Bending and Free Vibration Analysis of Porous-Functionally-Graded(PFG)Beams Resting on Elastic Foundations

The bending and free vibration of porous functionally graded(PFG)beams resting on elastic foundations are analyzed.The material features of the PFG beam are assumed to vary continuously through the thickness according to the volume fraction of components.The foundation medium is also considered to be linear,homogeneous,and isotropic,and modeled using the Winkler-Pasternak law.The hyperbolic shear deformation theory is applied for the kinematic relations,and the equations of motion are obtained using the Hamilton’s principle.An analytical solution is presented accordingly,assuming that the PFG beam is simply supported.Comparisons with the open literature are implemented to verify the validity of such a formulation.The effects of the elastic foundations,porosity volume percentage and span-to-depth ratio are finally discussed in detail.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method,the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method(ICCM).The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate.The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method.From the continuity condition of the vibration displacement function at the cutout,the transition matrix between the two coordinate systems is constructed,and the mass and stiffness matrices are completely obtained.As a result,the calculation is simplified and the computational efficiency of the solution is improved.In this paper,numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods,and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded(FG)beams supported by a viscoelastic Winkler-Pasternak foundation is presented.The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation,which were not considered in most literature on this subject,and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven(ε-D)and stress-driven(σ-D)two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered,which can address both the stiffness softening and toughing effects due to scale reduction.The generalized differential quadrature method(GDQM)is used to solve the complex eigenvalue problem.After verifying the solution procedure,a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained.Subsequently,the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.

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 viscousfluid 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 Poincarémaps.Also,the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

Dynamic characteristics of multi-span spinning beams with elastic constraints under an axial compressive force

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier series,which ensures the continuity of the derivative at the boundary and enhances the convergence.The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation.The efficiency and accuracy of the present method are validated in comparison with the finite element method(FEM)and other methods.The effects of the boundary spring stiffness,the number of spans,the spinning velocity,and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied.The results show that the present method can freely simulate any boundary constraints without modifying the solution process.The elastic range of linear springs is larger than that of torsion springs,and it is not affected by the number of spans.With an increase in the axial compressive force,the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger,while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary.

Free vibration and buckling analysis of polymeric composite beams reinforced by functionally graded bamboo fbers

Natural fibers have been extensively researched as reinforcement materials in polymers on account of their environmental and economic advantages in comparison with synthetic fibers in the recent years.Bamboo fibers are renowned for their good mechanical properties,abundance,and short cycle growth.As beams are one of the fundamental structural components and are susceptible to mechanical loads in engineering applications,this paper performs a study on the free vibration and buckling responses of bamboo fiber reinforced composite(BFRC)beams on the elastic foundation.Three different functionally graded(FG)layouts and a uniform one are the considered distributions for unidirectional long bamboo fibers across the thickness.The elastic properties of the composite are determined with the law of mixture.Employing Hamilton’s principle,the governing equations of motion are obtained.The generalized differential quadrature method(GDQM)is then applied to the equations to obtain the results.The achieved outcomes exhibit that the natural frequency and buckling load values vary as the fiber volume fractions and distributions,elastic foundation stiffness values,and boundary conditions(BCs)and slenderness ratio of the beam change.Furthermore,a comparative study is conducted between the derived analysis outcomes for BFRC and homogenous polymer beams to examine the effectiveness of bamboo fibers as reinforcement materials,demonstrating the significant enhancements in both vibration and buckling responses,with the exception of natural frequencies for cantilever beams on the Pasternak foundation with the FG-◇fiber distribution.Eventually,the obtained analysis results of BFRC beams are also compared with those for carbon nanotube reinforced composite(CNTRC)beams found in the literature,indicating that the buckling loads and natural frequencies of BFRC beams are lower than those of CNTRC beams.

Vibration Characteristics and Power Transmission of Coupled Rectangular Plates with Elastic Coupling Edge and Boundary Restraints

