
MATTEO STROZZI
Professore Associato Dipartimento di Scienze e Metodi dell'Ingegneria

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2024
 NonlocalStrainGradientBased Anisotropic Elastic Shell Model for Vibrational Analysis of SingleWalled Carbon Nanotubes
[Articolo su rivista]
Strozzi, Matteo; Elishakoff, Isaac E.; Bochicchio, Michele; Cocconcelli, Marco; Rubini, Riccardo; Radi, Enrico
abstract
2023
 A Comparison of Shell Theories for Vibration Analysis of SingleWalled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model
[Articolo su rivista]
Strozzi, Matteo; Elishakoff, Isaac E.; Bochicchio, Michele; Cocconcelli, Marco; Rubini, Riccardo; Radi, Enrico
abstract
In the present paper, a comparison is conducted between three classical shell theories as applied to the linear vibrations of singlewalled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders, and Flügge shell theories. The actual discrete SWCNT is modelled by means of a continuous homogeneous cylindrical shell considering equivalent thickness and surface density. In order to take into account the intrinsic chirality of carbon nanotubes (CNTs), a molecular based anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and a complex method is applied to solve the equations of motion and to obtain the natural frequencies. Comparisons with the results of molecular dynamics simulations available in literature are performed to check the accuracy of the three different shell theories, where the Flügge shell theory is found to be the most accurate. Then, a parametric analysis evaluating the effect of diameter, aspect ratio, and number of waves along the longitudinal and circumferential directions on the natural frequencies of SWCNTs is performed in the framework of the three different shell theories. Assuming the results of the Flügge shell theory as reference, it is obtained that the Donnell shell theory is not accurate for relatively low longitudinal and circumferential wavenumbers, for relatively low diameters, and for relatively high aspect ratios. On the other hand, it is found that the Sanders shell theory is very accurate for all the considered geometries and wavenumbers, and therefore, it can be correctly adopted instead of the more complex Flügge shell theory for the vibration modelling of SWCNTs.
2023
 A Comparison of Signal Analysis Techniques for the Diagnostics of the IMS Rolling Element Bearing Dataset
[Articolo su rivista]
Sacerdoti, Diletta; Strozzi, Matteo; Secchi, Cristian
abstract
In this paper, a comparison of signal analysis techniques for the diagnostics of rolling element bearings is carried out. Specifically, the comparison is performed in terms of fault detection, diagnosis and prognosis techniques with regards to the first rolling element bearing dataset released by NASA IMS Center in 2014. As for fault detection, it is obtained that RMS value, Kurtosis and Detectivity, as statistical parameters, are able to properly detect the arising of the fault on the defective
bearings. Then, several signal processing techniques, such as deterministic/random signal separation, timefrequency and cyclostationary analyses are applied to perform fault diagnosis. Among these techniques, it is found that the combination of Cepstrum PreWhitening and Squared Envelope Spectrum, and Improved Envelope Spectrum, allow the faults to be correctly identified on specific bearing components. Finally, the Correlation, Monotonicity and Robustness of the previous statistical parameters are computed to identify the most accurate tools for bearing fault prognosis.
2023
 A European Researchers’ Night project on mechanical vibrations for high school students
[Relazione in Atti di Convegno]
Cocconcelli, Marco; Fonte, Cosimo; Grosso, Pasquale; Mottola, Giovanni; Strozzi, Matteo; Rubini, Riccardo
abstract
The present works were conceived to be exhibited during the 2022 European Researchers’ Night (ERN 2022), at the University of Modena and Reggio Emilia. The idea is to illustrate the key concepts of mechanical vibration through the use of 3D models and virtual simulation analysis. The paper is directed to high school students planning to enroll in a mechanical engineering bachelor’s degree, in order to approach or consolidate some fundamental concepts of mechanical vibration. Topics not easy to explain, such as the natural frequencies of a body, could be presented more effectively using physical models. Mathematical formalism will be kept to a minimum, as it is beyond the scope of this paper.
2022
 Applicability and Limitations of Ru’s Formulation for Vibration Modelling of DoubleWalled Carbon Nanotubes
[Articolo su rivista]
Strozzi, Matteo
abstract
2022
 Applicability and limitations of Donnell shell theory for vibration modelling of doublewalled carbon nanotubes
[Articolo su rivista]
Strozzi, Matteo; Elishakoff, Isaac E.; Manevitch, Leonid I.; Gendelman, Oleg V.
abstract
2022
 Modal analysis and condition monitoring for an electric motor through MEMS accelerometers
[Relazione in Atti di Convegno]
Mottola, Giovanni; Grosso, Pasquale; Fonte, Cosimo; Strozzi, Matteo; Rubini, Riccardo; Cocconcelli, Marco
abstract
2022
 Nomograms: history, properties and applications
[Abstract in Atti di Convegno]
Mottola, Giovanni; Strozzi, Matteo; Rubini, Riccardo; Cocconcelli, Marco
abstract
2022
 Nonlocal anisotropic elastic shell model for vibrations of doublewalled carbon nanotubes under nonlinear van der Waals interaction forces
[Articolo su rivista]
Strozzi, M.; Smirnov, V. V.; Pellicano, F.; Kovaleva, M.
abstract
In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of doublewalled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric singlewalled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via LennardJones potential and He's formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are reexpanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found.
2022
 Open Issues in Continuum Modeling of Carbon Nanotubes
[Abstract in Atti di Convegno]
Strozzi, Matteo; Fonte, Cosimo; Rubini, Riccardo; Cocconcelli, Marco
abstract
2022
 Reliability of a resistance spot welding process based on characteristics parameters
[Relazione in Atti di Convegno]
Strozzi, Matteo; Grosso, Pasquale; Mottola, Giovanni; Rubini, Riccardo
abstract
2022
 Simulation of the vibration signal of cycloidal drives: preliminary results
[Abstract in Atti di Convegno]
Grosso, Pasquale; Strozzi, Matteo; Rubini, Riccardo; Cocconcelli, Marco
abstract
2021
 Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of DoubleWalled Carbon Nanotubes
[Articolo su rivista]
Strozzi, Matteo; Gendelman, Oleg V.; Elishakoff, Isaac E.; Pellicano, Francesco
abstract
The applicability and limitations of simplifiedmodels of thin elastic circular cylindrical shells for linear vibrations of doublewalled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thinshell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beamlike modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shelllike modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shelllike modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semimoment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.
2021
 Breather arrest in a chain of damped oscillators with Hertzian contact
[Articolo su rivista]
Strozzi, M.; Gendelman, O. V.
abstract
Breather propagation in a damped oscillatory chain with Hertzian nearestneighbour coupling is investigated. The breather propagation exhibits an unusual twostage pattern. The first stage is characterized by powerlaw decay of the breather amplitude. This stage extends over finite number of the chain sites. Drastic drop of the breather amplitude towards the end of this finite fragment is referred to as breather arrest. At the second stage, the breather exhibits very small amplitudes with hyperexponential decay. Numeric results are rationalized by considering a simplified model of two damped linear oscillators coupled by Hertzian contact forces. Initial excitation of one of these oscillators results in a finite number of beating cycles in the system. This simplified model reliably predicts main features of the breather arrest. More general coupling potentials and effect of precompression on the breather propagation are also discussed.
2021
 Detectivity: A combination of Hjorth's parameters for condition monitoring of ball bearings
[Articolo su rivista]
Cocconcelli, M.; Strozzi, M.; Cavalaglio Camargo Molano, J.; Rubini, R.
abstract
Hjorth's parameters are statistical timedomain parameters used in signal processing and introduced by Bo Hjorth in 1970. These parameters are Activity, Mobility and Complexity. They are related to the variance of the signal and of its subsequent derivatives. They are commonly used in the analysis of electroencephalography (EEG) signals for feature extraction, but also in the tactile signal analysis in robotic area. In this paper, Hjorth's parameters are applied to vibration signals for fault detection in ball bearings. In particular, two openaccess datasets are used: the NASA bearing dataset of the University of Cincinnati and the Polytechnic of Turin rolling bearing dataset. In the first part of the paper the three parameters are used for monitoring the health of the bearings along their lifetime, proving their effectiveness in condition monitoring. In the second part, a new parameter is introduced, named Detectivity, that merges the information carried by Hjorth's parameters in a single value and is suitable for continuous monitoring of machines.
