Nonlinear Natural Frequencies Research Articles

Nonlinear thermo-electro-mechanical free vibrations of sandwich nanocomposite beams bonded with sensor layers considering pyroelectricity

An examination on the linear as well as nonlinear thermo-electro-mechanical free vibrations of a sandwich piezoelectric beam is performed in this research. Core layer has been strengthened with graphene platelets. Two piezoelectric layers as sensors are bonded to the bottom and top surfaces of the core layer. The formulation is developed for beams with general immovable end supports. A nonlinear integro-partial differential equation along with its nonlinear boundary conditions is discharged presuming a second order approximation for the nonlinear normal strain. The direct technique of multiple scales treats the nonlinear governing integro-partial differential equation along with its associated nonlinear boundary conditions. Subsequently, an analytical closed form relation is released for the estimation of amplitude dependent nonlinear natural frequencies. It is shown that the strengthening of a beam that is experiencing a thermal environment during its work with the graphene platelets becomes dysfunctional since it promotes the chance of static instability occurrence. On the other hand, it is seen that the pyroelectricity can postpone the static instability possibility. Furthermore, the pyroelectricity alleviates the hardening nonlinearity. Moreover, at large amplitude of vibrations, the influences of the temperature developing, the pyroelectricity, and the graphene platelet weight fraction decline.

Approximating nonlinear response to harmonic base-excitation using fixed-interface quasi-static modal analysis

The recently developed quasi-static modal analysis (QSMA) technique approximates the amplitude dependent modal characteristics of large-scale finite element models with frictional contact at preloaded interfaces. The nonlinear natural frequencies and damping ratios offer insight into the dynamic behavior of assembled structures, particularly the amount of frictional damping and stiffness loss introduced from an isolated modal oscillation due to the joints. This work presents an approach to approximate the nonlinear steady-state response of a base-excited structure using the nonlinear frequency and damping estimates from QSMA. A Hurty/Craig-Bampton transformation allows for the base-excited system to be expressed in terms of the fixed-interface modal coordinates, thus allowing the extension of previously developed analytical approximations using the prescribed base acceleration and the nonlinear, fixed-interface modes from QSMA. The accuracy and relevance of this non-intrusive approach rely on the applicability of Masing's hypothesis within QSMA to approximate the hysteresis curves from the initial loading curve. The limitations are investigated by considering both Masing and non-Masing joint models. A comparison to reference solutions demonstrates the accuracy of the approximation to efficiently obtain nonlinear steady-state solutions of large-scale models with frictional interfaces for cases that satisfy Masing's hypothesis.

Analysis of a Clapping Vibration Energy Harvesting System in a Rotating Magnetic Field.

This technical note proposes a clapping vibration energy harvesting system (CVEH system) installed in a rotating system. This device includes a rotating wheel, a drive shaft that rotates the wheel, and a double elastic steel sheet fixed on the drive shaft. One of the free ends of the steel is fixed with a magnet, and the free end of the other elastic steel is fixed with a PZT patch. We also install an array of magnets on the periphery (rim) of the wheel. The rim magnets repulse the magnet on the elastic steel sheet of the transmission shaft, causing the elastic steel to oscillate periodically, and slap the piezoelectric patch installed on the other elastic steel sheet to generate electricity. In this study, the authors’ previous study on the voltage output was improved, and the accurate nonlinear natural frequency of the elastic steel was obtained by the dimensional analysis method. By adjusting the rotation speed of the wheel, the precise frequency was controlled to accurately excite the energy harvesting system and obtain the best output voltage. A simple experiment was also performed to correlate with the theoretical model. The voltage and power output efficiencies of the nonlinear frequency to linear frequency excitation of the CVEH system can reach 15.7% and 33.5%, respectively. This study confirms that the clapping VEH system has practical power generation benefits, and verifies that nonlinear frequencies are more effective than linear frequencies to excite the CVEH system to generate electricity.

Nonlinear vibrations of Timoshenko nanobeam using stress driven nonlocal theory

As properties of nonlinear systems can introduce particular responses not present in linear systems, results from linear models can be inaccurate or misleading. Therefore, studying nonlinear vibrations is of extreme importance in structural vibration analysis. Nonlinear vibrations of a Timoshenko nanobeam were analyzed for the first time using the stress driven nonlocal theory (SDM) and the strain gradient theory (SGT). Nonlinear equations of motion were solved for simply supported- simply supported, clamped-clamped and clamped- simply supported beams using the homotopy analysis method. The results indicated that as the dimensionless characteristic parameter increased, the nonlinear natural frequency increased accordingly. High agreement was observed between the results from a linear frequency of Timoshenko beam model based on the SGT and those from previous research.

