Nonlinear Free Vibration Response Research Articles

Multiquadric Collocations for Nonlinear Vibration of Functionally Graded Plates of Ti–6Al–4V Metal

While Finite Element Method (FEM) has proven reliable for analyzing linear or nonlinear stresses in materials, structures, and fluid flows, its reliance on mesh generation can escalate project simulation costs. In contrast, mesh- free techniques offer an alternative by directly generating system algebraic equations for the entire issue domain, bypassing the need for a preset mesh. This study focuses on investigating the nonlinear free vibration response of shear-deformable Functionally Graded Material (FGM) plates. These plates are expected to exhibit varying material properties, following a power-law distribution linked to the volume fractions of constituent materials as plate thickness increases. Utilizing shear deformation theories and von Karman nonlinear kinematics, the study seeks a mathematical solution for the real physical issues encountered in these plates. To address these challenges, a meshless method employing the Multi Quadric Radial Basis Function (MQRBF) is employed for evaluation. The computed results reveal insights into how different material properties and amplitudes influence the nonlinear free vibration response of composite plates. Comparing these outcomes to prior research demonstrates promising progress in this area.

Analytical Framework for Tension Characterization in Submerged Anchor Cables via Nonlinear In-Plane Free Vibrations

Submerged tensioned anchor cables (STACs) are pivotal components utilized extensively for anchoring and supporting offshore floating structures. Unlike tensioned cables in air, STACs exhibit distinctive nonlinear damping characteristics. Although existing studies on the free vibration response and tension identification of STACs often employ conventional Galerkin and average methods, the effect of the quadratic damping coefficient (QDC) on the vibration frequency remains unquantified. This paper re-examines the effect of bending stiffness on the static equilibrium configuration of STACs, and establishes the in-plane transverse free motion equations considering bending stiffness, sag, and hydrodynamic force. By introducing the bending stiffness influence coefficient and the Irvine parameter, the exact analytical solutions of symmetric and antisymmetric frequencies and modal shapes of STACs are derived. An improved Galerkin method is proposed to discretize the nonlinear free motion equations ensuring the accuracy and applicability of the analytical results. Additionally, this paper presents an analytical solution for the nonlinear free vibration response of the STACs using the improved averaging method, along with improved frequency formulas and tension identification methods considering the QDC. Through a case study, it is demonstrated that the improved methods introduced in this paper offer higher accuracy and wider applicability compared to the conventional approaches. These findings provide theoretical guidance and reference for the precise dynamic analysis, monitoring, and evaluation of marine anchor cable structures.

Nonlinear vibrations analysis of two-directional functionally graded porous cylindrical shells resting on elastic substrates in a thermal environment based on Donnell nonlinear shell theory

The primary objective of this study is to investigate the nonlinear free vibrational characteristics of temperature-dependent two-directional functionally graded porous (TDFGP) cylindrical shells resting on elastic substrates in a thermal environment. To accomplish this, the thermomechanical equations are derived based on the Donnell nonlinear shell theory framework in conjunction with the von Kármán assumption. Two-directional functionally graded porous cylindrical shell models have mechanical properties that can change smoothly and continuously across the length and thickness of the shell. Additionally, it is assumed that the internal porosities in the matrix materials can be dispersed into two independent patterns, either even or uneven porosity distribution. The nonlinearity in free vibration assessed via the nonlinear-to-linear frequency ratio concerning the central deflection amplitude can be gained employing the Galerkin discretization approach and modified Poincare–Lindstedt (P-L) method. The accuracy and effectiveness of the present analytical model are indicated through comparison with existing solutions. Finally, some comprehensive parametric investigations are carried out to gain insight into the impacts of several factors on the nonlinear free vibration characteristics of structures under different conditions. The results of this article demonstrate that parameters such as gradient indices, volume fraction, distribution pattern of porosity, geometric parameters, and ambient temperature rise significantly influence the structure’s nonlinear frequency and free vibration response.

