Partial Differential Equations Of Motion Research Articles

Polarization force driven FHD flow over a permeable disc with geothermal viscosity

In this paper, we investigate the Bödewadt boundary layer flow of ferrous oxide-based electrically non-conducting Nanofluid considering the influence of thermal radiation and viscous dissipation over a permeable disc with slip conditions. Here, viscosity is taken as a function of temperature and depth. The equation of energy incorporates the radiative heat flux as described by the Rosseland approximation. The Von Kármán Model alters the constitutive nonlinear coupled partial differential equations of motion, which include slip boundary conditions. The final system of equations in PDE is solved computationally using a shooting approach in MATLAB with ode45, and the results are elucidated, through graphs and tables. The effects of all the above-considered physical entities including the geothermal viscosity on the velocity profile and temperature distribution over the disc are examined. An exhaustive analysis of all the above-investigated results is presented for the record. It is observed that the geothermal viscosity with considered boundary conditions retards the motion of the fluid causing decay in the net molecular movement and the thermal diffusion.

Mechanical behavior of porous functionally graded plates

In this study, the effect of porosity distribution pattern on the bending and free vibration analysis of porous FG plates is studied by using sinusoidal shear deformation theory. This theory account for sinusoidal variation of transverse shear strain through the depth of the plate and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The material properties of the plate and the porosities within the plate are considered to vary continuously through the thickness direction according to the volume fraction of constituents defined by the modified rule of the mixture, this includes porosity volume fraction with four different types of porosity distribution over the cross-section. The governing partial differential equation of motion for the bending and free vibration analysis is obtained using sinusoidal shear deformation theory. An analytical solution is presented for the governing equation. Results of the presented solution are compared and validated by the available results in the literature. Moreover, the effects of material and porosity distribution and geometrical parameters on bending and free vibrational properties are investigated.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

A semi-analytical methodology for predicting the vibroacoustic response of functionally graded nanoplates under thermal loads

In this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of nanoplates made of functionally graded materials considering the nonlocal strain gradient theory is presented. It is presumed that the properties of the functionally graded nanoplate are in the form of the simple power law scheme and continuous along the thickness, under nonlinear temperature rise and incident oblique acoustic wave as well as the first-order shear deformation theory. For this purpose, applying Hamilton’s principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the von Kármán nonlinear strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to the Duffing equation. Then, applying the homotopy analysis method, the equation of the transverse movement of the nanoplate is solved semi-analytically to obtain the nonlinear frequencies. Lastly, the nonlinear vibration and acoustic response of functionally graded nanoplate are investigated by considering the variation of the significant factors such as material gradient indices, nonlocal parameters, length scale parameters, aspect ratio, dimensionless amplitude, nonlinear temperature changes, and sound characteristics of the functionally graded nanoplate.

Free vibration Analysis of Beams with Functional Gradient Materials with Cracks

The article presents the development of a dynamic model for a functionally graded material (FGM) beam incorporating cracks. Initially, it assumes an exponential distribution of material properties along the thickness direction of the beam and simulates the opening crack using a zero-mass rotational spring model. This approach enables the calculation of bending stiffness and local flexibility at the cracked section. Subsequently, drawing upon Timoshenko beam theory and von Kármán geometric nonlinear theory, the study formulates the energy equation of the beam and establishes the partial differential control equations for the cracked FGM using Hamilton's principle. The method of separation of variables is employed to discretize the partial differential motion equations into ordinary differential motion equations. Beam functions serve as mode functions, whose unknown coefficients are determined according to the boundary and continuity conditions, thereby yielding the natural frequencies and mode shapes of the cracked FGM beam. Numerical analysis is conducted to evaluate the impact of boundary conditions, the relative position of cracks, and the length-to-thickness ratio on the natural frequencies of the cracked FGM beam.

Analysis of bending vibrations of a three-layered pre-twisted sandwich beam with an exact dynamic stiffness matrix

This paper presents the novel development of the dynamic stiffness matrix for a three-layered symmetric pre-twisted sandwich beam (PTSB), aiming to investigate its free vibration characteristics. The outer layers of the beam are modeled using the Euler–Bernoulli theory, while the core is assumed to deform solely in shear. The boundary conditions and governing partial differential equations of motion are derived based on Hamilton's principle. By applying harmonic variations of displacements, the governing equations of motion are expressed as a tenth-order equation, which is solved to obtain the desired dynamic stiffness matrix. To compute the natural frequencies of in-plane and out-of-plane free vibration for both uniform and PTSBs, the Wittrick–Williams algorithm is employed. The computed frequencies are compared with the results obtained by other authors as well as those obtained from ABAQUS simulations. Various vibration modes of uniform and twisted sandwich beams are plotted and thoroughly discussed. Interestingly, contrary to straight symmetric sandwich beams, the results indicate that flexural displacements in pre-twisted symmetric sandwich beams exhibit coupling in two planes. Additionally, although there are minor changes in vibration frequencies, the mode shapes undergo significant transformations as the pre-twist angle is altered.

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.