2020
 Condition Monitoring Techniques of Ball Bearings in Nonstationary Conditions
[Relazione in Atti di Convegno]
Strozzi, M.; Rubini, R.; Cocconcelli, M.
abstract
Frequently, the Industry suggests nontrivial problems and new fields of research for the Academy. This is the case of the ball bearing diagnostics in directdrive motors. Directdrive motors are brushless motors fully controlled by the drive system. Thanks to an encoder or a resolver mounted on the shaft, they can perform complex motion profiles, such as polynomials or splines, including reverse rotation of the shaft. The main advantage of directdrive motors is the removal of cams or gearboxes afterwards motor with a consequent strong reduction of economic and maintaining costs. Indeed, their main drawback is the difficulty to make diagnostics on the bearings. Regarding bearing diagnostics, most of the techniques present in literature are based on the search of faultcharacteristic frequencies in the vibration spectrum of the motor. These fault frequencies are linearly dependent on the rotational frequency of the shaft if it is supposed constant. However, in directdrive motors the rotational speed changes continuously and consequently the fault frequencies are meaningless. The paper reports a brief overview of some techniques for the condition monitoring of ball bearings in nonstationary conditions used by the Authors in the case of a packaging machine working under variable speed. The techniques adopted include an improved version of the computed order tracking, the crosscorrelation function and three supervised learning approaches: artificial neural networks, artificial immune systems and support vector machines.
2020
 Condition monitoring and reliability of a resistance spot welding process
[Relazione in Atti di Convegno]
Strozzi, Matteo; Cocconcelli, Marco; Rubini, Riccardo
abstract
The reliability of a resistance spot welding (RSW) process is studied monitoring the quality of the corresponding
welding points. Each welding point is uniquely represented by a specific resistance characteristic curve over time.
Five learning resistance characteristic curves, the good quality of the related welding points was experimentally
verified by means of a nondestructive technique, are selected as a reference to check the quality of welding points
related to different process resistance characteristic curves. A first estimate of the quality of the welding point is
made comparing the corresponding process resistance characteristic curve with the learning maximum, minimum
and average resistance characteristic curves. Both good quality and defective (glued or squeezed) welding points
are observed. In order to more correctly identify the quality level of each welding point, two different parameters
comparing the related process resistance characteristic curve with the learning average resistance characteristic
curve are applied. First, the residual resistance, as the difference at each instant of time between the two resistance
characteristic curves, is considered. Then, the Euclidean distance, as the geometric distance at each instant of time
between the two resistance characteristic curves, is adopted. Finally, the trend of the quality of the welding points
as their number increases for welding electrodes with a fixed number of dressings is investigated.
2020
 Efficiency and Durability of DLCCoated Gears
[Relazione in Atti di Convegno]
Barbieri, M.; Iarriccio, G.; Pellicano, F.; Strozzi, M.; Zippo, A.
abstract
This paper presents an experimental study on spur gears. Gears with and without tungstencarbide coatings (WC/C) are compared in terms of efficiency, durability and vibration performance. In order to carry out the experiments, a test rig including two electric motors/brakes is described. Gears are designed for this specific experimental campaign, so that the number of teeth, the selected materials and thermochemical treatments are optimal to investigate gear efficiency and durability. The experimental procedure allows for a simultaneous evaluation of efficiency and dynamic transmission error by varying the rotational velocity of the gear pair. An additional investigation has been performed for varying load, so that a complete characterization of the effect of WC/C coating on gear performance is presented.
2020
 Metodologie non distruttive per l’individuazione di difetti su sanitari in ceramica: indagine sperimentale.
[Altro]
Castagnetti, Davide; Cocconcelli, Marco; Spaggiari, Andrea; Strozzi, Matteo; Dragoni, Eugenio; Rubini, Riccardo
abstract
Metodologie non distruttive per l’individuazione di difetti su sanitari in ceramica: pianificazione sperimentale, prove sperimentali, analisi dei risultati, proposta di parametri identificativi dei difetti.
2020
 Motor current cyclicnonstationarity analysis for bearing diagnostic
[Relazione in Atti di Convegno]
D'Elia, G.; Cocconcelli, M.; Strozzi, M.; Mucchi, E.; Dalpiaz, G.; Rubini, R.
abstract
The Motor Current Signature Analysis (MCSA) is a research area focused on the diagnosis of components of electric motors based on postprocessing of the current signal mainly. In particular, the bearing diagnostics is based on two different assumptions: the fault on the bearing causes a vibration of the shaft it supports, so there is an air gap variation between stator and rotor causing a modulation in the current signal; the fault on the bearing hinders the rotation of the shaft, so it can be modeled as an additional loading torque that the motor satisfies increasing the current signal. In this paper, a cyclicnonstationarity analysis of the motor current is used to assess the status of ballbearings in servomotors, running at variable speed. Both speed of the motor and motor current are provided by the control loop of the servomotor, that is no external sensors are used. The cyclic nature of the application allows an average of the cycliccyclic order maps to increase the signaltonoise ratio. The proposed technique is successfully applied to both healthy and faulty bearings.
2020
 Nonlinear Vibration of Continuous Systems 2020
[Articolo su rivista]
Pellicano, Francesco; Strozzi, Matteo; Avramov, Konstantin V.; Sinniah, Ilanko
abstract
2020
 Nonlinear normal modes, resonances and energy exchange in singlewalled carbon nanotubes
[Articolo su rivista]
Strozzi, M.; Smirnov, V. V.; Manevitch, L. I.; Pellicano, F.
abstract
The nonlinear resonance interaction and energy exchange between bending and circumferential flexure modes in singlewalled carbon nanotubes is studied. First, the results of an analytical model of the resonance interaction between the considered nonlinear normal modes previously developed are reported. This approach was based on a reduced form of the Sanders–Koiter thin shell theory obtained by using simplifying hypotheses on the shell deformations. The analytical model predicted that the nonlinear resonance interaction leads to energy localization in a certain coherence domain over the carbon nanotube surface within a specific range of the initial oscillation amplitude. Then, a numerical model of the resonance interaction between the analysed nonlinear normal modes in the framework of the complete Sanders–Koiter thin shell theory is reported. Numerical simulations are performed to verify the energy localization phenomenon over the carbon nanotube surface and to compute the threshold values of the initial oscillation amplitude giving rise to energy localization. Finally, from the comparison between the two different approaches, it is obtained that the results of the numerical model for the threshold values of the nonlinear energy localization confirm with very good accuracy the predictions of the analytical model.
2020
 Preliminary orthotropic elastic model for the study of natural frequencies and mode shapes of a 3D printed Onyx thin circular cylindrical shell
[Articolo su rivista]
Strozzi, M.; Giacomobono, R.; Rubini, R.; Cocconcelli, M.
abstract
The linear vibrations of a 3D printed Onyx thin circular cylindrical shell are considered. A model based on SandersKoiter shell theory and orthotropic elastic constitutive equations is adopted to obtain elastic strain and kinetic energy. The deformation of the middle surface of the shell is described in terms of longitudinal, circumferential and radial displacements, which are expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction of the shell. Freefree boundary conditions are considered. The RayleighRitz method is applied to calculate approximate natural frequencies and mode shapes. An isotropic elastic model is first adopted to obtain initial reference values for natural frequencies and mode shapes of the 3D printed shell. An experimental modal analysis is then performed to verify the accuracy of the initial isotropic elastic model and to find exact values for natural frequencies and mode shapes of the 3D printed shell. A more effective orthotropic elastic model is finally applied assuming different values of Young’s modulus along the longitudinal and circumferential directions of the shell. A parametric analysis is carried out by assuming a constant circumferential Young’s modulus and varying the longitudinal Young’s modulus. The goal is to minimise the difference between analytical and experimental results, in order to identify the actual orthotropy degree of the 3D printed shell.
