Galerkin Procedure Research Articles

Nonlinear vibration of full-filled fluid corrugated sandwich functionally graded cylindrical shells

This article proposes a semianalytical approach for the nonlinear free and forced asymmetric vibration of corrugated sandwich functionally graded cylindrical shells containing fluid under harmonic radial load. The corrugation is considered with two types: trapezoidal corrugation and round corrugation. By using the Donnell shell theory, taking into account the geometrical nonlinearity of von Kármán and an equivalent model for corrugated structures, the governing equations of shell are established. The nonlinear motion equations are reduced to nonlinear differential equation in terms of time by applying the Galerkin procedure. The numerical investigations for the nonlinear dynamic response of cylindrical shells are obtained by using the fourth-order Runge–Kutta method. Numerical results show the very large effects of corrugation and fluid on the natural frequency and nonlinear vibration behavior of shells.

Efficient analysis method and design approach for broadband reflect‐arrays with isotropic/artificial anisotropic substrates

Plane-wave scattering by metallic grating with artificial anisotropic substrate is a problem solved in this study for the design of broadband reflectarrays. For determination of the reflection characteristics of such structures, a generalised transmission line modelling is used for evaluating the dyadic Green's function. This function is utilised in a summation equation used for calculating the induced electric currents on the reflectarray's metallic grating. For solving the obtained summation equation, the Periodic Method of Moments with Galerkin procedure is applied. Such an analysis method helps to compute the phase diagram needed to design of reflectarray. The efficiency of this method is assessed by considering its cost-time and rate of convergence versus space harmonics. In the design step, a new phase realisation approach is implemented for minimising the frequency dispersion's adverse effects. This approach optimises the arrangement of metallic elements on the reflectarray's aperture and decreases the dependency of broadband reflectarray design to the element phase behaviour. As a result, this technique leads to lower side lobe level and better frequency response than traditional approaches. To verify the results of the developed solution, two different broadband reflectarrays for X- and Ku-bands are fabricated and measured.

A Modified Fourier Series Solution for a Thermo-Acoustic Tube with Arbitrary Impedance Boundaries

Rijke tube is a benchmark model widely used in thermo-acoustic community. As an alternative to existing modeling methods, this work proposes a modified Fourier series solution for modal characteristic analyses of a one dimensional (1D) thermo-acoustic system. The proposed modeling framework allows the consideration of arbitrary impedance boundaries owing to the special feature of the Fourier expansion series enriched by boundary smoothing polynomial terms. Thermo-acoustic Helmholtz governing equation coupled with a first-order heat release model is discretized through Galerkin procedure. Thermo-acoustic modal parameters are obtained by solving a standard quartic matrix characteristic equation, different from conventionally used root searching based on a transcendental equation. Numerical examples are presented to validate the proposed model through comparisons with results reported in the literature. Influences of boundary impedance are analyzed. Results reveal a quantitative relationship between the thermo-acoustic instability and heat source position with respect to the acoustic mode shapes. Results also show the existence of a sensitive zone, in which the thermo-acoustic modal behavior of the impedance-ended (IE) tube shows drastic changes with the boundary impedance. Meanwhile, a stable zone can be achieved upon a proper setting of the boundary impedance through suitable combination of the real and imaginary parts of the impedance.

Effect of Throughflow on the Convective Instabilities in an Anisotropic Porous Medium Layer with Inconstant Gravity

The significance of inconstant gravity force and uniform throughflow on the start of convective movement in an anisotropic porous matrix is investigated numerically utilizing large-term Galerkin procedure. The porous layer is acted to uniform upright throughflow and inconstant downward gravitational force which changes with the height from the layer. In this study, two types of gravity field digression were examined: (a) linear and (b) parabolic. It is found that the throughflow parameter Pe, the thermal anisotropy parameter η and gravity deviation parameter λ postpone the beginning of convective activity, whereas the mechanical anisotropy parameter ξ rapids the onset of convective activity. The dimension of the convection cells enhances on enhancing the thermal anisotropy parameter η, the mechanical anisotropy parameter ξ and gravity deviation parameter λ while, the throughflow parameter Pe decreases the extent of the convective cells. It is also noted that the structure with linear variation of gravity force is more stable.

