Nonlinear Forced Vibration Research Articles

Nonlinear forced vibrations of initially curved rectangular single layer graphene sheets: An analytical approach

Abstract Geometrical imperfections and residual stresses are two inevitable aspects in studying of graphene sheets (GSs) which despite of their vital influence on the forced nonlinear vibrational behavior they have not been considered in previous research works. Hence, this work is an attempt to fill this important gap for the first time. Utilizing Eringen's nonlocal plate theory, assuming von Karman nonlinear terms in strain-displacement relations and employing calculus of variations, we have derived partial differential equations (PDEs) of the displacement field. The function of initial curvature was assumed as the first mode-shape of the simply-supported rectangular plates with a variable mode-coefficient. Next, Galerkin decomposition method has been used to achieve the equation of motion and for nonlinear vibration analysis time multiple scales method (TMS) was applied. The nonlinear frequency response relation has been analytically achieved. Afterwards, comparing to molecular dynamic (MD) results presented in previous researches we validated natural frequencies of single layer graphene sheets (SLGSs) in different sizes. Furthermore, nonlinear frequency response curves and time histories based on TMS, were in an excellent agreement with numerical method results. Therefore, the influences of external pressure, nonlocal parameter, geometric imperfection and residual stress have been presented comprehensively. It was observed that both imperfection and stress intensify the softening behavior in the system. Interestingly, for a certain controllable magnitude of the imperfection, it has been illustrated that the generated softening term counteracts with the hardening due to geometrical nonlinearity and consequently the nonlinear system behaves as a linear one.

Nonlinear free and forced vibrations of fractional modeled viscoelastic FGM micro-beam

In this paper, nonlinear vibrations of fractional viscoelastic functionally graded micro-beam are studied based on the fractional calculus. The Modified Couple Stress Theory (MCST) is utilized for considering the effect of the micro-scale. The micro-beam is modeled based on the Euler-Bernoulli theory and the Von Karman’s nonlinear strain relations. The viscoelastic part of the micro-beam is considered using the fractional Kelvin-Voigt viscoelastic model. Functionally graded properties vary along the thickness based on a power-law function. The small-scale effects of the micro-beam are modeled using the MCST. Hamiltons principle is used to derive the governing equation of motion. For the solution, the Finite Difference Method (FDM) and the Finite Element Method (FEM) are employed. The FDM is used for discretizing the time domain, and the FEM is used for discretizing the space domain. Effects of the fractional-order, microstructure parameters, and Functionally Graded Material (FGM) properties on the time response of the viscoelastic micro-beam are analyzed numerically. Numerical results show that the fractional-order modeling of the viscoelastic micro-beam can change the behavior of the structure, and increasing the fractional-order, can increase the damping of the system. Numerical simulations also suggest that the effect of the fractional derivative order on the responses of free and forced vibrations is different because there is an important correlation between the fractional-order derivative and the excitation frequency. The results of this paper can be used for modeling the damping of the viscoelastic structures.

Nonlinear dynamic responses of beamlike truss based on the equivalent nonlinear beam model

The main goal of this paper is to develop an equivalent nonlinear beam model (ENBM) for forced vibration analysis of the nonlinear beamlike truss (NBT) and its numerical implementation for describing the nonlinear behaviors of the periodic NBT. For the equivalent dynamic modeling method based on the energy equivalence principle of the truss structures, most of the existing approaches focus on studying the linear characteristic which cannot satisfy the nonlinear dynamic analytical requirement. Here, we extend the equivalent dynamic modeling to nonlinear forced vibration responses of the large NBT civil engineering structure with two pinned ends. We introduce the geometric nonlinearity into the equivalent dynamic model using the von Karman nonlinear strain-displacement relationship to imitate the nonlinear behaviors of the beamlike truss. On the basis of the Hamilton principle, the fourth-order governing partial differential equations of motion of the ENBM are obtained and solved utilizing the Galerkin method. It is shown that the ENBM accurately captures the nonlinear dynamic response of the beamlike truss subjected to external load by using the first four order modes. In order to showcase the efficiency and accuracy of the ENBM, time histories, phase maps and fast Fourier transform frequency spectra are investigated for the NBT and the ENBM under different external excitation magnitudes and frequencies. Comparisons between results of the ENBM and those obtained from the finite element simulations of the NBT show good agreements and identify the periodic motions of them. Furthermore, the effect of the external loading and damping parameters on the frequency-response and force-response curves is discussed. In addition, computing time of the proposed ENBM are compared with that of the full-scale finite element model in ANSYS so as to highlight the significant less computational cost of the equivalent nonlinear dynamic modeling. Particularly, the proposed ENBM can achieve the nonlinear mechanism of the NBT in analytical method and design the low-order control law conveniently.

