Matteo Strozzi - Academia.edu (original) (raw)

Papers by Matteo Strozzi

In this paper, the nonlinear vibrations of functionally graded (FGM) circular cylindrical shells ... more 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 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; this allows the travellingwave 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.

The nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. T... more The nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. The Sanders-Koiter 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.

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elast... more The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elast... more The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM... more In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter 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 travelling-wave 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 power-law 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 re-expanded 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 (power-law exponent N 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.

In this paper, the energy localization phenomena in low-frequency nonlinear oscillations of singl... more In this paper, the energy localization phenomena in low-frequency nonlinear oscillations of single-walled carbon nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the Sanders-Koiter 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 the nonlinear formulation.

Vibrations of Single-Walled Carbon Nanotubes for various boundary conditions are considered in th... more Vibrations of Single-Walled Carbon Nanotubes for various boundary conditions are considered in the framework of the Sanders-Koiter 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 Sanders-Koiter 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.

In this paper, the low-frequency nonlinear oscillations and energy localizations of Single-Walled... more In this paper, the low-frequency nonlinear oscillations and energy localizations of Single-Walled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied within the framework of the Sanders-Koiter 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 Runge-Kutta numerical method. The analytical model considers a reduced form of the shell theory assuming small circumferential and tangential shear deformations. The Galerkin pro-cedure 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 analyti-cal and numerical values of the localization threshold is investigated in nonlinear field.

In this paper, the effect of the companion mode participation on the nonlinear vibrations of func... more In this paper, the effect of the companion mode participation on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is ap-plied to model the nonlinear dynamics of the system in the case of finite amplitude of vibra-tion. 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 charac-terize the nonlinear response when the shell is subjected to an harmonic external load. A con-vergence analysis is carried out by considering a different number of axisymmetric and asymmetric modes. The present study is focused on modelling the nonlinear travelling-wave response of the shell in the circumferential direction with the companion mode participation.

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded cy-l... more In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded cy-lindrical shells is analyzed. The Sanders-Koiter 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 (power-law 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.

In this paper, the low-frequency nonlinear oscillations and energy localization of Single-Walled ... more In this paper, the low-frequency nonlinear oscillations and energy localization of Single-Walled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the Sanders-Koiter 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 Runge-Kutta 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.

The nonlinear dynamics of Single-Walled Carbon Nanotubes is analysed. The Sanders-Koiter elastic ... more The nonlinear dynamics of Single-Walled Carbon Nanotubes is analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

The present work is concerned with the analysis of low-frequency linear vibrations of SWNTs: two ... more The present work is concerned with the analysis of low-frequency linear vibrations of SWNTs: two approaches are presented: a fully analytical method based on a simplified theory and a semi-analytical method based on the theory of thin shells.
The semi-analytical approach (shortly called “numerical approach”) is based on the Sanders-Koiter shell theory and the Rayleigh-Ritz 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. The Rayleigh-Ritz method is then applied to obtain numerically approximate natural frequencies and mode shapes.
The second approach is based on a reduced version of the Sanders-Koiter 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 fourth-order 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 (free-free, clamped-free) with extreme accuracy. Both methods are suitable for a forthcoming extension to multi-walled nanotubes and nonlinear vibrations.

In this paper an experimental study on circular cylindrical shells subjected to axial compressive... more 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.

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM... more In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. 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 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 re-expanded 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.

The low-frequency oscillations and energy localization of Single-Walled Carbon Nanotubes (SWNTs) ... more The low-frequency oscillations and energy localization of Single-Walled Carbon Nanotubes (SWNTs) are studied in the framework of the Sanders-Koiter 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 [1] uses in the linear analysis a double mixed series expansion for the displacement fields based on Chebyshev polynomials and harmonic functions. The Rayleigh-Ritz method is applied to obtain approximate natural frequencies and mode shapes. In the nonlinear analysis, the three displacement fields are re-expanded 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 Runge-Kutta numerical method.
The analytical model [2] considers a reduced version of the Sanders-Koiter 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 fourth-order 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 [3].
The concept of energy localization in SWNTs is introduced, which is a strongly nonlinear phenomenon. The low-frequency 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 zone-boundary and nearest nonlinear normal modes leading to the confinement of the vibration energy in one part of the system [4].
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.

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elast... more The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

In this paper, the nonlinear vibrations of functionally graded (FGM) cylindrical shells under dif... more In this paper, the nonlinear vibrations of functionally graded (FGM) cylindrical shells under different constituent volume fractions and configurations are analyzed. The Sanders-Koiter 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 Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered, allowing for the travelling-wave response of the FGM shell. The functionally graded material considered is made of stainless steel and nickel; the properties are graded in the thickness direction, according to a real volume fraction power-law distribution. In the nonlinear model, the shells are subjected to an external radial excitation. Nonlinear vibrations due to large amplitude excitation are considered. Specific modes are selected in the modal expansions; a dynamical nonlinear system is 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; amplitude-frequency curves are obtained. Convergence tests are carried out considering a different number of asymmetric and axisymmetric modes. The effect of the material distribution on the natural frequencies and nonlinear responses of the shells is analyzed.

We present the results of analytical study and Molecular Dynamics simulation of low energy nonlin... more We present the results of analytical study and Molecular Dynamics simulation of low energy nonlinear non-stationary dynamics of single-walled carbon nanotubes (CNTs). New phenomena of intensive energy exchange between different parts of CNT and weak energy localization in a part CNT are analytically predicted in the framework of the continuum shell theory. These phenomena take place for CNTs of finite length with medium aspect ratio under different boundary conditions. Their origin is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the MD simulation of simply supported CNTs.

