Nonlinear Vibration Frequency Research Articles

Vibrational sum frequency generation (VSFG) spectroscopy of water adsorption on surfaces of yttria-stabilized cubic zirconia (YSZ).

Yttria-stabilized zirconia (YSZ) is found in a wide range of applications, from solid-oxide fuel cells to medical devices and implants. A molecular-level understanding of the hydration of YSZ surfaces is essential for optimizing its performance and durability in these applications. Nevertheless, only a limited amount of literature is available about the surface hydration of YSZ single crystals. In this study, we employ surface-sensitive non-linear vibrational sum frequency generation spectroscopy to investigate the hydration of YSZ(100), (110), and (111) single crystal substrates under ambient laboratory conditions. Three types of hydroxyl groups were identified at all three YSZ-D2O interfaces: (i) hydroxyls on the metal sites of Zr or Y resulting from the dissociative chemisorption of water, (ii) hydroxyls from proton adsorption to O sites formed from water dissociation, and (iii) hydroxyl groups as part of the physisorbed water at the interface.

Thermomechanical Nonlinear Vibration of Axially Loaded Functionally Graded Material Cylindrical Panels Including Porosity

For the first time, the combined influences of axial compressive load, porosity, geometric imperfection, elastic edge restraint, and elevated temperature on the nonlinear free vibration of functionally graded material (FGM) cylindrical panels are investigated in this paper. Unlike previous studies, the structural model considered in this paper includes practical situations and conditions arising in the design process and application of FGM cylindrical panels. The properties of material constituents are assumed to be temperature-dependent, and the effective properties of porous FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on first-order shear deformation theory, taking into account von Kármán–Donnell nonlinearity and initial geometric imperfection. Analytical solutions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to derive a time differential equation containing both quadratic and cubic nonlinear terms. This differential equation is numerically integrated employing the fourth-order Runge–Kutta scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies of porous FGM cylindrical panels. The study reveals that natural frequencies are strongly decreased and that frequency nonlinearity is more significant due to axial compressive loads.

An efficient Volterra series approach for identifying nonlinear system vibration responses to an excitation

This paper focuses on the issue of nonlinear vibration responses identification of nonlinear systems. An efficient algorithm is presented, in which the nonlinear vibration system under studied is decomposed into a multiple-input/single-output (MISO) linear system with a series of power characterized inputs based on Volterra series, and the nonlinear output responses of different orders are identified by taking power spectra operation to the input and output data, revealing the contributions of each order nonlinearity to the output of the system. Compared to the existing method, the input signal of the system required in the presented approach need not remain unchanged in waveform and adjustable in magnitude. To verify the approach, a classic Duffing–Van der Pol oscillator was simulated, obtaining results very close to the fourth order Runge–Kutta method. Finally, an experiment analysis was carried out, in which the vibration transmission properties of a bolt connection were tested when the bolt was tight and loose, respectively. The results proved that the approach is effective in identifying nonlinear vibration frequency responses.

Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment

This study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length.

Nonlinear free vibration analysis of multi-directional functionally graded porous sandwich plates

Multi-directional functionally graded materials (MFGMs) have attracted significant research attention due to their advantages over one-directional FGMs. In MFGM structures, material properties can be tailored to grade in the required direction, overcoming practical problems like excessive temperature gradients or extreme deflections. This paper aims to investigate the nonlinear free vibration of MFG porous sandwich plates to improve their application in sandwich structures. The two outer layers of the plates are composed of three-directional functionally graded material (3D-FGM), with a bi-directional functionally graded material (2D-FGM) core layer. Additionally, the porosity distribution within the material matrix is assumed to be either even or uneven across the plate thickness. A higher-order finite element model based on Shi's plate theory and the von Kármán assumption is developed. The nonlinear free vibration frequencies are determined through the maximum vibrational amplitude using an iterative algorithm with a displacement control strategy. The accuracy and effectiveness of the proposed model are demonstrated through a comparison with published data. The results show that the vibration response is significantly influenced by various parameters. Specifically, increasing the material gradient indexes in the thickness direction enhances the frequency ratio, while increasing the gradient indexes in the length and width directions reduces it. Higher porosity coefficients in the core layer decrease the frequency ratio, whereas higher pore coefficients in the outer layers increase it. The additional knowledge gained from this study can help with future analysis and design procedures related to the nonlinear responses of these complex structures.