A new model for non-linear vibration of functionally graded porous nano-Beam based on non-local curvature and strain gradient tensors

In this paper, non-linear free vibration analysis of nano-beam has been studied. The non-local strain gradient theory and curvature tensor are used to show the size effect. The length scale parameter expresses the effect of strain gradient tensor in the non-local strain gradient theory. However, the aim of this article is to show the simultaneous effect of curvature and strain gradient tensors in non-linear vibration of functionally graded porous nano-beams. The effect of curvature tensor is demonstrated with the curvature tensor dependent parameter. Considering non-linear Von Kármán strains and Euler–Bernoulli beam theory, the governing vibrational equation of FG porous nano-beams are derived using Hamilton’s principle in the presence of strain gradient and curvature tensors simultaneously. The non-linear differential equation is extracted by using Galerkin’s method and the non-linear natural frequency of nano-beam is obtained according to Hamiltonian approach. Results represent the simultaneous effects of the length scale and curvature tensor dependent parameters on dimensionless non-linear natural frequencies. Also effects of different parameters such as non-local parameter, length scale parameter, porosity volume index, and power-law index are discussed in the presence and absence of the curvature tensor dependent parameter. Also, the beginning points of stiffness-hardening and stiffness-softening of nano-beam are always constant values in the non-local strain gradient theory, whereas considering the curvature tensor changes the beginning points of stiffness-hardening and stiffness-softening. The results are also compared with previous researches for validation.

Application of the Higher-Order Hamilton Approach to the Nonlinear Free Vibrations Analysis of Porous FG Nano-Beams in a Hygrothermal Environment Based on a Local/Nonlocal Stress Gradient Model of Elasticity.

Nonlinear transverse free vibrations of porous functionally-graded (FG) Bernoulli–Euler nanobeams in hygrothermal environments through the local/nonlocal stress gradient theory of elasticity were studied. By using the Galerkin method, the governing equations were reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency was then established using the higher-order Hamiltonian approach to nonlinear oscillators. A numerical investigation was developed to analyze the influence of different parameters both on the thermo-elastic material properties and the structural response, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, mixture parameter, and the amplitude of the nonlinear oscillator on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.

On combination of all small-scale theories for nonlinear free vibrations and thermal buckling of nanobeams under thermal loading

Nonlinear free vibration characteristics of nanobeams exposed to thermal loading and surface effects are studied in this paper. Within the framework of the Euler–Bernoulli beam theory in conjunction with the nonlocal strain gradient theory, a size-dependent model for nanobeams subjected to thermal loadings has been developed with considering the von Kármán nonlinearity. In the current investigation, size dependencies are captured in the nonlinear free vibrations of nanobeams by using the strain gradient tensor and the curvature tensor. Moreover, the surface elasticity theory is employed to study the effects of surface elasticity on the dynamic behaviors of nanobeams. Governing equations of nonlocal strain gradient nanobeams are obtained using Hamilton’s principle. The influences of nonlocal parameter, length scale parameter resulting from the strain gradient theory, surface elasticity, and temperature on the nonlinear natural frequency of nanobeams are discussed in detail. Moreover, the hardening-softening behaviors of nanoscale beams are investigated in the presence of the curvature tensor. To validate the results of the proposed model for nanobeams, the results obtained in terms of natural frequencies are compared to those from available well-known references.

Nonlinear free and forced vibrations of a fiber-reinforced dielectric elastomer-based microbeam

This paper investigates the nonlinear free and forced vibrations of a fiber-reinforced dielectric elastomer-based microbeam using the Euler–Bernoulli beam theory. The nonlinear material properties have been taken into account based on the original Gent hyperelastic approach. Also, the generalized von Kármán theory is utilized for obtaining nonlinear strains. We assumed that the elastomeric microbeam is incompressible. The system’s free energy fibrous component has been formulated using the Holzapfel-Gasser-Ogden model, which is appropriate to the two families of collagen fibers with the same mechanical properties. An electric part is connected to the system to aim to excite the microbeam considering two types of voltage: a DC (static) and a DC+AC (dynamic) voltage. Governing equations of motion have been derived using Hamilton’s principle. Subsequently, the reduced-order vibration model of the fiber-reinforced microbeam has been obtained by using the Galerkin method. We have calculated the nonlinear natural frequencies (free vibration) via the Hamiltonian scheme. Also, for the forced vibration, the shooting method combined with the arc-length continuation method has been applied to explore the resonant response of the system. The results show that fiber reinforcement affects the performance of the dielectric elastomers. Eventually, the present study can be considered to develop biological structures such as arterial walls and soft tissues because of their fibrous structures.

