Nonlinear Vibrations Of Beams Research Articles

Parametric study on nonlinear vibration of FG-GNPRC dielectric beam with Kelvin–Voigt damping

Internal damping plays an essential role in the structural behaviors of composites. This paper adopts Kelvin–Voigt model to investigate the effects of the internal damping on nonlinear vibration of functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC) beam with dielectric properties. Within the framework of Timoshenko beam theory, governing equations for the vibration of the damped beam are developed by incorporating the internal damping in terms of energy. The governing equations are discretized by differential quadrature (DQ) method after state–space​ transformation and then solved numerically by direct iteration. The validated results demonstrate that the nonlinear natural frequency of the structure is highly dependent on the shear proportionality constant of the Kelvin–Voigt internal damping. It is found when the beam is subjected to higher voltage, the functionally graded distribution of the reinforcing filler enables the composite structure to be more stable compared to uniform distribution. Moreover, the nonlinearity of the structure is more sensitive to DC voltage for composite beams with smaller slenderness ratio and thickness.

Modified harmonic balance method for forced nonlinear vibration of beam on an elastic foundation

A modified harmonic balance method has been presented and adopted for the nonlinear vibration problem of beams on an elastic foundation. By this method, a nonlinear algebraic equation along with a set of linear algebraic equations are required to be solved, which reduces computation effort. The results determined by the present method are compared with the corresponding results obtained by the existing method. Besides, the effect of various parameters on nonlinear vibrational behavior is examined.

Nonlinear free vibration and flexural analysis of hyperelastic beam utilizing a meshless method based on radial basis function

This paper investigates nonlinear free and flexural analysis of hyperelastic beams. The constitutive relations of the hyperelastic beam were derived using the neo-Hookean strain energy function and Timoshenko beam theory. Also, the nonlinear governing equations and nonlinear natural boundary conditions were derived using Hamilton’s principle. The meshless collocation method based on the multiquadric radial basis function (MQ-RBF) was utilized to discretize the nonlinear governing equations. Also, the arc-length algorithm was used to solve the nonlinear system of equations. To validate the results of the meshless method, different boundary conditions (clamped–clamped, simply supported–simply supported, clamped–simply supported, and clamped-free) were examined, and the results obtained from the meshless method were compared with those of the finite element method in Abaqus finite element software. The results show that the maximum and minimum differences between meshless and finite element methods occur in clamped and free-boundary conditions, respectively. The results also show that the meshless method based on the MQ-RBF has good accuracy compared to the finite element method for bending and free vibration analysis of hyperelastic beams.

Linear and nonlinear vibrations of strain gradient beams with rough surface

This paper aims to study the effect of surface roughness on the linear and nonlinear free vibration behaviors of microbeams based on the strain gradient theory. Linear and nonlinear equations of motion and boundary conditions are derived by using the Hamilton’s principle. A closed-form solution of strain gradient beams (SGBs) is obtained by solving a general sixth-order differential equation. Furthermore, by utilizing the linear mode and Galerkin’s method, an exact solution for nonlinear vibration is also obtained for SGBs. A detailed parametric study is performed to study the influences of surface roughness, scale parameter, beam thickness, and boundary condition on the linear and nonlinear vibration behaviors of SGBs. The present study may provide help for optimal designing and fabricating micro-/nano-resonators.

Softening-spring nonlinearity in large amplitude vibration of unsymmetric double-layer lattice truss core sandwich beams

In recent years, many novel sandwich structures with multilayer or graded lattice truss cores that exhibited superior structural performance have been proposed. However, only limited works have studied the nonlinear behavior of sandwich structures with unsymmetric lattice truss cores. This paper aims to provide an analytical study on the nonlinear vibration of the unsymmetric double-layer lattice truss core sandwich beams (LTCSBs). The double-layer LTCSB is designed to be unsymmetric and possesses varying material property and structural geometry in each layer. In this study, six unsymmetric cases of LTCSB classified as two categories according to the midplane locations are considered. Subsequently, an analytical model for the unsymmetric double-layer LTCSB is developed based on the Allen’s model and von Kármán nonlinear theory. The axial displacement of the midplane of LTCSB is considered in the analytical model, therefore the proposed model is more generalized compared with previous models for the symmetric double-layer LTCSB. The Ritz method with a direct iterative procedure is applied to solve the nonlinear governing equations and determine the nonlinear frequencies for the unsymmetric double-layer LTCSB. Finally, the effects of six unsymmetric cases of LTCSBs on the nonlinear frequency ratio versus amplitude curve under three different types of boundary conditions are discussed detailly. An interesting phenomenon of softening-spring nonlinearity is found for hinged–hinged and clamped–hinged sandwich beams with large bending–extension coupling.

