Nonlinear Vibrations Of Beams Research Articles

Nonlinear free vibrations of a structurally tailored anisotropic pre-twisted thin-walled beam subjected to large deformations

It is essential to understand the nonlinear free vibration behaviour of a thin-walled composite structure as they find their applications in harsher environments prone to large deformations. Further, these structures made of fibrous materials, have directional properties for different fibre orientations resulting in a diversified and a complex nonlinear behaviour. In this regard, present work provides a comprehensive mathematical formulation on nonlinear vibration of pre-twisted thin-walled anisotropic box beam. The mathematical model is derived as coupled (flap-lag-extension-torsion) model considering shear deformation and green's strain tensor for large displacements in order to study the influence of nonlinear couplings on the dynamic behaviour. The derived energy equations are presented depending on degree of nonlinearity. These nonlinear equations those derived are nondimensionalized and solved with in the frame work of energy method using classical Ritz approximation. An iterative method is used to solve the nonlinear equations pertaining displacement terms. This nonlinear behaviour is evaluated for two important types of structural tailoring techniques available for thin-walled composite beams namely Circumferentially Asymmetric Stiffness (CAS) and Circumferentially Uniform Stiffness (CUS). It is found that CAS layup experiences significantly severe nonlinear effects compared to CUS due to the presence of 3rd degree nonlinear coupling terms. The variation of nonlinear frequency ratios over different ply angles showing the influence of nonlinearity on fibres orientation is presented for the first time for the two ply lay ups. The influence of degree of nonlinearity on the accuracy of the nonlinear frequencies is evaluated and presented. Moreover, the impact of nonlinearity on the pre-twisted beam is presented for different pre-twist angles.

Asymptotic analysis of geometrically nonlinear beam vibrations: Kirchhoff and Bolotin equations

AbstractThe paper analyzes various approximate models of geometrically nonlinear vibrations of a beam. In practice, simplified equations are often based on the quasi‐static Kirchhoff hypothesis—neglecting axial inertia. This hypothesis is justified with the prescribed end‐displacements of the beam in the axial direction. Under dead loading, quasi‐static Kirchhoff hypothesis results in a linear equation. The corresponding approximate equations obtained in this paper are based on the asymptotic procedure. The ratio of bending stiffness to reduced tensile/compressive stiffness is taken as a small parameter. Axial inertia is taken into account in the equation of the first approximation. Introduced by V.V. Bolotin concept “nonlinear inertia” is discussed. The most common errors in using the quasi‐static Kirchhoff hypothesis are analyzed.

Nonlinear vibration of sandwich beams made of FGM faces and FGP core under multiple moving loads using a quasi-3D theory

This study aims to investigate nonlinear dynamic behavior of sandwich beams constituted by functionally graded material (FGM) faces and functionally graded porous (FGP) core under the action of multiple moving loads. The energy equations based on a quasi-3D beam theory considering several functions of shear deformation were established for producing the nonlinear equations of motion used for describing dynamic responses of beams with various boundary conditions. By using the hybrid processes of the Newton-Raphson iteration procedure, the time-integration approach of Newmark-β, and the Gram-Schmidt-Ritz method, the new results of nonlinear analysis were explored and proposed. Several prime parameters such as the porous coefficient, sandwich thickness ratio, power law index and others that significantly affect the dynamic response of the beams were taken into investigations. It can be concluded that beams under one moving load have larger dynamic deflections compared to that under many moving loads. This is because the distance between the moving loads can produce a bending moment against the beam deformation.

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

Analytical Study of Nonlinear Flexural Vibration of a Beam with Geometric, Material and Combined Nonlinearities

This article explores the nonlinear vibration of beams with different types of nonlinearities. The beam vibration was modeled using Hamilton’s principle, and the equation of motion was solved using method of multiple time scales. Three models were developed assuming (a) geometric nonlinearity, (b) material nonlinearity and (c) combined geometric and material nonlinearity. The material nonlinearity also included both third and fourth nonlinear elasticity terms. The frequency response equation of these models were further evaluated quantitatively and qualitatively. The models capture the hardening effect, i.e., increase in resonant frequency as a function of forcing amplitude for geometric nonlinearity, and the softening effect, i.e., decrease in resonant frequency for material nonlinearity. The model is applied on the first three bending modes of the cantilever beam. The effect of the fourth-order material nonlinearity was smaller compared to the third-order term in the first mode, whereas it is significantly larger in second and third mode. The combined nonlinearity models shows a discontinuous frequency shift, which was resolved by utilizing a set of transition assumptions. This results in a smooth transition between the material and geometric zones in amplitude. These parametric models allow us to fine tune the nonlinear response of the system by changing the physical properties such as geometry, linear and nonlinear elastic properties.