Thermal Backbone Curves of Nanocomposite Beams Reinforced with Graphene Platelet on Elastic Foundation

In this paper, an examination on the backbone curves of nanocomposite beams reinforced with graphene platelet (GPL) on elastic foundation exposed to a temperature increment is accomplished. By means of the Hamilton principle and in the framework of a third-order shear deformation beam theory, i.e. Reddy’s beam theory (RBT), along with the von Karman nonlinear strains, the nonlinear motion equations are disclosed. The Halpin–Tsai micromechanical model is exploited in order to release the effective modulus of elasticity of the nanocomposite beam while the thermal expansion coefficient, Poisson’s ratio and the mass density are estimated resorting to the rule of mixtures. Elastic foundation, and uniform temperature (UD) rising impacts are incorporated during the mechanical modeling of the nanocomposite beam. The Ritz scheme together with an iterative procedure reveals the nonlinear natural frequency associated to an assumed deflection in order to sketch the corresponding backbone curve. The outcomes are validated in comparison with the available results. Some case studies are established for the sake of clarifying the impressions of the distribution pattern of the GPL, the beam length to its thickness ratio, the weight fraction of the GPL, the elastic foundation, the boundary condition type, and the temperature changes on the first backbone curve of the nanocomposite beam. It is elucidated that the increment of the weight fraction of the GPL with X/O distribution pattern decreases/increases the hardening behavior of clamped–clamped (C-C) and simply-supported beam, while the softening behavior of a clamped-free (C-F) nanocomposite beam is independent of the division pattern, and the weight fraction of the GPL. Moreover, the temperature increment unlike the elastic foundation develops the hardening behavior of the backbone curves of simply supported and C-C nanocomposite beams. Although the backbone curve associated to a C-F nanocomposite beam is invariant with respect to the temperature, the elastic foundation develops the softening trend of C-F nanocomposite beams.

Third order nonlinear vibration of viscoelastic circular microplate based on softening and hardening nonlinear viscoelastic foundation under thermal loading

The nonlinear free vibration of the viscoelastic solid and annular circular microplate is investigated. Nonlinear viscoelastic symmetric and asymmetric foundations are considered. Thermal loading is assumed as constant value. The nonlinear equations of the microplate are derived by Hamilton principle based on third order shear deformation theory (TSDT) and modified couple stress theory (MCST). The finite element methods (FEM) and Newton repeated algorithm are employed to obtain the nonlinear natural frequency. The obtained results are validated with the literatures. Then, the effects of various parameters such as, length scale parameter, outer radius to thickness ratio, maximum deflection to thickness ratio, softening and hardening nonlinear foundation, symmetric and asymmetric nonlinear viscoelastic foundations, different boundary conditions, structure and foundation damping are conducted. The obtained results are compared with several literature and a close agreement are observed between them. The results show that the symmetric and asymmetric nonlinear viscoelastic foundations affect more the nondimensional nonlinear frequency ratio. Reduction effect of temperature difference on MCST nondimensional nonlinear frequency ratio is higher than that of classical theory. The results can be applied micro and nanomachines and smart systems.

Analysis of the vibration of a cracked ferromagnetic rectangular plate in a transverse magnetic field

The vibration of a cracked rectangular plate that is subject to an in-plane force and a transverse magnetic field and is simply supported at all its edges is considered. Based on the von Kármán plate theory, the equation of motion, including an in-plane force, a transverse magnetic field, and a damping coefficient, is derived. This study focused only on the first mode of natural frequency in the vibration. The nonlinear natural frequency is affected by the crack length and aspect ratio. These results are observed to be similar to those reported in the literature. The equations of vibration are reduced to an ordinary differential equation by assuming mode shape and using the Galerkin method. The natural frequency is determined using an ordinary different equation. Using the Runge–Kutta method, the amplitude–time, and velocity–amplitude relationships are determined. The effects of several vibrational parameters—such as the in-plane force, transverse magnetic field, damping coefficient, and crack length—on a cracked plate are considered and discussed in this study. The vibration of plate has been one of the interesting fields of research. Although vibration characteristics of cracked plate have extensively been performed, and have given rise to notable scientific achievements, the vibration of ferromagnetic crack plate is yet to be explored. The derivation in this study determines the vibration of a cracked rectangular plate in a magnetic field and is presented to bring out the effects of crack ratio on vibration.