Nonlinear dynamic behavior of single-layered black phosphorus with an attached mass

The investigation of black phosphorus-based (BP)-based mass sensors provides theoretical support for the development of mass-detection devices. This study examines the non-linear dynamic behavior of a rectangular single-layered BP (SLBP) with an attached mass through the utilization of molecular dynamics (MD) simulations and a nonlinear orthotropic plate model (NOPM) with a concentrated mass. The results indicate that significant deformation of an SLBP with an attached mass necessitates consideration of geometric nonlinearity, although the attached mass does not affect the deformation. Additionally, this paper discusses the impact of the attachment mass and the amplitude of harmonic force on the non-linear forced vibration of the SLBP. It is observed that as the attachment mass increases, the nonlinear vibration resonance frequency decreases, while the peak amplitude increases. Furthermore, the thermal nonlinear vibration of SLBP with an attached mass has been investigated, revealing that an increase in the attached mass leads to a decrease in the nonlinear vibration frequency but an increase in the amplitude of the nonlinear vibration of SLBP with an attached mass. Overall, comparison with MD simulation results, this investigation suggests that NOPM with a concentrated mass effectively describes the nonlinear vibration behavior of an SLBP with an attached mass, providing theoretical support for designing such devices to detect attached masses.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Nonlinear vibration analysis of geometrically imperfect FGM Timoshenko beams with porosity and elastically restrained ends in thermal environments

This paper aims to analyze the combined influences of pore imperfection, initial geometric imperfection, elasticity of tangential constraints of ends, elastic foundations and elevated temperature on the nonlinear vibration of functionally graded material (FGM) beams. The pores are assumed to distribute into FGM according to even and uneven patterns and effective properties of porous FGM are estimated using a modified rule of mixture. Motion equations of geometrically imperfect beams are established within the framework of Timoshenko beam theory incorporating von Kármán nonlinearity and foundation interaction. Analytical solutions are assumed to satisfy simply supported and clamped conditions of beam ends and Galerkin method is applied to derive a nonlinear time ordinary differential equation. The frequencies of nonlinear free vibration are determined employing fourth-order Runge-Kutta numerical integration scheme. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies. The results reveal that frequency nonlinearity is more significant for beams with tangentially restrained ends, especially at elevated temperatures. It is also found that geometric imperfection increases the linear frequency and lowers ratio of nonlinear-to-linear frequencies.

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.

Nonlinear vibration analysis of a double curved shallow sandwich shell in which the core made of three-phase nanocomposite and the two-outer layer of electromagnetic materials

This paper investigates the nonlinear vibrations of a double curved shallow sandwich shell with a three-phase nanocomposite core and two outer electromagnetic layers considering the thermal environment's influence. Relying on the Halpin–Tsai model, the core is assumed to be made of some composite plies incorporating three constituents: carbon nanotubes (CNTs) or graphene nanoplatelets (GPLs) as nanocomposites, fibers and polymeric matrix (resin). Governing equations calculating the nonlinear vibration of the double curved shallow shell were obtained by combining Reddy's first-order shear deformation theory (FSDT) and the Von Kármán geometrical nonlinearity. Later, this system of equations is transformed into the nonlinear ordinary differential expressions by utilizing the Galerkin procedure. Thus, the effects of material properties, temperature, geometrical properties, elastic foundations to the frequency of linear and nonlinear free vibration, and amplitude–frequency relations of the double curved shallow sandwich shell were discussed. The results demonstrate that the free vibration frequency of the shell increases if the magnetic field enhances, whereas that decreases when the voltage expands in both models as CNTs and GPLs. Simultaneously, the shallow shell has a better ability to force load compared to a plate with the same dimensions. These outcomes can make a profit in the design and manufacturing of new smart structures applied in energy harvesting, and medicine.

Nonlinear Forced Vibration of Functionally Graded Graphene-Reinforced Composite (FG-GRC) Laminated Cylindrical Shells under Different Boundary Conditions with Thermal Repercussions

