Thin Shell Theory Research Articles

On the modelling of stability problems for thin cylindrical shells with two-directional micro-periodic structure

Thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a micro-periodic structure in circumferential and axial directions (biperiodic shells) are considered. The aim of this paper is to investigate the effect of a periodicity cell size on the stationary and dynamical stability of such shells. In order to take into account the length-scale effect in some special stability problems, an averaged non-asymptotic biperiodic shell model derived by means of the known tolerance modelling procedure is applied. Governing equations of this averaged model have constant coefficients depending also on a microstructure size, contrary to the starting exact shell equations with periodic, non-continuous and highly oscillating coefficients (the well-known governing equations of Kirchhoff-Love second-order theory of thin elastic shells). The effect of a cell size on critical forces and on frequency equation being a starting point in the analysis of parametric vibrations and dynamical stability is studied. It will be shown that the micro-periodic heterogeneity of the shells leads to the fourth-order ordinary differential frequency equation, which can be treated as a certain generalization of the known Mathieu second-order equation. It reduces to the Mathieu equation provided that the length-scale effect is neglected.

Cuticle and skin cell walls have common and unique roles in grape berry splitting

The skin protects a fruit from environmental stresses and supports the fruit’s structure. Failure of the skin leads to fruit splitting and may compromise commercial production for fruit growers. The mechanical properties of the cuticle and skin cell walls might influence the splitting susceptibility of fleshy fruits. Thin shell theory and fracture mechanics were utilized in this study to target the potential factors contributing to splitting susceptibility. The study analyzed the structure of the cuticle and epidermis in ripening grape berries and examined the temporal dynamics of berry splitting. Cuticular waxes were partially removed, and skin cell walls were manipulated using wall stiffening and loosening solutions that altered reactions involving hydrogen peroxide. A more than twofold difference in cuticle thickness among grape cultivars did not account for their differences in splitting resistance. However, while removing predominantly epicuticular wax did not alter the berries’ splitting resistance, their surface appearance and increasing yield strength following partial wax removal support the notion that cuticular waxes contribute to berry mechanical properties. Immersing berries in H2O2-based cell wall loosening solutions increased the splitting probability and accelerated berry splitting, whereas cell wall stiffening solutions decreased the splitting probability and delayed berry splitting. These results showed that both cuticle and skin cell walls contribute to the mechanical properties of grape berries and to their splitting resistance. The results also suggest that the two current explanations for fruit splitting, the critical turgor model and the zipper model, should be viewed as complementary rather than incompatible.

Buckling Behavior of Thin-Walled Stainless-Steel Lining Wrapped in Water-Supply Pipe under Negative Pressure

This paper presents a study about the buckling behavior of thin stainless-steel lining (SSL) for trenchless repair of urban water supply networks under negative pressure. The critical buckling pressure and displacement (p–δ) curves, temperature changing curves, hoop and axial strain of the lining monitoring section and the strain changes with system pressure (p–ε) of the lining under the action of different diameters, different lining wall thickness and different ventilation modes were obtained through five groups of full-scale tests. The variation principles of the post-buckling pressure and the reduction regularity of the flowing section of the lining were further investigated. By comparing different pipeline buckling models and introducing thin-shell theory, the buckling model of liner supported by existing pipe was established. The comparison between the test results and thin-shell theory indicates that one of the significances of the enhancement coefficient k value is to change the constraint condition of the aspect ratio, l/R, thus increasing the critical buckling pressure of the lining. Finally, an improved lining buckling prediction model (enhancement model) is presented. A previous test is used as a case study with the results showing that the enhanced model is able to predict critical buckling pressure and lobe-starting amount of the liner, which can provide guidance for trenchless restoration of the liner with thin-walled stainless steel.