Nonlinear vibration and acoustic radiation of an internally resonant buckled beam

Although the nonlinear vibration of buckled beam has been thoroughly studied, its acoustic radiation does not get much attention. This paper deals with the nonlinear vibro-acoustic problem of an internally resonant buckled beam immersed in an infinite fluid. A finite element method based on an arbitrary Lagrangian-Eulerian framework is adopted to analyze the nonlinear interaction between the large-deformed buckled beam and the finite-amplitude acoustic waves in the fluid. The effects of acoustic pressure excitation, external harmonic excitation, and internal resonance on the vibro-acoustic responses of the buckled beam are discussed in the first and second primary resonances. It is found that the amplitude of the acoustic pressure can be comparable to the amplitude of the external harmonic force. This leads to an increase in the critical harmonic excitation amplitude for the occurrence of quasi-periodic responses and an increase in the second mode frequency component of quasi-periodic responses compared to the case neglecting the acoustic pressure excitation on the beam. Saturation phenomena are discovered in the second mode frequency components of beam displacement and acoustic pressure. Due to the 2:1 internal resonance, different time-frequency characteristics, quadrupole acoustic directivity patterns with low radiation efficiency, and dipole acoustic directivity patterns with high radiation efficiency can be observed at different harmonic excitation amplitudes and frequencies. Understanding these phenomena will help design loudspeakers and recognize sound sources.

Non-linear vibration and bifurcation analysis of Euler-Bernoulli beam under parametric excitation

This paper presents an analysis of the non-linear vibrations of beams, which play a crucial role in various industrial and construction structures. Understanding the transverse vibrations of beams and accurately determining their frequency response is essential for achieving optimal design and structural performance. The novelty of this study lies in conducting a transverse non-linear vibration analysis of a three-dimensional beam while considering the effect of mid-plane elongation. By incorporating this aspect into the analysis, the study aims to provide deeper insights into the dynamic behavior of beams subjected to non-linear effects. A multiple-time scale approach has been adopted to conduct this research. To verify the accuracy of the method as well as the accuracy of the outcomes gained from this method, a contrast has been made with the 4th-order Runge-Kutta technique, which indicates that the results obtained are acceptable. The frequency response of the beam indicates the presence of a phenomenon of splitting into two non-linear branches during the three-dimensional vibrations of the beam, as well as a hardening state in the frequency response as a result of stretching the middle plane of the beam. Furthermore, a parametric study was conducted in which different parameters were examined to determine the starting point of non-linear bifurcation. As a result, the damping coefficient and resonance deviation parameter are two factors that affect the preference for critical bifurcation over safe bifurcation. Furthermore, the stretching of the middle plane results in a higher non-linear term coefficient in the vibration equations of the beam, which increases the oscillation frequency of the beam.

The Approximate Analysis of Higher-Order Frequencies of Nonlinear Vibrations of a Cantilever Beam With the Extended Galerkin Method

Abstract The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.

Parametric study on damped nonlinear vibration of FG-GPLRC dielectric beam with edge crack

Parametric study on damped nonlinear vibration of FG-GPLRC dielectric beam with edge crack

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Beam vibrations with allowance for deformation effects in the transverse direction are modeled using a nonlinear Sobolev-type differential equation, for which the Cauchy problem is investigated in the space of continuous functions. Conditions for solution blow-up on a finite time interval are considered.