Size-dependent vibrations of thermally pre/post-buckled FG porous micro-tubes based on modified couple stress theory

Based on the modified couple stress theory, an analytical approach is presented to study vibrations of thermally pre/post-buckled functionally graded (FG) tubes. The case of simply supported FG micro-tubes with uniform distributed porosity surrounded by nonlinear elastic medium in uniform temperature field is analyzed. Thermomechanical properties are functionally graded across the radius of cross-section of the tube by means of a power law function. The traction free boundary conditions on the inner and outer surfaces of the structure are satisfied. Basic formulations are constructed using the higher-order shear deformation tube theory and the von Kármán type of nonlinear strain-displacement relationships. The governing equations of motion are derived using the Hamilton principle. The dimensionless equations of motion are solved analytically by employing the two-step perturbation technique and the Galerkin procedure. An explicit closed-form solution is obtained to analyze the size-dependent linear and nonlinear free vibrations of micro-tubes in thermal environment. The numerical results are verified by comparison with the existing data in literature for tubes without couple stress effect. Novel numerical results are revealed in the parametric studies to show the large amplitude vibration responses of thermally pre/post-buckled FG porous micro-tubes. The effects of microstructural length scale parameter, functionally graded patterns, porosity distribution parameter, nonlinear elastic foundation, temperature dependence, and geometrical properties of the structure are investigated.

Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters

In this article, vibration of viscoelastic axially functionally graded (AFG) moving Rayleigh and Euler–Bernoulli (EB) beams are investigated and compared, aiming at a performance improvement of translating systems. Additionally, a detailed study is performed to elucidate the influence of various factors, such as the rotary inertia factor and axial gradation of material on the stability borders of the system. The material properties of the beam are distributed linearly or exponentially in the longitudinal direction. The Galerkin procedure and eigenvalue analysis are adopted to acquire the natural frequencies, dynamic configuration, and instability thresholds of the system. Furthermore, an exact analytical expression for the critical velocity of the AFG moving Rayleigh beams is presented. The stability maps and critical velocity contours for various material distributions are examined. In the case of variable density and elastic modulus, it is demonstrated that linear and exponential distributions provide a more stable system, respectively. Furthermore, the results revealed that the decrease of density gradient parameter and the increase of the elastic modulus gradient parameter enhance the natural frequencies and enlarge the instability threshold of the system. Hence, the density and elastic modulus gradients play opposite roles in the dynamic behavior of the system.

Dynamical analysis of spinning functionally graded pipes conveying fluid with multiple spans

In this paper, a dynamical model of spinning multi-span pipes conveying fluid is proposed and the transverse natural and resonant frequencies and mode characteristics of such system are explored. The pipe body is considered to be composed of functionally graded materials (FGMs), in which a power law is used to govern the distribution of material properties along the pipe wall thickness. The partial differential equations (PDEs) governing two transverse motions of the pipe are derived by the extended Hamilton principle, in which the contributions of the FGM and intermediate supports are highlighted. The PDEs are discretized by the Galerkin procedure and the eigensystem theorem is applied to find the numerical solutions. The results show that various frequency characteristics can be attainable by use of different materials and mixing patterns. Attachments of intermediate supports can heighten the rigidity and improve the stability of spinning FG pipes conveying fluid, which are consequently used as “stabilizers” for the slender drill strings. Also, the mode characteristics of different spans will determine the locations of vibration amplitude of the pipes.

