Cantilevered Shells Research Articles

Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation

Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation

Chaotic snap-through vibrations of bistable asymmetric deployable composite laminated cantilever shell under foundation excitation and application to morphing wing

The realization of the dynamic snap-through behaviors provides the new design ideas in the fields of the morphing aircraft and piezoelectric energy harvesting. This paper studies 1:2 internal resonance, nonlinear vibrations and chaotic snap-through phenomena of the bistable asymmetric composite laminated (BSCL) cantilever shell with the lower-frequency and higher-frequency primary resonances for the first time. The transverse foundation excitation subjects to the fixed end of the bistable cantilever shell. The perturbation analysis of two-degrees-of-freedom nonlinear ordinary differential equations is carried out by using the first-order approximate multiple scale method. The analytical results of the frequency-amplitude and force-amplitude response curves are obtained under the small foundation excitation. The obtained results reveal that the BSCL cantilever shell exhibits the double-jumping characteristics when 1:2 internal and primary resonances occur. There is a continuous energy exchange back and forth between two modes of the bistable laminated cantilever shell. As the foundation excitation increases, the BSCL cantilever shell exhibits saturation phenomenon. Numerical simulations are finished to further investigate the effects of the large excitation on the chaotic, quasi-periodic and snap-through vibrations for the BSCL cantilever shell. The vibration experiment is carried out to investigate the internal resonance and dynamic snap-though motions of the BSCL cantilever shell. Using the snap-though behaviors of the BSCL cantilever shell, we obtain the morphing structure of the aircraft wing.

Data-driven uncertainty quantification and sensitivity studies in free vibration behavior of bio-inspired helicoidal laminated composite cylindrical shells

The uncertainty in the ply orientation of bio-inspired helicoidal laminated composite cylindrical shells is quantified. The frequencies obtained from the higher-order shear deformation theory are trained to formulate a surrogate model in the multi-output Gaussian Process Regression algorithm framework, and the stochastic analysis is then carried out using the Monte Carlo simulation method. For simply supported and clamped shells, uncertainties in the top and bottom-most layers widely affect the free vibration behavior of the shells compared to the inner-most or central layers, whereas for cantilever shells, opposite trend is observed. The inclusion of noise widely affects the free vibration behavior.

Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell: an experimental and numerical study

Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell: an experimental and numerical study

Non-linear dynamics of cantilevered circular cylindrical shells with thickness stretch, containing quiescent fluid with small-amplitude sloshing

Studies on nonlinear vibrations of circular cylindrical shells containing fluid are mostly limited to thin simply supported shells, such that no results regarding the behavior of thick cantilevered shells with shear and thickness deformations can be found in the literature. In this article, for the first time, thin and thick circular cylindrical shells with clamped-free boundary conditions are modeled according to the third-order shear deformation theory with thickness stretch. The contained liquid satisfies the Laplace equation and linearized boundary conditions are imposed at the free surface and at the contact with the shell. Using Lagrange equations, the governing differential equations of the system in terms of shell-fluid interaction are derived. In linear free vibration analysis, it was found that increasing the fluid level inside the shell and decreasing the shell thickness, cause a significant rise in the fluid free-surface wave elevation. This increase is such that limits the application of linear sloshing theory. It was observed that the fluid presence can change the nonlinear behavior from softening to hardening type and intensifies the shell thickness deformation. In addition, the presence of contained liquid reduces the circumferential dynamic contraction of the shell caused by large amplitude vibrations.

