Partial Differential Equations Of Motion Research Articles

Large-amplitude nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes

The present study discusses the nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes (FG-CNTs). Mechanical properties of the plate have been continuously changed along the thickness of the plate by considering four different distribution patterns of CNTs. Based on the first-order shear deformation theory (FSDT) and von Ḱarḿan’s geometrical nonlinearity, the governing equations of the plate are derived using Hamilton’s principle. Galerkin discretization technique is employed to convert the partial differential equations of motion to ordinary differential equations. Afterward, the nonlinear time-dependent equations of the plate were analytically solved using the multiple time scales method. Eventually, the effect of CNT parameters, the geometrical ratios and rotational speed on the nonlinear frequency of the plate have been studied. For verification purposes, the present numerical results have been compared with those available in the literature with evident agreement. The present research may give benchmark solutions and some guidelines for designing FG-CNTs rotating plates.

Joint Resonance Analysis in Multiple Modes of Soft Ferromagnetic Rectangular Thin Plate

In this article, the nonlinear principal and internal resonance properties of a soft ferromagnetic rectangular thin plate are investigated in a magnetic field environment. The nonlinear partial differential equation of motion of a soft ferromagnetic rectangular thin plate is derived under the effect of homogeneous simple harmonic excitation. The system of nonlinear differential equations with multiple degrees of freedom is established by the assumed one-sided fixed trilateral simply support condition using the Galerkin’s method. The system of nonlinear differential equations is solved by the multiscale method to obtain the response of two modes under the simple harmonic force at the principal and internal resonance. The numerical results of the system response show that when the frequency of the simple harmonic force is close to one of the modes (first-order or second-order mode) causing it to resonate, the other mode will also resonate internally. The magnetic field can have an inhibiting effect on the resonant response of the system and also affect the kinematic state of the system. The internal resonance provides a mechanism for transferring energy from a high mode to a lower mode.

Linear and Nonlinear Dynamics Responses of an Axially Moving Laminated Composite Plate-Reinforced with Graphene Nanoplatelets

The subharmonic resonances of an axially moving graphene-reinforced laminated composite plate are studied based on the Galerkin and multiscale methods. Graphene nanoplatelets (GPLs) are added into matrix material which acts as the basic layer of the plate, and a graphene-reinforced nanocomposite plate is thus obtained. Different GPL distribution patterns in the laminated plate are considered. The Halpin–Tsai model is selected to predict the physical properties of the nanocomposite. Hamilton’s principle is utilized to conduct the dynamic modeling of the plate and the von Kármán deformation theory is used. The velocity is assumed to be a combination of constant and harmonically varied velocities. The natural frequencies of the linear system with constant velocity can be obtained using the eigenvalues of the coefficient matrix of the ordinary differential equations after the governing partial differential equations of motion are discretized through the Galerkin method. The instability regions of the linear system and the amplitude–frequency relations of the nonlinear system considering the harmonically varied velocity are obtained based on the multiscale analysis. The effect of GPL reinforcement on the subharmonic resonances of the linear and nonlinear systems is analyzed in detail.

Formulation and free vibration analysis of spinning Timoshenko composite beams using nonlinear geometrically exact theory

AbstractIn this study, a set of generalized nonlinear equations of motion for Timoshenko composite shafts is derived using the geometrically exact approach. These partial differential equations of motion are discretized by Galerkin method and the free vibration of an orthotropic spinning shaft with hinged‐hinged boundary conditions is investigated using the multiple‐scale method. We find that in the case of two mode discretization, composite couplings can have significant linear and nonlinear effects on the behavior of the shaft and that these effects were not predictable in the classical model. Also, in the investigating of composite layering we find that the effects of the fibers angle as well as the composite couplings can be noticeable, and these effects are greater in the case of the asymmetrical layering.

