Nonlinear Frequency Ratio Research Articles

Semi-analytical Solution for Nonlinear Transverse Vibration Analysis of an Euler–Bernoulli Beam with Multiple Concentrated Masses Using Variational Iteration Method

In this article, linear and nonlinear transverse vibration of Euler–Bernoulli beams with multiple concentrated masses have been investigated using variational iteration method (VIM), and the effects of concentrated mass on natural frequencies and mode shapes have been taken into account as well. VIM is a powerful method with high convergence which gives analytical solution to linear problems and is capable of being extended to present semi-analytical solutions to nonlinear ones. The proper choice of Lagrange’s multiplier and Initial Function can increase the convergence speed. In this study, along with presenting the suitable trend for determining these two functions, the obtained frequencies in linear and nonlinear states are compared with those calculated from other methods, and the accuracy and convergence speed of this procedure are examined. It is concluded that with increasing the intensity of concentrated mass, linear natural frequency and the ratio of nonlinear frequency to linear one will decline.

Nonlinear vibration analysis of electro-hygro-thermally actuated embedded nanobeams with various boundary conditions

In this manuscript, a non-classical beam model is proposed based on the Eringen’s nonlocal elasticity theory for nonlinear vibration analysis of magneto–electro–hygro–thermal piezoelectric functionally graded (PFG) nanobeams rested in elastic foundation. The Hamilton’s principle is employed to derive the nonlinear motion equation and related boundary conditions within the framework of Euler–Bernoulli beam model with von-Karman type nonlinearity. The power–law distribution model is utilized to explain the continuous variation of material properties through the thickness of FG nanobeam. The Galerkin-based method is employed to discretize the nonlinear partial differential motion equations into a set of time-dependent ordinary differential equations. The numerical results are given to investigate the influence of various parameters such as nonlocal parameter, amplitude ratio, aspect ratio, power–law index, external voltage, temperature change, magnetic field, moisture effect and Winkler–Pasternak elastic foundations on the nonlinear frequency ratio of PFG nanobeam for various boundary conditions. It is expressly shown that the nonlinear frequency ratio of PFG nanobeam is significantly influenced by these effects. Some of the numerical results are compared with well-known literature, which are proved to be in good agreement. The presented results can serve as benchmarks for future nonlinear analysis of PFG nanobeams.

A simulation of atomic force microscope microcantilever in the tapping mode utilizing couple stress theory

The nonlinear vibration behavior of a Tapping mode atomic force microscopy (TM-AFM) microcantilever under acoustic excitation force has been modeled and investigated. In dynamic AFM, the tip–surface interactions are strongly nonlinear, rapidly changing and hysteretic. First, the governing differential equation of motion and boundary conditions for dynamic analysis are obtained using the modified couple stress theory. Afterwards, closed-form expressions for nonlinear frequency and effective nonlinear damping ratio are derived utilizing perturbation method. The effect of tip connection position on the vibration behavior of the microcantilever are also analyzed. The results show that nonlinear frequency is size dependent. According to the results, an increase in the equilibrium separation between the tip and the sample surface reduces the overall effect of van der Waals forces on the nonlinear frequency, but its effect on the effective nonlinear damping ratio is negligible. The results also indicate that both the change in the distance between tip and cantilever free end and the reduction of tip radius have significant effects on the accuracy and sensitivity of the TM-AFM in the measurement of surface forces. The hysteretic behavior has been observed in the near resonance frequency response due to softening and hardening of the forced vibration response.

Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model

This paper comprehensively studies the nonlinear vibration of functionally graded nano-beams resting on elastic foundation and subjected to uniform temperature rise. The small-size effect, playing an essential role in the dynamical behavior of nano-beams, is considered here applying the innovative stress driven nonlocal integral model due to Romano and Barretta. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Karman strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural frequency for four different boundary conditions is then established employing the Homotopy Analysis Method. The nonlinear natural frequencies, evaluated according to the stress-driven nonlocal integral model, are compared with those obtained by Eringen differential model. Finally, the effects of different parameters such as length, elastic foundation parameter, thermal loading and nonlocal characteristic parameter are investigated. The emergent results establish that when the nonlocal characteristic parameter increases, the nonlinear natural frequencies obtained by the stress-driven nonlocal integral model reveal a stiffness-hardening effect. On the other hand, Eringen's differential law reveals a stiffness-softening effect excepting the case of cantilever nano-beam. Also, increase in temperature and the elastic foundation parameter leads to increase in the nonlinear frequency ratios in Eringen differential model but decrease in the frequency ratios in the stress-driven nonlocal integral model.