A nonlinear model of the non-isothermal slip flow between two parallel plates

We study a mathematical model describing the steady-state non-isothermal flow of a viscous fluid between two parallel plates under the Navier slip boundary condition. The flow is driven by an applied pressure gradient. The dependence of the viscosity, thermal conductivity and slip coefficients on the temperature is taking into account. The model under consideration is a boundary value problem for a system of nonlinear coupled ordinary differential equations. We give the weak formulation of this problem and establish sufficient conditions for the existence and uniqueness of a weak solution. To construct weak solutions, we propose an algorithm based on the Galerkin procedure, methods of the topological degree theory, and compactness arguments. Moreover, explicit expressions for the velocity and the temperature on the plates are obtained.

Buckling Behavior of FG-CNT Reinforced Composite Conical Shells Subjected to a Combined Loading.

The buckling behavior of functionally graded carbon nanotube reinforced composite conical shells (FG-CNTRC-CSs) is here investigated by means of the first order shear deformation theory (FSDT), under a combined axial/lateral or axial/hydrostatic loading condition. Two types of CNTRC-CSs are considered herein, namely, a uniform distribution or a functionally graded (FG) distribution of reinforcement, with a linear variation of the mechanical properties throughout the thickness. The basic equations of the problem are here derived and solved in a closed form, using the Galerkin procedure, to determine the critical combined loading for the selected structure. First, we check for the reliability of the proposed formulation and the accuracy of results with respect to the available literature. It follows a systematic investigation aimed at checking the sensitivity of the structural response to the geometry, the proportional loading parameter, the type of distribution, and volume fraction of CNTs.

Global Existence and Energy Decay of Solutions to a Coupled Wave and Petrovsky System with Nonlinear Dissipations and Source Terms

In this paper, we consider the nonlinearly damped nonlinear coupled wave and Petrovsky system $$\begin{aligned}&u'' + \Delta ^{2} u+ av + g_{1}(u')=f_{1}(u,v), \\&v'' - \Delta v+ au + g_{2}(v')=f_{2}(u,v). \end{aligned}$$We prove the global existence of solutions by means of the stable set method in $$H_{0}^{2}(\Omega )\times H^1_0(\Omega )$$ combined with the Faedo–Galerkin procedure. Furthermore, we establish a more general energy decay of solutions when the nonlinear dissipative terms $$g_{1},g_{2}$$ do not necessarily have a polynomial growth near the origin.

Dynamic response of cable-stayed bridges due to sudden failure of stays: the 3D problem

This paper studies analytically the problem of the sudden failure of a number of stays through a suitable mathematical model, based on the analytical method exposed by authors in previous publications and extended in this study through a 3D analysis. The analysis is carried out by the modal superposition method, and the gathered equations of the problem are solved through the Galerkin procedure and the Duhamel’s Integrals. Characteristic examples are solved and useful diagrams and plots are drawn, while interesting results are obtained.

Flutter and bifurcation instability analysis of fluid-conveying micro-pipes sandwiched by magnetostrictive smart layers under thermal and magnetic field

Fluid-conveying micro/Nano structures are key tools in MEMS and NEMS applications especially for drug delivery systems to attack a specific tumor like cancer cells. Vibrational characteristics of such tools play a crucial role in delivering efficient and reliable performance in various applications. As a result, vibration and instability control of such systems is of great importance. Vibration and instability response of magnetostrictive sandwich cantilever fluid-conveying micro-pipes is investigated in this paper utilizing smart magnetostrictive layers as actuators. Euler–Bernoulli beam model together with modified couple stress theory (MCST) are used to model the problem. As main properties of these smart layers, magnetic intensity effect, velocity feedback gain and thermal effects are taken into account in the modeling. The governing equation is extracted employing Hamilton’s principle. Extended Galerkin procedure is applied to discretize the governing equation and obtain the eigenvalue problem which is solved straightforwardly to reach the eigenvalues. Afterwards, eigenvalue diagrams are studied to analyze the vibrational characteristics and possible instabilities (flutter and bifurcation) occurring in first three modes of the system. Throughout this analysis, the role of various intrinsic properties of the magnetostrictive layers on the critical flow velocity and frequency is studied in detail. The numerical results show a good ability for the used smart layers to control the instability of fluid-conveying micro-pipes. Therefore, these sandwich structures may be helpful for achieving a novel design for such systems.