2019
 Analysis of NASA Bearing Dataset of the University of Cincinnati by Means of Hjorth’s Parameters
[Relazione in Atti di Convegno]
CAVALAGLIO CAMARGO MOLANO, Jacopo; Strozzi, Matteo; Rubini, Riccardo; Cocconcelli, Marco
abstract
2019
 Interazioni di risonanza e localizzazioni di energia in nanotubi di carbonio
[Abstract in Atti di Convegno]
Andrisano, A. O.; Manevitch, L. I.; Pellicano, Francesco; Strozzi, Matteo
abstract
2019
 Investigation on apparently related modes in experimental modal analysis
[Abstract in Atti di Convegno]
Giacomobono, Roberto; Rubini, Riccardo; Cocconcelli, Marco; Strozzi, Matteo
abstract
2019
 Nonlinear Resonance Interaction between Conjugate Circumferential Flexural Modes in SingleWalled Carbon Nanotubes
[Articolo su rivista]
Strozzi, M.; Pellicano, F.
abstract
This paper presents an investigation on the dynamical properties of singlewalled carbon nanotubes (SWCNTs), and nonlinear modal interaction and energy exchange are analysed in detail. Resonance interactions between two conjugate circumferential flexural modes (CFMs) are investigated. The nanotubes are analysed through a continuous shell model, and a thin shell theory is used to model the dynamics of the system; freefree boundary conditions are considered. The RayleighRitz method is applied to approximate linear eigenfunctions of the partial differential equations that govern the shell dynamics. An energy approach, based on Lagrange equations and series expansion of the displacements, is considered to reduce the initial partial differential equations to a set of nonlinear ordinary differential equations of motion. The model is validated in linear field (natural frequencies) by means of comparisons with literature. A convergence analysis is carried out in order to obtain the smallest modal expansion able to simulate the nonlinear regimes. The time evolution of the nonlinear energy distribution over the SWCNT surface is studied. The nonlinear dynamics of the system is analysed by means of phase portraits. The resonance interaction and energy transfer between the conjugate CFMs are investigated. A travelling wave moving along the circumferential direction of the SWCNT is observed.
2019
 Nonlinear dynamic stability of cylindrical shells under pulsating axial loading via Finite Element analysis using numerical time integration
[Articolo su rivista]
Rizzetto, Fabio; Jansen, Eelco; Strozzi, Matteo; Pellicano, Francesco
abstract
Nonlinear dynamic stability investigations for isotropic and composite cylindrical shells under pulsating axial loading are carried out through Finite Element analysis using numerical time integration. In particular, im portant characteristics of the geometrically nonlinear behaviour are systematically studied through Finite Element analysis. The results of the Finite Element analysis are compared with results obtained in earlier studies using semianalytical procedures. In order to facilitate the evaluation and the comparison of these two com plementary approaches, a modal projection procedure has been developed for the Finite Element analysis. Critical dynamic loads and frequencyresponse curves for isotropic and composite shells under pulsating loading obtained with the Finite Element analysis using numerical time integration are shown to be generally in good qualitative agreement with the results of earlier semianalytical work. The analysis of the modal amplitude achieved via the modal projection procedure also makes it possible to study the interactions between con tributing modes and to observe and interpret interesting phenomena such as the occurrence of travelling waves in the circumferential direction of the shell.
2019
 Nonlinear vibration of continuous systems
[Articolo su rivista]
Pellicano, F.; Strozzi, M.; Avramov, K. V.
abstract
Continuous systems, such as beams, membranes, plates, shells, and other structural/mechanical components, represent fundamental elements of mechanical systems in any field of engineering: Aerospace, Aeronautics, Automation, Automotive, Civil, Nuclear, Petroleum, and Railways.
The modern designer is required to optimize structural elements to improve the performancetocost ratio, produce lightweight machines, and improve the efficiency. Such optimizations easily lead to a magnification of vibration/dynamic problems such as resonances, instabilities, and nonlinear behaviors. Therefore, the development of new methods of analysis, testing, and monitoring is greatly welcome.
This special issue focuses on sharing recent advances and developments of theories, algorithms, and applications that involve the dynamics and vibrations of continuous systems.
The contributions to this special issue include innovative theoretical studies, advanced numerical simulations, and new experimental approaches to investigate and better understand complex dynamic phenomena; more specifically, methods and theories for beams, membranes, plates, and shells; numerical approaches for structural elements; fluidstructure interaction; nonlinear acoustics; identification, diagnosis, friction models, and vehicle dynamics.
Seventeen contributions have been received from all over the world: Canada, China, Kazakhstan, Italy, Macau, Spain, and USA. This shows the generalized interest on the topic.
The following short description of the special issue content is organized by grouping the contributions in coherent subtopics.
2018
 Damping oriented design of thinwalled mechanical components by means of multilayer coating technology
[Articolo su rivista]
Catania, G.; Strozzi, M.
abstract
The damping behaviour of multilayer composite mechanical components, shown by recent research and application papers, is analyzed. A local dissipation mechanism, acting at the interface between any two different layers of the composite component, is taken into account, and a beam model, to be used for validating the known experimental results, is proposed. Multilayer prismatic beams, consisting of a metal substrate and of some thin coated layers exhibiting variable stiffness and adherence properties, are considered in order to make it possible to study and validate this assumption. A dynamical model, based on a simple beam geometry but taking into account the previously introduced local dissipation mechanism and distributed viscoelastic constraints, is proposed. Some different application examples of specific multilayer beams are considered, and some numerical examples concerning the beam free and forced response are described. The influence of the multilayer system parameters on the damping behaviour of the free and forced response of the composite beam is investigated by means of the definition of some damping estimators. Some effective multicoating configurations, giving a relevant increase of the damping estimators of the coated structure with respect to the same uncoated structure, are obtained from the model simulation, and the results are critically discussed.
2018
 Linear vibrations of triplewalled carbon nanotubes
[Articolo su rivista]
Strozzi, Matteo; Pellicano, Francesco
abstract
In this paper, the linear vibrations of triplewalled carbon nanotubes (TWNTs) are investigated. A multiple elastic thin shell model is applied. The TWNT dynamics is studied in the framework of the Sanders–Koiter shell theory. The van der Waals interaction between any two layers of the TWNT is modelled by a radiusdependent function. The shell deformation is described in terms of longitudinal, tangential and radial displacements. Simply supported, clamped and free boundary conditions are applied. The three displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the tangential variable. The Rayleigh–Ritz method is applied to obtain approximate natural frequencies and mode shapes. The present model is validated in the linear field by means of comparisons with data from the literature. This study is focused on determining the effect of geometry and boundary conditions on the natural frequencies of TWNTs.
2018
 Nonlinear vibrations and energy exchange of singlewalled carbon nanotubes. Radial breathing modes
[Articolo su rivista]
Strozzi, Matteo; Smirnov, Valeri V.; Manevitch, Leonid I.; Pellicano, Francesco
abstract
In this paper, the nonlinear vibrations and energy exchange of singlewalled carbon nanotubes (SWNTs) are analysed. The SandersKoiter shell theory is used to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are applied. The resonant interaction between radial breathing (axisymmetric) modes (RBMs) is analysed. An energy method, based on the Lagrange equations, is considered in order to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is then solved applying the implicit RungeKutta numerical method. The present model is validated in linear field comparing the RBM natural frequencies numerically predicted with data reported in the literature from experiments and molecular dynamics simulations. The nonlinear energy exchange between the two halves along the SWNT axis in the time is studied for different amplitudes of initial excitation applied to the two lowest frequency resonant RBMs. The influence of the SWNT aspect ratio on the numerical value of the nonlinear energy beating period under different boundary conditions is analysed.
2017
 Dynamic imbalance of highspeed planetary gears
[Articolo su rivista]
Masoumi, Asma; Barbieri, M; Pellicano, F; Zippo, A; Strozzi, M
abstract
A nonlinear 2D lumped mass model of a singlestage spur planetary gear system with timevarying mesh stiffness, bearing compliance and nonsmooth nonlinearity due to backlash is taken into account. The timevarying meshing stiffness is evaluated by means of a nonlinear finite element model, through an accurate evaluation of global and local tooth deformation. The nonlinear dynamic behaviour of the system is analysed over a reasonable range of rotation speed and torque. The possibility of occurrences of different dynamic phenomena and instability of the system with respect to the bearing compliance and operating parameters are also evaluated. The possibility of dynamic imbalance of equallyspaced planetary gears in the presence of chaotic regimes is discussed. Such imbalance may lead to unexpected highlevel stresses on bearings and gears.The effect of tooth profile modification at the sunplanet and ringplanet meshes on the vibration behaviour of the planetary gear system is also investigated in this paper. In order to avoid modification on the ring gear, both tip and root reliefs are considered for sun and planet gears.