The assessment of electric voltage influence on vibrational behavior of intelligent sandwich plates with GO-reinforced metal foam core and piezoelectric actuator face sheets

In this research, it is tried to study the effect of the external electric voltage on the vibrational behavior of a smart sandwich plate made of graphene oxide-strengthened metal foam nanocomposite core and two piezoelectric face sheets for the first time. The sandwich plate is modeled utilizing first-order shear deformation theory. The effective material properties of the core layer are estimated with the aim of the blend of Halpin-Tsai micromechanical model and rule of the mixture while considering different graded graphene oxide distribution patterns. Additionally, various porosity distributions are regarded in the metal matrix of the core layer. The partial differential equations of motion are derived using Hamilton’s principle and the derived governing equations are solved through Navier’s solution. To perform an accurate verification, the obtained results are compared with different literature investigations. Afterward, the influence of various important variants including external electric voltage, aspect ratio of plate, core to face sheet thickness ratio, different porosity distribution patterns, porosity coefficient, nanofillers weight fraction, and various distribution patterns of graphene oxides on the dimensionless frequency of smart sandwich plate in the framework of a group of tables and figures.

Dynamic and instability analysis of single walled carbon nanotubes with geometrical imperfections resting on elastic medium in a magneto-thermally-electrostatic environment with impact of Casimir force using homotropy perturbation method

The discovery of carbon nanotubes (CNTs) has renewed a major chapter in the field of physics, chemistry, mechanics and materials science owing to their high-quality possession of: excellent tensile strength, high conductivity, high aspect ratio, thermally stable and high chemical stability. In this work, dynamic and instability analysis of single walled carbon nanotube with geometrical imperfection resting on elastic medium in a magneto-thermally-electrostatic environments with impact of Casimir force. However, Eringen nonlocal theory and Hamilton principles are used to develop the nonlinear governing partial differential equations of motions and the governing equations of motion is converted into a duffing equation using Galerkin decomposition method and subsequently, the duffing equation is solve using Homotropic Perturbation Method (HPM), where dynamic responses are obtained. The results obtain depicted that, the effects of magnetic term, thermal term and Pasternak type foundation on dimensionless amplitude-frequency ratio for fixed-fixed and fixed-simple supports make the investigation novelty as it can be used as reference in future study. Finally, the deflection curves show how the compression zone is augmented using Casimir and electrostatic forces and the results obtained shows reasonable accurate.

Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán’s nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young’s modulus and mass density as well as Poisson’s ratio, respectively. Hamilton's principle, TSDT, and von-Kármán’s strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam’s geometric parameters, boundary conditions, and GPL’s distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples.

Nonlinear forced vibrations of functionally graded three-phase composite cylindrical shell subjected to aerodynamic forces, external excitations and hygrothermal environment

In this paper, the new functionally graded three-phase composite cylindrical shell is assumed as a common structure in the carrier rocket in the future, and we creatively study the nonlinear forced vibration of this cylindrical shell considering the interaction of different factors in the complex operating environment, including the aerodynamic forces, external excitations, and hygrothermal environment. Based on the first-order shear deformation theory, Von-Karman geometric nonlinear theory, and Hamilton's principle, we derive the nonlinear partial differential equations of motion of the functionally graded three-phase composite cylindrical shell. Considering the axisymmetry of the perfect circular shell, there is a 1:1 internal resonance between the conjugate modes of this cylindrical shell. On this basis, the nonlinear forced vibration of the cylindrical shell is investigated by a combination of Galerkin's method and the pseudo-arc length continuation method. Matcont toolbox can directly solve the ordinary differential equations to obtain the nonlinear frequency response curves. The method can effectively obtain both stable and unstable solutions, avoiding the mathematical difficulties encountered in the formulation process, and facilitating the study of the effects of parametric variables on the resonance response in complex environments. The results show that the variation of material parameters and the complex environment have important effects on the nonlinear resonance response of functionally graded three-phase composite cylindrical shell.

Nonlinear phenomena in vibrations of embedded carbon nanotubes conveying viscous fluid

Various nonlinear phenomena such as bifurcations and chaos in the responses of carbon nanotubes (CNTs) are recognized as being major contributors to the inaccuracy and instability of nanoscale mechanical systems. Therefore, the main purpose of this paper is to predict the nonlinear dynamic behavior of a CNT conveying viscous fluid and supported on a nonlinear elastic foundation. The proposed model is based on nonlocal Euler–Bernoulli beam theory. The Galerkin method and perturbation analysis are used to discretize the partial differential equation of motion and obtain the frequency-response equation, respectively. A detailed parametric study is reported into how the nonlocal parameter, foundation coefficients, fluid viscosity, and amplitude and frequency of the external force influence the nonlinear dynamics of the system. Subharmonic, quasi-periodic, and chaotic behaviors and hardening nonlinearity are revealed by means of the vibration time histories, frequency-response curves, bifurcation diagrams, phase portraits, power spectra, and Poincaré maps. Also, the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