Nonlinear vibration behavior of doubly-curved functionally graded piezoelectric microshells in thermal environments

In this paper, nonlinear vibrational behaviors of doubly-curved functionally graded piezoelectric (FGP) microshells in thermal environments have been studied. The doubly-curved microshell has been modeled by using modified couple stress and thin shell theories. The microshell has two material constituents which are piezoelectric materials and their gradation is described using power-law functions. Temperature fields are considered to be uniform, linear, and nonlinear. A closed-form of nonlinear vibration frequency has been obtained by analytically solving the governing equations using the Harmonic balance method. It will be demonstrated that nonlinear vibrational frequency of doubly-curved microshell relies on couple stress influence, material gradation, curvature radius, center deflection, and electrical field.

Standing wave drift of hemispherical resonator with quality factor non-uniformity under a ring electrode excitation

Hemispherical Resonator Gyroscope (HRG) is a classical high precision Coriolis Vibration Gyroscope (CVG), which performs attitude estimation of carrier by detecting the precession of standing wave of resonator, thus, the drift of standing wave of resonator has a great influence on the output accuracy of gyroscope, where the quality factor non-uniformity of resonator is one of main error sources. Ring electrode is a classical excitation structure of HRG because the standing wave can precess freely under its excitation, which makes the gyroscope have more accurate scale factor, larger measurement range and better dynamic characteristics. In this paper, the equations of motion of an ideal resonator excited by a ring electrode are derived by the elastic thin shell theory and Lagrange mechanical principle, then the corresponding equivalent mechanical model is established. According to the “average method”, it can be seen that the ideal resonator excited by the ring electrode works in integral mode, and any position in the circumferential direction of resonator can be a working point, which means that the quality factor non-uniformity has a great effect on the drift of standing wave. Therefore, the equations of motion of resonator with quality factor non-uniformity under the ring electrode excitation are deduced by the equivalent mechanical model, and the drift model of standing wave is established by the “average method”, it can be found that both the amplitude of quality factor non-uniformity and angle between the “inherent damping axis” and antinode axis of standing wave can affect the drift rate of standing wave. Moreover, the drift model indicates that if the input angular rate does not reach the threshold, the precession angular rate of standing wave will appear “self-locking” phenomenon, that is, the gyroscope will lose the integral effect.

Thermally vibrational analyses of functionally graded graphene nanoplatelets reinforced funnel shells with different complex shapes surrounded by elastic foundation

The paper investigates nonlinear dynamic behavior of functionally grade graphene nanoplatelets reinforced (FG-GPLRC) shell with radius as a high-order function using the classical thin shell theory (CTST) and the Von Karman-Donnell geometrical nonlinearity assumption. Different from cylindrical or conical shells, variations of radius are formulated as quadratic or/and cubic functions, generating three types of funnel shells. To verify the computation, the fundamental frequencies are compared with the previous literatures and Finite Element Analysis. The effects of various parameters including geometries, materials, temperature are considered. Obtained results of the shell promisingly make a significant contribution to various fields of engineering.

Effect of longitudinal joint on train-induced vibrations from subway shield-driven tunnels in a homogeneous half-space

With the rapid increase of urban rail transit networks, the environmental problems induced by vibrations from subway traffic become more and more prominent. Most subway tunnels are constructed by the shield machine, which are assembled by tunnel linings connected by joints. This study investigates the effect of the tunnel longitudinal joints on the train-induced vibrations from a subway shield-driven tunnel in a homogeneous half-space. The shield-driven tunnel is considered as an elastic lining ring connected by longitudinal joints, while the ground is simplified as a homogeneous elastic half-space. The general solution for the tunnel and the ground are derived by using the thin-shell theory and the elastic continuum theory, respectively. By satisfying the boundary conditions on the tunnel-ground interface, the tunnel is coupled to the ground in a half-space domain. The responses during the running of a metro train are calculated by coupling the tunnel-ground model with the vehicle-track model. The longitudinal joint has a significant impact on ground vibrations, which increases the running RMS values of the vertical velocity by 20% ∼ 70% and of the horizontal verticity by 10% ∼ 20%. As the stiffness of the longitudinal joint decreases, the ground vibrations increase.