In this paper, the nonlinear vibrations of functionally graded (FGM) circular cylindrical shells ... more 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 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; this allows the travellingwave 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.

The nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. T... more The nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. The Sanders-Koiter 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.

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elast... more The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elast... more The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM... more In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter 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 travelling-wave 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 power-law 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 re-expanded 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 (power-law exponent N 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.

In this paper, the energy localization phenomena in low-frequency nonlinear oscillations of singl... more In this paper, the energy localization phenomena in low-frequency nonlinear oscillations of single-walled carbon nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the Sanders-Koiter 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 the nonlinear formulation.

Vibrations of Single-Walled Carbon Nanotubes for various boundary conditions are considered in th... more Vibrations of Single-Walled Carbon Nanotubes for various boundary conditions are considered in the framework of the Sanders-Koiter 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 Sanders-Koiter 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.

In this paper, the low-frequency nonlinear oscillations and energy localizations of Single-Walled... more In this paper, the low-frequency nonlinear oscillations and energy localizations of Single-Walled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied within the framework of the Sanders-Koiter 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 Runge-Kutta numerical method. The analytical model considers a reduced form of the shell theory assuming small circumferential and tangential shear deformations. The Galerkin pro-cedure 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 analyti-cal and numerical values of the localization threshold is investigated in nonlinear field.

In this paper, the effect of the companion mode participation on the nonlinear vibrations of func... more In this paper, the effect of the companion mode participation on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is ap-plied to model the nonlinear dynamics of the system in the case of finite amplitude of vibra-tion. 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 charac-terize the nonlinear response when the shell is subjected to an harmonic external load. A con-vergence analysis is carried out by considering a different number of axisymmetric and asymmetric modes. The present study is focused on modelling the nonlinear travelling-wave response of the shell in the circumferential direction with the companion mode participation.

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded cy-l... more In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded cy-lindrical shells is analyzed. The Sanders-Koiter 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 (power-law 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.

In this paper, the low-frequency nonlinear oscillations and energy localization of Single-Walled ... more In this paper, the low-frequency nonlinear oscillations and energy localization of Single-Walled Carbon Nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the Sanders-Koiter 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 Runge-Kutta 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.

The nonlinear dynamics of Single-Walled Carbon Nanotubes is analysed. The Sanders-Koiter elastic ... more The nonlinear dynamics of Single-Walled Carbon Nanotubes is analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

The present work is concerned with the analysis of low-frequency linear vibrations of SWNTs: two ... more The present work is concerned with the analysis of low-frequency linear vibrations of SWNTs: two approaches are presented: a fully analytical method based on a simplified theory and a semi-analytical method based on the theory of thin shells.
The semi-analytical approach (shortly called “numerical approach”) is based on the Sanders-Koiter shell theory and the Rayleigh-Ritz 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. The Rayleigh-Ritz method is then applied to obtain numerically approximate natural frequencies and mode shapes.
The second approach is based on a reduced version of the Sanders-Koiter 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 fourth-order 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 (free-free, clamped-free) with extreme accuracy. Both methods are suitable for a forthcoming extension to multi-walled nanotubes and nonlinear vibrations.

In this paper an experimental study on circular cylindrical shells subjected to axial compressive... more 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.

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM... more In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. 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 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 re-expanded 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.

The low-frequency oscillations and energy localization of Single-Walled Carbon Nanotubes (SWNTs) ... more The low-frequency oscillations and energy localization of Single-Walled Carbon Nanotubes (SWNTs) are studied in the framework of the Sanders-Koiter 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 [1] uses in the linear analysis a double mixed series expansion for the displacement fields based on Chebyshev polynomials and harmonic functions. The Rayleigh-Ritz method is applied to obtain approximate natural frequencies and mode shapes. In the nonlinear analysis, the three displacement fields are re-expanded 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 Runge-Kutta numerical method.
The analytical model [2] considers a reduced version of the Sanders-Koiter 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 fourth-order 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 [3].
The concept of energy localization in SWNTs is introduced, which is a strongly nonlinear phenomenon. The low-frequency 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 zone-boundary and nearest nonlinear normal modes leading to the confinement of the vibration energy in one part of the system [4].
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.

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elast... more The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter 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 Rayleigh-Ritz 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 re-expanded 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.

In this paper, the nonlinear vibrations of functionally graded (FGM) cylindrical shells under dif... more In this paper, the nonlinear vibrations of functionally graded (FGM) cylindrical shells under different constituent volume fractions and configurations are analyzed. The Sanders-Koiter 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 Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered, allowing for the travelling-wave response of the FGM shell. The functionally graded material considered is made of stainless steel and nickel; the properties are graded in the thickness direction, according to a real volume fraction power-law distribution. In the nonlinear model, the shells are subjected to an external radial excitation. Nonlinear vibrations due to large amplitude excitation are considered. Specific modes are selected in the modal expansions; a dynamical nonlinear system is 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; amplitude-frequency curves are obtained. Convergence tests are carried out considering a different number of asymmetric and axisymmetric modes. The effect of the material distribution on the natural frequencies and nonlinear responses of the shells is analyzed.

We present the results of analytical study and Molecular Dynamics simulation of low energy nonlin... more We present the results of analytical study and Molecular Dynamics simulation of low energy nonlinear non-stationary dynamics of single-walled carbon nanotubes (CNTs). New phenomena of intensive energy exchange between different parts of CNT and weak energy localization in a part CNT are analytically predicted in the framework of the continuum shell theory. These phenomena take place for CNTs of finite length with medium aspect ratio under different boundary conditions. Their origin is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the MD simulation of simply supported CNTs.