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

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

INTERACTION BETWEEN FORCED AND PARAMETRIC VIBRATIONS OF A PIPELINE

The bending vibrations of a pipeline under the influence of the variable pressure of the transported fluid and the vertical vibrational movement of the supports are studied. The first vibration mode is considered. Nonlinear forced parametric vibrations and their interaction are studied. The results of approximate analytical and numerical solutions are compared.

Nonlinear vibration and chaos of a moving flexible graphene smart electronic web

Flexible graphene smart electronics are a kind of electronic device made of silver or other inorganic material and a flexible graphene web substrate. Because of their flexibility and excellent electronic properties, flexible graphene smart electronics have become a research topic of interest. However, during the mass production process, the high-speed transmission of the continuous tension state is susceptible to external disturbances, which seriously restricts the improvement of the flexible graphene smart electronic printing speed. Therefore, studying the dynamic characteristics of moving flexible graphene electronic materials is of great significance for improving their quality and production efficiency. This paper considers a roll-to-roll smart RFID label graphene web as the research object. Based on the energy functional variational theory, the Bubnov-Galerkin method is used to eliminate the residual value of the dynamic Von Karman partial differential equation, and a discrete forced nonlinear vibration control equation for a moving flexible graphene smart electronic web is established. The Melnikov method is used to obtain the Melnikov function of the orbit parameter for the vibration control equation and solve the conditions of the chaos criterion. Using the fourth-order Runge-Kutta method, the nonlinear vibration phase-plane portraits, Poincare maps, time history diagram, and power spectra of the moving flexible graphene smart electronic web are obtained. The effects of the web moving speed, external damping, and initial system conditions on nonlinear dynamic behaviour are calculated, and the stable and unstable working regions of the moving flexible graphene smart electronic web are obtained. This research provides a theoretical basis for the dynamic stability and vibration control of a moving graphene smart electronic web.

Nonlinear forced vibration of bilayer van der Waals materials drum resonator

The interlayer shear effect could play a crucial role in the dynamic behaviors of the nanoelectromechanical system (NEMS) resonators based on multilayer van der Waals (vdW) materials. However, the interlayer shear effect is rarely considered in the nonlinear forced vibrations of this kind of resonators. In this study, the nonlinear static and dynamics behaviors of drum resonators based on double layer MoS2 (DLMoS2) are investigated using molecular dynamics simulations and a nonlinear circular sandwich plate model (NCSPM), respectively. The interlayer shear effect and the geometrical nonlinearity are systematically considered in the NCSPM. Our results show that comparing with drum resonators based on single layer MoS2 (SLMoS2), the DLMoS2 drum resonators exhibit higher fundamental frequency. The load–deflection curve of DLMoS2 shows clear hardening-type nonlinearity, and the deflections of DLMoS2 are smaller than that of SLMoS2 under the same uniform load per layer. In particular, both frequency sweep excitation simulations and the NCSPM show that the hardening-type nonlinearity of the DLMoS2 drum resonators is largely reduced due to the interlayer shear effect. Furthermore, DLMoS2 with different twist angle exhibits different interlayer shear strengths; thus, the nonlinear characteristic could be controlled by changing the twist angles in the DLMoS2 drum resonators. The present study should be of great help for designing NEMS resonators based on such vdW materials.