Nonlinear micromechanically analysis of forced vibration of the rectangular-shaped atomic force microscopes incorporating contact model and thermal influences

The linear and nonlinear forced vibrations of microprobe in contact-mode atomic force microscopes are presented based on a rectangular beam model of modified couple stress theory. The Hamilton principle is employed to derive governing PDEs for the first and second modes transformed into nonlinear ODEs by the Galerkin procedure coupled with time multiple scales method. The important effect of linear tip inertia is regarded despite of others. The impact of damping, thermal loading, and constant contact stiffness on the amplitude and phase angle-frequencies are investigated. Location of a concentrated tip mass with tip-sample interaction are modeled by delta Dirac functions. Communicated by Francesco Tornabene.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Analysis of the Effect of a Gyrotropic Anisotropy on the Phase Constant and Characteristic Impedance of a Shielded Microstrip Line

In this work, we present an analytical modeling of a highly complex medium-based shielded microstrip line. The study aims at a numerical evaluation of the characteristic impedance and the dispersion characteristics of the dominant hybrid mode in the microstrip line printed on an anisotropic medium. The newly considered complex anisotropy has a full 3×3 tensor form of permittivity and permeability. The study is based on the derivation of the Green's functions of the general complex-medium-based structure in the Fourier domain. The spectral Method of Moments (MoM) and the Galerkin's procedure are combined to solve the resulting homogeneous system of equations. The effect of the gyrotropic anisotropy on the phase constant and the characteristic impedance is particularly investigated. Original and interesting numerical results are obtained and discussed. Our results are found to be in good agreement with available isotropic case data reported in literature.

Buckling and postbuckling of advanced grid stiffened truncated conical shells with laminated composite skins

Abstract A theoretical approach is presented to derive an explicit formula for buckling load and postbuckling path of advanced grid stiffened conical shells (stiffeners with laminated composite skins). Different types of fiber paths of grids including stringer, ring, and helical are considered. The simply supported truncated conical shell with imperfection is subjected to axial compression. Basic formulations are constructed using the classical theory of shells and von Karman type of nonlinear strain-displacement relationships. The equilibrium and compatibility equations are solved by Galerkin procedure, and an explicit relation is obtained to predict the equilibrium paths. Results for different conditions are verified by comparison to existing data in the literature. Novel results are revealed in the parametric study to show the effects of different fiber paths, geometrical conditions, and lay-up configurations on the buckling and postbuckling of the advanced grid stiffened structures. As a key finding, the helical and ring fibers of grids are more effective than stringer on the buckling load and postbuckling path. Especially when high number of ribs are considered.

Size-dependent instability of organic solar cell resting on Winkler–Pasternak elastic foundation based on the modified strain gradient theory

The present study employs the modified strain gradient theory (MSGT) in conjunction with the refined shear deformation plate theory to explore the buckling behaviour of simply supported and clamped OSC. The Winkler-Pasternak elastic foundation is implemented to idealise the foundation. The size-dependent effect of the OSC is captured by the three length scale parameters within the MSGT. The Hamilton principle is used to derive the equations of motion and the boundary conditions, and the Galerkin procedure is subsequently implemented to obtain the critical buckling load. Subsequently, the framework is extended to the thermally induced buckling behaviour, and three types of temperature rise patterns, namely uniform, linear and nonlinear temperature variations, along the thickness of the OSC are considered. Several verification studies are conducted to illustrate the accuracy of the present method. Besides, size-dependent material properties are taken into consideration during the numerical experiments. Thorough studies are conducted to demonstrate the difference between critical buckling loads obtained from the MSGT, the modified couple stress theory (MCST), and the classical plate theory (CPT) models. Furthermore, the effects of length scale parameter (h/l), the aspect ratio (a/b), the length-to-thickness ratio (a/h) and the Winkler-Pasternak elastic foundation parameters on the buckling behaviour of the OSC are also revealed by the numerical results.

