Vibration of black phosphorus nanotubes via orthotropic cylindrical shell model

Black phosphorus nanotubes(BPNTs)may have good properties and potential applications.Determining thevibration property of BPNTs is essential for gaining insight into the mechanical behaviour of BPNTs and designingoptimized nanodevices.In this paper,the mechanical behaviour and vibration property of BPNTs are studied viaorthotropic cylindrical shell model and molecular dynamics(MD)simulation.The vibration frequencies of twochiral BPNTs are analysed systematically.According to the results of MD calculations,it is revealed that thenatural frequencies of two BPNTs with approximately equal sizes are unequal at each order,and that the naturalfrequencies of armchair BPNTs are higher than those of zigzag BPNTs.In addition,an armchair BPNTs witha stable structure is considered as the object of research,and the vibration frequencies of BPNTs of differentsizes are analysed.When comparing the MD results,it is found that both the isotropic cylindrical shell modeland orthotropic cylindrical shell model can better predict the thermal vibration of the lower order modes of thelonger BPNTs better.However,for the vibration of shorter and thinner BPNTs,the prediction of the orthotropiccylindrical shell model is obviously superior to the isotropic shell model,thereby further proving the validity ofthe shell model that considers orthotropic for BPNTs.

Vibration and Sound Radiation of Cylindrical Shell Covered with a Skin Made of Micro Floating Raft Arrays Excited by Turbulence

To reduce the vibration and sound radiation of underwater cylindrical shells,a skin composed of micro floating raft arrays and a compliant wall is proposed in this paper.A vibroacoustic coupling model of a finite cylindrical shell covered with this skin for the case of turbulence excitation is established based on the shell theories of Donnell.The model is solved with the modal superposition method to investigate the effects of the structural parameters of micro floating raft elements on the performance of reducing vibration and sound radiation of the cylindrical shell of this skin.The results indicate that increasing the stiffness ratio,damping ratio,mass ratio,or decreasing the interval betweenmicro floating raft elements can improve the vibration and sound radiation reduction performance of this skin over the frequency range 0∼2000 Hz.Moreover,the mean quadratic velocity level and sound radiation power level of the finite cylindrical shell with this skin can be reduced by 12.00 dB and 9.65 dB respectively compared to the finite cylindrical shell with homogeneous viscoelastic coating in the frequency range from0∼2000Hz,implying a favorable performance of this skin for reducing the vibration and sound radiation of cylindrical shells.

The semi-analytical modeling and vibration reduction analysis of the cylindrical shell with piezoelectric shunt damping patches

By considering electromechanical coupling, a unified dynamic model of the cylindrical shell with the piezoelectric shunt damping patch(PSDP) is created. The model is universal and can simulate the vibration characteristic of the shell under different states including the states in which PSDP cannot be connected, partially connected, and completely connected to the shunt circuit. The equivalent loss factor and elastic modulus with frequency dependence are proposed to consider the electrical damping effect of resistance shunt circuits. Moreover, the semi-analytical dynamic equation of the cylindrical shell with PSDP is derived by the Lagrange equation. An experimental test is carried out on the cylindrical shell with PSDP to verify the vibration suppression ability of PSDP on the cylindrical shell and the correctness of the proposed model. Furthermore, the parameter analysis shows that determining the appropriate resistance value in the shunt circuit can achieve a good vibration suppression effect.

Numerical study on the dynamic fracture of explosively driven cylindrical shells

Research on the expansion and fracture of explosively driven metal shells has been a key issue in weapon development and structural protection.It is important to study and predict the failure mode,fracture mechanism,and fragment distribution characteristics of explosively driven metal shells.In this study,we used the finite element-smoothed particle hydrodynamics(FE-SPH)adaptive method and the fluid-structure interaction method to perform a three-dimensional numerical simulation of the expansion and fracture of a metal cylindrical shell.Our method combined the advantages of the FEM and SPH,avoiding system mass loss,energy loss,and element distortion;in addition,the proposed method had a good simulation effect on the interaction between detonation waves and the cylindrical shell.The simulated detonation wave propagation,shell damage morphology,and fragment velocity distribution were in good agreement with theoretical and experimental results.We divided the fragments into three regions based on their shape characteristics.We analyzed the failure mode and formation process of fragments in different regions.The numerical results reproduced the phenomenon in which cracks initiated from the inner surface and extended to the outer surface of the cylindrical shell along the 45°or 135°shear direction.In addition,fragments composed of elements are identified,and the mass and characteristic lengths of typical fragments at a stable time are provided.Furthermore,the mass and size distribution characteristics of the fragments were explored,and the variation in the fitting results of the classical distribution function under different explosion pressures was examined.Finally,based on mathematical derivation,the distribution formula of fragment velocity was improved.The improved formula provided higher accuracy and could be used to analyze any metal cylindrical shells with different length-to-diameter ratios.