Nonlinear dynamical characteristics of carbon nanotube-reinforced composite beams with piezoelectric actuators and elastically restrained ends under thermo-electro-mechanical loads

Nonlinear free vibration and dynamical responses of carbon nanotube (CNT) reinforced composite beams with surface-bonded piezoelectric layers and tangentially restrained ends under thermo-electro-mechanical loads are investigated in this paper. The properties of constitutive materials are assumed to be temperature-dependent and effective properties of nanocomposite are estimated using an extended rule of mixture. Unlike previous studies, the present work considers the effects of tangentially elastic constraints of two ends on the nonlinear dynamic characteristics of hybrid beams. Motion equation is established within the framework of Euler-Bernoulli beam theory taking into account von Kármán nonlinearity. Analytical solution is assumed to satisfy simply supported boundary conditions and Galerkin procedure is employed to obtain a time ordinary differential equation including both quadratic and cubic nonlinear terms. This differential equation is numerically solved employing fourth-order Runge-Kutta scheme to determine the frequencies of nonlinear free vibration and nonlinear transient response. Parametric studies are executed to examine numerous influences on the nonlinear dynamical characteristics of hybrid nanocomposite beams. The study reveals that tangential constraints of ends substantially effect the frequencies and dynamic response of the beam, especially at elevated temperatures. The results also indicate that nonlinear dynamic responses can be controlled effectively by means of piezoelectric actuators and elasticity of tangential constraints of ends should be considered in design of piezo-CNTRC beams.

Hygrothermal Effects on Vibration Response of Porous FG Nanobeams Using Nonlocal Strain Gradient Theory Considering Thickness Effect

In this work, the thickness-dependent size effect on nonlinear free vibration response of a Porous Functionally Graded (PFG) nanobeam in a hygrothermal environment is examined via nonlocal strain gradient theory (NSGT). Through the depth of the beam, hygro-thermal loadings are uniformly distributed. The mechanical properties of the beam are assumed to vary continuously in the thickness direction according to the power law expression. The equations of motion are derived using Hamilton’s principle with consideration of von-Kármán nonlinearity. The nanobeam vibration responses are obtained with the aid of the Galerkin technique and the Hamiltonian approach. A vital point of this work is to explore the effects of power law index, slenderness ratio, porosity volume fraction, and hygrothermal changes on the vibration behavior of PFG nanobeams. The results indicate how the thickness effect and porosity volume fraction significantly change the size-dependent vibration response of the FG nanobeam under hygrothermal conditions. The paper findings can be applied to improve the mechanical performance of PFG nanobeams.

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.

Reexamination for linear and nonlinear free vibration of porous sandwich cylindrical shells reinforced by graphene platelets

This paper reexamines the linear and nonlinear free vibration responses of porous sandwich cylindrical shells with graphene platelet (GPL) reinforcements surrounded by an elastic medium under temperature conditions. The graphene platelets reinforced composite (GPLRC) core is assumed to be multilayers, and each layer may have different values of porosity coefficient to achieve a piece-wise functionally graded pattern. By introducing an inhomogeneous model instead of the equivalent isotropic model (EIM), the Young’s moduli along with the shear modulus of porous GPLRC core are predicted through a generic Halpin–Tsai model in which the porosity is included. Thermo-mechanical properties of porous GPLRC core and metal face sheets are assumed to be temperature dependent. Motion equations of porous sandwich cylindrical shells reinforced by GPLs are formulated based on the Reddy’s third order shear deformation theory. In the modeling, von Kármán nonlinear strain-displacement relationships, shell-foundation interaction and thermal effect are also taken into account. The analytical solution for the nonlinear vibration problem is obtained by applying a two-step perturbation approach. Numerical studies are performed to compare the results obtained from the present model and the EIM. The results reveal that, in most cases, the difference of the fundamental frequency between the two models is over 10% and in that case the EIM is not suitable for the linear free vibration analysis of porous sandwich cylindrical shells reinforced by GPLs, whereas for the nonlinear-to-linear frequency ratio curves, the difference between the two models is less than 2% when the non-dimensional vibration amplitude reaches 1.0 and in that case the EIM may be valid in the analysis for the same shell.