Design, Modeling, and Experimental Evaluation of a Compact Piezoelectric XY Platform for Large Travel Range.

A novel piezoelectric XY platform driven by a single actuator was presented for a large travel range and compact structure. The actuator operated at the inertial mechanism and moved the output platform step-by-step. A dynamic model of the piezoelectric actuator was established based on the Timoshenko beam theory and Galerkin procedure, which was used to aid the structure design. The dynamic model of the platform system was established based on the dynamic model of the actuator and the LuGre friction model. A prototype was fabricated and its experimental system was established; the total size was 100 × 100 × 93.5 mm3 and the travel range was 15 × 15 mm2. The measured stepper motions agreed well with the simulation results, and the correctness of the dynamic model was confirmed. The proposed platform achieved maximum speeds of 2.13 and 3.11 mm/s along the axes X and Y , respectively, and a carrying capacity of 20 kg was achieved. Furthermore, the closed-loop control experiments, including the positioning resolution and the sinusoidal trajectory tracking, were carried out, and a positioning resolution better than [Formula: see text] and a tracking error rate of 4% were achieved, which revealed the potential of the proposed piezoelectric platform in field of manipulating heavy objects with submicrometer accuracy and large travel range, especially for some specific fields including microparticle manipulation, ultraprecision manufacturing, and optical device posture adjustment where large travel range, high accuracy, and multidimension are expected.

Nonlinear planar secondary resonance analyses of suspended cables with thermal effects

This paper deals with a geometrically nonlinear model of suspended cables in thermal environments. In particular, the secondary resonant behaviors with thermal effects are investigated and discussed analytically and numerically. First, considering the influence of temperature changes, a dimensionless coefficient is generated. The nonlinear dynamic equations of motion in thermal environments are established, and the governing PDEs are reduced to a set of ODEs by using the Galerkin procedure. Four super and sub-harmonic resonant cases are studied, and frequency responses equations and steady-state solutions for each case are obtained. Extensive numerical results show that the hardening/softening behaviors, the response amplitudes, the number of nontrivial solutions, the range of resonant responses, the jump points, and the peak values are all closely connected to temperature. In some specific warming/cooling conditions, the nonlinear system just shows linear vibration behaviors or even just opposite spring characteristics. The time–displacement curves and phase portraits are extremely sensitive to temperature changes, but the number of cluster points in Poincaré sections seem to be independent of temperature changes.

Towards a physical model of Antarctic sea ice microstructure including biogeochemical processes using the extended Theory of Porous Media

AbstractPredictive numerical climate models include an ocean modeling system as an important module, which comprises the components of ocean and sea ice physics as well as the coupled biogeochemical processes occuring in the ocean and sea ice, respectively. Whereas large‐scale, physical sea ice models are well represented, a new thrust for climate modeling is to include the sea ice biogeochemistry (BGC) as a strong influencing factor. For that reason, a demanding challenge is to develop a numerical model for the sea ice microstructure, which considers all relevant physical parts, like the evolution of salinity, thermodynamic properties and mechanical deformation, as well as the biological and chemical conversion processes. The model should be able to quantify phytoplankton biomass, nutrient sources and utilization as well as carbon content and export in connection to the underlying physical and biogeochemical driven mechanisms. A thermodynamically consistent derived continuum mechanical model is developed and prepared by means of a standard GALERKIN procedure to be implemented and solved with the finite element method (FEM). The governing equations and constitutive relations are set up in the framework of the homogenization approach of the extended Theory of Porous Media (eTPM).