Nonlinear Forced Vibration Research Articles

Nonlinear primary responses of a bilateral supported X-shape vibration reduction structure

The nonlinear forced vibration responses of a bilateral supported X-shape vibration reduction structure are researched. The vertical and rotational displacements of the upper and lower platforms of the structure are taken into account and the nonlinear dynamic model of the structure with four degrees of freedom is established by applying the Lagrange’s equation. The validate experimental study of the structure is carried out and the acceleration response measured from the experiment is compared with the numerical result to verify the correctness of the dynamic model. The steady state amplitude-frequency responses of the system are obtained via the incremental harmonic balance (IHB) method and validated by using the Runge-Kutta numerical simulation. From the analysis, it can be seen that the primary responses of the vertical and rotational displacements of the structure exhibit softening type of nonlinearity. The coupled resonance of the vertical and the rotational displacements of the system can be observed when the system resonances around the fundamental and second natural frequencies. Besides, parametric investigations are performed to analyze the effects of the viscous damping coefficient, the external excitation amplitude, the angle between the rod and the platform, and the stiffness coefficient of the support spring on the resonance properties of the X-shape vibration reduction structure.

Nonlinear forced vibrations of supported pipe conveying fluid subjected to an axial base excitation

The present work focuses on the nonlinear forced vibrations of a supported pipe conveying fluid under an axial base excitation, devoting to exploring the effects of internal flow velocity and axial base excitation on the nonlinear dynamical behaviors of the pipe. Based on Hamilton's principle, the nonlinear coupled governing equations of the pipe system are introduced and discretized into a set of coupled ordinary differential equations by the aid of a Galerkin method. The resulting equations are then solved by a fourth-order Runge-Kutta integration algorithm. Numerical results indicate that the excitation amplitude and internal flow velocity have a combined effect on the dynamic responses of the pipe. When the internal flow velocity is below the critical value for buckling instability, an axial base excitation with small, moderate or large amplitude can only induce oscillations of the pipe in the axial direction, while an axial excitation with very large amplitude, interestingly, could induce the oscillations of the pipe in both axial and transverse directions. When the internal flow velocity is beyond the critical value for buckling instability, however, an axial base excitation with small amplitude can make the pipe oscillate around a positive or a negative static/dynamic equilibrium position in a wide range of excitation frequencies. In addition, a strong axial base excitation with large amplitude can make the pipe conveying a supercritical fluid oscillate around both positive and negative equilibrium positions for relatively large excitation frequencies. Indeed, various dynamical behaviors including periodic, multi-periodic, quasi-periodic and chaotic oscillations can be observed in many cases.

Size-dependent Nonlinear Forced Vibration Analysis of Viscoelastic/Piezoelectric Nano-beam

In this paper, the nonlinear forced vibration of isotropic viscoelastic/ piezoelectric Euler-Bernoulli nano-beam is investigated. For this purpose, the consistent couple stress theory is utilized for modeling the viscoelastic/piezoelectric nano-beam. Hamilton’s principle is also employed to obtain the governing equations of motion. Further, the Galerkin method is used in order to convert the governing partial differential equations to a nonlinear second-order ordinary differential one, and then multiple scale method is used to solve motion equation.

Nonlinear forced vibrations of multi-scale epoxy/CNT/fiberglass truncated conical shells and annular plates via 3D Mori-Tanaka scheme

In the context of classic conical shell formulation, nonlinear forced vibration analysis of truncated conical shells and annular plates made of multi-scale epoxy/CNT/fiberglass composites has been presented. The composite material is reinforced by carbon nanotube (CNT) and also fiberglass for which the material properties are defined according to a 3D Mori-Tanaka micromechanical scheme. By utilizing the Jacobi elliptic functions, the frequency-deflection curves of truncated conical shells and annular plates related to their forced vibrations have been derived. The main focus is to study the influences of CNT amount, fiberglass volume, open angle, fiber angle, truncated distance and force magnitude on forced vibrational behaviors of multi-scale truncated conical shells and annular plates.