Porosity effects on the dynamic response of arbitrary restrained FG nanobeam based on the MCST

Abstract In this study, two different general eigenvalue problems for nanobeams made of functionally graded material with pores in their sections according to Rayleigh beam theory using modified couple stress theory are established. Fourier sine series and Stokes transformation are used for the solution. First, the partial differential equation of motion of the problem is discretized into an ordinary differential equation. Then, the Fourier sine series of infinite series is substituted into this ordinary differential equation to determine the Fourier coefficient. Using the force boundary conditions of the system, Stokes’ transformation is performed at both ends to include elastic spring parameters. The unknown displacement terms are discretized to form two eigenvalue problems. By solving these eigenvalue problems, vibration frequencies for different boundary conditions can be found analytically. The variations of some parameters are discussed in a series of graphs.

Functionally graded doubly-curved shell with temperature dependent material properties and surface-mounted MEE layers

In the present work, the geometrically nonlinear behavior of a simply supported functionally graded doubly curved shell with a surface-mounted magneto-electro-elastic layer is investigated based on Donnell’s theory. Kirchoff’s assumption and von Kármán’s strain-displacement relationship are used to obtain the governing differential equations. The through-the-thickness shell temperature is computed using Fourier’s heat conduction equation. Elastic material properties are assumed to be temperature-dependent and follow the power-law distribution function of the constituent ceramic-metal volume fractions. The magneto-electro-elastic layer is polarized along the thickness direction, which is subjected to electric and magnetic potentials. Gauss’s laws for electrostatics and magnetostatics are satisfied for magneto-electric materials. Nonlinear partial differential equations of motion are reduced to ordinary differential equations by introducing a stress function and using a single-term time-dependent assumed displacement function. Nonlinear free vibration response is obtained by direct numerical integration. After comparing the derived model with published literature results, numerical studies on the effects of applied electric and magnetic coupled fields of surface-mounted magneto-electro-elastic layers on shell vibration are reported. The nonlinear frequency ratios with respect to specified displacement for both spherical and cylindrical shells with thin Piezoelectric BaTiO 3 and Piezomagnetic CoF e 2 O 4 panels across material compositions, the radius of curvature, and thermal conduction are studied.

Nonlinear vibroacoustic response and sound transmission loss analysis of functionally graded doubly curved shallow shells

This paper presents the semi-analytical nonlinear vibroacoustic and sound transmission loss behaviors of functionally graded doubly curved shallow. The model is developed by considering the simple power law scheme of material distribution in the thickness direction, under thermal load and incident oblique plane sound wave, as well as Donnell’s nonlinear shallow shell theory. By defining the stress function in terms of the force resultants and applying differential operators to these resultants, the coupled compatibility equation and nonlinear equation of motion are obtained only with regard to the stress function and displacement components. The methods of inverse differential operator and average form approach are then utilized to solve the particular and homogeneous solutions of the stress function associated with different boundary conditions. Based on this solution and the use of the Galerkin method with trigonometric mode shape functions, the nonlinear partial differential equation of motion is turned into the Duffing equation. Next, the method of multiple scales is implemented to capture the frequency of the shell corresponding to transverse motion. Finally, the nonlinear vibration and acoustic responses of shells are studied by considering the variation of important parameters such as aspect ratio, dimensionless amplitude, volume fraction power of functionally graded material, phase portrait, sound transmission loss, velocity, average mean square velocity of drive point, and the sound power level of the shells.

