Stefano Lenci

Mathematical Analysis of a Bonded Joint with a Soft Thin Adhesive

This paper considers the problem of two adherents joined by a soft thin adhesive along their common surface. Using the asymptotic expansion method, the authors obtain a simplified model in which the adhesive is treated as a material surface and is replaced by returning springs. The authors show weak and strong convergence of the exact solution toward the solution of the limit problem. The singularities of the limit problem are analyzed, and it is shown that typically they are logarithmic. Furthermore, the authors investigate the phenomenon of boundary layer by studying the correctors, the extra terms, which must be added to the classical asymptotic expansion to verify the boundary conditions. The correctors show that, contrary to the adherents, in the adhesive there are power-type singularities, which are at the base of the failure of the assemblage.

Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam

This work deals with the problem of controlling the nonlinear dynamics, in general, and the dynamic pull-in, in particular, of an electrically actuated microbeam. A single-well softening model recently proposed by Gottlieb and Champneys [1] is considered, and a control method previously proposed by the authors is applied. Homoclinic bifurcation, which triggers the safe basin erosion eventually leading to pull-in, is considered as the undesired event, and it is shown how appropriate controlling superharmonics added to a reference harmonic excitation succeed in shifting it towards higher excitation amplitudes. An optimization problem is formulated, and the optimal excitation shape is obtained. Extensive numerical simulations aimed at checking the effectiveness of the control method in shifting the erosion of the safe basin are reported. They highlight good performances of the control method beyond theoretical expectations.

Optimal control of homoclinic bifurcation: Theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikovs theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.

Optimal Control of Nonregular Dynamics in a Duffing Oscillator

A method for controlling nonlinear dynamics and chaos previouslydeveloped by the authors is applied to the classical Duffing oscillator.The method, which consists in choosing the best shape of externalperiodic excitations permitting to avoid the transverse intersection ofthe stable and unstable manifolds of the hilltop saddle, is firstillustrated and then applied by using the Melnikov method foranalytically detecting homoclinic bifurcations. Attention is focused onoptimal excitations with a finite number of superharmonics, because theyare theoretically performant and easy to reproduce. Extensive numericalinvestigations aimed at confirming the theoretical predictions andchecking the effectiveness of the method are performed. In particular,the elimination of the homoclinic tangency and the regularization offractal basins of attraction are numerically verified. The reduction ofthe erosion of the basins of attraction is also investigated in detail,and the paper ends with a study of the effects of control on delayingcross-well chaotic attractors.

Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks

A method for controlling nonlinear dynamics and chaos, previously developed by the authors, is applied to the rigid block on a moving foundation. The method consists in modifying the shape of the excitation in order to eliminate, in an optimal way, the heteroclinic intersections embedded in the system dynamics. Two different cases are examined: (i) generic block under small perturbations and (ii) slender block under generic perturbations, and they are investigated analytically either by a perturbation analysis (former case) or exactly (latter case). Two different strategies are proposed: (i) one-side control, which consists in eliminating the intersections of a single heteroclinic connection, and (ii) global control, which consists in simultaneously eliminating the intersections of both heteroclinic connections. The best excitations permitting the maximum distance between stable and unstable manifolds are determined in both cases. Finally, some numerical investigations aimed at highlighting meaningful aspects of system response under controlled (optimal) and noncontrolled (harmonic) excitations are performed.

Analysis of a crack at a weak interface

The problem of two elastic half-planes joined along the common part of their boundary by a cracked weak interface is considered. The central part of the joint is detached, while in the remaining part there is a continuous distribution of springs which assures continuity of stress which is proportional to the displacement gap. The adherents are homogeneous and isotropic, while the interface is allowed to be orthotropic with principal directions normal and tangential to the interface, respectively. The body is subjected to constant normal and tangential loads applied at infinity and at the crack faces. Using classical solutions for elastic half-planes as Green functions, the integral equation governing the problem is obtained and solved numerically. Attention is paid to the analysis of the solution around the crack tip, and an asymptotic estimate showing that the derivative of the solution is logarithmically unbounded is obtained analytically. Accordingly, it is shown that there may exist, at most, logarithmic stress singularities. It is further shown how, contrary to the case of perfect bonding, stress singularities are not related to the normal propagation of the crack, but possibly to the crack deviation. The crack propagation is analyzed by the energy Griffith criterion, and it is shown that some drawbacks of linear elastic fracture mechanics disappear in the case of weak interface.

Nonlinear dynamics of an electrically actuated imperfect microbeam resonator: experimental investigation and reduced-order modeling

We present a study of the dynamic behavior of a microelectromechanical systems (MEMS) device consisting of an imperfect clamped–clamped microbeam subjected to electrostatic and electrodynamic actuation. Our objective is to develop a theoretical analysis, which is able to describe and predict all the main relevant aspects of the experimental response. Extensive experimental investigation is conducted, where the main imperfections coming from microfabrication are detected, the first four experimental natural frequencies are identified and the nonlinear dynamics are explored at increasing values of electrodynamic excitation, in a neighborhood of the first symmetric resonance. Several backward and forward frequency sweeps are acquired. The nonlinear behavior is highlighted, which includes ranges of multistability, where the nonresonant and the resonant branch coexist, and intervals where superharmonic resonances are clearly visible. Numerical simulations are performed. Initially, two single mode reduced-order models are considered. One is generated via the Galerkin technique, and the other one via the combined use of the Ritz method and the Pade approximation. Both of them are able to provide a satisfactory agreement with the experimental data. This occurs not only at low values of electrodynamic excitation, but also at higher ones. Their computational efficiency is discussed in detail, since this is an essential aspect for systematic local and global simulations. Finally, the theoretical analysis is further improved and a two-degree-of-freedom reduced-order model is developed, which is also capable of capturing the measured second symmetric superharmonic resonance. Despite the apparent simplicity, it is shown that all the proposed reduced-order models are able to describe the experimental complex nonlinear dynamics of the device accurately and properly, which validates the proposed theoretical approach.

Competing dynamic solutions in a parametrically excited pendulum: Attractor robustness and basin integrity

The competing solutions of a planar pendulum parametrically excited by the vertical motion of the pivot are investigated in terms of both attractor robustness and basin integrity. Two different measures are considered to highlight how the integrity of the system is modified by changing the amplitude of the excitation. Various attractors, both in-well and out-of-well, are considered, and the integrity profiles of each of them are determined. A detailed discussion of the interaction and mutual erosion of the various attractors is performed by clarifying the role of the two complementary measures, and the most relevant characteristics of the erosion profiles are interpreted in terms of the underlying topological mechanisms involving local or global bifurcations.

Chaotic vibrations of the duffing system with fractional damping

We examined the Duffing system with a fractional damping term. Calculating the basins of attraction, we demonstrate a broad spectrum of non-linear behaviour connected with sensitivity to the initial conditions and chaos. To quantify dynamical response of the system, we propose the statistical 0-1 test as well as the maximal Lyapunov exponent; the application of the latter encounter a few difficulties because of the memory effect due to the fractional derivative. The results are confirmed by bifurcation diagrams, phase portraits, and Poincaré sections.

AN IMPERFECT MICROBEAM UNDER AN AXIAL LOAD AND ELECTRIC EXCITATION: NONLINEAR PHENOMENA AND DYNAMICAL INTEGRITY

This work deals with the nonlinear dynamics of a microelectromechanical system constituted by an imperfect microbeam under an axial load and an electric excitation. The device is characterized by a...