Von Karman Theory Research Articles

Nonlinear vibration behaviours of foam-filled honeycomb sandwich cylindrical shells: Theoretical and experimental investigations

In this study, the nonlinear vibration behaviours of foam-filled honeycomb sandwich cylindrical shells (FHSCSs) are investigated theoretically and experimentally. Initially, a time-domain minimum residual (TDMR) method is first adopted for the analysis of nonlinear free and forced vibrations of the FHSCSs, which is based on a dynamical model developed, with the effective material parameters of the foam-filled honeycomb core being obtained by employing the Gibson method and the Hamilton equivalence technique. Based on the high-order shear deformation shell principle and von Karman's theory, the nonlinear natural frequencies and resonant responses are solved using the Broyden iterative approach and TDMR method. Furthermore, several literature results are utilized for the rough verification of such a method. Meanwhile, the FHSCS specimens are made manually and comprehensive tests with various base excitation energies are undertaken to provide a thorough validation of the current method. It can be found that the largest calculation errors of natural frequencies and resonant responses are 4.9 and 12.3%, respectively. Finally, the influences of the core form and base excitation amplitudes on the nonlinear vibration behaviours of the FHSCSs are examined. It can be noticed that the foam-filled FHSCS structure exhibits a more excellent anti-vibration capability compared to the non-foam-filled structure. The solving method, fabrication technique, and valuable findings in the current study highlight the path for the design and implementation of such advanced shells.

On the boundary conditions for a thin circular plate conjugated to a massive body

The problem of deformation under the action of uniform pressure of a circular plate coupled with a massive base is considered, while the condition for the coupling of the plate with the base is modeled using boundary conditions of the generalized elastic embedding type, i.e. the relationship between the bending moment and forces at the edge of the plate with displacements and rotation angles through the compliance matrix. The main goal of the work is to study the influence of the elasticity of the embedding on the elastic response of the plate. The solution to the problem was obtained in the formulation of the linear theory of plates, the theory of membranes in the approximation of homogeneity of longitudinal forces, and the Foppl — von Karman theory, also in the approximation of the assumption of homogeneity of longitudinal forces. The values of the coefficients of the compliance matrix were obtained using the finite element method for the auxiliary problem and compared with the values of the coefficients obtained for related problems by analytical methods. Numerical results were obtained for an aluminum wafer on a silicon base. The obtained solution was compared with the solution obtained for the rigid embedment condition for all three models used. It is shown that in the case of large deflections (several plate thicknesses), taking into account the compliance of the embedment becomes essential.

Parametric resonance of an axially accelerating viscoelastic membrane with a fractional model

In this paper, the parametric resonance of an axially accelerating viscoelastic membrane is examined with a fractional model. The membrane is transported with a time-dependent velocity. The viscoelastic model incorporating fractional derivative into the Kelvin-Voigt relation is considered. The governing equation is formulated from Newton's second law and von-Karman theory, and solved via Galerkin truncation and the multiple-scale method. Furthermore, the frequency-responses of the nonlinear resonance are estimated by eliminating the secular terms, and the Routh-Hurwitz criterion is employed to identify the stability and bifurcation of the steady-state responses. Finally, the numerical results are used to analyze the effects of the fractional order, viscoelastic coefficient and time-dependent velocity on the frequency- responses, solution stability and bifurcation points. The novelty of this study is to introduce a fractional viscoelastic model and time-dependent velocity to analyze the resonance stability of axially accelerating viscoelastic membranes in a roll-to-roll system. The results of this investigation are expected to be useful for quantifying the relative influence of the system parameters in the roll-to-roll manufacturing process.