Machine learning aided uncertainty analysis on nonlinear vibration of cracked FG-GNPRC dielectric beam

Functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC) have shown significant potential for the development of high-performance and multifunctional materials and structures. Conducting an accurate analysis of the nonlinear vibration of FG-GNPRC dielectric beams can accelerate their application in various engineering fields. To investigate the nonlinear vibration of FG-GNPRC dielectric beams, the effective material properties of a cracked composite beams are evaluated using effective medium theory (EMT) and the rule of mixture. Three ML models adopted in this work, i.e., artificial neural network (ANN), random forest (RF), and AutoGluon (AG), are used to capture the complex relationship between the systematic inputs (i.e., parameters for materials and structure, the attributes of electrical field, etc.) and frequency ratio of the cracked FG-GNPRC beams. Moreover, shapley additive explanations (SHAP) is employed for the analysis of the nonlinear vibration of FG-GNPRC dielectric beams considering the influence of multi-factor coupling and uncertainties. The SHAP analysis suggests that the uncertainty of structural parameters has the greatest impact on the nonlinear vibration of FG-GNPRC dielectric beams, particularly when the structure is subjected to a voltage of 40 V.

Nonlinear vibrations of beams with fractional derivative elements crossed by moving loads

This paper addresses the evaluation of the time-domain response of nonlinear beams endowed with a fractional derivative element crossed by moving loads. Nonlinearities originate from the assumption of moderately large displacements of the beam. Following a Galerkin-type solution procedure, beam transversal displacement is represented in terms of the linear modes of vibration and time-dependent generalized coordinates. A novel step-by-step integration scheme, labeled improved pseudo-force method (IPFM), is developed for the numerical solution of the set of coupled nonlinear fractional differential equations governing the time-dependent generalized coordinates. The proposed procedure stems from the extension of a recently developed step-by-step scheme for the dynamic analysis of fractional single-degree-of-freedom systems. The IPFM involves the following main steps: i) to apply the Grünwald–Letnikov approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response as pseudo-forces; iii) to handle nonlinearities by performing iterations at each time step.Numerical results are presented to assess the accuracy of the IPFM as well as to investigate the influence of the fractional derivative order and coefficient on nonlinear beam vibrations under moving loads.

Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.

Nonlinear vibration of SMA hybrid composite beams actuated by embedded pre-stretched SMA wires with tension-bending coupling effect

Theoretical and numerical analyses for nonlinear vibrations of the shape memory alloy hybrid composite(SMAHC) beam actuated by the pre-stretched shape memory alloy(SMA) fibers with tension-bending coupling effect are presented in this paper. Under the sudden action of thermal temperature, a recovery force and moment are caused in the SMAHC beam due to the phase transition in the SMA fibers, exciting the nonlinear vibration of the SMAHC beam. The one-dimensional Brison constitutive model was adopted to express the influence of the phase transition temperature, initial volume fraction of martensite, and pre-strain on the phase transition in the shape memory alloy. On this basis, the constitutive law of the SMAHC beam is built by considering the tension-bending coupling effect and the phase transition in the SMAHC beam. The equation of motion for nonlinear vibration of the SMAHC beam with geometric nonlinearity and material nonlinearity is derived from the Hamilton principle and is discretized into matrix form using the Galerkin method. According to the algorithm of the NewMark-β method, numerical solutions are computed by developing Matlab program and are compared with the results from ABAQUS analysis. It shows that the frequencies and amplitudes of the theoretical calculations are very consistent with those obtained from ABAQUS analysis. In addition, both the theoretical study and ABAQUS simulation indicates that the high frequency for longitudinal vibration is not observed in the spectrum diagram of transverse vibration, while the low frequencies for transverse vibration are captured in the longitudinal vibration. With the established Matlab program, parametric study is carried out to investigate the effect of the temperature and the location of the SMA fibers on the vibration responses of the SMAHC beam. The results from this study indicate that the transverse vibration of SMAHC beams is characterized as one of low frequency, while the longitudinal vibration has the characteristics of high frequency. Especially in the case of nonlinear vibration, not only the vibration of high frequency but also the vibration of low frequency observed in the transverse vibration will occur in the longitudinal direction of the SMAHC beam. Furthermore, the double and triple fundamental frequencies of the transverse vibration are excited in the nonlinear vibration of the SMAHC beam. The results of this paper can provide a theoretical basis for the evaluation of the safety and stability of smart composite structures.