The key aim of this research is to develop an analytical model for the forced vibration of graphene-reinforced composite (GRC) cylindrical shell with viscous damping and thermal environment effects under several boundary conditions. The prescribed mechanical and thermal characteristics of the GRC layer are evaluated utilizing a micromechanical extended Halpin–Tsai technique. Further, the governing equations are determined to be grounded on the shear deformation theory (SDT) with the von Kármán-form in terms of the geometric nonlinearity. The basic nonlinear partial differential governing equation is transformed into a nonlinear ordinary differential equation with the Galerkin-based method. The resulting ordinary governing differential equations are systematically solved based on the multiple scales scheme in order to obtain the nonlinear forced vibration frequency response of the laminated GRC cylindrical shell in the presence of damping impact and subjected to multiple boundary conditions. The validation of the obtained expressions is achieved by comparing them with the available literature data where the comparison revealed a clear consistency between them. Moreover, a parametric investigation is provided to illustrate the impacts of the distribution of graphene layers, elastic foundation coefficients, damping ratios, temperature effects and dimensionless radial excitation amplitude on the forced nonlinear amplitude-frequency ratio responses for a functionally-graded graphene-reinforced composite (FG-GRC)-layered cylindrical shells. The analytical outcomes indicate that the different distribution types of graphene, elastic foundation coefficients, damping ratios, temperature, and radial excitation parameters have a noteworthy impact on the frequency–amplitude behaviors of an FG-GRC-layered cylindrical shell. In addition, the new insights gained from this research might contribute to a deeper understanding of the nonlinear forced vibration responses in subsequent analysis and design techniques for giving appropriate benchmark findings.

Nonlinear Vibration and Stability Analysis of Rotating Functionally Graded Piezoelectric Nanobeams

Presented herein is an investigation for the nonlinear vibration and stability analysis of rotating functionally graded (FG) piezoelectric nanobeams based on the nonlocal strain gradient theory. The present model can be regarded as a simplified version for the rotating nanowire of biomechanical nanogenerators. The Hamilton principle is used to derive nonlinear equations of motion and their related boundary conditions, which are then discretized to form a set of algebraic equations. Accordingly, the nonlinear vibration frequencies and buckling loads of the nanobeams can be determined by an iterative method. A parametric study of rotational velocity, nonlocal parameter, material length parameter, power-law index, and electrostatic voltage on the dynamic stability behavior of such nanobeams is also presented. In the cantilever case, increasing the nonlocal parameter and material length parameter can result in a stiffness-hardening effect that is unaffected by rotational velocity and the material length parameter to nonlocal parameter ratio. Yet, this has not been reported previously. More importantly, incorporating the effect of geometric nonlinearity on the dynamic responses and stability results of the nanobeams is indispensable. In particular, new observations for the coupling effect of vibration amplitude and power-law index on the electric potential effect are useful for the design of rotating microelectromechanical devices.

On the primary resonance of laminated moderately-thick plates containing of heterogeneous nanocomposite layers considering nonlinearity

In this study, the primary resonance of laminated plates consisting of heterogeneous nanocomposite layers is investigated comparatively within first order shear deformation theory (FSDT) and classical lamination plate theory (CLPT). One of the features of the work is the generalization of the FSDT for homogeneous anisotropic laminated plates to laminated functionally graded anisotropic nanocomposite plates. The mechanical properties of the matrix reinforced with carbon nanotube (CNT), and fundamental relations of laminated moderately thick plates consisting of heterogeneous anisotropic nanocomposite layers are modeled theoretically within FSDT using von Kármán type nonlinear theory. It then derives equations of motion in the form of nonlinear partial differential equations (NL-PDEs) within FSDT. NL-PDEs equations are transformed into nonlinear ordinary differential equations (NL-ODEs) by Galerkin method and are solved by multi-scale method. The nonlinear forced vibration frequency as the function of amplitude at primary resonance within both theories are obtained for the first time. In addition, backbone curve and nonlinear frequency/linear frequency ratio are found. The reliability and accuracy of the proposed formulation is verified by comparing with the results in the literature. Finally, the effects of external excitation, non-linearity, and variation of CNT patterns on the forced vibration frequencies are examined.

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.

An analysis of nonlinear thickness vibration frequencies of multi-layered film bulk acoustic resonators

The fast reduction of the physical size of film bulk acoustic wave resonators as a layered structure implies the intensification of the electric field which can induce large deformation in the functioning state of devices as a circuit element. Consequently, the nonlinear behavior of the resonator and accompanying properties are to be included and evaluated in the development and optimization for performance improvement. With this objective, the nonlinear formulation of a multilayered film bulk acoustic resonator is presented for the analysis of vibration frequencies and mode shapes with the consideration of larger mechanical deformation. The dominantly linear relationship between the voltage or deformation and frequency is obtained to understand the nonlinear behavior and properties which have been subjected to extensive research analytically and experimentally to satisfy the application needs in all modes of communications and network technology.