Nonlinear forced vibrations analysis of imperfect stiffened FG doubly curved shallow shell in thermal environment using multiple scales method

This study investigates the non-linear vibrations of stiffened imperfect functionally graded double-curved shallow shells, as rested on nonlinear elastic foundations. The shells are exposed to external harmonic excitation and are placed in the thermal situations. The modeling of shells is derived according to the classical shell theory and the non-linear geometric von Kármán relationships. It is considered that the distribution of material properties changes along the thickness direction based on a power law index. The smeared stiffener technique is considered to model the stiffened shells. An approximation, according to Galerkin’s approach, is utilized to reduction of the shell governing equations into the non-linear coupled ordinary differential relations. The ODE equations are analytically solved and analyzed through the perturbation methodology for investigating the resonance behavior of shells. Simulation results are reported to examine the influences of stiffeners, initial imperfection, foundation coefficients, thermal environment, and geometrical characteristics on the non-linear primary resonance response of doubly curved shallow shells. Also, the nonlinear dynamic behaviors are analyzed by numerical methods through the bifurcation diagrams, and the nonlinear dynamical behaviors of the shell for different value of parameters are examined.

Nonlinear forced vibrations of nanocomposite-reinforced viscoelastic thick annular system under hygrothermal environment

Mechanical systems especially annular disks have many applications in different fields such as engineering, agriculture, and medicine. In this regard, the large amplitude vibrations of multi-scale hybrid nanocomposite annular system (MHCAS) under hygrothermal environment and subjected to mechanical loading. Also, the structure is covered with elastic foundation. The matrix material is reinforced with carbon nanotubes or carbon fibers at the nano- or macro-scale, respectively. The displacement strain of nonlinear vibration of the multi-scale laminated disk via third-order shear deformation theory and using von Karman nonlinear shell theory is obtained. Hamilton’s principle is employed to establish the governing equations of motion, which is solved by the generalized differential quadrature method and perturbation method. Finally, the results show that increasing the value of the and parameters and the rigidity of the boundary conditions lead to increase in the frequency of the structure. Besides, when the value of the nonlinearity parameter () is positive and negative, the dynamic behavior of the system tends to have hardening and softening behaviors, respectively, and could not be seen any effects from parameter on the maximum amplitudes of resonant vibration of the MHCAS. The last but not the list by increasing the rigidity of the structure, the system can be susceptible to have unstable responses. The presented outputs can be used in the structural health monitoring.

Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation

In the present paper, nonlinear forced vibrations of an axial moving nanobeam which is vertically influenced by an external harmonic excitation and gravity is analyzed by considering the effects of linear damping. Considering certain assumptions, a nonlinear Euler-Bernoulli beam theory is developed. With the implementation of the nonlocal elasticity theory, the governing integro-partial-differential equation is obtained by using the Hamilton principle. The multiple scale method is employed to obtain a steady-state response for the size-dependent viscoelastic nanobeam with fixed-free boundary conditions. Subsequently, the trivial and non-trivial steady-state response and the bifurcation point types are examined. Finally, the effects of damping coefficient and nonlocal parameter on stability and bifurcation of trivial and non-trivial solutions are studied. It is found that the effect of nonlocal parameter on the steady-state response and the bifurcation point types is quite important.

Nonlinear vibration of an elastically connected double Timoshenko nanobeam system carrying a moving particle based on modified couple stress theory

Based on the modified couple stress and non-classical Timoshenko beam theories, the nonlinear forced vibration of an elastically connected double nanobeam system subjected to a moving particle is assessed here. This system is assumed to be resting on an elastic medium. Hamilton’s principle and the Galerkin method are applied to govern the equations of system motion and corresponding boundary conditions and to solve these equations, respectively. The numerical study reveals that by applying the nonlinear and modified couple stress theories the system is predicted stiffer than what is obtained through linear and classical theories. To determine the effects of different parameters like the material length scale, the elastic stiffness modulus of the interlayer and medium, and the velocity of the moving particle on the system’s vibration, a parametric study is performed. The material length scale has a significant effect on the dynamic response of the system, indicating that the classical theory cannot predict the dynamic behavior of nanosize beam systems. The elastic stiffness modulus of both the interlayer and medium and the velocity of the moving particle have considerable effects on the dynamic deflections of the double nanobeam system.