Theory of thin shells as a spatial two-dimensional continuum in an oblique system of coordinates

In contrast to the known approaches in the theory of shells, based on certain assumptions, and the traditional reduction of a three-dimensional problem to a two-dimensional one, in this paper, static and geometric relations are obtained from the position of the interaction of linear forces and moments in a spatially curved two-dimensional continuum. This approach was applied by E. Reissner, but in an orthogonal grid and for physical components. Relations are presented in tensor, vector, and scalar forms.

Random vibration of composite saddle membrane structure under the impact loading

Large-span composite membrane structure is light in weight and small in stiffness, which may produce violent random vibration under the random impact, such as wind, rain, hail and windborne debris. In order to accurately understand the random vibration response of membrane structure, the dynamics of composite saddle membrane structure under the stochastic impact load is studied in this paper. The governing equation of random vibration is established based on the von Karman large deflection theory of thin shell and the shallow membrane assumption. The orthotropy, damping and geometrical nonlinearity are considered for the composite membrane. The stochastic pulse of the impact load is imposed based on the statistical data of impact velocity difference. The perturbation method is used to solve the stochastic vibration equation, and the statistical moments of the displacement are derived from the Galerkin method, which is then validated against the numerical simulations. The comparison shows that the analytical solution can well predict the random dynamic response of saddle membrane structure under the impact load. Finally, it is discovered by the parametric studies that the average displacement reduces with the increase of membrane pretension and arch span ratio, and grows with the increase of impact velocity difference.

Vibration analysis of ribbed and ring-stiffened FG cylindrical shell under hydrostatic pressure

This paper proposed an energy method, based on the Donnell thin shell theory, to derive the vibration characteristic equations of underwater ribbed and ring-stiffened functionally graded (FG) cylindrical shells, and the characteristic equations are solved by MATLAB to obtain their natural frequencies. Through comparative analysis with the existing literature, the correctness and effectiveness of the calculation method are verified. By using numerical examples, the effects of the shell size, material composition, rib section length to width ratio, and rib arrangement on the natural frequency are analyzed, and it has been proved that the aforementioned parameters are the key parameters influencing natural frequency. It is found that the natural frequency of the FG cylindrical shell increases with the decrease in the length–diameter ratio of the shell, increases with the increase in the shell thickness–diameter ratio, and increases with the decrease in the rib spacing. When the cylindrical shell is short and thin, the addition of longitudinal ribs can significantly change the natural frequency of the cylindrical shell.

Free and forced vibration analysis of 3D graphene foam truncated conical microshells

This paper analyzes the free and forced vibrations of three-dimensional graphene foam (3D-GrF) truncated conical microshells. Different distributions of the 3D-GrF are considered. In accordance with Love’s thin shell theory and the modified couple stress theory, governing equations which consider the size effect are established by the Hamilton’s principle. Then, vibration characteristics of 3D-GrF truncated conical microshells are studied through the Galerkin’s method. A series of results show that the apex angle, the 3D-GrF distribution, the foam coefficient, the circumferential wave number and the material length scale parameter play important roles on vibration characteristics of the conical microshells.

On the free vibrations of joined grid-stiffened composite conical-cylindrical shells

The present study aims to investigate the vibrational characteristics of joined stiffened conical-cylindrical composite shells using experimental, numerical and analytical techniques. The stiffness contribution of the shells and helical stiffeners have been superimposed by means of a smeared approach in order to calculate the total stiffness coefficients of the whole structure. The governing equations have been derived according to Donnell’s shallow thin shell theory and then solved using power series technique. Applying boundary conditions along with continuity conditions at the interface section of the shell segments, the natural frequencies and mode shapes of the grid-stiffened joined conical-cylindrical shells are obtained through an eigenvalue problem. The experimental modal analysis has been carried out on the real specimens made of monolithic and wound glass fibers using a specially-designed filament winding device. For further validation, 3-D models have been built in ABAQUS CAE software and analyzed using finite element analysis. In order to verify the present model, several comparisons have been made which indicate a good agreement between the analytical, numerical and experimental results. Furthermore, the effects of several design parameters such as the cone semi-vertex and stiffeners’ orientation angles on the modal parameters have been thoroughly investigated.