Nonlinear vibration of annular radially graded plate subjected to temperature at one edge

An annular plate graded in radial direction made of ceramic and metal having intrinsic geometric imperfection subjected to mechanical and thermal loading is studied for nonlinear vibration response. The gradation of material in radial direction is represented by a simple power law. The Hamilton principle along with Von Karman nonlinear strain displacement relation & FSDT employed for developing governing nonlinear equation of motion. The finite element formulation of radially graded imperfect heated plate is solved by displacement control direct iteration method. The analysis reveals a significant reduction in the hardening stiffness of the plate because of the increase of plate-edge temperature. The results also shows that the plate undergoes initial bending deformation due to the plate-edge temperature that substantially influence the dynamic behaviour of the plate. The geometric imperfection shows substantial impact on the dynamic response of the plate when its inner-edge is heated as compared to that for heated outer-edge of the plate. The nonlinear free and forced vibration response for different volume fraction of involved materials is also explored.

Finite strain-based theory for the superharmonic and subharmonic resonance of beams resting on a nonlinear viscoelastic foundation in thermal conditions, and subjected to a moving mass loading

We address the nonlinear free vibration, superharmonic and subharmonic resonance response of homogeneous Euler–Bernoulli beams resting on nonlinear viscoelastic foundations, under a moving mass and an abrupt uniform temperature rise. The nonlinear differential equation of motion stemming from the Hamiltonian principle and Finite Strain Theory is discretized according to a Galerkin decomposition method, and is solved by means of a multiple time scale method. A comparison between the Finite Strain theory and the Von-Karman approach is discussed, accounting for the effect of temperature rise, linear and nonlinear coefficients of the elastic foundation on the nonlinear vibration history and phase trajectory. At the same time, we check for the sensitivity of the frequency response of the system in superharmonic and subharmonic resonance for different input parameters, namely, location, velocity, and magnitude of the moving load, temperature rise and elastic foundation.

Relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates

In this paper, the relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates is studied based on Reddy higher order shear deformation theory and von Kármán geometric nonlinearity assumption. Kelvin–Voigt viscoelastic model is adopted to describe the viscoelasticity of the plate. According to the geometric equations and constitutive equations, the elastic potential energy, kinetic energy and virtual work done by the external force and viscous damping force are obtained. Furthermore, the nonlinear governing differential equations of the system are achieved by using Hamilton’s principle. On the basis of the harmonic balance method and Runge–Kutta method, the nonlinear free vibration responses and nonlinear forced vibration responses of the system are presented, respectively. The major innovation of the manuscript is that the nonlinear frequency ratio derived from the nonlinear free vibration of the system, is in good agreement with the resonant frequency ratio obtained from the nonlinear forced vibration of the system. Also, the viscous damping coefficient has a little effect on the nonlinear frequency ratio.

Nonlinear free vibration analysis of functionally graded plates and shell panels using quasi-3D higher order shear deformation theory

This paper presents the nonlinear free vibration behavior and response of functionally graded plates and shell panels under various boundary conditions. The displacement formulation followed uses a sinusoidal non-polynomial quasi-3D higher order shear deformation theory with six variables and, the parabolic transverse stress distribution functions are used to model stresses in the thickness direction. Present formulation employs the Green-Lagrange nonlinear strains along with Sander’s approximation to account for larger flexural response which could be experienced by the structure. An eight noded C0 continuous isoparametric finite element is used for numerical implementation of nonlinear finite element solver developed in MATLAB. Developed numerical formulation is employed to study and analyze the nonlinear free vibration response of the spherical, cylindrical and hyperboloid shell panels and the obtained results are seen to be consistent and good agreement with the literature thus establishing the effectiveness and accuracy of the present work.

Nonlinear Bending and Vibration Analysis of Imperfect Functionally Graded Microplate with Porosities Resting on Elastic Foundation Via the Modified Couple Stress Theory

This paper represents the nonlinear bending and free vibration analysis of a simply supported imperfect functionally graded (FG) microplate resting on an elastic foundation based on the modified couple stress theory and the Kirchhoff plate theory (KPT) together with the von-Kármán’s geometrical nonlinearity. The FG microplates with even and uneven distributions of porosities are considered. Analytical solutions for the nonlinear bending and free vibration are obtained. Comparing the obtained results with the published one in the literature shows the accuracy of the current solutions. Numerical examples are further presented to investigate the effects of the material length scale parameter to thickness ratio, the length to thickness ratio, the power-law index and the elastic foundation on the nonlinear bending and free vibration responses of the FG microplate.