Haar Wavelet Method for Nonlinear Vibration of Functionally Graded CNT‐Reinforced Composite Beams Resting on Nonlinear Elastic Foundations in Thermal Environment

This paper presents a simple and effective approach based on the Haar wavelet discretization method (HWDM) for the nonlinear vibration analysis of carbon nanotube‐reinforced composite (CNTRC) beams resting on a nonlinear elastic foundation in a thermal environment. Material properties are assumed to be functionally graded (FG) in the thickness direction and temperature‐dependent and are evaluated through the extended rule of mixture. Based on the first‐order shear deformation beam theory in conjunction with the von Kármán nonlinearity, the nonlinear governing equations of CNTRC beams on nonlinear elastic foundations are derived as well as the related boundary conditions. Moreover, the initial thermal stress due to uniform temperature rise is considered in this study. To evaluate the nonlinear natural frequencies, the obtained equations are discretized into a set of nonlinear algebraic equations through HWDM and then the direct iteration technique is employed to solve the resulting algebraic equations. The convergence and comparison studies are carried out, and the results indicate that the numerical rate of convergence of the proposed method is in agreement with the convergence theorem, and good accuracy of the present results is observed. Studies on the effects of different parameters such as CNT volume fraction, distribution type of CNTs, foundation stiffness coefficients, boundary condition, slenderness ratio, temperature rise, and initial thermal stress on the linear frequencies and the nonlinear frequency ratios are also reported. This study offers an insight into the vibration behaviors of CNTRC beams resting on nonlinear elastic foundation subjected to the uniform temperature rise.

Nonlinear high-order dynamic stability of AL-foam flexible cored sandwich beam with variable mechanical properties and carbon nanotubes-reinforced composite face sheets in thermal environment

In this paper, the nonlinear dynamic stability analysis of sandwich beam including AL-foam flexible core and carbon nanotubes-reinforced composite face sheets subjected to axial periodic load are investigated by using generalized differential quadrature method. The flexible core of sandwich beam is made of Aluminum alloy foam with variable mechanical properties in the thickness direction. With considering the high-order geometrical nonlinearity in the core and face sheets, the high-order sandwich panel theory and modified couple stress theory are employed for AL-foam flexible core and face sheets, respectively. The governing nonlinear partial differential equations of dynamic stability are derived from the Hamilton’s principle and then discretized by using generalized differential quadrature method to convert them into a linear system of Mathieu–Hill equations. These formulations lead to nine partial differential equations which are coupled in axial and transverse deformations. The boundaries of the instability region are achieved by Bolotin’s method and are illustrated in the dimensionless nonlinear excitation frequency (Ω NL) and excitation frequency ratio (Ω NL/Ω L) to load amplitude plane. A parametric study is carried out to investigate the influence of some important parameters such as slenderness ratio, face sheet thickness, temperature rise, carbon nanotube volume fraction, static load factor, coefficients of Pasternak foundation, and end supports on the nonlinear dynamic instability characteristics of AL-foam core sandwich beam. The numerical results show that with temperature increasing, the nonlinear excitation frequency (Ω NL) and width of corresponding unstable zone decrease, but dynamic frequency ratio (Ω NL/Ω L) and associated unstable region increase. With an increase in the application of sandwich structures for compressible core in advanced industries such as spacecraft, high-speed aircraft, naval vessels, transportation, and automobiles, a further interest in the problem-involving dynamic instability of structures has resulted. Because of their applications, sandwich structures are frequently exposed to periodic axial compressive forces and so the dynamic instability has been a very important topic in structural dynamics and is of practical importance in different engineering industries.

An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates

In this article, an analytical solution to the moderately large amplitude transverse vibration of thin functionally graded micro-plates (FGMPs) is presented based on a practical approach. The size-dependent nonlinear governing equation is obtained in conjunction with the Kirchhoff’s plate and modified couple stress theories. The material properties of functionally graded (FG) micro-plates are varied according to the Reddy’s model. The employed non-classical theory contains one material length scale parameter to capture the size effects. The highly nonlinear governing equation is solved by means of homotopy analysis method to obtain accurate analytic approximations. The both of simply supported and clamped micro-plates with immovable edges are considered. Comparison of the present results with earlier studies wherever possible confirms the reliability and effectiveness of the present formulation for the design purpose. Furthermore, the effects of different parameters such as material gradient index, length scale parameter, and aspect ratio on the nonlinear frequency ratio are investigated.