Coupled-plate structures are widely used in the practical engineering such as aeronautical,civil and naval engineering etc.Limited works can be found on the vibration of the coupled-plate structure due to the increased mathematical complexity compared with the single plate structure.In order to study analytically the vibration characteristics and power transmission of the coupled-plate structure,an analytical model consisting of three coupled plates elastically restrained along boundary edges and elastically coupled with arbitrary angle is considered,in which four groups of springs are distributed consistently along each edge of the model to simulate the transverse shearing forces,bending moments,in-plane longitudinal forces and in-plane shearing forces separately.With elastic coupling condition and general boundary condition of both flexural and in-plane vibrations taken into account by setting the stiffness of corresponding springs,the double Fourier series solution to the dynamic response of the structure was obtained by employing the Rayleigh-Ritz method.In order to validate the model,the natural frequency and velocity response of the model are firstly checked against results published in literatures and the ANSYS data,and good agreement was observed.Then,numerical simulation of the effects of several relevant parameters on the vibration characteristics and power transmission of the coupled structure were performed,including boundary conditions,coupling conditions,coupling angle,and location of the external forces.Vibration and energy transmission behaviors were analyzed numerically.The results show that the power transmission can be significantly influenced by the boundary restraints and the location of excitation.When the excitation is located at the central symmetry point of the model,the energy flow shows a symmetrical distribution.Once the location deviates from the central symmetry point,the power circumfluence occurs and the vortex energy field is formed at high frequency.

Conformal analysis of fundamental frequency of vibration of simple-supported elastic ellipse-plates with concentrated substance

Aimed at calculating the fundamental frequency of vibration of special-shaped, simple-supported elastic plates, Conformal Mapping theory is applied, and the mathematical method of trigonometric interpolation with interpolation points mutual iterative between odd and even sequences in boundary region is provided, as well as the conformal mapping function which can be described by real number region between complicated region and unit dish region is carried out. Furthermore, in the in-plane state of constant stress, vibrating function is completed by unit dish region method for simple-supported elastic plates with concentrated substance of complicated vibrating region, and the coefficient of fundamental frequency of the plate is analyzed. Meanwhile, taking simple-supported elastic ellipse-plates as an example, the effects on fundamental frequency caused by eccentric ratio, the coefficient of constant in-plane stress, as well as the concentrated substance mass and positions are analyzed respectively.

On the Elastic Vibration Model for High Length-Diameter Ratio Rocket with Attitude Control System

An elastic vibration model for high length diameter ratio spinning rocket with attitude control system which can be used for trajectory simulation is established. The basic theory of elastic dynamics and vibration dynamics were both used to set up the elastic vibration model of rocket body. In order to study the problem more conveniently, the rocket's body was simplified to be an even beam with two free ends. The model was validated by simulation results and the test data.

First-principles study of the structural,elastic,electronic,optical,and vibrational properties of intermetallic Pd_2Ga

The structural, elastic, electronic, optical, and vibrational properties of the orthorhombic Pd2Ga compound are investigated using the norm-conserving pseudopotentials within the local density approximation in the frame of density functional theory. The calculated lattice parameters have been compared with the experimental values and found to be in good agreement with these results. The second-order elastic constants and the other relevant quantities, such as the Young＇s modulus, shear modulus, Poisson＇s ratio, anisotropy factor, sound velocity, and Debye temperature, have been calculated. It is shown that this compound is mechanically stable after analysing the calculated elastic constants. Furthermore, the real and imaginary parts of the dielectric function and the optical constants, such as the optical dielectric constant and the effective number of electrons per unit cell, are calculated and presented. The phonon dispersion curves are derived using the direct method. The present results demonstrate that this compound is dynamically stable.

Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium

A mathematical model for nonlocal vibration and buckling of embedded two-dimensional(2 D) decagonal quasicrystal(QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2 D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional(3 D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories.Numerical examples are provided to display the effects of the quasiperiodic direction,length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence,and medium elasticity on the vibration frequency and critical buckling load of the 2 D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate.This feature is useful since the frequency and critical buckling load of the 2 D decagonal QCs as coating materials of plate structures can now be tuned as one desire.

NONLINEAR VIBRATION FOR MODERATE THICKNESS RECTANGULAR CRACKED PLATES INCLUDING COUPLED EFFECT OF ELASTIC FOUNDATION

Based on Reissner plate theory and Hamilton variational principle, the nonlinear equations of motion were derived for the moderate thickness rectangular plates with transverse surface penetrating crack on the two-parameter foundation. Under the condition of free boundary, a set of trial functions satisfying all boundary conditions and crack＇s continuous conditions were proposed. By employing the Galerkin method and the harmonic balance method, the nonlinear vibration equations were solved and the nonlinear vibration behaviors of the plate were analyzed. In numerical computation, the effects of the different location and depth of crack, the different structural parameters of plates and the different physical parameters of foundation on the nonlinear amplitude frequency response curves of the plate were discussed.

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for the elastic dynamics system,the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method.Through numerical calculation,curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained,and three edges clamped and the other edge elastically restrained versus the spring constant,locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained.Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.