Nonlinear free vibration and transient responses of porous functionally graded magneto-electro-elastic plates

Nonlinear free vibration and transient responses of porous functionally graded magneto-electro-elastic plates

Nonlinear free and forced vibration of porous piezoelectric doubly-curved shells based on NUEF model

In this paper, the nonlinear free and forced vibration of porous piezoelectric doubly-curved shells resting on visco-Pasternak foundation is performed within the framework of the nonuniform electric field (NUEF) model and von Kármán geometric nonlinearity assumption. Three types of porosity distribution are adopted to study the effect of porosity on the nonlinear dynamic responses of the current system. According to Hamilton’s principle, the nonlinear governing differential equations are obtained. For the nonlinear free vibration of porous piezoelectric doubly-curved shells, an analytical solution is achieved by utilizing the harmonic balance method. For the nonlinear forced vibration of the present system, the numerical solution is obtained by means of Runge–Kutta method. Through numerical examples, the relationship between nonlinear free vibration responses and nonlinear forced vibration responses is explored under different amplitudes of external excitation, width to thickness ratios, types of porosity distribution, electric field models and elastic foundation parameters. The main novelty of the present study is that the nonlinear frequency ratio that is calculated based on the nonlinear free vibration, is in excellent agreement with the resonant frequency ratio which is determined on the basis of the nonlinear forced vibration.

Influence of porosity distribution on nonlinear free vibration and transient responses of porous functionally graded skew plates

This article deals with the investigation of the effects of porosity distributions on nonlinear free vibration and transient analysis of porous functionally graded skew (PFGS) plates. The effective material properties of the PFGS plates are obtained from the modified power-law equations in which gradation varies through the thickness of the PFGS plate. A nonlinear finite element (FE) formulation for the overall PFGS plate is derived by adopting first-order shear deformation theory (FSDT) in conjunction with von Karman’s nonlinear strain displacement relations. The governing equations of the PFGS plate are derived using the principle of virtual work. The direct iterative method and Newmark’s integration technique are espoused to solve nonlinear mathematical relations. The influences of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the PFGS plate for different skew angles are studied in various parameters. The effects of volume fraction grading index and skew angle on the plate’s nonlinear dynamic responses for various porosity distributions are illustrated in detail.

Analysis and active control of nonlinear vibration of composite lattice sandwich plates

This paper is devoted to investigate the nonlinear vibration characteristics and active control of composite lattice sandwich plates using piezoelectric actuator and sensor. Three types of the sandwich plates with pyramidal, tetrahedral and Kagome cores are considered. In the structural modeling, the von Karman large deflection theory is applied to establish the strain–displacement relations. The nonlinear equations of motion of the structures are derived by Hamilton’s principle with the assumed mode method. The nonlinear free and forced vibration responses of the lattice sandwich plates are calculated. The velocity feedback control (VFC) and H∞ control methods are applied to design the controller. The nonlinear vibration responses of the sandwich plates with pyramidal, tetrahedral and Kagome cores are compared. The influences of the ply angle of the laminated face sheets, the thicknesses of the lattice core and face sheets and the excitation amplitude on the nonlinear vibration behaviors of the sandwich plates are investigated. The correctness of the H∞ control algorithm is verified by comparing with the experiment results reported in the literature. The controlled nonlinear vibration response of the sandwich plate is computed and compared with that of the uncontrolled structural system. Numerical results indicate that the VFC and H∞ control methods can effectively suppress the large amplitude vibration of the composite lattice sandwich plates.