Investigation of flexoelectric effect on nonlinear forced vibration of piezoelectric/functionally graded porous nanocomposite resting on viscoelastic foundation

Investigation of flexoelectric effect on nonlinear forced vibration of piezoelectric/functionally graded porous nanocomposite is the objective of this study. The nanocomposite is exposed to electric voltage and external parametric excitation. First, a functionally graded porous core nanoplate is modeled and then two piezoelectric layers are glued with core. It is also rested on a visco-Pasternak foundation. Second, to derive governing equation of motion, two theories including Mindlin and Kirchhoff plate theories and Hamilton’s principle are utilized. In the next step, to obtain and solve ordinary differential equation, Galerkin technique and multiple time scales method are used, respectively. At the end, modulation equation of piezoelectric/functionally graded porous nanocomposite for both primary and secondary resonances is obtained and discussed. Emphasizing the effect of piezoelectric and flexoelectric, von Karman nonlinear deformation and parametric external excitation are simultaneously taken into account. It is found that electric voltage has no effect on the performance of piezoelectricity and flexoelectricity of the material on vibration behavior. The results of this study can be useful as benchmark for the next investigations in field of energy harvesting systems and piezoelectric structures.

Nonlocal and strain gradient effects on nonlinear forced vibration of axially moving nanobeams under internal resonance conditions

Based on the nonlocal strain gradient theory (NSGT), a model is proposed for an axially moving nanobeam with two kinds of scale effects. The internal resonance-accompanied fundamental harmonic response of the external excitation frequency in the vicinities of the first and second natural frequencies is studied by adopting the multivariate Lindstedt-Poincare (L-P) method. Based on the root discriminant of the frequency-amplitude equation under internal resonance conditions, theoretical analyses are performed to investigate the scale effects of the resonance region and the critical external excitation amplitude. Numerical results show that the region of internal resonance is related to the amplitude of the external excitation. Particularly, the internal resonance disappears after a certain critical value of the external excitation amplitude is reached. It is also shown that the scale parameters, i.e., the nonlocal parameters and the material characteristic length parameters, respectively, reduce and increase the critical amplitude, leading to a promotion or suppression of the occurrence of internal resonance. In addition, the scale parameters affect the size of the enclosed loop of the bifurcated solution curves as well by changing their intersection, divergence, or tangency.

Nonlinear dual frequency excited vibration of viscoelastic graphene sheets exposed to thermo-magnetic field

Nonlinear forced vibration of a single layer graphene sheets (SLGSs) resting on a visco-Pasternak foundation and exposed to a dual frequency excitation and a thermo-magnetic field is the main objective of the present study. To obtain this goal, based on nonlocal elasticity theory, Kelvin–Voigt model, nonlinear strain-displacement relations are used to model the geometrical nonlinearity and governing equation of motion is derived. Then, applying Galerkin technique, the partial differential equation is transformed to the ordinary differential one. Derived equation of motion is analyzed and solved using multiple time scales method. Finally, modulation equation under sub-harmonic and super-harmonic stimulation are studied. Emphasizing the effect of nonlinearity, great attention is given to dual frequency excitation and results for nonlinear frequency response with respect to amplitude, the phase angle and force amplitude for SLGS are also plotted. At the end, results of this article are compared with results in the other researches. The results emphasize that the multi-frequency excitation intensifies resonance behavior and jump phenomenon in SLGS.

Stabilization of nonlinear vibrations of carbon nanotubes using observer-based terminal sliding mode control

This article is concerned with suppression of nonlinear forced vibration of a single-wall carbon nanotube conveying fluid based on the nonlocal elasticity theory and Euler–Bernoulli beam theory. Electrostatic actuation is considered as the control force for the suppression of carbon nanotube. Based on Galerkin approach, the governing nonlinear partial differential equation is reduced to an ordinary one. Since the sliding mode controller (SMC) does not assures finite time system stabilization and also causes chattering in the control input and consequently vibration in the system, terminal sliding mode controller (TSMC) is developed for the stabilization of carbon nanotube based on a disturbance observer. TSMC and disturbance observer suppress the vibrations of nanotube in the presence of external disturbances caused by the internal flow. Numerical simulation results are presented to illustrate the effectiveness and performance of the proposed control scheme in comparison to similar approaches. Simulation results show that the proposed control method successfully stabilizes the uncertain system in a finite time.

