Nonlinear Vibrations Of Beams Research Articles

Nonlinear vibrations of double beams connected by a spring and dashpot

Nonlinear vibrations of double beams connected by a spring and dashpot

Non-linear dynamic instability analysis of thin-walled stiffener beam subjected to uniform harmonic in-plane loading

Present analysis deals with nonlinear flexural-torsional vibration and dynamic instability of thin-walled stiffener beam with open section subjected to harmonic in-plane loading. The static and dynamic components of the applied harmonic in-plane loading are assumed to vary uniformly. A set of nonlinear partial differential equations (PDEs) describing the vibration of system is derived. Using Galerkin's method, these partial differential equations are reduced into coupled Mathieu equations. The steady state response of the system is determined by solving the condition for a non-trivial solution. The principal regions of parametric resonance are determined using the method suggested by Bolotin. The numerical results are presented to investigate the effect of aspect ratios, boundary conditions and static load factor on the frequency-amplitude responses and instability regions.

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

PurposeRecently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.Design/methodology/approachIn this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.FindingsThe theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.Originality/valueA new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

Nonlinear Vibration and Stability Analysis of Beam on the Variable Viscoelastic Foundation

The aim of this study is the investigation of the large amplitude deflection of an Euler-Bernoulli beam subjected to an axial load on a viscoelastic foundation with the strong damping. In order to achieve this purpose, the beam nonlinear frequency has been calculated by homotopy perturbation method (HPM) and Hamilton Approach (HA) and it was compared by the exact solutions for the different boundary conditions such as simple-simple, clamped-simple and clamped-clamped which showed a good accuracy in results. In addition, to find the deflection of the nonlinear Euler-Bernoulli beam, the problem has been solved based on homotopy perturbation method and modified differential transform method (MDTM) and finally, the results were compared by Rung-Kutta exact solutions. The derived deflection results by two mentioned methods had a good agreement with the exact RK4 solutions. By considering the paper results, buckling force is increased for each case permanently by increase in the boundary rigidity for a constant value of system amplitude (A). As a final comparison, in based on paper results, the buckling force is arisen by increasing the system amplitude for each case.

Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation

The purpose of this research is to study the nonlinear free vibration and post-buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams resting on a nonlinear elastic foundation. Uniformly and functionally graded distributions of single walled carbon nanotubes as reinforcing phase are considered in the polymeric matrix. The modified form of rule of mixture is used to estimate the material properties of CNTRC beams. The governing equations are derived employing Euler-Bernoulli beam theory along with energy method and Hamilton\'s principle. Applying von Karman\'s strain-displacement assumptions, the geometric nonlinearity is taken into consideration. The developed governing equations with quadratic and cubic nonlinearities are solved using variational iteration method (VIM) and the analytical expressions and numerical results are obtained for vibration and stability analysis of nanocomposite beams. The presented comparative results are indicative for the reliability, accuracy and fast convergence rate of the solution. Eventually, the effects of different parameters, such as foundation stiffness, volume fraction and distributions of carbon nanotubes, slenderness ratio, vibration amplitude, coefficients of elastic foundation and boundary conditions on the nonlinear frequencies, vibration response and post-buckling loads of FG-CNTRC beams are examined. The developed analytical solution provides direct insight into parametric studies of particular parameters of the problem.

Nonlinear vibration of viscoelastic beams described using fractional order derivatives

The problem of non-linear, steady state vibration of beams, harmonically excited by harmonic forces is investigated in the paper. The viscoelastic material of the beams is described using the Zener rheological model with fractional derivatives. The constitutive equation, which contains derivatives of both stress and strain, significantly complicates the solution to the problem. The von Karman theory is applied to take into account geometric nonlinearities. Amplitude equations are obtained using the finite element method together with the harmonic balance method, and solved using the continuation method. The tangent matrix of the amplitude equations is determined in an explicit form. The stability of the steady-state solution is also examined. A parametric study is carried out to determine the influence of viscoelastic properties of the material on the beam's responses.