On size-dependent large-amplitude free oscillations of FGPM nanoshells incorporating vibrational mode interactions

At nanoscale, surface free energies of the atoms located on the free surfaces of structures significantly affect their mechanical characteristics. In this study, nonlinear large-amplitude free vibration response of nanoshells prepared from functionally graded porous materials (FGPM) is investigated by taking into account surface stress size effects and vibrational mode interactions. Non-classical shell model is constructed on the basis of the Gurtin–Murdoch type of the surface theory of elasticity having the capability of capturing surface stress size dependency. The accuracy of nonlinear vibration analysis is improved by incorporating the interaction of the main vibration mode and the first, third and fifth symmetric oscillation modes. Moreover, the closed-cell Gaussian-Random field scheme is put to use to extract the mechanical characteristics of FGPM nanoshell. Multiple timescales technique is then applied to achieve surface stress elastic-based nonlinear frequency of FGPM nanoshell analytically for different interactions between vibrational modes. It is revealed that by incorporating the interactions of the main vibration mode and higher symmetric oscillation modes, the behavior of the backbone curves belongs to the nonlinear free oscillation response of FGPM nanoshells changes from hardening to softening schema. It is found that when only the main vibration mode is taken into account, surface elasticity effects makes an enhancement in the significance of the hardening schema. However, by considering the interactions of higher symmetric oscillation modes, surface elasticity effects makes a reduction in the significance of the softening schema.

Surface stress effect on the nonlinear free vibrations of functionally graded composite nanoshells in the presence of modal interaction

As one of the innovative materials, functionally graded (FG) composite materials have the capability to vary microstructure and design attributes from one side to other representing the desired material properties. The prime aim of this work is to analyze the surface stress effect on the nonlinear free vibration response of FG cylindrical nanoshells incorporating various modal interactions. To this end, the Gurtin–Murdoch theory of elasticity together with the von Karman geometrical nonlinearity is implemented to the classical shell theory to construct an efficient size-dependent shell model. In order to take the modal interactions between the main oscillation mode and various symmetric vibration modes, the lateral deflection of the FG nanoshell is expressed as combination of the simple main vibration mode and convergent symmetric modes. Thereafter, the solution of problem is considered as the summation of the homogenous and particular parts to put the Galerkin technique to use. Finally, the multiple time-scales method is employed to achieve analytical expression for the surface elastic-based frequency response of FG nanoshells. It is displayed that in the presence of modal interaction, by increasing the shell deflection, the value of the frequency ratio decreases while in the absence of modal interaction, it enhances.

Snap back testing of unbonded post-tensioned concrete wall systems

Unbonded Post-Tensioned (UPT) precast concrete systems have been shown to provide excellent seismic resistance. In order to improve understanding of the dynamic response of UPT systems, a series of snap back tests on four UPT systems was undertaken consisting of one Single Rocking Wall (SRW) and three Precast Wall with End Columns (PreWEC) systems. The snap back tests provided both a static pushover and a nonlinear free vibration response of a system. As expected the SRW exhibited an approximate bi-linear inertia force-drift response during the free vibration decay and the PreWEC walls showed an inertia force-drift response with increased strength and energy dissipation due to the addition of steel O-connectors. All walls exhibited negligible residual drifts regardless of the number of O-connectors or the post-tensioning force. When PreWEC systems of the same strength were compared the inclusion of further energy dissipating O-connectors was found to decrease the measured peak wall acceleration. Both the local and global wall parameters measured at pseudo-static and dynamic loading rates showed similar behaviour, which demonstrates that the dynamic behaviour of UPT walls is well represented by pseudo-static tests. The SRW was found to have Equivalent Viscous Damping (EVD) between 0.9-3.8% and the three PreWEC walls were found to have maximum EVD of between 14.7-25.8%.

Effects of van der Waals force on nonlinear vibration of electromechanical integrated electrostatic harmonic actuator

An electromechanical integrated electrostatic harmonic actuator is promising for the miniaturization of electromechanical devices. As the dimensions of the actuator decrease, the effects of the van der Waals force become obvious. In this study, by considering the nonlinearity of electrostatic and van der Waals forces, nonlinear vibration equations of the flexible ring of an electrostatic harmonic actuator are deduced. Using these equations, the nonlinear free vibration and nonlinear forced response of the actuator are investigated. The effects of the van der Waals force on the nonlinear vibration of the flexible ring are analyzed. Results show that these effects of the van der Waals force are relatively obvious under some conditions and should be considered.