Nonlinear dynamics of fractional viscoelastic PET membranes with linearly varying density

In this paper, the nonlinear dynamics of fractional viscoelastic polyethylene terephthalate (PET) membranes with linearly varying density are elaborated. The viscoelasticities of the PET membranes are characterized with the fractional Kelvin–Voigt model, and the density distribution is considered a linear fluctuation in the lateral direction. The geometrically nonlinear formulation is established with the von-Karman theory and the Hamilton principle. The Bubnov–Galerkin method is used to solve the resulting fractional differential equations, and a refined numerical scheme combining the finite difference with a discrete Caputo-type fractional derivative approximation is developed. Comparison examples are provided to substantiate the accuracy and effectiveness of the present method. A detailed parametric analysis is performed to illuminate the effects of the fractional order, density coefficient and various system parameters on the time response of the viscoelastic PET membranes. This study paves the way for fractional modelling and vibrational analysis of viscoelastic substrates in flexible electronics manufacturing.

Nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support

Roll-to-roll manufacturing is the primary means of flexible electronics to facilitate scale-up production, and the web as the printed substrate is guided by the intermediate roller in this process. This paper aims to investigate the nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support. This support is modeled as a nonlinear elastic spring with linear and nonlinear stiffness, and the equation of motion of the web attached with nonlinear elastic support is derived from the D’Alembert principle and von Karman theory. The resulting equation is reduced to the two-degree ordinary differential equations via the Galerkin truncation, the superharmonic resonance responses of the web system are obtained by the multi-scale method, and the bifurcation analysis and stability are analyzed by the Runge-Kutta numerical method. The results indicate that the linear and nonlinear stiffnesses have a significant effect on amplitude-frequency responses and chaotic motion. This study provides an exploration of vibration behaviors of the web in flexible manufacturing, thereby laying the foundation for the improvement of fabrication productivity.

Low-velocity impact resistance of all composite cylindrical shell panels with a foam filled honeycomb core: Theoretical and experimental investigation

The low-velocity impact resistance of all composite cylindrical shell panels (CCSPs) with a foam filled honeycomb core (FFCHC) is studied experimentally and theoretically. Initially, a dynamical model of the FFCHC-CCSPs is proposed, with the equivalent material properties of the core being determined by combining the Hamilton equivalence theory and the Gibson theory. By using the virtual work principle and quasi-static method, together with the von Karman's theory and the high order shear deformation shell theory, the constitutive relationship related to each damage case is derived. Also, by considering the influence of collapse of the FFCHC, the failure criteria and damage modes are modified. The impact displacement, load and impact energy absorption are solved by using the elastoplastic contact principle and energy principle. Several numerical results from the literature are employed for preliminary validation of the model. Furthermore, the FFCHC-CCSP specimens with and without polyurethane foam are fabricated and the detailed measurements associated with different impact energies are performed to provide a solid validation of the present model as well as to evaluate the structural impact resistance. Finally, the influences of the thickness of FFCHC, wall thickness and side length of honeycomb cell and density of foam on the impact resistance of the FFCHC-CCSP structures are discussed, with some useful conclusions being refined for better impact suppression design and manufacture of such advanced shells.

Nonlinear vibrations of doubly curved micropanels reinforced by graphene nanoplates in length direction under non-uniform thermal loading

Structures made of composites are quite common in different industries entailing automotive, marine, and aircraft industries, thanks to their favorable lightweight and strength. This study investigated the nonlinear vibrations of the various shapes of size-dependent graphene nanoplatelets reinforced micropanels under non-uniform thermal loading. In diverse ways in the axial direction, the material characteristics of graphene nanoplates (GPLs) alter. Taylor’s series expansion terms in curvature coordinates are used to illustrate the displacement field. Nonlinear von Karman theory and Hamiltonian principle work together to derive the nonlinear form of the pertinent equations and boundary conditions. Strain gradient theory with consideration for inherent length scale characteristics is investigated to account for size effects. By discretizing the spatial domain equations, deriving Duffing-type equations, and using Kronecker and Hadamard products, the governing equations and numerous nonlinear boundary edges are resolved. Using the techniques and answers outlined above, the results section is produced. There are four sections for this research. The findings are cross-checked with those from other studies that have been published in the first section. The effects of geometric and physical factors are shown in Parts 2, 3, and 4 on the nonlinear to linear frequency curve, the nonlinear frequency curve, and the sensitivity of the nonlinear frequency-to-size effect (SNFSE).