Primary and secondary resonances in pipes conveying fluid with the fractional viscoelastic model

Nonlinear forced vibrations of a fractional viscoelastic pipe conveying fluid exposed to the time-dependent excitations is investigated in the present work. Attention is focused in particular on the primary and secondary resonances with the Kelvin–Voigt fractional order constitutive relationship model. The nonlinear geometric partial differential equations due to stretching effect have been expressed by assumptions with Von Karman’s strain-displacement relation and Euler–Bernoulli beam theory. Viscoelastic fractional model for damping and stiffness, and also plug flow model for fluid flow are considered to derive the equation of motion. Based on the Galerkin truncation, the coupled Fluid-Solid interaction nonlinear equation transferred to ordinary differential equations. The method of multiple scales is adopted to analyze steady-state solutions for the primary, superharmonic, and subharmonic resonances. Finally, the detailed parametric studies on the nonlinear dynamic behavior are discussed. Results delineate that the fractional derivative order and the retardation time have significant effects on the oscillation exhibited for different values of flow velocity.

Nonlinear forced vibrations of sandwich smart nanobeams with two-phase piezo-magnetic face sheets

The present research is devoted to examine nonlinear forced vibration behavior of sandwich nanobeams with piezo-magnetic face sheets and rested on a hardening elastic substrate based on ordinary differential nonlocal elasticity. A uniform harmonic load is applied to the sandwich nanobeam leading to forced oscillations. The face sheets are constructed from two phases called piezoelectric and magnetic phases. In the face sheets, different percentages of the piezoelectric phase is considered for the first time. It will be shown that vibration behavior of the nanobeam is affected by the amount of piezoelectric phase in the face sheets. Via an analytical procedure, the ordinary nonlinear governing equation is solved to express the amplitude-frequency relationship. It is also shown that the effect of electric and magnetic fields on forced vibrations of sandwich nanobeam rely on the amount of piezoelectric content. Also, forced vibration characteristics of the nanobeam are prominently affected by the nonlocality, hardening foundation and applied harmonic load.

Nonlinear free and forced vibrations of porous sigmoid functionally graded plates on nonlinear elastic foundations

This paper focuses on the nonlinear free and forced vibrations of porous sigmoid functionally graded material plates resting on nonlinear elastic foundations. Two types of porosity distributions, even and uneven, were considered. A nonlinear three-parameter foundation model was employed to estimate the plate-foundation interactions. The material properties of the plates, described by the sigmoid distribution law, were assumed to be graded in the thickness direction. All four edges of the plates were simply supported and had no in-plane displacements. Based on a higher-order shear deformation plate theory and general von Kármán-type equation, the equations of motion with the effects of nonlinear elastic foundations were developed. The equations of motion were solved by an improved perturbation technique to determine the nonlinear frequencies and dynamic responses of the plates. The numerical illustrations are presented in both tabular and graphical forms to show the effects of the nonlinear foundation parameters, pore volume fraction, and material volume fraction on the nonlinear vibration and dynamic responses of the plates.

The effect of bi-axial in-plane loads on nonlinear dynamics of micro-plates under harmonic excitation

Based on the theories of the first-order shear deformation (FSDT) plate and the modified couple stress, a nonlinear micro-plate model is developed for steady state dynamics of harmonically excited micro-plates subjected to two-sided in-plane tension or compressive loads. The governing equations of motion and corresponding boundary conditions are derived using Hamilton's principle. The semi-analytical method of Galerkin's is utilized to discretize the governing equations of motion into ordinary differential equations. By performing complexification-averaging method and arc-length continuation techniques, semi-analytical approximations for nonlinear steady-state response is developed. For micro-plate with post-buckling deformation coexisting attractors are observed. These attractors have different nonlinear behaviors such as multi frequency, period doubling, intermittency, quasi-periodic and chaotic behaviors. The effects of different parameters such as in-plane load, load ratio, the material length scale parameter and aspect ratio on the nonlinear forced vibration are examined.