2017
 Experimental identification of FGM shell properties (aimeta 2017)
[Relazione in Atti di Convegno]
Zippo, A.; Pellicano, F.; Barbieri, M.; Strozzi, M.; Masoumi, A.
abstract
Functionally gradient materials (FGMs) have attracted a growing interest as advanced structural materials because of their heatresistance properties. In this paper, an experimental study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of Polyethylene terephthalate (PET) is presented: to obtain functional gradient proprieties the PET shell had been exposed at a thermal temperature gradient in the range of its glass transition temperature of 79°C. The setting up of the experiment is explained and deeply described along with the thermal characterisation of the specimen. The linear and the nonlinear dynamic behaviour have been investigated. The shell behaviour is also investigated by means of a finite element model, in order to enhance the comprehension of experimental results.
2017
 Modal localization in vibrating circular cylindrical shells
[Relazione in Atti di Convegno]
Pellicano, F.; Zippo, A.; Barbieri, M.; Strozzi, M.
abstract
The goal of the present paper is the analysis of the effect of geometric imperfections in circular cylindrical shells. Perfect circular shells are characterized by the presence of double shelllike modes, i.e., modes having the same frequency with modal shape shifted of a quarter of wavelength in the circumferential direction. In presence of geometric imperfections, the double natural frequencies split into a pair of distinct frequencies, the splitting is proportional to the level of imperfection. In some cases, the imperfections cause an interesting phenomenon on the modal shapes, which present a strong localization in the circumferential direction. This study is carried out by means of a semianalytical approach compared with standard finite element analyses.
2017
 Multilayer composite beam modelling and optimization for high speed mechanical applications
[Abstract in Atti di Convegno]
Catania, Giuseppe; Strozzi, Matteo
abstract
2017
 Nonlinear optical vibrations of singlewalled carbon nanotubes
[Articolo su rivista]
Manevitch, L. I.; Smirnov, V. V.; Strozzi, M.; Pellicano, F.
abstract
We demonstrate the new specific phenomenon of the longtime resonant energy exchange in the carbon nanotubes (CNTs) in the two optical branches  the Circumferential Flexure Mode (CFM) and Radial Btreathing Mode (RBM). It is shown that the modified nonlinear SchrÃ¶dinger equation, obtained in the framework of nonlinear elastic thin shell theory, allows to describe the CNT nonlinear dynamics connected with considered frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the principal difference between nonlinearity effects. If the nonlinear resonant interaction of the lowfrequency modes in the CFM branch leads to the energy capture in the some domain of the CNT, the same interaction in the RBM branch does not appear any tendency to the energy localization. The reason of such a distinction is the difference of the nonlinear terms in the equations of motion. If the CFMs are specified by the soft power nonlinearity, the RBM dynamics is determined by the hard gradient nonlinearity. Moreover, in contrast to CFM the importance of nonlinearity in the case of RBM oscillations decreases with increasing of the length to radius ratio. The numerical integration of the thin shell theory equations confirms the results of the analytical study.
2017
 Nonlinear optical vibrations of singlewalled carbon nanotubes.
[Relazione in Atti di Convegno]
Manevitch, L. I.; Smirnov, V. V.; Strozzi, M.; Pellicano, F.
abstract
We demonstrate a new specific phenomenon of the longtime resonant energy exchange in carbon nanotubes (CNTs), which is realized by two types of optical vibrations, the Circumferential Flexure Mode (CFM) and the Radial Breathing Mode (RBM). We show that the modified nonlinear Schrdinger equation, obtained in the framework of the nonlinear theory of elastic thin shells, allows us to describe the nonlinear dynamics of CNTs for specified frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the qualitative difference of nonlinear effects for these branches. While the nonlinear resonant interaction of the lowfrequency modes in the CFM branch leads to energy capture in some domains of the CNT, the same interaction in the RBM branch does not demonstrate any tendency for energy localization. The reason lies in the distinction in the nonlinear terms in the equations of motion. While CFMs are characterized by soft polynomial nonlinearity, RBM dynamics is characterized by hard gradient nonlinearity. Moreover, in contrast to the CFM, the importance of nonlinearity in the case of RBM oscillations decreases as the length to radius ratio increases. Numerical integration of the equations of thin shell theory confirms the results of the analytical study.
2017
 Numerical simulation and experimental validation of normal strain distribution and pitting phenomenon in industrial gears
[Relazione in Atti di Convegno]
Strozzi, M.; Barbieri, M.; Zippo, A.; Pellicano, F.
abstract
In this paper, the normal strain distribution and pitting phenomenon on gears are investigated by means of numerical finite element analyses and experimental activities. In the first part of the paper, results of experimental tests for the investigation of the pitting phenomenon on gears are reported. These durability tests are made at a specific nominal load and far from the resonance. The experimental data are collected periodically from two triaxial accelerometers placed on the gear shafts. After a short time, a visible pitting phenomenon arises on the gear teeth, where the contact pattern is perfectly centered (due to the high lead crown imposed on the teeth) and the wear pattern is consistent with FE simulations. In the second part of the paper, numerical finite element studies on the normal strain distribution at the base of the gear teeth during the contact are reported. These analyses are made at the same nominal load of the previous pitting analyses and at very low rotational speed (static analyses). A peak of normal strain at the base of the contact tooth is found around the contact time, preceded and followed by a low constant value of normal strain. The numerical results are validated by comparisons with experimental tests carried out in the same operating conditions and placing strain gauges at the tooth base of the gears.
2017
 Numerical study on nonlinear vibrations, energy exchange and resonant interactions in single walled carbon nanotubes
[Relazione in Atti di Convegno]
Strozzi, M.; Barbieri, M.; Zippo, A.; Pellicano, F.
abstract
In this paper, the nonlinear vibrations, energy exchange and resonant interactions of singlewalled carbon nanotubes (SWNTs) are investigated. The SandersKoiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The circumferential flexural modes (CFMs), radial breathing modes (RBMs) and beamlike modes (BLMs) are studied. A numerical model of the SWNT dynamics is proposed. The three displacement fields are expanded in the nonlinear field by using approximate linear eigenfunctions. An energy method based on the Lagrange equations is used to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is solved using the implicit RungeKutta numerical method. The nonlinear energy exchange along the SWNT axis is analysed for different initial excitation amplitudes. The resonant interactions between CFMs, RBMs and BLMs are investigated. The transition from energy beating to energy localization in the nonlinear field is studied.
2017
 Vibration Localization of Imperfect Circular Cylindrical Shells
[Relazione in Atti di Convegno]
Pellicano, Francesco; Zippo, Antonio; Barbieri, Marco; Strozzi, Matteo
abstract
2017
 Vibration of functionally graded cylindrical shells
[Relazione in Atti di Convegno]
Zippo, A.; Pellicano, F.; Barbieri, M.; Strozzi, M.
abstract
Functionally gradient materials (FGMs) have attracted a growing interest as advanced structural materials because of their heatresistance properties. In this paper, an experimental study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of Polyethylene terephthalate (PET) is presented: to obtain functional gradient proprieties the PET shell had been exposed at a thermal temperature gradient in the range of its glass transition temperature of 79°C. The setting up of the experiment is explained and deeply described along with the thermal characterisation of the specimen. The linear and the nonlinear dynamic behaviour have been investigated. The shell behaviour is also investigated by means of a finite element model, in order to enhance the comprehension of experimental results.