Analytical determination of non-local parameter value to investigate the axial buckling of nanoshells affected by the passing nanofluids and their velocities considering various modified cylindrical shell theories

We analytically determine the nonlocal parameter value to achieve a more accurate axial-buckling response of carbon nanoshells conveying nanofluids. To this end, the four plates/shells' classical theories of Love, Fl ¨ugge, Donnell, and Sanders are generalized using Eringen's nonlocal elasticity theory. By combining these theories in cylindrical coordinates,a modified motion equation is presented to investigate the buckling behavior of the nanofluid-nanostructure-interaction problem. Herein, in addition to the small-scale effect of the structure and the passing fluid on the critical buckling strain,we discuss the effects of nanoflow velocity, fluid density(nano-liquid/nano-gas), half-wave numbers, aspect ratio, and nanoshell flexural rigidity. The analytical approach is used to discretize and solve the obtained relations to study the mentioned cases.

Sanders＇ Mid-long Cylindrical Shell Theory and its Application to Ocean Engineering Structures

The cylindrical shell is one of the main structural parts in ocean engineering structures.These cylinders are mostly of medium length,which means that the radius of the cross section is significantly smaller than the length of the cylindrical shell.From the viewpoint of the shell theory,they belong to the mid-long cylindrical shell category.To solve mechanical problems on this kind of structure,especially a cracked cylindrical shell,analysis based on shell theory is necessary.At present the generally used solving system for the mid-long cylindrical shell is too complicated,difficult to solve,and inapplicable to engineering.This paper introduced the Sanders' mid-long cylindrical shell theory which reduces the difficulty of the solution process,and will be suitable for solving problems with complicated boundary conditions.On this basis,the engineering applications of this theory were discussed in conjunction with the problem of a mid-long cylindrical shell having a circumferential crack.The solution process is simple,and the closed form solution can usually be found.In practical engineering applications,it gives satisfactory precision.

AN APPLICATION OF THEORETICAL SOLUTIONS ABOUT CYLINDRICAL SHELLS WITH SMALL OPENINGS

Stress concentrations about thin cylindrical shells with small openings are reconsidered front a nerv angle. There is a sort of special internal relation between theoretical solutions about cylindrical shells,vith large openings and one,with small openings. Using this relation, the extent of applying the theory about small openings to engineering practice is estimated again, thus an idea of how to use this theory and a nerv appraisal of the application of theoretical solutions about cylindrical shells with small openings to engineering practice are given.

Dynamic Buckling of Elasto-Plastic Cylindrical Shells Under Axial Fluid-Solid Impact Load

The dynamic buckling of elasto-plastic cylindrical shells under axial fluid-solid impact is investigated theoretically. A simplified liquid- gas- structure model is given to approximately imitate the problem. The basic equation of the structure is derived from a minimum principle in dynamics of elasto-plastic continua at finite deformation, and the flow theory of plasticity is employed. The liquid is incompressible and the gas is compressed adiabatically. A number of numerical results are presented and the characteristics of the buckling behavior under fluid-solid impact are illustrated.

Bending and vibration analyses of a rotating sandwich cylindrical shell considering nanocomposite core and piezoelectric layers subjected to thermal and magnetic fields

The bending and free vibration of a rotating sandwich cylindrical shell are analyzed with the consideration of the nanocomposite core and piezoelectric layers subjected to thermal and magnetic fields by use of the first-order shear deformation theory （FSDT） of shells. The governing equations of motion and the corresponding boundary conditions are established through the variational method and the Maxwell equation. The closed-form solutions of the rotating sandwich cylindrical shell are obtained. The effects of geometrical parameters, volume fractions of carbon nanotubes, applied voltages on the inner and outer piezoelectric layers, and magnetic and thermal fields on the natural frequency, critical angular velocity, and deflection of the sandwich cylindrical shell are investigated. The critical angular velocity of the nanocomposite sandwich cylindrical shell is obtained. The results show that the mechanical properties, e.g., Young＇s modulus and thermal expansion coefficient, for the carbon nanotube and matrix are functions of temperature, and the magnitude of the critical angular velocity can be adjusted by changing the applied voltage.

Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

An analytical method for the three-dimensional vibration analysis of a functionally graded cylindrical shell integrated by two thin functionally graded piezoelectric （FGP） layers is presented. The first-order shear deformation theory is used to model the electromechanical system. Nonlinear equations of motion are derived by considering the von Karman nonlinear strain-displacement relations using Hamilton＇s principle. The piezoelectric layers on the inner and outer surfaces of the core can be considered as a sensor and an actuator for controlling characteristic vibration of the system. The equations of motion are derived as partial differential equations and then discretized by the Navier method. Numerical simulation is performed to investigate the effect of different para- meters of material and geometry on characteristic vibration of the cylinder. The results of this study show that the natural frequency of the system decreases by increasing the non-homogeneous index of FGP layers and decreases by increasing the non-homogeneous index of the functionally graded core. Furthermore, it is concluded that by increasing the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increase considerably.

Parametric Analysis of A Submerged Cylindrical Shell Subjected to Shock Waves

In this study, an FEM-SBFEM （scaled boundary finite element method） coupling procedure proposed by Fan et al. （2005） is adopted to obtain the dynamic responses of a submerged cylindrical shell subjected to plane step or exponential acoustic shock waves. The coupling procedure can readily be applied to three-dimensional problem, however for clarity, the problems to be presented are hmited to two-dimeusional domain. In the analyses, the cylindrical shell is modeled by simple beam elements （using FEM）, while the effects of the surrounding infinite fluid is modeled by the SBFEM. In it, no free surface and seabed are involved. Compared with Fan and his co-authors＇ works, the FEM-SBFEM coupling procedure is further verified to be feasible for shock waves by benchmark examples. Furthermore, parametric studies are performed and presented to gain insight into effects of the geometric and material properties of the cylindrical shell on its dynamic responses.

Vibration and Acoustic Radiation from Orthogonally Stiffened Infinite Circular Cylindrical Shells in Water

Based on the motion differential equations of vibration and acoustic coupling system for thin elastic shells with ribs, by means of the Fourier integral transformation and the Fourier inverse transformation, as well as the stationary phase method, an analytic solution, which has satisfying computational effectiveness and precision, is derived for the solution to the vibration and acoustic radiation from a submerged stiffened infinite circular cylinder with both ring and axial ribs. It is easy to analyze the effect of stiffening supports in the acoustic radiation field by use of the formulas obtained by the presented method and corresponding numerical computation. It is shown that the axial-stiffeners can improve the mechanical and acoustical characteristics. Moreover, the present method can be used to study the acoustic radiation mechanism of the type of structure.

Dynamic behavior of single-layer latticed cylindrical shells subjected to seismic loading

The single-layer latticed cylindrical shell is one of the most widely adopted space-fl'amed structures.In this paper,free vibration properties and dynamic response to horizontal and vertical seismic waves of single-layer latticed cylindrical shells are analyzed by the finite element method using ANSYS software.In the numerical study,where hundreds of cases were analyzed,the parameters considered included rise-span ratio,length-span ratio,surface load and member section size.Moreover,to better define the actual behavior of single-layer latticed shells,the study is focused on the dynamic stress response to both axial forces and bending moments.Based on the numerical results,the effects of the parameters considered on the stresses are discussed and a modified seismic force coefficient method is suggested.In addition,some advice based on these research results is presented to help in the future design of such structures.

BUCKLING AND POSTBUCKLING OF LAMINATED THIN CYLINDRICAL SHELLS UNDER HYGROTHERMAL ENVIRONMENTS

The influence of hygrothermal effects on the buckling and postbuckling of composite laminated cylindrical shells subjected to axial compression is investigated using a micro-to-macro-mechanical analytical model. The material properties of the composite are affected hy the variation of temperature and moisture, and are hosed on a micromechanical model of a laminate. The governing equations are based on the classical laminated shell theory, and including hygrothermal effects. The nonlinear prebuckling deformations and initial geometric imperfections of the shell were both taken into account. A boundary layer theory of shell buckling was extended to the case of laminated cylindrical shells under hygrothermal environments, and a singular peturbation technique was employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical shells under different sets of environmental conditions. The influences played by temperature rise, the degree of moisture concentration, fiber volume fraction, shell geometric parameter, total number of plies, stacking sequences and initial geometric imperfections are studied.