Anti-Dead-Zone Integral Sliding Control and Active Vibration Suppression of a Free-floating Space Robot with Elastic Base and Flexible Links

Under the conditions of joint torque output dead-zone and external disturbance,the trajectory tracking and vibration suppression for a free-floating space robot(FFSR)system with elastic base and flexible links were discussed.First,using the Lagrange equation of the second kind,the dynamic model of the system was derived.Second,utilizing singular perturbation theory,a slow subsystem describing the rigid motion and a fast subsystem corresponding to flexible vibration were obtained.For the slow subsystem,when the width of deadzone is uncertain,a dead-zone pre-compensator was designed to eliminate the impact of joint torque output dead-zone,and an integral sliding mode neural network control was proposed.The integral sliding mode term can reduce the steady state error.For the fast subsystem,an optimal linear quadratic regulator(LQR)controller was adopted to damp out the vibration of the flexible links and elastic base simultaneously.Finally,computer simulations show the effectiveness of the compound control method.

The Interface Stress Field in the Elastic System Consisting of the Hollow Cylinder and Surrounding Elastic Medium Under 3D Non-Axisymmetric Forced Vibration

The paper develops and employs analytical-numerical solution method for the study of the time-harmonic dynamic stress field in the system consisting of the hollow cylinder and surrounding elastic medium under the non-axisymmetric forced vibration of this system.It is assumed that in the interior of the hollow cylinder the point-located with respect to the cylinder axis,non-axisymmetric with respect to the circumferential direction and uniformly distributed time-harmonic forces act.Corresponding boundary value problem is solved by employing of the exponential Fourier transformation with respect to the axial coordinate and by employing of the Fourier series expansion of these transformations.Numerical results on the frequency response of the interface normal stresses are presented and discussed.

Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells

The free vibration characteristics of fluid-filled functionally graded cylindrical shells buried partially in elas- tic foundations are investigated by an analytical method. The elastic foundation of partial axial and angular dimen- sions is represented by the Pasternak model. The motion of the shells is represented by the first-order shear defor- mation theory to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shells are composed of stainless steel and silicon nitride. Material prop- erties vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The governing equation is obtained using the Rayleigh-Ritz method and a variation approach. The fluid is described by the classical potential flow theory. Numerical examples are presented and compared with existing available results to validate the present method.

On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells

Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially symmetric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervious and an impervious surface. Results of previous works are obtained as a particular case of the present study.

Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory

In this paper, the free vibration of magneto- electro-elastic （MEE） nanoplates is investigated based on the nonlocal theory and Kirchhoff plate theory. The MEE nanoplate is assumed as all edges simply supported rectan gular plate subjected to the biaxial force, external electric potential, external magnetic potential, and temperature rise. By using the Hamilton＇s principle, the governing equations and boundary conditions are derived and then solved analytically to obtain the natural frequencies of MEE nanoplates. A parametric study is presented to examine the effect of the nonlocal parameter, thermo-magneto-electro-mechanical loadings and aspect ratio on the vibration characteristics of MEE nanoplates. It is found that the natural frequency is quite sensitive to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the thermal loading.

Transverse Vibration Analysis of Single-Walled Carbon Nanotubes Embedded in an Elastic Medium Using Bernoulli-Fourier Method

Based on the Timoshenko beam theory and Bernoulli-Fourier method, a single-elastic beam model is developed for transverse vibrations of single-walled carbon nanotubes under additional axial load, which includes the effects of the elastic medium around them. Explicit expressions are derived for the natural frequencies and transversal responses of simply supported single-walled carbon nanotubes. The influence of addition axial load and the properties of elastic medium on the vibrations are discussed. The results showed that the effects of addition axial load on the lower natural frequencies of single-walled carbon nanotubes are sensitive to the lower vibration modes and the stiff elastic medium. The lower natural frequencies depend on the axial load;they become smaller with increasing axial load and vary with the vibration modes. In addition, except for the first mode, the effects of the axial load on the stiff elastic medium are considerably greater than those on the flexible one. However, the constants of the elastic medium have little effect on the first mode. The critical axial buckling stress and strain for simply-supported single-walled carbon nanotubes are also obtained.

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅰ)—RECTANGULAR PLATES WITH FOUR CLAMPED EDGES AND WITH THREE CLAMPED EDGES

In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided.MRT is a simple, convenient and general method for solving the steady stale responses of rectangular plates under various harmonic disturbing forces.The paper contains three parts: (I) rectangular plates with four damped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates.We arc going to publish them one after another.