A Discrete Model for Nonlinear Vibration of Beams Resting on Various Types of Elastic Foundations

This paper presents a discrete physical model to approach the problem of nonlinear vibrations of beams resting on elastic foundations. The model consists of a beam made of several small bars, evenly spaced. The bending stiffness is modeled by spiral springs, and the Winkler soil stiffness is modeled using linear vertical springs. Concentrated masses, presenting the inertia of the beam, are located at the bar ends. Finally, the nonlinear effect is presented by the axial forces in the bars, assumed to behave as longitudinal springs, due to the change in their length induced by the Pythagorean Theorem. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing any modification in the mass matrix, the stiffness matrix, and the nonlinearity tensor to be made separately. Therefore, once the model is established, various practical applications may be performed without the need of going through all the formulation again. The study of the nonlinear behavior makes the solution of the movement equation rise in complexity. By considering this discrete model and using the linearization method, one can achieve an idealized approach to this nonlinear problem and obtain quite easily approximate solutions.

Geometrically nonlinear transverse vibrations of Bernoulli-Euler beams carrying a finite number of masses and taking into account their rotatory inertia

The objective of this paper is to establish the theoretical formulation of the problem of nonlinear transverse vibrations of Bernoulli–Euler beams carrying a finite number of masses at arbitrary positions, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. The method used is based Hamilton’s principle and spectral analysis for nonlinear free vibrations exhibiting large displacement amplitudes. The problem is reduced to solution of a nonlinear algebraic system using numerical or analytical methods. This has been previously applied to nonlinear transverse vibrations of continuous structures such as beams, plates and shells, to nonlinear longitudinal vibrations of 2-dof and multi-dof systems and to nonlinear transverse vibrations of N-dof systems. The nonlinear algebraic system has been solved using an approximate explicit method developed previously (The so-called second formulation) leading to the nonlinear fundamental mode shape of beams carrying a finite number of masses and to the corresponding backbone curves.

Geometrically nonlinear, steady state vibration of viscoelastic beams

The problem of geometrically non-linear steady state vibrations of beams excited by harmonic forces is considered in this paper. The beams are made of a viscoelastic material defined by the classic Zener rheological model - the simplest model that takes into account all the basic properties of real viscoelastic materials. The constitutive stress-strain relationship for this type of material is given as a differential equation containing derivatives of both stress and strain. This significantly complicates the solution to the problem. The von Karman theory is applied to describe the effects of geometric nonlinearities of beam deformations. The equations of motions are derived using the finite element methodology. A polynomial approximation of bending moments is used. The order of basis functions is set so as to obtain a coherent approximation of moments and displacements. In the steady-state solution of equations of motion, only one harmonic is taken into account. The matrix equations of amplitudes are derived using the harmonic balance method and the continuation method is applied for solving them. The tangent matrix of equations of amplitudes is determined in an explicit form. The stability of steady-state solution is also examined. The resonance curves for beams supported in a different way are shown and the results of calculation are briefly discussed.

Nonlinear vibrations of periodic beams

Geometrically nonlinear vibrations of beams with properties periodically varying along the axis are investigated. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain governing equations with constant coefficients. The proposed model describes dynamics of the beam with the effect of microstructure size. In an example, an analysis of undamped forced nonlinear vibrations of the periodic beam is shown. Moreover, the results obtained for undamped free vibrations of periodic beams by the tolerance model are justified by those results from the finite element method. These results can be used as a benchmark in similar problems.