Parametric study of damping characteristics of rotating laminated composite cylindrical shells using Haar wavelets

In this paper, the damping characteristics of rotating laminated composite cylindrical shells considering the frequency dependent damping of composite materials are studied. Based on the Love thin shell theory, Hamilton principle is used to obtain the governing equations of the composite cylindrical shell, which are converted into ordinal differential equations by the separation of variables. The dynamical equations and corresponding boundary conditions are discretized by Haar wavelet method and then the standard eigenvalue equation is obtained. The damping with frequency dependence of composite materials is taken into account by the complex modulus method. The eigenvalue equation is solved by iterative calculation to obtain the modal frequency and damping characteristics of rotating composite cylindrical shells. The correctness of present analysis is validated by comparing the predicted results with those given in the literature. The effects of frequency, rotation speed, lamination and geometric parameters on damping properties of the composite shell under four typical boundary conditions are further analyzed. Numerical results show that it is important to consider the frequency dependence of damping of composite materials, especially for the analysis of lower order modes. There are forward and backward damping respectively corresponding to the forward travelling wave and backward travelling wave modes, and the forward wave motion is more stable. The rotating composite cylindrical shell may lose its stability under certain geometric parameters due to the influences of rotation and the inherent material damping.

Dynamic Characteristics of Microring Driven by the Symmetrically Distributed Electrostatic Force

This paper aims at investigating the dynamic characteristics of a microring driven by dual arch electrodes because they are basic elements of microelectrostatic motors. The dual arch electrodes surround the periphery of the microring and are arranged symmetrically to the center of the ring. The electrodes are fixed while the microring is flexible. The electrostatic force will deform the microring, while the deflection of the microring changes the gap between the microring and the electrodes, thereby changing the electrostatic force. Therefore, this is an electromechanical coupling effect. The nonlinear partial-differential equation that governs the motion of the microring is derived based on thin shell theory. Then, based on the assumption of small deflection, the nonlinear governing equation is linearized by truncating the higher-order terms of the Taylor series expansion of the nonlinear electrostatic force. After that, the linearized governing equation is discretized into a set of ordinary differential equations using Galerkin method in which the mode shape functions of the ring are adopted. The influences of the structural damping of the microring and the span of the arch electrodes on the forced response and dynamical stabilities of the microring are investigated. The results show that the damping ratio has a great influence on the system instability during high-frequency excitation. The unstable region of the system can increase with the increase of the electrode span; the response amplitude can also be increased within a certain range.

Nonlinear dynamics of rectangular nano-shells

Mathematical model of non-linear vibrations of shallow, elastic, isotropic nano-shells with rectangular base subjected to transverse sign-variable load are constructed. Based on Kirchhoff-Love thin shell theory with von Kármán nonlinear strains and the modified couple stress theory (MCST), size-dependent governing equations and corresponding boundary conditions are established through Hamilton’s principle. The governing PDEs are reduced to ODEs by the second-order Finite Difference Method (FDM). The obtained system of equations is solved by Runge–Kutta methods of second order accuracy. The Cauchy problem is solved by the Runge–Kutta fourth-order method. We analyzed the convergence of these solutions depending on the step of integration over time and spatial coordinate. It was revealed, that taking into account nano-effects increases area of harmonic vibrations and leads to the appearance of as chaotic and hyperchaotic vibrations. The carried out numerical experiment shows, that the transition of vibrations from harmonic to chaotic follows to Feigenbaum’s scenario. In particular, to analyze the character type of vibration computation of largest Lyapunov exponents are employed. We found that hyperchaotic vibrations are characterized by two positive Lyapunov exponents and chaotic vibration by one positive Lyapunov exponent. For nano-shells, this phenomenon was discovered for the first time. Lyapunov exponents spectra estimated by different algorithms, including Wolf’s, Rosenstein’s, Kantz’s, and Sawada. Numerical examples of the theoretical investigations are given.