Buckling analysis of super-light composite cylinders with auxetic core and isotropic facing sheets with variable thickness: An analytical approach

An analytical method is presented to determine the axial buckling load of composite cylindrical shells with honeycomb core layer and nonuniform thickness using the first-order shear deformation theory, the nonlinear von Karman theory, and the Hooke law relations. The composite shell consists of two isotropic inner and outer layers with non-uniform thicknesses and one honeycomb core with constant thicknesses and adjustable Poisson's ratio. The equilibrium equations are a system of nonlinear differential equations with variable coefficients, and they are derived by employing the virtual work principle. The equilibrium equations have been solved analytically using the matched asymptotic expansion method of the perturbation technique. The stability equations extracted using the adjacent criterion include a system of linear homogenous equations with variable coefficients. By solving these equations, the buckling load is obtained analytically. A parametric study investigates the effects of different geometrical and mechanical parameters such as the honeycomb structure on the buckling load. The metamaterial honeycomb core layer has an adjustable Poisson's ratio, and the effect of negative/positive Poisson's ratio has been studied on the buckling behavior. The accuracy of the presented method is studied by comparing the results with the finite element method and some other references in special cases.

Behaviour of a non-linear mesh cylindrical shell as an element of mems and nems

A new mathematical model of a non-linear mesh cylindrical shell behaviour in the temperature field under normal distributed load is constructed. The construction of the mathematical model takes into account the Kirchhoff-Love kinematic model and the Duhamel-Neumann hypothesis. The scaling effect is taken into account by the modified moment theory of elasticity. It is assumed that the displacement and rotation fields are not independent. Geometric nonlinearity is taken into account according to T. von Karman's theory. The equations of motion of the smooth shell, boundary and initial conditions are derived from the variational Ostrogradsky-Hamiltonian principle. The lattice structure of the shell was modelled by the continuum theory of G. I. Pshenichny. This allowed us to replace the regular rib system by a continuous layer. The equilibrium conditions of a rectangular element were used to write down the relations connecting stresses occurring in an equivalent smooth shell with stresses in the ribs. The Lagrange multiplier method has been used to determine the physical relationships for the mesh shell. By means of the method of establishment the study of features of the shell's plasticity has been carried out and the frequencies of natural linear vibrations depending on the mesh geometry have been obtained.

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

This paper presents the application of an exponential version of the harmonic balance method to the analysis of steady state vibration of geometrically nonlinear systems. A detailed description of the method and of the corresponding numerical procedure is provided. The von Karman theory is used to describe the effects of geometric nonlinearity. The material of the beams is modelled with the help of the Zener model using the fractional calculus. The problem is solved using an exponential version of the harmonic balance method. In the above-mentioned version, the complex calculus is used in contrast to the ordinary harmonic balance method, where the steady state vibrations are described with the help of the trigonometric functions. It significantly simplifies derivation of the amplitude equations. Moreover, the exponential version of the harmonic balance method is more elegant in comparison with the ordinary one. A detailed derivation of the amplitude equations is presented. The modified continuation method is proposed to solve the nonlinear amplitude equations and to determine the response curves. Moreover, the results of the exemplary calculation are presented and compared with known results in order to justify the efficiency and the correctness of the proposed approach.

Primary and secondary resonance responses of fractional viscoelastic PET membranes

This paper considers nonlinear vibration of fractional viscoelastic Polyethylene Terephthalate (PET) membranes subjected to external excitation. Particular attention has been paid to the effects of parameters associated with the fractional model on the primary and secondary resonance responses. Here, the fractional Kelvin–Voigt constitutive relation is adopted to describe the viscoelastic material. Based on von-Karman theory and Newton’s second law, the governing equation is derived. The Galerkin technique is utilized to discretize the nonlinear equation of motion. The multiple-scale method is employed to assess the forced responses of the membrane system under external excitation. Numerical results reveal that the fractional viscoelastic model strongly affects the resonance response of membranes. In this regard, these results offer insight into further study on the vibration characteristics of fractional viscoelastic membranes and also determine the stable operating regions of the moving system to avoid divergent instability in the fabrication of flexible electronics.