Geometrically nonlinear vibration analysis of multiferroic composite plates and shells

In this article, a layerwise shear deformation theory is incorporated for geometrically nonlinear vibration (GNV) analysis of multiferroic composite plates and doubly curved shells. The coupled constitutive equations involving ferroelastic, ferroelectric and ferromagnetic properties of multiferroic composite materials along with the total potential energy principle are utilized to derive the finite element formulation for the multiferroic or magneto-electro-elastic (MEE) plates/shells. The electric and the magnetic potentials are assumed to vary linearly in the transverse direction. The electric and magnetic potential distribution in the plate/shell is computed by using the Maxwell's electromagnetic relations. The significance of geometric nonlinearity has been considered using the von Kármán nonlinear strain-displacement relations. Importance of curvature aspect ratio, curvature ratio and the thickness aspect ratio on the nonlinear frequency ratios of the multiferroic/MEE doubly curved shells has been investigated. The backbone curves for multiferroic plates and shells have been studied by considering various aspect ratios. Impact of layer stacking sequence, boundary conditions and coupled fields on the central deflection and nonlinear frequency ratio of the multiferroic plates and shells have been investigated.

Nonlinear static and vibration analysis of Euler-Bernoulli composite beam model reinforced by FG-SWCNT with initial geometrical imperfection using FEM

In this paper, the nonlinear static and free vibration analysis of Euler-Bernoulli composite beam model reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNTs) with initial geometrical imperfection under uniformly distributed load using finite element method (FEM) is investigated. The governing equations of equilibrium are derived by the Hamilton's principle and von Karman type nonlinear strain-displacement relationships are employed. Also the influences of various loadings, amplitude of the waviness, UD, USFG, and SFG distributions of carbon nanotube (CNT) and different boundary conditions on the dimensionless transverse displacements and nonlinear frequency ratio are presented. It is seen that with increasing load, the displacement of USFG beam under force loads is more than for the other states. Moreover it can be seen that the nonlinear to linear natural frequency ratio decreases with increasing aspect ratio (h/L) for UD, USFG and SFG beam. Also, it is shown that at the specified value of (h/L), the natural frequency ratio increases with the increasing the values amplitude of waviness while the dimensionless nonlinear to linear maximum deflection decreases. Moreover, with considering the amplitude of waviness, the stiffness of Euler-Bernoulli beam model reinforced by FG-CNT increases. It is concluded that the R parameter increases with increasing of volume fraction while the rate of this parameter decreases. Thus one can be obtained the optimum value of FG-CNT volume fraction to prevent from resonance phenomenon.

Nonlinear vibration analysis of composite laminated trapezoidal plates

Nonlinear vibration characteristics of composite laminated trapezoidal plates are studied. The geometric nonlinearity of the plate based on the von Karman's large deformation theory is considered, and the finite element method (FEM) is proposed for the present nonlinear modeling. Hamilton's principle is used to establish the equation of motion of every element, and through assembling entire elements of the trapezoidal plate, the equation of motion of the composite laminated trapezoidal plate is established. The nonlinear static property and nonlinear vibration frequency ratios of the composite laminated rectangular plate are analyzed to verify the validity and correctness of the present methodology by comparing with the results published in the open literatures. Moreover, the effects of the ply angle and the length-high ratio on the nonlinear vibration frequency ratios of the composite laminated trapezoidal plates are discussed, and the frequency-response curves are analyzed for the different ply angles and harmonic excitation forces.

Nonlinear free vibration analysis of nanobeams under magnetic field based on nonlocal elasticity theory

In the present study, we perform the nonlinear free vibration analysis of nanobeams under longitudinal magnetic field based on Eringen’s nonlocal elasticity and Euler-Bernoulli beam theory. To capture the small scale effect of the nanobeams, we adopt the nonlocal beam model with the nonlocal parameter. We use the Hamilton’s principle and von-Kármán’s nonlinear strain-displacement relationships to derive the governing equations of nanobeams subject to magnetic field. In order to solve the equations, we utilize the Galerkin method and He’s variational method to come up with an approximate analytical solution for the nonlinear frequency of the nanobeams under magnetic field. In the numerical results, the nonlinear frequency ratio is presented for various values of the dimensionless nonlocal parameter and the dimensionless amplitude. Finally, we investigate the effect of nonlocal parameter on the nonlinear frequency ratio; moreover, we study and discuss the effect of magnetic field on the nonlinear free vibration behavior of nanobeams.

Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach

In this study, a novel size-dependent beam model is proposed for nonlinear free vibration of a functionally graded (FG) nanobeam with immovable ends based on the nonlocal strain gradient theory (NLSGT) and Euler-Bernoulli beam theory in conjunction with the von-Kármán's geometric nonlinearity. It is assumed the material properties of the nanobeam changes continuously in the thickness direction according to simple power-law form. To remove the stretching and bending coupling due to the unsymmetrical material variation along the thickness, the formulation of the problem is developed based on a new reference surface. The Hamilton's principle is utilized to derive the equations of the motion and the corresponding boundary conditions. The partial nonlinear differential equation describes the nonlinear vibration of FG nanobeam is reduced to an ordinary nonlinear differential equation with cubic nonlinearity via Galerkin's approach under the assumption that the axial inertia is negligible. A closed-form solution is obtained for nonlinear frequency by the novel Hamiltonian approach, and some illustrative numerical examples are given in order to study the effects of the strain gradient length scale, the nonlocal parameters, vibration amplitude and various material compositions on the ratio of nonlinear frequency to linear frequency (the nonlinear frequency ratio).

Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates

In the present work, a nonlocal continuum model is developed for the nonlinear free vibration of size-dependent magneto-electro-elastic nanoplates subjected to external electric and magnetic potentials. Using the nonlocal elasticity theory and Hamilton’s principle, the nonlinear differential equations of motion and corresponding boundary conditions are derived. The effect of geometric nonlinearity is taken into account based on the von Kármán’s assumptions. Various non-classical plate theories are introduced by considering two additional scale parameters. The coupled nonlinear differential equations are solved analytically using a perturbation technique. Closed-form solutions are obtained for the nonlinear natural frequencies, critical external electric voltages and critical magnetic potentials of magneto-electro-elastic nanoplates with immovable and movable edges. The present nonlocal continuum model and method of solution are validated by comparing the results with available results in the literature. It is found that the natural frequencies of magneto-electro-elastic nanoplates can be tuned by adjusting the values of external electric and magnetic potentials. The nonlinear frequency ratio decreases by considering the effect of length scale on the compatibility equation.

Nonlinear Vibration Response of Shear Deformable Functionally Graded Plate Using Finite Element Method

This paper presents the free vibration behavior of functionally graded (FG) plate with von Karman nonlinearity. Hamilton principle is employed to establish the governing equation of motion of FG plate using higher order shear deformation theory (HSDT). The established governing equation also satisfied the traction free boundary conditions on the top and bottom faces of the plate. The gradation of material properties in the thickness direction of functionally graded plate is assumed according to a power law distribution. Results are obtained by employing an efficient C0 finite element with seven degrees of freedom (DOFs) per node. Convergence and comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. The effect of nonlinear kinematics, thickness ratio, amplitude ratio and the volume fraction ratio on the vibration characteristics of FGM plate is investigated. It is noticed that the nonlinear frequency ratio is greatly influenced by geometric configuration and nonlinearity exist in stress-strain relationship.

Adomian-modified decomposition method for large-amplitude vibration analysis of stepped beams with elastic boundary conditions

ABSTRACTThe main objective of this paper is to apply an Adomian modified decomposition method for solving large amplitude vibration analysis of stepped beams with various general and elastic boundary conditions. Damaged or imperfect supports of beams can be modeled by using elastic boundary conditions composing of translational and rotational springs. For the beams subjected to dynamic severe loading, it is important to include the nonlinear term of axial stretching force developed by the large vibration amplitude in the governing equation for more accurate design. By using the method, the convergence studies for linear and nonlinear vibration analyses of stepped beams are shown for determining an appropriate number of terms in the solutions. The accuracy of the present results is validated numerically by comparing with some available results in the literature. New results of nonlinear frequency ratios of stepped beams with different boundary conditions are presented and discussed in detail. Aspects of step ratio, step location, boundary conditions, vibration amplitudes, etc., which have significant impact on linear and nonlinear frequencies of such beams are taken under investigation.

Non-linear free vibration analysis of point supported laminated composite skew plates

The element-free Galerkin (EFG) method is employed to analyze the large amplitude free vibration of point supported laminated composite skew plates. The geometrical non-linearity is considered based on the von Karman’s assumptions and the point support boundary conditions are satisfied through the use of Lagrange multiplier method and orthogonal transformation technique. Assuming a periodic solution and applying the weighted residual method, the non-linear governing equations are used, and the problem is solved by direct iteration technique. The obtained results are first validated against the available solutions of simply supported and clamped plates, and then the point supported plates are dealt with. To this end, four types of point support arrangements are considered and non-linear frequency ratios for different numbers of point supports, amplitude ratios, skew angles, aspect ratios and fiber orientations are calculated which may serve as benchmarks for future research.