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

This paper presents the application of an exponential version of the harmonic balance method to the analysis of steady state vibration of geometrically nonlinear systems. A detailed description of the method and of the corresponding numerical procedure is provided. The von Karman theory is used to describe the effects of geometric nonlinearity. The material of the beams is modelled with the help of the Zener model using the fractional calculus. The problem is solved using an exponential version of the harmonic balance method. In the above-mentioned version, the complex calculus is used in contrast to the ordinary harmonic balance method, where the steady state vibrations are described with the help of the trigonometric functions. It significantly simplifies derivation of the amplitude equations. Moreover, the exponential version of the harmonic balance method is more elegant in comparison with the ordinary one. A detailed derivation of the amplitude equations is presented. The modified continuation method is proposed to solve the nonlinear amplitude equations and to determine the response curves. Moreover, the results of the exemplary calculation are presented and compared with known results in order to justify the efficiency and the correctness of the proposed approach.

Influence of hygro-thermal effects on the geometrically nonlinear free and forced vibrations of piezoelectric functional gradient beams with arbitrary number of concentrated masses

Influence of hygro-thermal effects on the geometrically nonlinear free and forced vibrations of piezoelectric functional gradient beams with arbitrary number of concentrated masses

Modal analysis of viscoelastic three-dimensional rotating beam with generic tip mass

In this paper, the nonlinear dynamic modelling, free vibration and nonlinear analysis of three-dimensional viscoelastic Euler-Bernoulli beam undergoing hub motion incorporating substantial tip mass is accomplished. The kinetic, and potential energy of the system are derived in terms of velocities of general point on the link and center of gravity of the tip mass expressed in multi-floating co-ordinate systems. Hamilton's principle is used to obtain the governing equations of motion and linearly coupled boundary conditions. The material of the beam is considered as viscoelastic constituted of Kelvin-Voigt rheological model. The mass attached at the terminal end of the beam is assumed to have eccentricity in axial, lateral, as well as transverse directions. Free vibration analysis is performed on the linearized system model to obtain the transcendental eigenfrequency equation. Further, the dynamic equations of motion of beam are discretized using the obtained mode shapes and the response of system under rotary motion of hub is investigated. The steady state solutions and frequency response curves exhibiting bi-stable and tri-stable regions are obtained using method of multiple scales. The bifurcation diagrams of the system are studied for resonance conditions when the frequency of the rotary motion becomes equal or nearly equal to the normalized beam frequencies. The saddle node and pitchfork bifurcations exhibiting multiple solutions and jump phenomena are observed and investigated to avoid catastrophic failure of the system. The numerical simulations of modal parameters of the system, nonlinear characteristics, and their parametric dependency is discussed thoroughly.

Nonlinear vibration of third-order shear deformable FG-GPLRC beams under time-dependent forces: Gram–Schmidt–Ritz method

This study presents nonlinear vibration of functionally graded-graphene platelet-reinforced composite (FG-GPLRC) beams under various time-dependent forces. Their material distributions are characterized by continuous functions with four patterns of reinforcement which are uniform, linear and parabolic I and II. The third-order shear deformation theory is used to represent the displacement fields, while the geometric nonlinearity is based on the von Kármán assumption. The Gram–Schmidt–Ritz method is utilized with iteration process to obtain the linear and nonlinear results. Several effects such as weight fraction of graphene nanoplatelets, types of material distributions, beam geometry, etc. on nonlinear dynamic deflection of the beams are investigated. It is found that the beams reinforced by graphene nanoplatelets mostly near the top and bottom faces are stronger than those with other different patterns of reinforcement. The comparison between the responses of continuous and multi-layers FG-GPLRC beams is presented. Some new results of FG-GPLRC beams are given and discussed in details and they can be considered as a benchmark solution for future investigations.