Dynamic Snap-Through and nonlinear vibrations of bistable asymmetric Cross-Ply composite laminated cantilever shell under external excitation

The bistable asymmetric composite laminated structures have the broad applications in the design of the deformable aircrafts and energy harvester due to the unique nonlinear dynamic snap-through characteristics. This paper studies the dynamic snap-through phenomena and nonlinear vibrations of the bistable asymmetric cross-ply composite laminated cantilever shell under the external excitation. Reddy’s third-order shear deformation theory and Hamilton principle are applied to establish the differential dynamic governing equation of the bistable asymmetric composite laminated cantilever shell. Two-degree-of-freedom nonlinear dynamic system is obtained by employing Galerkin approach. The curvature-displacement relationship is given for the bistable asymmetric composite laminated cantilever shell. Chebyshev-Ritz is utilized to obtain both natural frequencies and mode shapes of the bistable asymmetric composite laminated cantilever shell. Considering the cantilever boundary condition, the free vibrations are analyzed by considering different factors of the bistable asymmetric composite laminated cantilever shell, such as the curing temperature, numbers of the layers and geometry parameters. It is observed that the dynamic snap-through behaviors and chaotic vibrations are found under varying the frequency and amplitude of the external excitation. Several validations are carried out among the obtained results with these of other references. The results of the numerical simulation and finite element software ABAQUS are in good agreement with these of the vibration experiment.

Автоколивання тришарової усіченої конічної оболонки із стільниковим заповнювачем, виготовленим адитивними технологіями

This paper presents a nonlinear mathematical model of self-vibrations of conical sandwich shells with a honeycomb core made by additive technologies. The vibrations of the structure are described by fifteen unknowns. Each layer of the structure is described by five unknowns: three projections of the displacements of the layer middle surface and two rotation angles of the middle surface normal. Displacement continuity conditions at the layer interfaces are used. The higher-order shear theory is used to describe the stress-strain state of the structure. The case of conical sandwich shell ? supersonic gas flow interaction is considered. Due to this interaction, self-vibrations of the shell structure are set up. In their analysis, the geometrical nonlinearity of the structure is accounted for. Motion equations of the structure are derived using the assumed-mode method, which uses the kinetic and the potential energy of the structure. The self-vibrations are represented as eigenmode expansions, which contain a set of generalized coordinates. A system of nonlinear autonomous ordinary differential equations in the generalized coordinates is derived. The self-vibrations are studied using a combination of the shooting technique and the parameter continuation method. Multipliers are calculated to analyze the stability of periodic vibrations and their bifurcations. The dynamic instability of the structure’s trivial equilibrium is studied by numerical simulation. For clamped-clamped and cantilever shells, the properties of their periodic, quasiperiodic, and chaotic motions are analyzed in detail.

Nonlinear dynamics of bistable composite cantilever shells: An experimental and modelling study

The nonlinear dynamics of cantilever bistable shells with asymmetric stable configurations is investigated. The possibility of maximizing the kinetic energy associated with snap-through motion offered by the considered cantilevered shells is pursued to enhance the sought-after energy harvesting capabilities. Through an experimental campaign under harmonic forcing the resonance scenarios around the stable configurations are analysed. The resulting nonlinear behaviour observed for high amplitude excitation is adopted to guide a reduced order model derivation. By combining the experimental response with two double-well oscillators derived from FE simulations, a double-well quintic oscillator capturing the observed softening behaviour is proposed. It involves a parameter identification phase in which a nonlinear damping model is introduced. A numerical continuation approach is adopted to discuss the local periodic solutions around each stable configuration in terms of resonance curves, bifurcation diagrams and basins of attraction. The global dynamics involving regular and chaotic dynamic regimes as well as snap-through motion is eventually described.