Large Amplitude Free Vibration of Nanobeams Subjected to Magnetic Field Based on Nonlocal Elasticity Theory

In the present study, nonlinear free vibration behavior of nanobeam subjected to magnetic field is investigated based on Eringen's nonlocal elasticity and Euler–Bernoulli beam theory. The Hamilton's principle is adopted to derive the governing equations together with Euler–Bernoulli beam theory and the von-Kármán's nonlinear strain–displacement relationships. An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam under magnetic field by using the Galerkin method and He's variational method. In the numerical results, the ratio of nonlinear frequency to linear frequency is presented. The effect of nonlocal parameter on the nonlinear frequency ratio is studied; furthermore, the effect of magnetic field on the nonlinear free vibration behavior of nanobeam is investigated.

Nonlinear free vibration of magneto-electro-elastic rectangular plates

Nonlinear free vibration of symmetric magneto-electro-elastic laminated rectangular plates with simply supported boundary condition is studied for the first time. The first order shear deformation theory considering the von Karman’s nonlinear strains is used to obtain the equations of motion, whereas Maxwell equations for electrostatics and magnetostatics are used to model the electric and magnetic behavior. Closed circuit electro-magnetic boundary condition at top and bottom surfaces of the plate is considered. Then, the nonlinear partial differential equations of motion are transformed into five coupled nonlinear ordinary differential equations by using the Galerkin method. Afterward, the obtained coupled ordinary differential equations are reduced to a single nonlinear differential equation with quadratic and cubic nonlinear terms. A perturbation method is used to solve the equation of motion analytically and a closed-form solution is obtained for the nonlinear frequency ratio. The results for natural frequency and nonlinear frequency ratio are compared with the available results for isotropic, laminated and piezo-laminated, and laminated magneto-electro-elastic plates and good agreement is found between the results of present study with the results of previously published papers. Several numerical examples are carried out to show the effects of different parameters on the nonlinear behavior of these hybrid plates.

Nonlinear in-plane free oscillations of suspended cable investigated by homotopy analysis method

An analytical solution for the nonlinear in-plane free oscillations of the suspended cable which contains the quadratic and cubic nonlinearities is investigated via the homotopy analysis method (HAM). Different from the existing analytical technique, the HAM is indeed independent of the small parameter assumption in the nonlinear vibration equation. The nonlinear equation is established by using the extended Hamilton\'s principle, which takes into account the effects of the geometric nonlinearity and quasi-static stretching. A non-zero equilibrium position term is introduced due to the quadratic nonlinearity in order to guarantee the rule of the solution expression. Therefore, the mth-order analytic solutions of the corresponding equation are explicitly obtained via the HAM. Numerical results show that the approximate solutions obtained by using the HAM are in good agreement with the numerical integrations (i.e., Runge-Kutta method). Moreover, the HAM provides a simple way to adjust and control the convergent regions of the series solutions by means of an auxiliary parameter. Finally, the effects of initial conditions on the linear and nonlinear frequency ratio are investigated.

Active control of large amplitude vibrations of smart magneto–electro–elastic doubly curved shells

This paper deals with the analysis of active constrained layer damping (ACLD) of large amplitude vibrations of smart magneto–electro–elastic (MEE) doubly curved shells. The constraining layer of the ACLD treatment is composed of the vertically/obliquely reinforced 1–3 piezoelectric composite (PZC). The constrained viscoelastic layer of the ACLD treatment is modeled by using the Golla–Hughes–McTavish method in the time domain. A three-dimensional finite element model of the overall smart MEE doubly curved shells has been developed taking into account the effects of electro–elastic and magneto–elastic couplings, while the von Karman type nonlinear strain displacement relations are used for incorporating the geometric nonlinearity. Influence of the curvature ratio, the curvature aspect ratio, the thickness aspect ratio on the nonlinear frequency ratios of the MEE doubly curved shells has been investigated. Effects of the location of the ACLD patches and the edge boundary conditions on the control of geometrically nonlinear vibrations of paraboloid and hyperboloid MEE shells have been studied. Particular attention has been paid to investigate the performance of the ACLD treatment due to the variation of the piezoelectric fiber orientation angle in the 1–3 PZC constraining layer of the ACLD treatment.