Nonlinear vibrations of beams with Bouc–Wen hysteretic boundary conditions

Nonlinear vibrations of beams with Bouc–Wen hysteretic boundary conditions

Nonlinear Thermo-Mechanical Response of Bi-Directional Functionally Graded Porous Beams with Initial Geometrical Imperfection

In this paper, the nonlinear vibration and buckling of bi-directional functionally graded (2D-FG) beam subjected to thermal loading are examined via the higher-order shear deformable beam theory incorporating the von Kármán geometric nonlinearity. Two types of initial defects including the internal porosity distribution and external geometrical imperfection shape are taken into consideration. The temperature-dependent material property varying along the length and thickness following the power-law function is employed, and the porosities inside the 2D-FG beam with even and uneven distributions are considered. The initial geometrical imperfection is described by the product of trigonometric and hyperbolic functions, which can capture both global and localized imperfection shapes. The discrete equations of motion for the 2D-FG beam are established by using the differential quadrature method (DQM), and the derived function is solved by an iterative procedure. The nonlinear thermo-mechanical vibration and buckling of 2D-FG beams under uniform, linear, and nonlinear temperature loads are systemically compared. Moreover, several key factors such as 2D-FG indexes, porosity distribution, geometrical imperfections, as well as boundary conditions are investigated in detail. The proposed model and obtained results can be used to guide the optimization design of multi-functional and multi-graded materials serviced under thermal environment.

Oscillation Attenuation in a Building-like Structure by Using a Flexible Vibration Absorber

This is a theoretical, numerical, and experimental study on the vibration attenuation capability of the dynamic response of a building-like structure using a dynamic vibration absorber in cantilever flexible beam configuration, taking into account gravitational effects associated with its mass. The dynamic model of the primary vibrating structure with the passive vibration control device is obtained using the Euler–Lagrange formulation considering the flexible vibration absorber as a generalized system of one degree of freedom. The application of the Hilbert transform to the frequency response function to determine the tuning conditions between this nonlinear flexible beam vibration absorber and the primary system is also proposed. In this fashion, Hilbert transform analysis is then carried out to show that nonlinearities present in the dynamic model do not significantly contribute to the performance of the implemented absorber. Therefore, it is valid to linearize the equations of motion to obtain the tuning condition in which the flexible vibration absorber can attenuate undesirable harmonic vibrations that are disturbing to the building-like flexible structure. Thus, the present study shows that the Hilbert transform can be applied to obtain tuning conditions for other configurations of dynamic vibration absorbers in nonlinear vibrating systems. Simulation and experimental results are included to demonstrate the efficient performance of the presented vibration absorption scheme.

An analytical approach to geometrically nonlinear free and forced vibration of piezoelectric functional gradient beams resting on elastic foundations in thermal environments

In this article, the geometrically nonlinear free and forced vibrations of a beam made of functional gradient materials are analytically analyzed, in which piezoelectric actuators are bound. Based on the Euler–Bernoulli beam theory and the nonlinear Von-Karman deformation field, the analytical solutions of the nonlinear free vibrations, and forced vibrations of a beam with two clamped ends and resting on an elastic foundation are predicted. Using Hamilton's principle and an approximate multimodal dominant mode method, the nonlinear governing equations are obtained. Due to the lack of forced vibration results of the functionally graded piezoelectric beam subjected to thermal loading and resting on elastic foundations, the results obtained by the current solution were compared with their counterparts. The good agreement between the current solution and equivalent data existing in the open literature proves the efficiency and accuracy of the current resolution analysis method. Finally, the new parametric studies covered in this research include the effects of several parameters such as elastic foundation, thermal loading, and thermal properties of constituent materials on the free and forced nonlinear vibration of the piezoelectric functional gradient beam.

Numerical analysis of nonlinear functionally graded piezoelectric energy harvester subjected to multi moving oscillators

In this paper, energy harvesting from nonlinear vibration of a functionally graded beam covered by a piezoelectric patch under multi-moving oscillators is studied. The material of both the substructure and the piezoelectric patch is assumed to be functionally graded in the thickness direction. A coupled system of equations considering Euler-Bernoulli beam theory and von-Karman nonlinearity as well as electromechanical coupling are derived using the generalized Hamilton’s principle. Finite element method as well as Newmark time integration scheme are used to solve the coupled nonlinear time dependent problem. The effects of different parameters including material distribution, velocity of the moving oscillators, piezoelectric patch thickness and load resistance on the output voltage and harvested power are investigated. Moreover, the effects of oscillator characteristics such as damping ratio and stiffness on the nonlinear behavior of the beam and harvested power are also studied. Results indicate that the aforementioned parameters have considerable effects on the harvested power. It is also shown that ignoring nonlinear effects may lead to erroneous and unacceptable results. To the best of authors’ knowledge, there is no study about energy harvesting from nonlinear vibration of beams under moving oscillators.

A Multimode Approach to Geometrically Nonlinear Free and Forced Vibrations of Multistepped Beams

The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.