On the nonlinear computational nanomechanics of microtubules using coupled generalized differential quadrature and multiple scales method

Up to now, some medical and scientific studies have been conducted to have therapy without any side effects or complications via prognosis about the behaviors of the microtubules. So, nonlinear frequency characteristics of microtubules inside axons with respect to various size effect component is investigated. MAP Tau proteins, as well as cytoplasm, are taken into account as a nonlinear foundation. Additionally, the Microtubules are considered to be a moderately thick cylindrical nanosheet which is curved. Von-Karman’s nonlinearity and the shell theory are employed to obtain the displacement strain associated with the substructures. The formulation, which is extracted via Hamilton’s principle, is solved by employing generalized differential quadrature method (GDQM) together with multiple scales approach (MSA). The influence of size, nonlinear foundation, and mode number are investigated for different end conditions. Based on presented numerical outcomes, the impact of cytoplasm surrounding on the ratio of vibrational frequency related to the system is not regular. Furthermore, microtubules’ nonlinear vibrational frequency responses depend on their physical environment condition in the cell; like the membrane and cell matrix. Additionally, it should be mentioned that microtubules can be utilized as biosensors to detect malignant tumors.

Vibrational analysis of pipes based on the drift-flux two-phase flow model

Linear and nonlinear free vibrations of supported pipes containing two-phase flow were modeled in different flow regimes by including gravitational force, structural and two-phase damping effects. To model the flow, the drift-flux model was utilized. Discretization of the dynamical equation was performed with the help of the Galerkin scheme. The linear vibrational frequency of the system was determined by solving the eigenvalue problem. Also, a mathematical closed-form expression for the nonlinear vibrational frequency is presented. For the validation purpose, theoretical and experimental data were obtained from literature and compared with the results of this work in various operating conditions. A detailed parametric analysis was also carried out to examine the influence of flow parameters, geometry, and physical properties on the dynamics of the system. It was concluded that by increasing the inner and outer pipe diameters, vibrational frequencies of the system decrease and increase, respectively. It was found that a lower liquid phase density leads to an improvement in the vibrational stability of the system. Besides, it was demonstrated that by increasing the void fraction/mixture velocity in the system, linear and nonlinear vibrational frequencies increase/decrease. Furthermore, a higher initial amplitude caused a larger nonlinear frequency shift. The outcomes of the current analysis can be applied as a frame to evaluate and optimize the performance of structures transporting two-phase flows.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Nonlinear Analyses of Porous Functionally Graded Sandwich Piezoelectric Nano-Energy Harvesters under Compressive Axial Loading

In this study, a sandwich piezoelectric nano-energy harvester model under compressive axial loading with a core layer fabricated of functionally graded (FG) porous material is presented based on the nonlocal strain gradient theory (NSGT). The von Karman type geometric nonlinearity and the axial loading were considered. The electromechanical governing equations were obtained using Hamilton’s principle. The nonlinear vibration frequencies, root mean square (RMS) voltage output and static buckling were obtained using the Galerkin method. The effects of different types of porous distribution, porosity coefficients, length scale parameters, nonlocal parameters, flexoelectricity, excitation frequencies, lumped mass and axial loads on the natural frequency and voltage output of nanobeams were investigated. Results show that the porous distributions, porosity coefficient of porous materials, the excitation frequencies and the axial load have a large effect on the natural frequency and voltage output of the sandwiched piezoelectric nanobeams. When the NSGT is considered, the critical buckling load depends on the values of the nonlocal parameters and strain gradient constants. In addition, the electromechanical conversion efficiency of the post-buckling process is significantly higher than that of the pre-buckling process. The flexoelectric effect can significantly increase the RMS voltage output of the energy harvester.

On the primary resonance of non-homogeneous orthotropic structures with viscous damping within shear deformation theory

This study is one of the first attempts on the nonlinear forced vibration behaviors of nonhomogeneous orthotropic (NHO) structural members with linear viscous damping at primary resonance within the shear deformation theory (SDT). First, mechanical properties of double curved systems consisting of NHO materials are mathematically modeled and nonlinear basic relations are established. Using these relations, nonlinear basic partial differential equations are derived and reduced to ordinary differential equations with second and third order nonlinearities by Galerkin procedure. Multiple-scales method is used to obtain the nonlinear forced vibration frequency–amplitude dependence of double curved NHO structural members with damping. After testing the correctness of the proposed methodology, the influences of non-homogeneity, damping, transverse shear deformations and anisotropy on nonlinear forced vibration frequencies for various structural members at the primary resonance are investigated and interpreted in detail.