Wavelength-dependent Image Quality Predictions and Other Considerations for Site Selection and Instrument Design

Stellar image motion measured differentially with small telescopes can be converted to image quality predictions for large telescopes at the same site. To demonstrate this methodology, multi-wavelength seeing distributions are predicted for Las Campanas Observatory based on the Differential Image Motion Monitor (DIMM) results from the Giant Magellan Telescope site survey. The observed seeing distribution at 0.5 μm is used to compute a family of predicted seeing distributions from 0.32 to 4.7 μm. Tokovinin’s formula is used to transform the DIMM seeing distributions (based on Kolmogorov theory) to the large telescope image sizes expected in the von Karman theory. The latter framework includes an outer scale, or largest size, for the turbulent eddies above a telescope. The impact of seeing distributions on site selection and on telescope and instrument design are discussed. Other considerations in selecting an excellent site are also listed and explained.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.

Nonlinear Dynamic Analysis of Plates Subjected to Explosive Loads

Blast loads have been increasingly studied in the past decades, especially regarding civil structures. Until recently, the negative phase of these loads has been disregarded, but studies concluded that the effect of suction must be included. In the case of plates, nonlinearity plays an important role, and the membrane effect should also be considered. This work focuses on the influence of nonlinearity in plates subjected to blast loads. Equations were developed for the calculation of blast load parameters, considering that positive and negative phases are approximated by the Friedlander equation and cubic polynomial, respectively. The plate is modeled as a SDOF system using von Karman's theory of large displacements. The development of the nonlinear dynamic differential equation is reviewed, considering a simply supported plate, and its solution is based on fourth order Runge-Kutta numerical method. A reference example is used as a benchmark and then parametric studies are conducted, in which the influence of scaled distance, mass of explosive, and the consideration or not of the negative phase is analyzed.

Tuned deep neural network for computer simulation of chaotic performance of the electromechanical annular system coupled with a piezoelectric patch

In this paper, an attempt is made to extend a nonlinear two-dimensional model for large oscillation and chaotic characteristics of the annular plate coupled with a piezoelectric patch subject to external excitation. For improving the mechanical properties, the core of the electrically structure is reinforced with graphene nanoplatelets. Von-Karman nonlinear theory and Hamilton’s principle are taken into consideration for the exact derivation of the general nonlinear governing equations and boundary conditions of the axisymmetric annular plate coupled with a piezoelectric patch. For developing a precise solution approach, the generalized differential quadrature element method, and Multiple scales technique, are eventually used. The results demonstrate that radius ratio, applied voltage, thickness to radius ratio, and harmonic load have an important role in the nonlinear large oscillation, and chaotic motion of the graphene platelets reinforced composite (GPLRC) annular plate coupled with a piezoelectric patch. The golden and fundamental result of the current research is that, as the thickness of the piezoelectric patch increases, the system's nonlinear frequency decreases, and the area of instability in responses and maximum amplitude or the peak point of the backbone curve of the smart annular plate increases.

Semi-Analytical Model to Predict the Elastic Post-Buckling Response of Axially Compressed Cylindrical Shells With Tailored Distributed Stiffness

Abstract This study introduces an approximate analytical model to predict the post-buckling response of cylinders with tailored non-uniform distributed stiffness. The shell's wall thickness, and thus its stiffness, is tailored so as to obtain multiple controlled elastic local buckling events when the cylinder is subjected to uniform axial compression. The proposed model treats cylinder segments of different stiffness as individual panels and combines their response by considering them as connected linear or nonlinear springs. The governing equations for the panels are formulated using von Karman's theory and solved by Galerkin's approximate method for a predefined radial deformation. Radial deformation functions are used to improve the model's accuracy, and results show that the model's accuracy increases significantly with the number of considered radial functions. The model's predicted axial response for different cylinders is compared with results from experiments on three-dimensional printed samples. Results indicate that this model accurately predicts the order of the buckling events, while the buckling forces from the model are higher than those measured experimentally.