2016
 Active vibration control of a composite plate
[Relazione in Atti di Convegno]
Zippo, A.; Barbieri, M.; Pellicano, F.; Strozzi, M.
abstract
A new active vibration control methodology is proposed and experimentally tested. The technique is applied to a honeycomb panel having a carbonfiber reinforced polymer (CFRP) outer skins and a polymerpaper core, subjected to an orthogonal disturbance, due to an electrodynamics exciter. The control is carried out by means of Macro Fibre Composite (MFC) actuators and sensors. MFC parches consist of rectangular piezoceramic rods sandwiched between layers of adhesive, electrodes and polyamide film. The MFC actuators and sensors are controlled by a programmable digital dSPACER controller board. The control algorithm proposed in this paper is based on the Positive Position Feedback (PPF) technique, Single Input  Single Output, MultiSISO and Multi Input Multi Output controls are applied in order to control the first four normal modes. The control appears to be robust and efficient in reducing vibration in linear (small am plitude) and nonlinear (large amplitude) vibrations regimes, although the structure under investigation exhibits a relativity high modal density, i.e. four resonances in a range of about 100Hz. The control strategy allows to effectively control each resonance both individually or simultaneously.
2016
 Dynamic imbalance of high speed planetary gears
[Relazione in Atti di Convegno]
Masoumi, A.; Barbieri, M.; Pellicano, F.; Zippo, A.; Strozzi, M.
abstract
A nonlinear 2D lumped mass model of planetary gear system with time varying mesh stiffness, bearing compliance and nonsmooth nonlinearity due to the backlash is taken into account. The time varying meshing stiffness is evaluated by means of a nonlinear finite element model, through an accurate evaluation of global and local tooth deformation. Nonlinear dynamic behaviour of the system is analyzed over a reasonable range of rotation speed and torque. Possibility of occurrences of different dynamic phenomena and instability of the system with respect to bearing compliance and operating parameters is evaluated as well. Bifurcation diagrams are extracted as well and for specific regimes, the nonlinear scenario of system is discussed using the spectra, phase portraits and Poincare maps. Periodic, quasiperiodic and chaotic regimes are found and discussed with respect to system parameters. The possibility of dynamic imbalance of equally spaced planetary gears in presence of chaotic regimes is discussed. Such imbalance may lead to unexpected high level stresses on bearings and gears.
2016
 Experimental investigation of dynamic behaviour of precompressed circular cylindrical shell
[Relazione in Atti di Convegno]
Zippo, Antonio; Pellicano, Francesco; Barbieri, Marco; Strozzi, Matteo
abstract
Circular cylindrical shells are very efficient structures that have many applications and plays as key elements in several engineering fields. Shells usually exhibit a complicated dynamic behaviours because the curvature will effectively couple the flexural and inplane deformations together as the three displacement fields simultaneously appear in each of the governing partial differential equations and boundary conditions. Therefore, it is understandable that the axial constraints can have direct effects on a predominantly radial modes. For instance, it has been shown that the natural frequencies for the circumferential modes of a simply supported shell can be noticeably modified by the constraints applied in the axial direction. In this paper the results of experimental tests on precompressed circular cylindrical shell will be presented: different combinations of preload and harmonic external axial load have been tested but for brevity only few results are shown.
2016
 Experiments on shells under base excitation
[Articolo su rivista]
Pellicano, Francesco; Barbieri, Marco; Zippo, Antonio; Strozzi, Matteo
abstract
The aim of the present paper is a deep experimental investigation of the nonlinear dynamics of circular cylindrical shells. The specific problem regards the response of circular cylindrical shells subjected to base excitation. The shells are mounted on a shaking table that furnishes a vertical vibration parallel to the cylinder axis; a heavy rigid disk is mounted on the top of the shells. The base vibration induces a rigid body motion, which mainly causes huge inertia forces exerted by the top disk to the shell. Inplane stresses due to the aforementioned inertias give rise to impressively large vibration on the shell. An extremely violent dynamic phenomenon suddenly appears as the excitation frequency varies up and down close to the linear resonant frequency of the first axisymmetric mode. The dynamics are deeply investigated by varying excitation level and frequency. Moreover, in order to generalise the investigation, two different geometries are analysed. The paper furnishes a complete dynamic scenario by means of: (i) amplitude frequency diagrams, (ii) bifurcation diagrams, (iii) time histories and spectra, (iv) phase portraits and Poincaré maps. It is to be stressed that all the results presented here are experimental.
2016
 Linear vibrations of multiwalled carbon nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco; Barbieri, Marco; Zippo, Antonio
abstract
In this paper, the linear vibrations of MultiWalled Carbon Nanotubes (MWNTs) are analysed. A multiple elastic shell model is considered. The shell dynamics is studied in the framework of the SandersKoiter shell theory. The van der Waals (vdW) interaction between two layers of the MWNT is modelled by a radiusdependent function. The shell deformation is described in terms of longitudinal, circumferential and radial displacements. Simply supported, clamped and free boundary conditions are considered. The three displacement fields are expanded by means of a double mixed series based on Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. The present model is validated in linear field by means of data derived from the literature. This study is focused on determining the effect of the geometry and boundary conditions on the natural frequencies of the MWNTs.
2016
 Modelling and Testing Techniques for Gear boxes Analysis and Optimization
[Relazione in Atti di Convegno]
Andrisano, A. O.; Pellicano, Francesco; Barbieri, Marco; Zippo, Antonio; Strozzi, Matteo
abstract
2016
 Natural Frequencies of TripleWalled Carbon Nanotubes
[Relazione in Atti di Convegno]
Andrisano, A. O.; Pellicano, Francesco; Strozzi, Matteo
abstract
2016
 Nonlinear Dynamics of PreCompressed Circular Cylindrical Shell Under Axial Harmonic Load: Experiments
[Relazione in Atti di Convegno]
Pellicano, Francesco; Zippo, Antonio; Barbieri, Marco; Strozzi, Matteo
abstract
2016
 Nonlinear dynamics of SWNTs. Energy beating and localization
[Abstract in Atti di Convegno]
Strozzi, Matteo; Manevitch, L. I.; Smirnov, V. V.; Pellicano, Francesco
abstract
2016
 Nonlinear optical vibrations of singlewalled carbon nanotubes. 1. Energy exchange and localization of lowfrequency oscillations
[Articolo su rivista]
Smirnov, V. V.; Manevitch, L. I.; Strozzi, M.; Pellicano, F.
abstract
We present the results of analytical study and molecular dynamics simulation of low energy nonlinear nonstationary dynamics of singlewalled carbon nanotubes (CNTs). New phenomena of intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT are analytically predicted in the framework of the continuum shell theory. Their origin is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the molecular dynamics simulation of simply supported CNTs.
2016
 Nonlinear vibrations and energy exchange of singlewalled carbon nanotubes. Circumferential flexural modes
[Articolo su rivista]
Strozzi, Matteo; Smirnov, Valeri V.; Manevitch, Leonid I.; Milani, Massimo; Pellicano, Francesco
abstract
In this paper, the nonlinear vibrations and energy exchange of singlewalled carbon nanotubes (SWNTs) are studied. The SandersâKoiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are considered. The circumferential flexural modes (CFMs) are investigated. Two different approaches based on numerical and analytical models are compared. In the numerical model, an energy method based on the Lagrange equations is used to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is solved by using the implicit RungeâKutta numerical method. In the analytical model, a reduced form of the SandersâKoiter theory assuming small circumferential and tangential shear deformations is used to get the nonlinear ordinary differential equations of motion, which are solved by using the multiple scales analytical method. The transition from energy beating to energy localization in the nonlinear field is studied. The effect of the aspect ratio on the analytical and numerical values of the nonlinear energy localization threshold for different boundary conditions is investigated.
2016
 Pitting and stress analysis of gears: A numerical and experimental study
[Relazione in Atti di Convegno]
Strozzi, M.; Barbieri, M.; Pellicano, F.; Zippo, A.
abstract
In this paper, the pitting phenomenon and stress distribution of gears are investigated by means of experimental activities and numerical finite element analyses. In the first part, results of experimental accelerated endurance tests for the investigation of the pitting phenomenon of gears are reported. These durability tests are made at a specific nominal load and far from the resonance. After a short time, a visible pitting phenomenon arises. In the second part, finite element numerical analyses for the evaluation of gear stresses are listed. The numerical analyses start from stressvibration correlations and dynamic factors obtained by a 2dof dynamic model; these results are used in the dynamic FEM simulations in order to calculate the maximum normal stress and the contact pressure on the contact tooth of the pinion vs. vibration amplitude for different dynamic factors.