Modal analysis of rotating Pre-twisted functionally graded sandwich blade

Modal analysis of rotating turbine blades is an inevitable step during the design process to avoid resonance conditions and related catastrophic failures. The need for such analysis arises because of vibration of supporting machine components like rotor and other dynamic forces involved. Mimicking actual structural and environmental conditions for the analysis to achieve close-form solution is very challenging, therefore approximations of real-time situations are adopted to analysis further. Here, functionally graded sandwich pre-twisted cantilever shell structure with metal core and functionally graded (metal/ceramic) face sheets is considered as thermal barrier coated turbine blades. The present work focuses on the modal analysis of rotating pre-twisted blade model under elevated temperature. Material gradation has been done using power-law distribution and Voigt’s model. The kinematics of the present model is based on the first-order shear deformation theory. The Hamilton principle is adopted to obtain the motion equations. Subsequently, the blade model is discretized and solved using 2D finite element approach via quadrilateral elements. The validation of the proposed work has been confirmed by comparing with the earlier published work. In addition, frequency responses are computed for different composite blade parameters such as twist angle, volume fractions, rotational velocity. The occurrence mode shape exchange phenomena with these parametric variations have been observed and discussed in detail.

Closed-form solution for free vibration of variable-thickness cylindrical shells rotating with a constant angular velocity

Based on the classical Donnell’s and Love’s shell theories, free vibration behavior of variable-thickness thin cylindrical shells rotating with a constant angular velocity is analyzed. The equations of motion and corresponding boundary conditions of rotating homogenous cylindrical shells with axisymmetric variation of thickness are derived using Hamilton’s principle. This formulation includes effects of initial hoop tension due to the centrifugal force as well as Coriolis and centrifugal accelerations. Considering the variation of stiffness coefficients in axial direction, the classical Love’s theory results in a coupled system of two second-order and one fourth-order partial differential equations with variable coefficients in terms of kinematic variables. The Galerkin’s method is used to obtain a closed-form expression of sixth-order frequency equation with coefficients in terms of rotation speed (Ω) for with boundary conditions. The natural frequencies of stationary and rotating cylindrical shells with constant thickness, as special cases, are validated with existing ones in the literature. Case studies are presented for three different patterns of thickness variation in forms of arbitrary power, axisymmetric modal and stepwise functions. It is seen that the influence of thickness variation on the natural frequencies and mode shapes is a function of different parameters including rotation speed, boundary conditions, and geometric ratios, especially this effect is much more pronounced for cantilever long shells with a high rotation speed. Also, differences in results predicted by the Donnell’s and Love’s theories become larger for long and cantilever shells in vibration mode shape corresponding to (m=1,n=2), i.e., the ovalization of cross section.

Multistable cantilever shells: Analytical prediction, numerical simulation and experimental validation

The numerical and experimental validation of multistable behavior of cantilever shells is addressed. The design of the laminated composite shells is driven by a recently proposed semi-analytical shell model, whose predictions are verified and critically examined by means of finite element simulations and stability tests on two manufactured demonstrators. In addition, the influence of the main design parameters on the shells stability scenario is discussed. Despite its simplicity, the reduced model allows to depict a fairly faithful picture of the stability scenario; therefore, it proves to be a useful tool in the early design stages of morphing shell structures.

Finite Strip Modeling of the Varying Dynamics of Shell-Like Structures During Machining Processes

The efficiency of the finite strip method (FSM) in modeling the varying dynamics of shell-like structures during machining operations is investigated. The workpiece is modeled as a shallow, helicoidal, cantilevered shell, and the natural modes are computed using FSM. In the FSM solution, the workpiece is discretized only in the chordwise direction, and the membrane and bending displacement fields of the shell in the spanwise direction are approximated by a set of basis functions that satisfy clamped-free boundary conditions. The displacement fields in the chordwise direction are approximated using polynomial functions. The efficiency of the presented FSM is investigated by comparing the computed natural vibration modes against the ones obtained using the finite element method (FEM). The FSM model was found to yield results of greater or comparable accuracy, even with up to 40% fewer degrees-of-freedom (DOFs). Also, the accuracy of the presented model is verified by comparing the predicted frequency response functions (FRFs) against the FRFs that were measured by conducting impulse hammer tests in various stages of machining a generic curved blade.

