of real objects more closely, the research subsequently focused on bodies with flat bases, typical of the vast majority of archaeological artifacts and vessels. The study of impact in this configuration required an explicit and detailed critical-scientific examination of the classical models of Housner and some subsequent contributions by Sinopoli and Augusti, starting from the fundamental principles of classical impact mechanics, for which the lectures by Levi-Civita and Amaldi served as a fundamental reference. Since, as already observed, Housner’s model can be reduced to that of an inverted pendulum, it was considered appropriate to first carry out a preliminary re-examination of the linearization and integration of the equation of motion of the simple pendulum. This step helped clarify several general issues related to the integration problem that the use of precompiled algorithms available in commercial software often leaves as inaccessible “black boxes.” The organization of the thesis into chapters is as follows:  Chapter 1 addresses the study of the dynamics of the planar semicircular disk on an oscillating rigid base, including the integration of the equations of motion and presenting several simulations of the response to impulsive and sinusoidal dynamic excitations.  Chapter 2 extends the model of Chapter 1 to the semi-ellipse, addressing the problem of the variability of the radius of curvature along the boundary of planar solids.  Chapter 3 is devoted to the development of numerical integration methods and the related problem of bounding the numerical error.  Chapter 4 focuses on the impact of bodies with flat bases, presenting a critical scientific review of the models of Housner, Sinopoli, and Augusti. This study leads to the development of a post-impulse motion diagram for rectangular bodies with flat bases, useful for distinguishing post-impact response mechanisms such as rocking and sliding as a function of the height-to-base ratio and the friction coefficient of the contacting materials.  Chapter 5 finally illustrates the results of the experimental campaign of metric, weight, and photogrammetric data acquisition for the Millefiori Cup conducted at the Restoration Laboratory of MANN. The Appendices accompanying the thesis collect brief theoretical summaries intended to support readers who may not have immediate access to all the complex algebra tools necessary to follow some of the more technical mathematical passages of the dissertation. In particular, Appendix A describes the time discretization method with midpoint collocation; Appendix B illustrates elementary model cases for the estimation of numerical error; Appendix C presents some foundations of complex algebra useful for bounding the error; and finally, Appendix D provides a compendium on the integration and linearization of the equation of the simple pendulum. At the end of the thesis, Appendix E presents the typographical conventions adopted, together with a list of the principal symbols used. The bibliography, including the references cited in this introduction, is organized by chapters.