Performance analysis of nonlinear piezoelectric energy harvesting system under bidirectional excitations

In this paper, a cantilever beam piezoelectric energy harvester model of nonlinear partial differential equation based on Hamilton's Principle is established. A piezoelectric device using a bimorph cantilever beam with a fixed is subjected to horizontal and vertical excitation. Based on the Euler-Bernoulli thin beam model assumptions and inextensible deformation, the analysis further includes the effects of geometric nonlinearity and damping nonlinearity. The reduced-order nonlinear partial differential equations of motion with electro-mechanical coupling distributed parameter are derived by using Galerkin method. By using the method of multiple scales and solving the equations of motion, the energy harvester in its fundamental first-order resonance response is analyzed and the amplitudes of fundamental transverse deflection, output voltage and output power are determined. The major influence aspects of excitation amplitude, linear damping coefficient, nonlinear damping coefficient, impedance, nonlinear electro-mechanical coupling coefficient on transverse deflection, voltage and power are analyzed. The result shows that the linear damping coefficient and resistance affect the initial threshold of parametric excitation. The combination of parametric excitation and direct excitation gives full play to the advantages of parametric excitation and improve the energy conversion efficiency of the energy harvesting system.

Frequency stability analysis of a cantilever viscoelastic CNT conveying fluid on a viscoelastic Pasternak foundation and under axial load based on nonlocal elasticity theory

AbstractIn this paper, the frequency stability analysis of a fluid conveying cantilever viscoelastic carbon nanotube (CNT) in the Kelvin–Voigt material model with slip boundary condition (BC) regime on a viscoelastic Pasternak foundation and under axial load has been performed based on nonlocal Euler–Bernoulli thin beam theory. The governing partial differential equation of motion (EOM) and its associated BCs have been derived using the Hamilton principle. The governing EOM has been converted into an ordinary differential equation using mode summation technique and solved by applying the extended Galerkin method. Then, the frequency stability analysis has been carried out for the vibrational response of CNT in the state‐space form. The obtained results have been validated with the literature works. The effect of changes of various parameters like elastic stiffness, shear stiffness and damping coefficients of foundation, nonlocal scale‐effect parameter of the nanotube, fluid flow Knudsen number, mass ratio, structural damping of the nanotube and applied axial force have been investigated in terms of frequency to predict the occurrence of different instability modes for the system. It is concluded that flutter and divergence instabilities in the vibration of the viscoelastic CNT are significantly affected by changes of different above‐mentioned parameters.

Effect of linear and nonlinear stiffness support on the dynamic response of high-speed railway bridges

In this paper, the effect of linear and nonlinear support vertical stiffness of elastic support on the dynamic response of the railway bridge is investigated. The bridge is modeled by a Euler-Bernoulli beam supported by linear and nonlinear supports stiffness. A constant moving load model HSLM-A is utilized to simulate the moving real train, and based on Hamilton’s principle, the governing partial differential equation of motion of the studied system is derived. A method to deal with the nonlinear boundary conditions is proposed, the fourth-order Runge-Kutta method in combination with the Galerkin method is used to obtain the dynamic response of the bridge. The finite difference method (FDM) is used to validate the proposed solution. The obtained results have shown that the dynamics of the system are governed essentially by a Duffing-like oscillator and proved that the elastic bearings and the cubic nonlinearities of supports have a significant effect on the global behavior of the railway bridge particularly at resonance.

Analytic Solution for Free Vibrations of Folded Plate Structures

In this work, a new high accuracy solution for vibrations of structures made of plate segments is presented. Each plate segment is deforming both in and out of its plane, and the edges common to two plate segments must have compatible displacements, rotations, shear forces, and moments. It is obtained by using carefully chosen series that solve the combined partial differential equations of motion, for in-plane and out-of-plane deformations for all possible combinations of edge conditions. The number of terms in the series is taken such that convergence is ensured to the number of digits as shown. Examples of the new solutions are given and compared with available solutions for the same cases.

Numerical Solution of Stagnation Point Flow of Non-Newtonian Fluid and Heat Transfer over a Porous Medium under Effect of Thermal Radiation and Magnetic Field

We study three key topics in this article, including Casson fluid flow at its stagnation point, heat transfer in the current magnetic field, and thermal radiation. Several equations related to coupled mass, momentum, and energy are controlled by the mathematical model. This is accomplished by using appropriate transformations to study dimensionless partial differential equations of motion, energy, and continuity. The Adam-Bashforth method is used to solve several nonlinear dimensionless partial differential equations. The analysis is based both on analytic and numerical representations of the impact of parameters which are relevant to the analysis, such as the Casson parameter β, porosity parameter K, magnetic field M, mixed convection parameter γ, and thermal radiation R. Finally, using the Adam-Bashforth approach, we are able to determine velocity profiles as well as temperature profiles.