2015
 Beating phenomenon and energy localization in SingleWalled Carbon Nanotubes
[Abstract in Atti di Convegno]
Strozzi, Matteo; Manevitch, Leonid I.; Pellicano, Francesco; Barbieri, Marco; Zippo, Antonio
abstract
In this paper, the lowfrequency nonlinear oscillations and energy localization of SingleWalled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the SandersKoiter nonlinear shell theory. The circumferential flexure vibration modes (CFMs) are considered. Simply supported, clamped and free boundary conditions are analysed. Two different approaches are compared, based on numerical and analytical models. The numerical model uses a double mixed series expansion for the displacement fields based on the Chebyshev polynomials and harmonic functions. The Lagrange equations are considered to obtain a set of nonlinear ordinary differential equations of motion which are solved using the implicit RungeKutta numerical method. The analytical model considers a reduced form of the shell theory assuming small circumferential and tangential shear deformations. The Galerkin procedure is used to get the nonlinear ordinary differential equations of motion, which are then solved using the multiple scales analytical method.
The natural frequencies of SWNTs obtained by considering the analytical and numerical approaches are compared for different boundary conditions. A convergence analysis in the nonlinear field is carried out for the numerical method in order to select the correct number of the axisymmetric and asymmetric modes providing the actual localization threshold. The effect of the aspect ratio on the analytical and numerical values of the localization threshold for SWNTs with different boundary conditions is investigated in the nonlinear field.
2015
 Dynamic modelling of gear pairs
[Relazione in Atti di Convegno]
Barbieri, Marco; Zippo, Antonio; Strozzi, Matteo; Serafini, Lorenzo; Pellicano, Francesco; Bonori, Giorgio
abstract
A clear understanding of the dynamics of gear pairs is important for many reasons. First of all, gear vibration is a main source of noise in gearboxes and vehicle trasmissions, secondly the torsional elasticity of the gear trasmission can produce relevant amplification of the contact force, and thus of the gear stress. Furthermore, gear vibrations are a useful parameter for gear monitoring and prognostics. In the present work, an overview of the models used to describe the dynamic behaviour of gear pairs will be presented, along with a comparison between a dynamic finite element model and different lumped parameter approaches. A correlation between the vibration transmitted to the gearbox, and thus easily measurable in a real application, and the local stresses in the gear pair will be drawn. The proposed approach is suitable to describe the effect of localized defects on the gear pair, such as tooth root cracks and pitted profiles, on the signal measured on a gearbox.
2015
 Dynamics and Stability of Carbon Nanotubes
[Abstract in Atti di Convegno]
Strozzi, Matteo; Barbieri, Marco; Zippo, Antonio; Pellicano, Francesco
abstract
The lowfrequency oscillations and energy localization of SingleWalled Carbon Nanotubes (SWNTs) are studied in the framework of the SandersKoiter shell theory. The circumferential flexure modes (CFMs) are analysed. Simply supported, clamped and free boundary conditions are considered. Two different approaches are proposed, based on numerical and analytical models. The numerical model uses in the linear analysis a double mixed series expansion for the displacement fields based on Chebyshev polynomials and harmonic functions. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations, which is solved by the RungeKutta numerical method. The analytical model considers a reduced version of the SandersKoiter shell theory obtained by assuming small circumferential and tangential shear deformations. These two assumptions allow to condense the longitudinal and circumferential displacement fields into the radial one. A nonlinear fourthorder partial differential equation for the radial displacement field is derived, which allows to calculate the natural frequencies and to estimate the nonlinearity effect. An analytical solution of this equation is obtained by the multiple scales method. The previous models are validated in linear field by means of comparisons with experiments, molecular dynamics simulations and finite element analyses retrieved from the literature. The concept of energy localization in SWNTs is introduced, which is a strongly nonlinear phenomenon. The lowfrequency nonlinear oscillations of the SWNTs become localized ones if the intensity of the initial excitation exceeds some threshold which depends on the SWNTs length. This localization results from the resonant interaction of the zoneboundary and nearest nonlinear normal modes leading to the confinement of the vibration energy in one part of the system. The value of the initial excitation corresponding to this energy confinement is referred to as energy localization threshold. The effect of the aspect ratio on the analytical and numerical values of the energy localization threshold is investigated; different boundary conditions are considered.
2015
 Energy localization in carbon nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Smirnov, Valeri V.; Manevitch, Leonid I.; Pellicano, Francesco; Shepelev, Denis S.
abstract
In this paper, the energy localization phenomena in lowfrequency nonlinear oscillations of singlewalled carbon nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the SandersKoiter shell theory. Simply supported and free boundary conditions are considered. The effect of the aspect ratio on the analytical and numerical values of the localization threshold is investigated in nonlinear field.
2015
 Nonlinear dynamics of carbon nanotubes
[Relazione in Atti di Convegno]
Andrisano, A. O.; Pellicano, F.; Strozzi, M.
abstract
2015
 Nonlinear oscillations of carbon nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco; Barbieri, Marco; Zippo, Antonio; Manevitch, Leonid I.
abstract
In this paper, the lowfrequency nonlinear oscillations and energy localizations of SingleWalled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied within the framework of the SandersKoiter thin shell theory. The circumferential flexure vibration modes (CFMs) are considered. Simply supported boundary conditions are investigated. Two different approaches are compared, based on numerical and analytical models. The numerical model uses a double series expansion for the displacement fields based on the Chebyshev polynomials and harmonic functions. The Lagrange equations are considered to obtain a set of nonlinear ordinary differential equations of motion which are solved using the implicit RungeKutta numerical method. The analytical model considers a reduced form of the shell theory assuming small circumferential and tangential shear deformations. The Galerkin procedure is used to get the nonlinear ordinary differential equations of motion which are solved using the multiple scales analytical method. The natural frequencies obtained by considering the two approaches are compared in linear field. The effect of the aspect ratio on the analytical and numerical values of the localization threshold is investigated in nonlinear field.
2014
 Eigenfrequencies and vibration modes of carbon nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Manevitch, Leonid I.; Smirnov, Valeri V.; Shepelev, Denis S.; Pellicano, Francesco
abstract
In 1991 Iijima discovered Carbon Nanotubes, he synthesised molecular carbon structures in the form of fullerenes and then reported the preparation of a new type of finite carbon structure consisting of needlelike tubes, the carbon nanotubes, described as helical microtubules of graphitic carbon.
Examples of applications of Carbon Nanotubes (CNTs) can be found in ultrahigh frequency nanomechanical resonators, in a large number of nanoelectromechanical devices such as sensors, oscillators, charge detectors and field emission devices. The reduction of the size and the increment of the stiffness of a resonator magnify its resonant frequencies and reduce its energy consumption, improving its sensitivity.
The modal analysis of carbon nanotubes is important because it allows to obtain the resonant frequencies and mode shapes, which influence the mechanical and electronic properties of the nanotube resonators.
A large number of experiments and atomistic simulations were conducted both on singlewalled (SWNTs) and multiwalled carbon nanotubes (MWNTs).
The present work is concerned with the analysis of lowfrequency linear vibrations of SWNTs: two approaches are presented: a fully analytical method based on a simplified theory and a semianalytical method based on the theory of thin walled shells.
The semianalytical approach (shortly called “numerical approach”) is based on the SandersKoiter shell theory and the RayleighRitz numerical procedure. The nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields, which are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the circumferential variable. The RayleighRitz method is then applied to obtain numerically approximate natural frequencies and mode shapes.
The second approach is based on a reduced version of the SandersKoiter shell theory, obtained by assuming small ring and tangential shear deformations. These assumptions allow to condense both the longitudinal and the circumferential displacement fields. A fourthorder partial differential equation for the radial displacement field is derived. Eigenfunctions are formally obtained analytically, then the numerical solution of the dispersion equation gives the natural frequencies and the corresponding normal modes.