Free and forced nonlinear vibrations of Bi-Directional functionally graded Euler–Bernoulli porous beams

In this study, the free and forced vibrations of a bi-directional functionally graded porous beam are investigated. The governing equations of motion are derived by Hamilton’s principle, and the reduced temporal equation of motion with cubic and quintic nonlinear terms is obtained using the Galerkin approach. Analytical solutions for the nonlinear natural frequencies in addition to the primary resonance response curves are established by the method of multiple scales. The effects of the axial and transverse functionally graded indexes, initial amplitude, porosity parameter, and the elastic and mass density ratios on the nonlinear frequencies and the forced responses are examined.

On size-dependent nonlinear forced dynamics of MRE-cored sandwich micro-pipes in presence of moving flow and harmonic excitation

This paper is dedicated to examine the size-dependent nonlinear forced dynamics of MRE-cored sandwich micro-pipes in existence of external distributed load and internal flow. For this aim, Euler-Bernoulli beam theory, considering von-Karman nonlinearity, together with the modified couple stress theory (MCST) are exploited to attain a mathematical formulation of the problem. Employing Galerkin’s approach with consideration of two axial and transverse modes, the obtained nonlinear partial differential equations are discretized, from which nonlinear ordinary equations are achieved. Thereafter, static condensation method is utilized in order to reduce the number of the obtained ordinary equations. The acquired nonlinear equations are then solved according to the method of multiple scales, which leads to nonlinear natural frequencies and frequency-response curves of the system. Throughout the numerical analysis, the effect of a smart MRE core existence on the system’s response, considering various boundary conditions, is explored. Additionally, a numerical investigation is implemented to highlight the influences of MRE-related intrinsic properties namely the intensity of the applied magnetic potential and MRE core thickness on the natural frequencies, frequency-response and load-response curves of the system. Totally, the substantial role of the MRE core and its intrinsic characteristics on the dynamic response of the system is explored, which implies that, this smart core could be a good candidate in such sandwich structures in order to control and boost the dynamic properties and response of the mechanical systems.

Large Amplitude Axisymmetric Free Vibration of a Closed Two-Dimensional Spherical Pressure Vessel with a Constrained Volume

The nonlinear axisymmetric vibrational modes of a closed two-dimensional spherical pressure vessel with a constrained volume are presented using the fundamental forms of surfaces. The strain energy of the spherical pressure vessel is derived as a quadratic function of the initial Eulerian, added, and total Lagrangian strains. Two-dimensional bilinear elements described in spherical polar coordinates with the symmetric boundary condition along both sides of the meridian line are used for investigating the nonlinear axisymmetric mode shapes of a spherical pressure vessel. Numerical results were obtained by a nonlinear finite element approach and verified in the case of an empty spherical shell for both lower and upper branches. This study showed that the spherical pressure vessel gave a higher nonlinear axisymmetric natural frequency than the empty spherical shell. The elastic modulus has a large effect on the nonlinear axisymmetric frequencies of both the lower and upper branches for the spherical pressure vessel. Changing the thickness and initial internal pressure significantly affects only the lower branch of the spherical pressure vessel at high mode numbers. The results also indicate that the ratios of the nonlinear natural frequencies of the lower to upper branches decrease when the vibrational modes increase.

Primary nonlinear damped natural frequency of dielectric composite beam reinforced with graphene platelets (GPLs)

Primary nonlinear damped natural frequency of dielectric composite beam reinforced with graphene platelets (GPLs)

Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam

Abstract In this present work, linear and non-linear vibration of multi-supported nanobeams, which are a fundamental part of the nano-electromechanical systems, is examined. To the best of the researchers’ knowledge, there is no study performed into multi-supported nanobeam in the literature. The governing equations of the system are obtained by dint of the Hamilton principle and solved via the perturbation technique which is divided linear and non-linear parts of the main equations. The natural frequencies and mode shapes are calculated from the linear problem. The non-linear natural frequencies and amplitude-phase modulation graphs are obtained from the non-linear equation. All equations are written in generalized form, and 3, 4 and 5 supported nanobeams are investigated in detail. The nonlocal coefficient, support number and position and end condition types are focused on. The three to one internal resonance cases are also investigated. It is occurred that the clamped-end conditions shift right in the hardening behaviour graphs more than the simply supported condition. Moreover, it is shown that the supported numbers play a significant role in natural frequency.