Large-amplitude dynamical behavior of multilayer graphene platelets reinforced nanocomposite annular plate under thermo-mechanical loadings

For the first time, the nonlinear forced vibration analysis of graphene nanoplatelets reinforced composite (GPLRC) annular plate under hygro-thermal environment and subjected to mechanical loading is presented. The GPLRC imperfect annular plate’s displacement fields are determined using third-order shear deformation theory (third-order SDT) and nonlinearity of vibration behavior of this structure is taken into account considering von Karman nonlinear shell model. Energy method known as Hamilton principle is applied to create the motion equations governed to the shell structures, while they are solved using generalized differential quadrature method (GDQM) as well as perturbation method (PM). Ultimately, the research’s outcomes reveal that increasing the value of the moisture change () and orientation angle parameter (), and the rigidity of the boundary conditions lead to an increase in the structure’s frequency. Besides, whenever the values of the nonlinear parameter () are positive and negative, the dynamic behavior of the plate tends to have hardening and softening behaviors, respectively, and could not be seen any effects from parameter on the maximum amplitudes of resonant vibration of the GPLRC imperfect annular plate. Last but not the list by decreasing the structure’s flexibility, the plate can be susceptible to have unstable responses.

On the size-dependent nonlinear dynamics of viscoelastic/flexoelectric nanobeams

In this study, size-dependent nonlinear forced vibration of viscoelastic/flexoelectric nanobeams has been investigated. By calculating enthalpy and kinetic energy and using Hamilton’s principle, the coupled governing equations of viscoelastic/flexoelectric nanobeams are derived along with dependent electrical and mechanical boundary conditions. Furthermore, to take the effects of the small scale into account, the nonclassical theory of continuous medium has been used and the Euler–Bernoulli beam model has been adopted to model the nanobeams. Finally, the governing equations are solved using numerical methods for distributed loaded and clamped–clamped boundary conditions. By comparing the results, it is determined that the parameters of the size effect and the viscoelastic medium effect can increase the vibrational frequency of the nanobeams. Also, the results show that the frequency of nanobeams outside of the viscoelastic medium strongly depends on the size-dependent parameters, and the increase in the length and thickness of the nanobeam decreases the frequency. The results also show that with the increasing flexoelectric effect, the amplitude of the nonlinear oscillation increases.

Nonlinear mode coupling in a passively isolated mechanical system

Recent studies in passively isolated systems have shown that mode coupling is desirable for better vibration suppression, thus refuting the long-standing rule of modal decoupling. However, these studies have ignored the nonlinearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear forced damped vibrations of a passively isolated ultra-precision manufacturing (UPM) machine. Experimental analysis is conducted to guide the mathematical formulation. The system comprises linearly and nonlinearly coupled in-plane horizontal and rotational motion of the UPM machine with quadratic nonlinear stiffness from the isolators. We present an analytical study using the method of multiple scales and the method of harmonic balance for different cases of external resonances, viz., the primary and the secondary resonances (superharmonic and combined resonances) with 1 : 2 internal resonance between the modes. We further validate our analytical findings using direct numerical integration and observe an excellent match. On extending our analysis, we observe the existence of subcritical, supercritical, and s-shaped bifurcation depending on the location of the isolators and the case of external resonance. Also, the saturation and quasi-saturation phenomenon are observed for the case of resonances close to the higher natural frequency and combined resonance, respectively. A parametric study is conducted to examine the effect of different parameters on the dynamics of the system, and consecutively the critical parameters of the system are identified for different cases of external resonance.