The methods are fully validated by comparing the natural frequencies of the SWNTs with data available in literature, namely: experiments, molecular dynamics simulations and finite element analyses. A comparison between the results of the numerical and analytical approach is carried out in order to check the accuracy of the last one.
It is worthwhile to stress that the analytical model allows to obtain results with very low computational effort. On the other hand the numerical approach is able to handle the most realistic boundary conditions of SWNTs (freefree, clampedfree) with extreme accuracy. Both methods are suitable for a forthcoming extension to multiwalled nanotubes and nonlinear vibrations.
2014
 Lowfrequency linear vibrations of singlewalled carbon nanotubes: Analytical and numerical models
[Articolo su rivista]
Strozzi, Matteo; L. I., Manevitch; Pellicano, Francesco; V. V., Smirnov; D. S., Shepelev
abstract
Lowfrequency vibrations of singlewalled carbon nanotubes with various boundary conditions are considered in the framework of the Sanders–Koiter thin shell theory. Two methods of analysis are proposed. The first approach is based on the Rayleigh–Ritz method, a double series expansion in terms of Chebyshev polynomials and harmonic functions is considered for the displacement fields; free and clamped edges are analysed. This approach is partially numerical. The second approach is based on the same thin shell theory, but the goal is to obtain an analytical solution useful for future developments in nonlinear fields; the Sanders–Koiter equations are strongly simplified neglecting inplane circumferential normal strains and tangential shear strains. The model is fully validated by means of comparisons with experiments, molecular dynamics data and finite element analyses obtained from the literature. Several types of nanotubes are considered in detail by varying aspect ratio, chirality and boundary conditions. The analyses are carried out for a wide range of frequency spectrum. The strength and weakness of the proposed approaches are shown; in particular, the model shows great accuracy even though it requires minimal computational effort.
2014
 Nonlinear Dynamics of SingleWalled Carbon Nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Manevitch, Leonid I.; Pellicano, Francesco
abstract
The nonlinear dynamics of SingleWalled Carbon Nanotubes is studied. The SandersKoiter elastic shell theory is applied. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. Free boundary conditions are considered. The total energy distribution of the system is studied by considering the combinations of different vibration modes. The effect of the companion mode participation on the energy distribution is analysed.
2014
 Nonlinear oscillations and energy localization in carbon nanotubes
[Relazione in Atti di Convegno]
Andrisano, Angelo Oreste; Manevitch, Leonid I.; Pellicano, Francesco; Strozzi, Matteo
abstract
In this paper, the lowfrequency nonlinear oscillations and energy localizations of SingleWalled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied within the framework of the SandersKoiter thin shell theory. The circumferential flexure vibration modes (CFMs) are considered. Simply supported boundary conditions are investigated. Two different approaches are compared, based on numerical and analytical models. The numerical model uses a double series expansion for the displacement fields based on the Chebyshev polynomials and harmonic functions. The Lagrange equations are considered to obtain a set of nonlinear ordinary differential equations of motion which are solved using the implicit RungeKutta numerical method. The analytical model considers a reduced form of the shell theory assuming small circumferential and tangential shear deformations. The Galerkin procedure is used to get the nonlinear ordinary differential equations of motion which are solved using the multiple scales analytical method. The natural frequencies obtained by considering the two approaches are compared in linear field. The effect of the aspect ratio on the analytic and numerical values of the localization threshold is investigated in nonlinear field.
2013
 Nonlinear dynamics of SingleWalled Carbon Nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Manevitch, Leonid I.; Pellicano, Francesco
abstract
The nonlinear vibrations of SingleWalled Carbon Nanotubes are analysed. The SandersKoiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The total energy distribution of the shell is studied by considering combinations of different vibration modes. The effect of the conjugate modes is analysed.
2013
 Nonlinear vibrations and energy conservation of SingleWalled Carbon Nanotubes
[Relazione in Atti di Convegno]
Zippo, Antonio; Strozzi, Matteo; Manevitch, Leonid I.; Pellicano, Francesco; Barbieri, Marco
abstract
The nonlinear vibrations of SingleWalled Carbon Nanotubes are analysed. The SandersKoiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The RayleighRitz method is applied in order to obtain approximate natural frequencies and mode shapes. Free boundary conditions are analysed. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions; an energy approach based on the Lagrange equations is considered in order to reduce the nonlinear partial differential equations to a set of nonlinear ordinary differential equations. Nondimensional parameters are considered. The total energy conservation of the system is verified by considering the combinations of different vibration modes. The effect of the companion mode participation on the nonlinear vibrations of the carbon nanotube is analysed.
2013
 Nonlinear vibrations and energy distribution of SingleWalled Carbon Nanotubes
[Relazione in Atti di Convegno]
Strozzi, Matteo; Manevitch, Leonid I.; Pellicano, Francesco
abstract
The nonlinear vibrations of SingleWalled Carbon Nanotubes are analysed. The SandersKoiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The RayleighRitz method is applied in order to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The energy distribution of the system is studied by considering combinations of different vibration modes. The effect of the conjugate modes participation on the energy distribution is analysed.
2013
 Nonlinear vibrations and energy distribution of carbon nanotubes
[Relazione in Atti di Convegno]
Andrisano, Angelo Oreste; Manevitch, Leonid I.; Pellicano, Francesco; Strozzi, Matteo
abstract
The nonlinear vibrations of SingleWalled Carbon Nanotubes are analysed. The SandersKoiter thin shell theory is applied in order to obtain the elastic strain and kinetic energy. The carbon nanotube deformation is described in terms of axial, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is then considered to obtain a set of nonlinear ordinary differential equations. The total energy distribution of the shell is studied by considering combinations of different vibration modes. The effect of the conjugate modes is analysed.
2013
 Nonlinear vibrations and energy distribution of carbon nanotubes
[Capitolo/Saggio]
Strozzi, Matteo; Manevitch, Leonid I.; Pellicano, Francesco; Smirnov, Valeri V.; Shepelev, Denis S.
abstract
The nonlinear vibrations of SingleWalled Carbon Nanotubes are analysed. The SandersKoiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The total energy distribution of the shell is studied by considering combinations of different vibration modes. The effect of the conjugate modes participation is analysed.
2013
 Nonlinear vibrations of functionally graded circular cylindrical shells subjected to harmonic external load
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco
abstract
The nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. The SandersKoiter theory is applied in order to model the nonlinear dynamics of the system. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered. Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to a harmonic external load. A convergence analysis is carried out to obtain the correct number of axisymmetric and asymmetric modes describing the actual nonlinear behaviour. The influence of the material distribution on the nonlinear response is analysed considering different configurations and volume fractions of the constituent materials. The effect of the companion mode participation on the nonlinear response of the shell is analysed.
2013
 Nonlinear vibrations of functionally graded cylindrical shells
[Articolo su rivista]
Strozzi, Matteo; Pellicano, Francesco
abstract
In this paper, the nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary condi tions are considered. The displacement fields are expanded by means of a double mixed series based on Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered; this allows the travelling wave response of the shell to be modelled. The model is validated in the linear field by means of data retrieved from the pertinent literature. Numerical analyses are carried out in order to characterise the nonlinear response when the shell is subjected to a harmonic external load; a convergence analysis is carried out by considering a variety of axisymmetric and asymmetric modes. The present study is focused on determining the nonlinear character of the shell dynamics as the geometry (thickness, radius, length) and material properties (constituent volume fractions and configurations of the constituent materials) vary.
2013
 Vibrations of Carbon Nanotubes: nonlinear models and energy distribution
[Relazione in Atti di Convegno]
Pellicano, Francesco; Strozzi, Matteo; Manevitch, Leonid I.
abstract
Vibrations of SingleWalled Carbon Nanotubes for various boundary conditions are considered in the framework of the SandersKoiter thin shell theory. A double series expansion of displacement fields, based on the Chebyshev orthogonal polynomials and harmonic functions, is used to analyse numerically the natural frequencies of shells having free or clamped edges. A reduced form of the SandersKoiter theory is developed by assuming small circumferential and shear deformations; such approach allows to determine an analytical solution for the natural frequencies. The numerical model is validated with the results of molecular dynamics and finite element analyses present in literature. The analytical model is validated by means of comparisons with the numerical approach. Nonlinear vibrations and energy distribution of carbon nanotubes are then considered.