Temperature-dependent vibrational behavior of bilayer doubly curved micro-nano liposome shell: Simulation of drug delivery mechanism

This article analyzes the vibrational behavior of the double-layered micro-nanosphere to simulate the drug delivery mechanism in the biological structure under the influence of the temperature environment and the viscoelastic substrate for the polar lipid fraction E (PLFE) liposome isotropic material model. In order to obtain the micro-nano structural equations of the double-layer spherical shell, the displacement-strain relations of the shell have been expressed according to the first-order shear deformation theory and the non-local strain gradient theory. The partial differential equations of motion have been obtained by applying Hamilton’s principle. The system of linear couple equations have been solved using the Galerkin method. After validating the model with the results of the articles available in the literature to determine the accuracy of the presented model, numerical results are presented to investigate the effects of various parameters such as the radius of curvature ratio to length, damping coefficient, Kelvin-Voight damping coefficient, boundary conditions and temperature on vibration frequency response are provided and discussed. Results of the current research indicates that natural frequency decreases as the damping coefficient related to the viscous effect between the liposome bilayers increase. Results of the current research can be used as a benchmark for drug delivery applications.

Geometrically nonlinear dynamic analysis of a damped porous microplate resting on elastic foundations under in-plane nonuniform excitation

This article uses the semi-analytical approach to study the combined nonlinear vibration and nonlinear response of a damped porous microplate under nonuniform periodic parametric excitation to understand the complete nonlinear dynamic behavior of the plate. The plate is supported by a Winkler-Pasternak elastic foundation and modeled using modified strain gradient and third-order shear deformation theories to simulate the small-scale effects and shear deformation, respectively. Using Hamilton’s principle, the governing partial differential equations of motion are derived and solved using Galerkin’s method to convert them into ordinary differential equations (ODEs). These ODEs are solved using a combined incremental harmonic balance (IHB) and arc-length continuation approaches to get the nonlinear vibration (frequency–amplitude curves). The same ODEs are solved using the Newmark-β technique to obtain the nonlinear response (time–amplitude curves). The effect of elastic foundation parameters and aspect ratio on mode shape is presented. The effect of parameters such as the porosity coefficient, type of porosity, Winkler-Pasternak elastic foundation parameters, different size-dependent theories, plate thickness, size of plate, damping coefficient, different loading profiles, and loading concentrations on the nonlinear vibration and nonlinear response is examined. Also, the dependence of initial displacements on the frequency–amplitude curves with respect to the excitation frequency is demonstrated with the help of time-amplitude curves.

On the pressure–deflection relations and instability of carbon-based composite nonlinear pipes

In this paper, for the first time, the nonlinear buckling of curved pipes is investigated for different distributions of single-walled carbon nanotubes (SWCNTs) in the matrix. The curved pipe is subjected to uniformly distributed transverse pressure and experiences uniform temperature rise. Additionally, the pipe is in one-sided contact with an elastic nonlinear medium. Defined thickness-dependent and temperature-dependent material properties of SWCNTs are taken from the results of molecular dynamics (MD) simulations, well-established in the literature on the subject. Five technologically justified types of CNT distributions are considered. Shen’s rule of the mixture is applied to determine the effective properties of CNT-based composites. To avoid determining shear correction factor, the higher-order shear deformation theory is used. The nonlinear components of the Green–Lagrange strain tensor and the principle of virtual displacements are applied to derive the nonlinear system of partial differential equations of motion of the curved pipe. The solution to formulated boundary value problem is obtained by the perturbation technique for both simply-supported and clamped-clamped boundary conditions. The pressure–deflection curves of the curved pipe are obtained and discussed. The investigation of snap-buckling intensity and upper/lower limit load of the curved pipe is presented and new insights for its nonlinear stability are presented. For instance, it is shown that the V-type of CNT distribution profile affects on lower deflection of the curved pipe while the higher deflection exists for the Λ-type profile.