Nonlinear dynamics of fluid conveying double-walled nanotubes incorporating surface effect: A bifurcation analysis

The aim of this work is to analyze the effect of the residual surface stress on the dynamic instability of the fluid conveying double-walled nanotubes (DWNTs). To achieve this goal, the theory of surface effect, nonlocal elasticity and nonlinear von Karman theory are combined which lead to new insight in bifurcations of fluid conveying DWNTs. The conveying fluid is also modeled and analyzed based on modified Navier–Stokes relation considering Knudsen number. In addition, Galerkin technique and multiple time scales method is applied to solve the equation. By means of the bifurcation analysis, it is shown that a discontinuous bifurcation is always accompanied by a jump. Finally, stable and unstable regions in dynamic instability of DWNTs are addressed. We close this paper by stating that the residual surface stress enhances the stiffness of DWNTs. Furthermore, the results indicate and emphasize that the nanofluid velocity as an external excitation can significantly affect the instability behaviors of the system.

Nonlinear vibration suppression of laminated composite conical shells on elastic foundations with magnetostrictive layers

Nonlinear vibration control of laminated composite conical shells surrounded by elastic foundations through magnetostrictive layers is presented in this paper. The shell modeled via the consideration of the effects of shear deformation and rotary inertia while von Karman theory of nonlinearity is also employed. In order to suppress the nonlinear vibration, velocity feedback control method is adopted. Modified Galerkin method is exerted to obtain the nonlinear ordinary differential equation of the system from equations obtained via Hamilton principle. The results of this study are acquired with a combination of modal analysis and multiple scales method. The asymptotical stability of the system around zero equilibrium point is investigated based on Lyapunov’s indirect method. The results for damping coefficients of nonlinear system and fundamental linear frequency as well as the curves of amplitude versus nonlinear frequency and nonlinear frequency ratio are obtained for different values of several parameters.

Effect of Initial Geometric Imperfections on Nonlinear Vibration of Thin Plate by an Asymptotic Numerical Method

In this work, the influence of initial geometric imperfections on geometrically nonlinear free vibrations of thin elastic plates has been investigated by an asymptotic numerical method. The nonlinear strain displacement relationship of von Karman theory is adopted to calculate the elastic strain energy. The harmonic balance approach and Hamilton’s principle are used to convert the equation of motion into an operational formulation. The nonlinear problem is transformed into a sequence of linear ones having the same stiffness matrix, which can be solved by a classical finite-element method. To improve the validity range of the power series, Pade approximants are incorporated and a continuation technique is also used to get the whole solution. Numerical results are discussed and compared to those available in the literature and convergence of the solution is shown for various amplitudes of imperfection.

Thermal buckling of laminated Nano-Composite conical shell reinforced with graphene platelets

Abstract A semi analytical approach based on the trigonometric expansion and generalized differential quadrature (TE-GDQ) method is implemented to investigate the thermal buckling of a functionally graded graphene platelet reinforced composite (FG-GPLRC) conical shell. Graphene platelets distribution is assumed randomly oriented and uniformly dispersed for each ply. Variation of the volume fraction of plies is modeled according to a step functionally graded medium. Effective material properties of the shell are calculated using the Halpin-Tsai micromechanical rule. Equilibrium equations are derived by means of the first order shear deformation theory (FSDT), nonlinear geometrical relation based on the von Karman theory, and the Donnell kinematics assumption. The pre-buckling of the shell under uniform thermal load is solved considering the linear membrane procedure. Afterward, the linearized stability equations are obtained utilizing the adjacent equilibrium criterion. Next, using the TE-GDQ method, stability equations convert to an eigenvalue problem. Finally, eigenvalue problem is solved and the critical temperature rise can be determined. After validation of the formulation and methods, several parametric studies are examined to research the effects of the volume fraction of GPLs, semi vertex angle, length-to-thickness ratio and other parameters on the critical temperature.