Nonlinear torsional vibrations of electromechanical coupling of diesel engine gear system and electric generator

New approach for analysis and detuning of nonlinear torsional vibrations of diesel electric generator gear system is suggested. Nonlinear magnetic field of the generator is calculated by finite element method to obtain the disturbing moments, which act on the gear system. Moreover, others disturbing moments, which arise due to operation in diesel engine cylinders, are accounted. Essential nonlinear model of torsional vibrations with sixteen degrees-of- freedom is derived. This model accounts the nonlinear moments of two elastic clutches and the backlashes of gear pairs. The comparison of the numerical simulations data with experimental results is performed to verify the obtained model. The approach for essential nonlinear free torsional vibrations calculations is suggested. The harmonic balanced method is the basis of this approach. Sensitivity theory is used to detune the nonlinear forced torsional vibrations from resonances. The results of nonlinear torsional vibrations analysis are discussed.

Nonlinear Vibrations and Stability of an Axially Moving Plate Immersed in Fluid

In this paper, the nonlinear forced vibrations and stability of an axially moving large deflection plate immersed in fluid are investigated. Based on von Karman’s large deflection plate theory and taking into consideration the influence of fluid–structure interaction, axial moving and axial tension, nonlinear dynamic equations are obtained by applying D’Alembert’s principle. These dynamic equations are further discretized into ordinary differential equations via the Galerkin method. The frequency–response curves of system are obtained and examined. Then numerical method is used to analyze the bifurcation behaviors of immersed plate. Results show that as the parameters vary, the system displays periodic, multi-periodic, quasi-periodic and even chaotic motion. Through the analysis on global dynamic characteristics of fluid–structure interaction system, rich and varied nonlinear dynamic characteristics are obtained, and various ways that lead to chaotic motion of the system are further revealed.

Comparative Analysis of Non-linear Periodic Response of Cross-ply and Angle-ply Laminated Composite Plates

Non-linear periodic response periodic response analysis of laminated structures is necessary for their optimal and safe dynamic design. The present work is intended to explore the influence of lamination scheme and fibre-orientation on the nonlinear forced vibration response of laminated plates. The analysis has been carried out using finite element. In present work, a comparative analysis of forced vibration response between cross-ply and angle-ply laminated composite plates have been explored. The effects of fiber orientation on non-linear forced vibration response characteristics have been analyzed. This finite element analysis has been done using shooting method along with continuation scheme in the time domain. The method employed is capable to get the complete stable and unstable branches past bifurcations. The dynamic behavior is explored using the temporal history of the response and phase plane plots. The method presented in the paper is computationally efficient and does not require apriori assumption on the participating modes, unlike harmonic balance and incremental harmonic balance methods.

Nonlinear forced vibration of smart multiscale sandwich composite doubly curved porous shell

This is the first research on the nonlinear vibration analysis of composite sandwich doubly curved shell with a flexible core integrated with a piezoelectric layer. By using the higher order shear deformable theory (HSDT) for the face sheets and the third-order polynomial theory for the flexible core, the strains and stresses are obtained. It is assumed a smart model including multiscale composite layers shell with a flexible core and magnetorheological layer (MR) that leading up by the nonlinearity of the in-plane and the vertical displacements of the core. Three-phase composites shells with polymer/Carbon nanotube/fiber (PCF) and polymer/Graphene platelet/fiber (PGF) and Shape Memory Alloy (SMA)/matrix either according to Halpin-Tsai model have been considered. The governing equations of multiscale shell have been derived by implementing Hamilton's principle and solved by multiple scale method. For investigating the correctness and accuracy, this paper is validated by other previous researches. Finally, the effect of different parameters such as temperature rise, various distributions pattern, magnetic fields and curvature ratio are explored in detail.