Dynamic modeling and output error analysis of an imperfect hemispherical shell resonator

A hemispherical resonator gyroscope (HRG) is very useful for inertial sensors in the 21st century. This paper proposes a novel dynamic model of an imperfect hemispherical shell resonator (DM-IHSR) with thickness nonuniformity and an output error model of the HRG (OEM-HRG) associated with frequency split. First, the DM-IHSR is established based on the thin shell theory of elasticity, and the thickness nonuniformity is described. The differential equations of vibratory motion of the IHSR are derived by leveraging Bubnov-Galerkin method, which reveals that frequency split is mainly dominated by the fourth harmonic of thickness nonuniformity. Then, by considering the characteristics of standing waves, the OEM-HRG is proposed, which discovers the effects of frequency split and the initial orientation angle on the HRG output error. Finally, the effect of thickness nonuniformity on the HRG output error is clarified. The quantitative relationship among the HRG output error, the fourth harmonic of thickness nonuniformity, and the initial orientation angle is determined, which gives machining accuracy ranges of these variables. The above dynamic models and machining accuracy ranges are of great significance for producing and processing high-precision HSRs and HRGs.

Modeling and Vibration Analysis of a Porous Rotational Shell Based on Biot Theory

Combining the Biot theory and classical elastic theory for thin shells, a new dynamic model of a thin fluid-saturated porous rotational shell is proposed. First-order ordinary differential control equations of the porous rotational shell are derived in the frequency domain. These equations are then solved by using the precise element method. The accuracy of this model has been verified by comparing with a vibration experiment. Moreover, the comparisons between the present model and two equivalent property models are carried out. Because the present approach considers the fluid-solid coupling effect and makes no assumptions for the fluid displacements, it is more accurate in the high-frequency range. Lastly, the dynamic characteristics of porous rotational shells are demonstrated by the proposed method.

Numerical Analysis of Cylindrical Shell under Axial Impact

The shell mainly resists the external load by the mid plane stress distributed uniformly along the thickness, rather than the bending stress varying along the thickness. Compared with the traditional flat plate, the shell can make full use of the material strength, so it has greater bearing capacity. In water conservancy projects, shells are widely used, such as double curvature flat shell gate, arch dam and so on. Thin shell theory is a classical theory in shell.

Собственные частоты изгибных колебаний углеродных нанотрубок

Using a molecular dynamics model with a reduced number of degrees of freedom, the natural frequencies of bending vibrations of carbon nanotubes (CNTs) of various diameters are calculated under plane strain conditions. It is shown that the theory of thin cylindrical shells provides high accuracy in estimating the frequencies of low-amplitude natural vibrations even for relatively small CNT diameters. It is shown that with an increase in the amplitude, the frequency of natural vibrations decreases, which is consistent with the data available in the literature. The results obtained are necessary for the design of terahertz resonators based on CNTs and high-precision mass and force nanosensors based on the effect of electromechanical coupling that CNTs exhibit.

NLMS Feedback Vibration Control of Open Cylindrical Shells With Active Constrained Layer Damping

A novel partial active constrained layer damping (ACLD) structure was proposed to reduce the vibration of thinwalled open cylindrical shells. Based on the Sanders thinshell theory and the Lagrange equation, the dynamic model for open cylindrical shells with the partial ACLD structure was established. According to the system state space form, an adaptive feedback controller was developed based on the normalized least mean square (NLMS) adaptive filter algorithm and the linear quadratic programming algorithm (LQR) to study the effects of control parameters on the open cylindrical shell midpoint dynamic characteristics and the control voltage. The numerical results demonstrate that, the NLMS feedback control method can ensure the effectiveness of the vibration attenuation of the open cylindrical shell under different control voltage frequencies, different filter orders and different adaptive step sizes. Increasing the adaptive step size and the filtering order can effectively improve the damping decrement but raise the control voltage overshoot, while increasing the filtering order and the frequency control voltage can reduce noise and disturbance to obtain better damping effects.