Periodic, quasi-periodic, and chaotic geometrically nonlinear forced vibrations of a shallow cantilever shell

Geometrically nonlinear forced vibrations of a cantilever shallow shell are analyzed. A finite degree of freedom nonlinear dynamical system is derived using the assumed mode method. The Neimark–Sacker bifurcations are detected close to the first principal resonance. The quasi-periodic vibrations, which originate from these bifurcations, are investigated numerically. These vibrations are transformed into chaotic motions as a result of the forcing frequency variation. Sub-harmonic vibrations with large amplitudes are analyzed in a wide forcing frequency range close to the second principal resonance.

Периодические, почти периодические и хаотические вынужденные колебания пологой консольной оболочки при геометрически нелинейном деформировании

Периодические, почти периодические и хаотические вынужденные колебания пологой консольной оболочки при геометрически нелинейном деформировании

Linear vibrations of cylindrical cantilever shells without imperfections

Cylindrical shells are commonly used as elements of rockets, aircrafts and others structures. The natural frequencies and eigenmodes of the linear cantilever cylindrical shells are very important to predict the dynamic behavior of complex engineering structures. The Rayleigh- Ritz method is applied to analyze the eigenfrequencies and the eigenmodes of the cantilever cylindrical shells. The Donnell’s and Sanders-Koiter shell theories with orthogonal polynomials are used to study the shell linear vibrations. The eigenfrequencies and the eigenmodes of the cantilever shell are investigated. The eigenfrequencies, which are obtained by these two theories, are close. The obtained results are compared with the data, obtained by software ANSYS. The properties of the conjugate eigenmodes are analyzed. The results of the analysis are compared with the data of finite element calculations.

Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory

ABSTRACTWe analyze static infinitesimal deformations of doubly curved shells using a third-order shear and normal deformable theory (TSNDT) and delineate effects of the curvilinear length/thickness ratio, a/h, radius of curvature/curvilinear length, R/a, and the ratio of the two principal radii on through-the-thickness stresses, strain energies of the in-plane and the transverse shear and normal deformations, and strain energies of stretching and bending deformations for loads that include uniform normal tractions on a major surface and equal and opposite tangential tractions on the two major surfaces. In the TSNDT the three displacement components at a point are represented as complete polynomials of degree three in the thickness coordinate. Advantages of the TSNDT include not needing a shear correction factor, allowing stresses for monolithic shells to be computed from the constitutive relation and the shell theory displacements, and considering general tractions on bounding surfaces. For laminated shells we use an equivalent single layer TSNDT and find the in-plane stresses from the constitutive relations and the transverse stresses with a one-step stress recovery scheme. The in-house developed finite element software is first verified by comparing displacements and stresses in the shell computed from it with those from either analytical or numerical solutions of the corresponding 3D problems. The strain energy of a spherical shell is found to approach that of a plate when R/a exceeds 10. For a thick clamped shell of aspect ratio 5 subjected to uniform normal traction on the outer surface, the in-plane and the transverse deformations contribute equally to the total strain energy for R/a greater than 5. However, for a cantilever shell of aspect ratio 5 subjected to equal and opposite uniform tangential tractions on the two major surfaces, the strain energy of in-plane deformations equals 95–98% of the total strain energy. Numerical results presented herein for several problems provide insights into different deformation modes, help designers decide when to consider effects of transverse deformations, and use the TSNDT for optimizing doubly curved shells.

Hydrothermoelastic Stability of Functionally Graded Circular Cylindrical Shells Containing a Fluid

The thermoelastic and hydroelastic stability of heated circular cylindrical shells made of functionally graded materials and interacting with an internal flow of an ideal compressible fluid was investigated. The effective properties of the material vary across the shell thickness according to a power law and depend on temperature. By way of a mathematical formulation the problem on dynamics the elastic structure, the classical theory of shells and the principle of virtual displacements are used. The radial temperature distribution is determined by solving the one-dimensional heat conduction equation. Behavior of the fluid is described using the potential theory. The corresponding wave equation, together with impermeability and boundary conditions, are transformed to a system of equations with the use of the Bubnov–Galerkin method. The solution of the problem, found by employing a semianalytical version of the finite-element method, is reduced to computing the complex eigenvalues of a coupled system of equations. A comparative analysis of the circular cylindrical shells is carried out at different boundary conditions and for different values of the consistency index of the functionally graded material. The effect of a thermal load on the critical speed of the loss of stability and of flow speed on the thermoelastic stability is estimated. It is shown that a flowing fluid has the greatest effect on the stability boundaries of heated cantilevered shells.