Flow-excited membrane instability at moderate Reynolds numbers

In this paper, we study the fluid–structure interaction of a three-dimensional (3-D) flexible membrane immersed in an unsteady separated flow at moderate Reynolds numbers. We employ a body-conforming variational fluid–structure interaction solver based on the recently developed partitioned iterative scheme for the coupling of turbulent fluid flow with nonlinear structural dynamics. Of particular interest is to understand the flow-excited instability of a 3-D flexible membrane as a function of the non-dimensional mass ratio ( $m^{*}$ ), Reynolds number ( $Re$ ) and aeroelastic number ( $Ae$ ). For a wide range of parameters, we examine two distinct stability regimes of the fluid–membrane interaction: deformed steady state (DSS) and dynamic balance state (DBS). We propose stability phase diagrams to demarcate the DSS and DBS regimes for the parameter space of mass ratio versus Reynolds number ( $m^{*}$ - $Re$ ) and mass ratio versus aeroelastic number ( $m^{*}$ - $Ae$ ). With the aid of the global Fourier mode decomposition technique, the distinct dominant vibrational modes are identified from the intertwined membrane responses in the parameter space of $m^{*}$ - $Re$ and $m^{*}$ - $Ae$ . Compared to the deformed steady membrane, the flow-excited vibration produces relatively longer attached leading-edge vortices which improve the aerodynamic performance when the coupled system is near the flow-excited instability boundary. The optimal aerodynamic performance is achieved for lighter membranes with higher $Re$ and larger flexibility. Based on the global aeroelastic mode analysis, we observe a frequency lock-in phenomenon between the vortex-shedding frequency and the membrane vibration frequency causing self-sustained vibrations in the dynamic balance state. To characterize the origin of the frequency lock-in, we propose an approximate analytical formula for the nonlinear natural frequency by considering the added mass effect and employing a large deflection theory for a simply supported rectangular membrane. Through our systematic high-fidelity numerical investigation, we find that the onset of the membrane vibration and the mode transition has a direct dependence on the frequency lock-in between the natural frequency of the tensioned membrane and the vortex-shedding frequency or its harmonics. These findings on the fluid-elastic instability of membranes have implications for the design and development of control strategies for membrane wing-based unmanned systems and drones.

Nonlinear free and forced vibration of carbon nanotubes conveying magnetic nanoflow and subjected to a longitudinal magnetic field using stress-driven nonlocal integral model

In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Kármán strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales.Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study, the first-order shear deformation theory (FSDT) is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite (FG-GPLRC). The vibration analyses of the FG-GPLRC truncated conical shell are presented. Considering the graphene platelets (GPLs) of the FG-GPLRC truncated conical shell with three different distribution patterns, the modified Halpin-Tsai model is used to calculate the effective Young’s modulus. Hamilton’s principle, the FSDT, and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell. The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell. Then, the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method. The effects of the weight fraction and distribution pattern of the GPLs, the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed. This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

Impact of blast and mechanical loads on the shear deformable stiffened sandwich plate with an auxetic core layer in thermal environment

By using the first order shear deformation theory (FSTD), this paper presents the results of the nonlinear dynamic behavior and natural frequencies of sandwich plate supported by elastic foundations in thermal environment and subjected to mechanical load and blast loading. This work takes advantage of the sandwich plate configuration with three layers: graphene platelet –reinforced composite (GPL) layer – auxetic layer – FGM layer, to analyze the dynamic and vibration problem, in which the auxetic core layer has a negative Poisson's ratios and the FGM layer is reinforced by stiffeners made of full metal or full ceramic depending on a situation of stiffeners at the metal-rich or ceramic-rich side of the plate respectively. Corresponding to the combination of material layers, the mechanical quantities of the problem are processed and calculated to suit the structure and reinforcement conditions. Numerical results are provided to explore the influences of geometrical parameters, elastic foundation parameters, GPL volume fraction, blast and mechanical loads on the nonlinear dynamic behavior and vibration of sandwich plate resting on elastic foundation and in thermal environment. In addition, the study is not only assumed that the material properties depend on environment temperature variation, but also considered the thermal stresses in the stiffeners, as well as considered the effect of imperfections in the original shape of the structure.