Nonlinear forced vibration of the magnetic gear system with the internal resonance

Considering the nonlinear magnetic coupling stiffness among components of the magnetic gear with intersecting axes and a straight SP (MGIASS), the nonlinear dynamic model and differential equations of the MGIASS system are established. Then, the expression of the forced response is obtained when the excited frequency is close to the torsional mode frequency of the input rotor. The forced responses analysis of the example system shows that: the response frequencies include the excited frequency and the double frequency. Multiple proximity between the natural frequencies directly affects the stability and resonance amplitude.

An analytical approach for nonlinear thermo-electro-elastic forced vibration of piezoelectric penta – Graphene plates

This paper deals with the nonlinear forced vibration of imperfect penta – graphene plates integrated with piezoelectric actuator layers. The plate is subjected to combination of mechanical, thermal, and electrical loadings. Based on the first order shear deformation plate theory, the governing equations are established taken into account the effect of the von Kármán type of geometrical nonlinearity, the Pasternak type elastic foundations, the damping and the piezoelectric – thermal effects. Four edges of the hybrid plate are assumed to be simply supported and immovable in the in-plane directions. The solution forms that satisfy the boundary conditions are assumed to be trigonometric. The closed form expressions of natural frequency, the frequency ratio – amplitude and the deflection amplitude – time curves are obtained by using the Galerkin and Runge – Kutta methods. The numerical results show positive effects of elastic foundations, negative effect of temperature increment and initial imperfection, considerable effect of geometrical parameters as well as small effect of applied voltage on the nonlinear forced vibration of piezoelectric penta – graphene plate.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.

Nonlinear forced vibration analysis of a rotating three-dimensional tapered cantilever beam

In this article, nonlinear forced vibration analysis is carried out for a rotating three-dimensional tapered cantilever beam subjected to a uniformly distributed load. Considering the effects of Coriolis terms, static axial deformation and geometric nonlinearity in modeling process, nonlinear partial motion equations of a rotating tapered Euler–Bernoulli beam are established by using Hamilton’s principle. Galerkin’s procedure is used to discretize the equations to obtain the dynamic response of the beam. Frequency responses, the time-history response, the phase diagram, and the Poincaré map are introduced to study the effects of the taper ratio, rotating velocity, radius of hub, and external excitation on the nonlinear resonances and detailed responses of the rotating three-dimensional tapered beam. Results show that the fundamental natural frequency increases with the increase of the taper ratio, radius of hub, and rotating velocity. Besides, by increasing the taper ratio and excitation amplitude and decreasing the rotating velocity and radius of hub, the nonlinearity and vibration amplitude of the rotating beam intensify.

Nonlinear forced vibration analysis of composite beam combined with DQFEM and IHB

In this paper, a forced vibration model of composite beams under the action of periodic excitation force considering geometric nonlinearity is proposed. For the strain–displacement relationship, Timoshenko beam theory is used, and the element and system matrices are developed using the differential quadrature finite element method. Each node has 3 degrees of freedom. The incremental harmonic balance method is used to solve the nonlinear forced vibration equation. In order to prove the validity of the proposed model, the solution of the Duffing equation is calculated using the analytical method and the proposed method. Next, linear forced vibration analysis of the beam made of isotropic material is performed and compared with the result of ABAQUS. The results are very close. Based on these comparisons, nonlinear vibration phenomena of composite beams are studied under the action of periodic forces.

Nonlinear forced vibration of sandwich plate with considering FG core and CNTs reinforced nano-composite face sheets

Nonlinear vibration of sandwich plate with functionally graded material (FGM) core and carbon nano tubes reinforced (CNTs) nano-composite layers by considering temperature-dependent material properties are studied in this paper. Base on Classical plate theory (CPT), the governing partial differential equations of motion for sandwich plate are derived using Hamilton principle. The Galerkin procedure and multiple scales perturbation method are used to find relation between nonlinear frequency and amplitude of vibration response. The dynamic responses of the sandwich plate are also investigated in both time and frequency domains. Then, the effects of nonlinearity, excitation, power law index of FG core, volume fraction of carbon nanotube, the function of material variations of FG core, temperature changes, scale transformation parameter and damping factor on the frequency responses are investigated.