Re-examination of nonlinear vibration, nonlinear bending and thermal postbuckling of porous sandwich beams reinforced by graphene platelets

This paper re-examines the nonlinear free vibration, nonlinear bending and thermal postbuckling behaviors of porous sandwich beams with graphene platelets (GPL) reinforcements rested on elastic foundations. We remove the equivalent isotropic model (EIM) and introduce an inhomogeneous model. The shear modulus together with the Young’s moduli for porous GPLRC core is determined by a generic Halpin–Tsai model with porosity. Mechanical properties of both porous GPLRC core and metal face sheets are temperature dependent. Based on the framework of higher order shear deformation beam theory, the motion equations of porous sandwich beams with GPL reinforcements are established. In the modeling, the von Kármán kinematic nonlinearity, beam-foundation interaction and thermal effect are considered. By employing the two-step perturbation method, the analytical solutions are obtained for the nonlinear vibration and nonlinear bending as well as thermal postbuckling problems. Numerical investigations are performed for comparing the results obtained from the EIM and the present model. The outcomes reveal that the EIM is invalid for linear free vibration, thermal buckling and postbuckling analyses of porous sandwich beams with GPL reinforcements, but the EIM is valid in some cases for nonlinear bending and nonlinear vibration analyses of the same beam.

Nonlinear transient response of sandwich beams with functionally graded porous core under moving load

Nonlinear transient response of sandwich beams possessing functionally graded porous core under the action of moving load was examined in this study. With this regard, Reddy's third-order shear deformation theory and the geometrical nonlinearity of von Kármán assumption were utilized to construct the governing equation system. The set of governing equations was solved by the Gram-Schmidt-Ritz method in conjunction with iterative procedures based on the time-integration of Newmark for the convergence in time and geometrical domains. According to the methodology, we received accurately stable results of nonlinear free and forced vibration of the sandwich beams. Several important parameters such as slenderness ratio, layer thickness's ratio, porous coefficient, types of porous distribution, etc. that have a significant impact on nonlinear deflection of the beams were taken into account. Based on numerical experiments, it can be disclosed that the sandwich beams carrying porous distribution in form of functionally graded materials are much better than the beams with uniform porous distribution in terms of strength and stiffness. The porous coefficient also plays an important role in changing the ability of loading resistance.

A Semi-Analytical Approach for the Linearized Vibration of Clamped Beams with the Effect of Static and Thermal Load

The geometric nonlinearity due to static and thermal load can significantly alter the vibration response of structures. This study presents a semi-analytical approach to illustrate the nonlinear vibration of clamped-clamped beams under static and thermal loads. The von Karman strain and Hamilton’s principle are employed to derive the nonlinear static equilibrium equation and nonlinear governing equation. The vibration equation’s coefficient is variable. The transfer-matrix method and local homogenization are used to solve the equation. The proposed method’s accuracy is validated by commercial software and literature. The numerical results indicate that uniform stress caused by thermal load only reduces the structural mode frequencies. The geometric nonlinearity of the structural static deformation affects both the mode frequencies and mode shapes. And the mode shapes cannot be approximated by harmonic functions. When the static deformation is significant, the structure’s local RMS response is substantially affected. The combined loads have a more significant impact on the acceleration response than the superposition of individual load effects.

Nonlinear vibrations of multiscale composite beams on a nonlinear softening foundation

This paper describes a new approach to evaluate the instability of beams on a softening elastic foundation. The basis of this approach is solving a nonlinear frequency problem. The beam is made of multiscale composite materials based on carbon nanotubes (CNTs) and long fibers. Homogenization methods are utilized in a sequential process to obtain the effective material properties of these composites. Furthermore, the thermal environment's effect on the beam's response is examined. Reddy's third-order shear deformation theory approximates the induced displacements in the composite beam. Also, the kinetic and strain energies of the beam are calculated based on the von Kármán nonlinear strains and linear thermoelasticity theories. A recently reported Lagrangian nodal weak formulation (LNWF) method is implemented to obtain the linear and nonlinear stiffness and mass matrices governing the beam's vibrations. The modified coefficients for nonlinear stiffness matrices are obtained using the residual Galerkin method across time. Finally, the nonlinear natural frequencies are calculated by employing the iterative displacement control strategy. In the results section, in addition to examining various parameters on nonlinear free vibrations, the effect of these parameters on the displacement required for the loss of stability of the beam placed on the softening foundation is also examined.

Control of nonlinear vibration of beams subjected to moving loads using tuned mass dampers

Control of nonlinear vibration of beams subjected to moving loads using tuned mass dampers