FREE VIBRATION OF PIEZOELECTRIC CYLINDRICAL SHELLS

Three displacement functions are introduced to represent each mechanical displacement according to the 3-D theory in this paper. By expanding the displacement functions and the electric potential in orthogonal series, the free vibration equation of piezoelectric cylindrical shells satisfying SS3 boundary conditions can be obtained. The equation was solved by utilizing Bessel functions with complex arguments. Results are presented graphically as well as in table, and compared with those of other references. Some frequencies that were missing in Ref. [9] are discovered.

Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities

The free thermal vibration of functionally graded material(FGM) cylindrical shells containing porosities is investigated. Both even distribution and uneven distribution are taken into account. In addition, three thermal load types, i.e., uniform temperature rise(UTR), nonlinear temperature rise(NLTR), and linear temperature rise(LTR), are researched to explore their effects on the vibration characteristics of porous FGM cylindrical shells. A modified power-law formulation is used to describe the material properties of FGM shells in the thickness direction. Love’s shell theory is used to formulate the straindisplacement equations, and the Rayleigh-Ritz method is utilized to calculate the natural frequencies of the system. The results show that the natural frequencies are affected by the porosity volume fraction, constituent volume fraction, and thermal load. Moreover,the natural frequencies obtained from the LTR have insignificant differences compared with those from the NLTR. Due to the calculation complexity of the NLTR, we propose that it is reasonable to replace it by its linear counterpart for the analysis of thin porous FGM cylindrical shells. The present results are verified in comparison with the published ones in the literature.

A NOVEL SEMI-ANALYTICAL METHOD FOR SOLVING ACOUSTIC RADIATION FROM LONGITUDINALLY STIFFENED INFINITE NON-CIRCULAR CYLINDRICAL SHELLS IN WATER

Based on the extended homogeneous capacity high precision integration method and the spectrum method of virtual boundary with a complex radius vector, a novel semi-analytical method, which has satisfactory computation efectiveness and precision, is presented for solving the acoustic radiation from a submerged infnite non-circular cylindrical shell stifened by longitudinal ribs by means of the Fourier integral transformation and stationary phase method. In this work, besides the normal interacting force, which is commonly adopted by some researchers, the other interacting forces and moments between the longitudinal ribs and the non-circular cylindrical shell are considered at the same time. The efects of the number and the size of the cross-section of longitudinal ribs on the characteristics of acoustic radiation are investigated. Numerical results show that the method proposed is more efcient than the existing mixed FE-BE method.

Acoustic scattering from a submerged cylindrical shell coated with locally resonant acoustic metamaterials

Using the multilayered cylinder model, we study acoustic scattering from a submerged cylindrical shell coated with locally resonant acoustic metamaterials, which exhibit locally negative effective mass densities. A spring model is introduced to replace the traditional transfer matrix, which may be singular in the negative mass region. The backscattering form function and the scattering cross section are calculated to discuss the acoustic properties of the coated submerged cylindrical shell.

Buckling of composite cylindrical shells with ovality and thickness variation subjected to hydrostatic pressure

The initial geometric imperfection is one of the primary factors affecting the buckling behaviors of composite cylindrical shells under hydrostatic pressure.In this study,ovality and thickness variations as two representative types of the geometric imperfections are considered.After measuring the geometric imperfections,a typical carbon fiber reinforced polymers(CFRP)cylindrical shell is tested to obtain the buckling pressure.The buckling behaviors of the shell sample are analyzed in combination with the strain responses.By using the nonlinear numerical analysis,the buckling shapes of the CFRP cylinder shells with different combinations of ovality and thickness variation are firstly discussed.The rules of influence of such imperfections on the buckling pressure are then obtained by nonlinear regression method.Finally,an empirical formula is proposed to predict the buckling pressure of the composite cylinder shells,and the calculated results from the formula are in good agreement with the numerical results.

Free vibration of circular cylindrical shell with constrained layer damping

Free vibration characteristics of circular cylindrical shell with passive constrained layer damping （PCLD）are presented. Wave propagation approach rather than finite element method, transfer matrix method, and Rayleigh-Ritz method is used to solve the problem of vibration of PCLD circular cylindrical shell under a simply supported boundary condition at two ends. The governing equations of motion for the orthotropic cylindrical shell with PCLD are derived on the base of Sanders＇ thin shell theory. Nu- merical results show that the present method is more effective in comparison with other methods. The effects of the thickness of viscoelastic core and constrained layer, the elastic modulus ratio of orthotropic constrained layer, the complex shear modulus of viscoelastic core on frequency parameter, and the loss factor are discussed.