A Multimode Approach to Geometrically Non-linear Forced Vibrations of Euler–Bernoulli Multispan Beams

The geometrically non-linear free and forced vibrations of a multi-span beam resting on an arbitrary number of supports and subjected to a harmonic excitation force is investigated. The theoretical model developed here is based on the Euler–Bernoulli beam theory and the von Karman geometrical non-linearity assumptions. Assuming a harmonic response, the non-linear beam transverse displacement function is expanded as a series of the linear modes, determined by solving the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton’s principle, the problem is reduced to a non-linear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure deflection function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence are determined. Considering the non-linear forced response, an approximate multimode approach has been used in the neighbourhood of the predominant mode, to obtain numerical results, for a wide range of vibration amplitudes. The effects on the non-linear forced dynamic response of the support number and locations, the excitation frequency and the level of the applied harmonic force (a centered point force or a uniformly distributed force) have been investigated and illustrated by various examples.

Modified Multi-level Residue Harmonic Balance Method for Solving Nonlinear Vibration Problem of Beam Resting on Nonlinear Elastic Foundation

Nonlinear vibration behavior of beam is an important issue of structural engineering. In this study, a mathematical modeling of a forced nonlinear vibration of Euler-Bernoulli beam resting on nonlinear elastic foundation is presented. The nonlinear vibration behavior of the beam is investigated by using a modified multi-level residue harmonic balance method. The main advantage of the method is that only one nonlinear algebraic equation is generated at each solution level. The computational time of using the new method is much less than that spent on solving the set nonlinear algebraic equations generated in the classical harmonic balance method. Besides the new method can generate higher-level nonlinear solutions neglected by previous multi-level residue harmonic balance methods. The results obtained from the proposed method compared with those obtained by a classical harmonic balance method to verify the accuracy of the method which shows good agreement between the proposed and classical harmonic balance method. Besides, the effect of various parameters such as excitation magnitude, linear and nonlinear foundation stiffness, shearing stiffness etc. on the nonlinear vibration behaviors are examined

Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads

Nonlinear free and forced vibration behavior of a porous functionally graded Euler-Bernoulli nanobeam subjected to mechanical and electrical loads is studied based on the nonlocal strain gradient elasticity theory. It is assumed that the porous functionally graded (FG) nanobeam is resting on a nonlinear foundation. Also, material properties of the nanobeam are assumed to vary in the thickness direction. Equations of motion are derived using Hamilton's principle. Galerkin method along with variation iteration method (VIM), Homotopy perturbation method (HPM), Hamiltonian approach method (HAM) and multiple scale method are employed to solve the governing equations based on clamped-clamped, simply-simply and clamped-simply boundary conditions. For verification purposes, the results of this study are compared with those of other studies. The effects of different parameters such as type of porosity distribution, nonlinear foundation, boundary conditions, electrical voltage and size effect parameters on the primary and secondary resonances were investigated. It was found that length-scale parameters have a crucial role on the nonlinear vibration behavior of such structures.

Size-dependent nonlinear forced vibration and dynamic stability of electrically actuated micro-plates

A size-dependent structural dynamic model that incorporates the effect of geometric nonlinearity is developed in this paper for the forced vibration and dynamic stability of thin rectangular micro-plates. The equations of motion for micro-plates are derived within the framework of classical plate theory, modified couple stress theory (MCST), and von Kármán geometric nonlinearity using Hamilton's principle. Galerkin method is used to convert the governing partial differential equations to a nonlinear second-order ordinary differential equation, which is solved by a Runge-Kutta method. The static instability analysis of the micro-plate is performed to determine the critical electrostatic voltages, and to avoid the pull-in instability. By tracking the static behavior of the microplate, and determining the electrostatic pull-in voltage, the frequency response curves are plotted. In dynamic response, primary, superharmonic, and subharmonic resonance are studied, and the frequency response equation is obtained for each case by the method of multiple scales. Further efforts are made to investigate the influence of size effect, electrical loading (DC and AC voltages), and excitation frequency on the static, and dynamic responses, critical AC voltages, and dynamic stability of micro-plates. It is found that the critical dynamic voltage is a function of the frequency of excitation force. It is shown that the stiffness of micro-plate decreases by increasing the constant DC voltage; however, the increase in the alternating AC voltage does not considerably affect the stiffness of the micro-plate.