Finite-element impact response of debonded composite turbine blades

This paper investigates on the transient behavior of debonded composite pretwisted rotating shallow conical shells which could be idealized as turbine blades subjected to low velocity normal impact using finite-element method. Lagrange's equation of motion is used to derive the dynamic equilibrium equation and the moderate rotational speeds are considered neglecting the Coriolis effect. An eight-noded isoparametric plate bending element is employed in the finite element formulation incorporating rotary inertia and effects of transverse shear deformation based on Mindlin's theory. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the impact parameters. The time-dependent equations are solved by using Newmark's time integration scheme. Parametric studies are performed to investigate the effects of triggering parameters like angle of twist, rotational speed, laminate configuration and location of debonding considering low velocity normal impact at the center of eight-layered graphite-epoxy composite cantilevered conical shells with bending stiff [Formula: see text], torsion stiff ([45°/-45°/-45°/45°]s) and cross-ply ([0°/90°/0°/90°]s) laminate configurations.

Influence of steady viscous forces on the non-linear behaviour of cantilevered circular cylindrical shells conveying fluid

In this study, the effect of steady viscous forces (skin friction and pressurization) on the non-linear behaviour and stability of cantilevered shells conveying fluid is investigated for the first time. These forces are obtained by using the time-mean Navier–Stokes equations and are modelled as initial loadings on the shell, which are in a membrane-state of equilibrium with in-plane stresses. The unsteady fluid-dynamic forces, associated to shell motions, act as additional loadings on this pre-stressed configuration; they are modelled by means of potential flow theory and obtained by employing the Fourier transform technique. The problem is formulated using the extended Hamilton's principle in which the shell model is geometrically non-linear and based on Flügge's thin shell assumptions. This model includes non-linear terms of mid-surface stretching and the non-linear terms of curvature changes and twist, as well. The displacement components of the shell are expanded by using trigonometric functions for the circumferential direction and the cantilevered beam eigenfunctions for the longitudinal direction. Axisymmetric modes are successfully incorporated into the solution expansion based on a physical approximation. The system is discretized and the resulting coupled non-linear ODEs are integrated numerically, and bifurcation analyses are performed using the AUTO program. Results show that the steady viscous effects diminish the critical flow velocity of flutter and extend the range of flow velocity over which limit cycle responses are stable. On the other hand, the non-linear terms of curvature changes and twist have very little effect on the dynamics. The system exhibits rich post-critical dynamical behaviour and follows a quasiperiodic route to chaos.

Nonlinear vibrations of a shell-shaped workpiece during high-speed milling process

This paper is focused on nonlinear dynamics of a shell-shaped workpiece during high speed milling. The shell-shaped workpiece is modeled as a double-curved cantilevered shell subjected to a cutting force with time delay effects. Equations of motion are derived by using the Hamilton principle based on the classical shell theory and von Karman strain-displacement relation. The resulting nonlinear partial differential equations are reduced to a two-degree-of-freedom nonlinear system by applying the Galerkin approach. The averaging method is used to obtain four-dimensional averaged equations for the case of foundational parametric resonance and 1:2 internal resonance. Using a numerical method, the dynamics of the cantilevered shell-shaped workpiece is studied under time-delay effects, parametric excitation, and forcing excitation. It is found that time-delay parameters have great impact on chaotic motion. With increasing amplitude of forcing and parametric excitations, the shell-shaped workpiece exhibits different dynamic behavior.