Analysis of the nonlinear forced vibration and stability of composite beams using the reduced-order model

The purpose of this paper is to analyze the nonlinear vibration of composite beams and the stability of the solution by using the reduced-order model and the incremental harmonic balance (IHB) method. Timoshenko beam theory is used to indicate the displacement of the beam element. Each nodal point has three degrees of freedom. Simplified homogenized beam theory is used to calculate the equivalent moduli of each ply of the composite beam. Element matrices are created using the weak form quadrature element method, and the equation of motion at the element is created using Lagrange’s equation. A system matrix is created using the element matrix assemble rule of the finite element method. In order to reduce the calculation time, a reduced-order model is used. The nonlinear forced vibration equation is solved using the IHB method. The results calculated using the non-reduced-order model and the reduced-order model are compared, and the results are very close. Based on this, the reduced-order model is used to analyze the nonlinear vibrations of composite beams at the first resonance point, and stability tests are conducted for the calculated solutions using the multivariable Floquet theory.

Nonlinear vibrations of beams with bilinear hysteresis at supports: interpretation of experimental results

Experimental vibration responses of beams with nonlinear boundary conditions were interpreted using a bilinear spring model. Two systems of beams were considered: one is a single rod and the second one is a cluster of parallel rods linked together. The boundary conditions at the beam ends, for both systems, were provided by spacer grids, which are specific supports used in nuclear industry. The experiments were conducted with the beams immersed in still water. The nonlinear boundary conditions imposed to the beams by the spacer grids were experimentally characterized. The boundary conditions changed from having a rotational linear stiffness at low vibration amplitudes to bilinear with hysteresis at higher vibration amplitudes. The bilinear stiffness was found to be of the softening type. A bilinear stiffness model with viscous damping was used to interpret the forced vibration response of the beams. The stiffness and damping ratio used in the model were identified at any excitation level by minimizing, in the frequency domain, the weighted distance between the set of experimental points and the corresponding numerical results. The equivalent viscous damping of the system grew with the vibration amplitude. Very good agreement between numerical and experimental results was obtained for both the beam systems studied in the frequency and force-displacement domains. The model was further refined by using a piecewise linear stiffness function with additional switching points and introducing nonlinear damping. The identified model was capable of reproducing the experimental results with accuracy and without the requirement to adjust the parameters at different excitation levels.

Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

Effects of physical parameters and time-delay coefficients on the amplitude and frequency bandwidth of saturation controller for a nonlinear beam vibration

This paper is focused on the effects of the physical parameters for saturation control bandwidth and the time-delay coefficient for the saturation control stability. The approximate analytical solutions are obtained when the primary external resonance and 1:2 internal resonance occur simultaneously. The effects of amplitude of excitation, nonlinear feedback coefficients and nonlinear coefficients on saturation control are investigated. It is shown that nonlinear feedback coefficients can enlarge the effective saturation control bandwidth and enhance the dynamic vibration absorber’s suppression. Two nonlinear coefficients enhance the system’s nonlinear phenomenon and remain the same saturation control regions. For the time-delay coefficient, the system begins to appear unstable saturation control regions with only a little time-delay. The increasing time-delay coefficients can enlarge the unstable saturation control regions, it should be avoided.

Nonlinear vibration of nonlocal strain gradient functionally graded beam on nonlinear compliant substrate

Beams supported on compliant substrate are used in nanostructures. Due to atomistic dimensions, mechanics of nanostructures are governed by molecular interactions, modeled by the Nonlocal (NL) and Strain Gradient (SG) continuum model. Due to their slender nature (owing to significant strength/stiffness of nano-materials), nanostructures are liable to geometric nonlinearity. Additionally, the supporting substrate may be nonlinear. However, nonlinear forced vibration of nanostructures is not researched well. This study investigates the geometrically nonlinear vibration of NL-SG beams on nonlinear substrate with shear interactions. The higher-order curvature, von Karman nonlinearity are included along with nonlinear Pasternak model for the substrate. The depth-wise functional gradation of material properties is also included. The governing equations of motion are derived following the variational method. A two-step perturbation is employed to obtain a closed-form solution. The free vibration nonlinear frequency, response time history, and forced vibration transmissibility characteristics are presented. The influence of important parameters is illustrated. Specifically, the dominant role of nonlinear bending and substrate stiffness is noted. The effect of NL and SG interactions is shown to significantly influence the vibration behavior. However, the effect of functional gradation of material is shown to be minor.