2012
 Experimental Study on Prestressed Circular Cylindrical Shell
[Relazione in Atti di Convegno]
Zippo, Antonio; Barbieri, Marco; Strozzi, Matteo; Errede, Vito; Pellicano, Francesco
abstract
In this paper an experimental study on circular cylindrical shells subjected to axial compressive and periodic loads is presented. Even though many researchers have extensively studied nonlinear vibrations of cylindrical shells, experimental studies are rather limited in number. The experimental setup is explained and deeply described along with the analysis of preliminary results. The linear and the nonlinear dynamic behavior associated with a combined effect of compressive static and a periodic axial load have been investigated for different combinations of loads; moreover, a non stationary response of the structure has been observed close to one of the resonances. The linear shell behavior is also investigated by means of a finite element model, in order to enhance the comprehension of experimental results.
2012
 Linear and nonlinear dynamics of a circular cylindrical shell under static and periodic axial load
[Relazione in Atti di Convegno]
Zippo, Antonio; Barbieri, Marco; Strozzi, Matteo; Errede, Vito; Pellicano, Francesco
abstract
In this paper an experimental study on circular cylindrical shells subjected to axial compres sive and periodic loads is presented. The setting of the experiment is explained and deeply described along with a complete analysis of the results. The linear and the nonlinear dynamic behaviour associated with a combined effect of compressive static and a periodic axial load has been considered and a chaotic response of the structure has been observed close to the resonance. The linear shell behaviour is also investigated by means of a theoretical and finite element model, in order to enhance the comprehension of experimental results, i.e. the natural frequencies of the system and their ratios.
2012
 Nonlinear vibrations of functionally graded cylindrical shells: Effect of companion mode participation
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco
abstract
In this paper, the nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analyzed. The SandersKoiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered, allowing for the travellingwave response of the shell. Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to an harmonic external load. A convergence analysis is carried out to obtain the correct number of axisymmetric and asymmetric modes describing the actual nonlinear behavior of the shells. The effect of the geometry on the nonlinear vibrations of the shells is analyzed, and a comparison of nonlinear amplitudefrequency curves of cylindrical shells with different geometries is carried out. The influence of the companion mode participation on the nonlinear response of the shells is analyzed; frequencyresponse curves with companion mode participation (i.e. the actual response of the shell) are obtained. The present model is validated in the linear field (natural frequencies) by means of data present in the literature.
2012
 Nonlinear vibrations of functionally graded cylindrical shells: Effect of the geometry
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco; Zippo, Antonio
abstract
In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The SandersKoiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. In the linear analysis, after spatial discretization, mass and stiff matrices are computed, natural frequencies and mode shapes of the shell are obtained. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions obtained by the linear analysis; specific modes are selected. The Lagrange equations reduce nonlinear partial differential equations to a set of ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response of the shell. A convergence analysis is carried out to determine the correct number of the modes to be used. The analysis is focused on determining the nonlinear character of the response as the geometry of the shell varies.
2012
 Nonlinear vibrations of functionally graded cylindrical shells: effect of the companion mode participation
[Relazione in Atti di Convegno]
Andrisano, Angelo Oreste; Pellicano, Francesco; Strozzi, Matteo
abstract
In this paper, the effect of the companion mode participation on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The SandersKoiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered. Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to an harmonic external load. A convergence analysis is carried out by considering a different number of axisymmetric and asymmetric modes. The present study is focused on modelling the nonlinear travellingwave response of the shell in the circumferential direction with the companion mode participation.
2011
 Effect of the boundary conditions on the vibrations of functionally graded shells
[Relazione in Atti di Convegno]
Andrisano, Angelo Oreste; Pellicano, Francesco; Strozzi, Matteo
abstract
In this paper, the effect of the boundary conditions on the nonlinear vibration of functionally graded circular cylindrical shells is analyzed. The SandersKoiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to an harmonic external load; different geometries and material distributions are considered. A convergence analysis is carried out in order to determine the correct number of the modes to be used; the role of the axisymmetric and asymmetric modes is carefully analyzed. The effect of the geometry on the nonlinear response is investigated; i.e. thickness and radius are varied; simply supported, clampedclamped and freefree shells are considered. The effect of the constituent volume fractions and the configurations of the constituent materials on the natural frequencies and nonlinear response are studied.
2011
 Effect of the geometry on the nonlinear vibrations of functionally graded cylindrical shells
[Relazione in Atti di Convegno]
Pellicano, Francesco; Strozzi, Matteo; Zippo, Antonio
abstract
In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The SandersKoiter theory is applied to model nonlinear dynamics of the system in the case of finite amplitude of vibration. Shell deformation is described in terms of longitudinal, circumferential and radial displacement fields; the theory considers geometric nonlinearities due to the large amplitude of vibration. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. Both driven and companion modes are considered, allowing for the travellingwave response of the shell. The functionally graded material is made of a uniform distribution of stainless steel and nickel, the material properties are graded in the thickness direction, according to a volume fraction powerlaw distribution.The first step of the procedure is the linear analysis, i.e. after spatial discretization mass and stiff matrices are computed and natural frequencies and mode shapes of the shell are obtained, the latter are represented by analytical continuous functions defined over all the shell domain. In the nonlinear model, the shell is subjected to an external harmonic radial excitation, close to the resonance of a shell mode, it induces nonlinear behaviors due to large amplitude of vibration. The three displacement fields are reexpanded by using approximate eigenfunctions, which were obtained by the linear analysis; specific modes are selected. An energy approach based on the Lagrange equations is considered, in order to reduce the nonlinear partial differential equations to a set of ordinary differential equations.Numerical analyses are carried out in order characterize the nonlinear response, considering different geometries and material distribution. A convergence analysis is carried out in order to determine the correct number of the modes to be used; the role of the axisymmetric and asymmetric modes carefully analyzed. The analysis is focused on determining the nonlinear character of the response as the geometry (thickness, radius, length) and material properties (powerlaw exponent and configurations of the constituent materials) vary; in particular, the effect of the constituent volume fractions and the configurations of the constituent materials on the natural frequencies and nonlinear response are studied.Results are validated using data available in literature, i.e. linear natural frequencies.
2011
 Nonlinear vibration of functionally graded cylindrical shells: effect of constituent volume fractions and configurations
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco
abstract
In this paper, the nonlinear vibration of functionally graded (FGM) cylindrical shells under different constituent volume fractions and configurations is analyzed. The SandersKoiter theory is applied to model nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. Displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. Both driven and companion modes are also considered, allowing for the travellingwave response of the shell. The functionally graded material considered is made of stainless steel and nickel, properties are graded in the thickness direction according to a real volume fraction powerlaw distribution. In the nonlinear model, shells are subjected to an external radial excitation. Nonlinear vibrations due to large amplitude of excitation are considered. Specific modes are selected in the modal expansions; a dynamical nonlinear system is then obtained. Lagrange equations are used to reduce nonlinear partial differential equations to a set of ordinary differential equations, from the potential and kinetic energies, and the virtual work of the external forces. Different geometries are analyzed; amplitudefrequency curves are obtained. Convergence tests are carried out considering a different number of asymmetric and axisymmetric modes. The present model is validated in linear field (natural frequencies) by means of data present in the literature.
2011
 Nonlinear vibrations of functionally graded circular cylindrical shells
[Relazione in Atti di Convegno]
Strozzi, Matteo; Pellicano, Francesco; Zippo, Antonio
abstract
In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded cy lindrical shells is analyzed. The SandersKoiter theory is applied to model nonlinear dynamics of the system in the case of finite amplitude of vibration. Shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to an harmonic external load; different geometries and material distribu tions are considered. A convergence analysis is carried out in order to determine the correct number of the modes to be used; the role of the axisymmetric and asymmetric modes is carefully analyzed. The analysis is focused on determining the nonlinear character of the response as the geometry (thickness, radius, length) and material properties (powerlaw exponent N and configurations of the constituent materials) vary. The effect of the constituent volume fractions and the configurations of the constituent materials on the natural frequencies and nonlinear response are studied.