Nonlinear dynamic analysis of damaged Reddy–Bickford beams supported on an elastic Pasternak foundation

Geometrically nonlinear free and forced vibrations of damaged high order shear deformable beams resting on a nonlinear Pasternak foundation are investigated in this paper. Equations of motion are derived for the beam which is under subjected combined action of arbitrarily distributed or concentrated transverse loading as well as axial loading. To account for shear deformations, the concept of high order shear deformation is used in comparison with the concept of first order shear deformation theory. Analyses are performed to investigate the effects of the specific stiffness of the foundation on the damaged beam frequencies and displacements with the aim of equalising the response of a damaged and an intact beam. According to that, functions of the foundation stiffness are determined depending on the location and size of the damage as a result of the possibility for the damaged beam to behave like one that is intact. An advanced p-version of the finite element method is developed for geometrically nonlinear vibrations of damaged Reddy–Bickford beams. The present study gives a clear view of the nonlinear dynamical behaviour of four types of beams according to high order shear deformation theory – an intact beam, a damaged beam, a damaged beam on an elastic foundation and intact beam on elastic foundation. The paper also presents the derivation of a new set of two nonlinear partial differential equations where only the transverse and axial displacements figure. The forced nonlinear vibrations problem is solved in the time domain using the Newmark integration method. Free vibration analysis carried out by harmonic balance and the use of continuation methods and backbone curves are constructed.

Analytical Approximation of Nonlinear Vibration of Euler-Bernoulli Beams

In this paper, the Homotopy Analysis Method (HAM) with two auxiliary parameters and Differential Transform Method (DTM) are employed to solve the geometric nonlinear vibration of Euler-Bernoulli beams subjected to axial loads. A second auxiliary parameter is applied to the HAM to improve convergence in nonlinear systems with large deformations. The results from HAM and DTM are compared with another popular numerical method, the shooting method, to validate these two analytical methods. HAM and DTM show excellent agreement with numerical results (the maximum errors in our calculations are about 0.002%), and they additionally provide a simple way to conduct a parametric analysis with different physical parameters in Euler-Bernoulli beams. To show the benefits of this method, the effect of different physical parameters on the amplitude is discussed for a cantilever beam with a cyclically varying axial load.

Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core

The nonlinear free vibration behavior of shear deformable sandwich porous beam is investigated in this paper within the context of Timoshenko beam theory. The proposed beam is composed of two face layers and a functionally graded porous core which contains internal pores following different porosity distributions. Two non-uniform functionally graded distributions are considered in this paper based on the equivalent beam mass, associated with a uniform distribution for purpose of comparison. The elastic moduli and mass density are assumed to vary along the thickness direction in terms of the coefficients of porosity and mass density, whose relationship is determined by employing the typical mechanical characteristic of an open-cell metal foam. The Ritz method and von Kármán type nonlinear strain-displacement relationships are applied to derive the equation system, which governs the nonlinear vibration behavior of sandwich porous beams under hinged or clamped end supports. A direct iterative algorithm is then used to solve the governing equation system to predict the linear and nonlinear frequencies which are presented by a detailed numerical study to discuss the effects of porosity coefficient, slenderness ratio, thickness ratio and to compare the varying porosity distributions and boundary conditions, providing a feasible way to improve the vibration behavior of sandwich porous beams.

Axial-transversal coupling in the free nonlinear vibrations of Timoshenko beams with arbitrary slenderness and axial boundary conditions.

The nonlinear free oscillations of a straight planar Timoshenko beam are investigated analytically by means of the asymptotic development method. Attention is focused for the first time, to the best of our knowledge, on the nonlinear coupling between the axial and the transversal oscillations of the beam, which are decoupled in the linear regime. The existence of coupled and uncoupled motion is discussed. Furthermore, the softening versus hardening nature of the backbone curves is investigated in depth. The results are summarized by means of behaviour charts that illustrate the different possible classes of motion in the parameter space. New, and partially unexpected, phenomena, such as the changing of the nonlinear behaviour from softening to hardening by adding/removing the axial vibrations, are highlighted.

Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

In this research, large amplitude free vibrations of a sandwich beam with stiff core and carbon nanotube (CNT)-reinforced face sheets are analysed. The distribution of CNTs across the thickness of the face sheets may be uniform or functionally graded. The equivalent single- layer theory of Timoshenko is used to construct the Hamiltonian of the beam under the von Karman type of geometrical nonlinearity assumptions. A uniform temperature field through the beam is also included in the formulation. The Ritz method with polynomial basis functions is used to discrete the equations of motion and establish the matrix representation of the governing equations. A nonlinear eigenvalue problem is obtained and solved using a standard continuation procedure. After validating the developed solution method and formulation, parametric studies are conducted to examine the influences of thermal environment, core thickness-to-face sheet thickness ratio, boundary conditions, amplitude of vibrations, CNTs volume fraction and their distribution pattern. It is concluded that an increase in the volume fraction of CNTs results in higher fundamental frequency and decreases the nonlinear-to-linear frequency ratio.

A weak form quadrature element method for nonlinear free vibrations of Timoshenko beams

Purpose– The purpose of this paper is to highlight the implementation of a recently developed weak form quadrature element method for nonlinear free vibrations of Timoshenko beams subjected to three different boundary conditions.Design/methodology/approach– The design of the paper is based on considering the geometrically nonlinear effects of axial strain, bending curvature, and shear strain. Then the quadrature element formulation of the beam is introduced.Findings– The efficiency of the method is demonstrated by a convergence study. Ratios of the nonlinear fundamental frequencies to the corresponding linear frequencies are extracted. Their variations with the ratio of amplitude to radius of gyration and the slenderness ratio are examined. The effects of the nonlinearity on higher order frequencies and mode shapes are also investigated.Originality/value– The computed results show fast convergence and compare well with available literature results.

Piezoelectric optimal delayed feedback control for nonlinear vibration of beams

An optimal delayed feedback control methodology is developed to mitigate the primary and super harmonic resonances of a flexible simply-simply supported beam with piezoelectric sensor and actuator. Stable vibratory regions of the feedback gains are obtained by using the stability conditions of eigenvalue equation. Attenuation ratio is used to evaluate the performance of vibration control by taking the proportion of peak amplitude of primary or super harmonic resonances for the suspension system with and without controllers. Optimal control parameters are obtained using an optimal method, which takes attenuation ratio as the objective function and the stable vibratory regions of the time delay and feedback gains as constraint conditions. The piezoelectric optimal controllers are designed to control the dynamic behaviour of the nonlinear dynamic system. It is found that the optimal feedback gains obtained by the optimal method result in a good control performance.

A differential quadrature procedure for linear and nonlinear steady state vibrations of infinite beams traversed by a moving point load

A differential quadrature procedure is proposed to study the steady state linear and nonlinear vibrations of an infinite beam resting on an elastic Winkler foundation and subjected to a moving point load. The governing nonlinear partial differential equation of motion of the beam is first expressed with respect to a moving coordinate system. This step reduces the governing nonlinear partial differential equation of motion of the beam to an ordinary nonlinear differential equation. This equation is then converted to a set of nonlinear algebraic equations by application of the differential quadrature method. The Newton–Raphson method is used to solve the resultant system of nonlinear algebraic equations. Issues related to implementation of infinite boundary conditions and modeling the point load are addressed. To accurately predict the dynamic behavior of the beam at high speeds of the moving point load, an efficient and robust absorbing boundary condition is also introduced. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison study with available analytical solutions in the literature is performed. Numerical results reveal that the proposed procedure can be used as an effective tool for handling nonlinear moving load problems on infinite domains.

Nonlinear harmonic vibration and stability analysis of a cantilever beam carrying an intermediate lumped mass

Nonlinear forced vibration of a cantilever beam with an intermediate lumped mass is studied, and the nonlinear governing equation of the vibrating beam using Euler–Lagrange method is derived. Two types of nonlinearities, including the inertial term and the elastic part, are included in the nonlinear differential equation of motion. The multiple scales method is adopted as the solution procedure for this problem and the conditions of the primary and secondary resonances of the system are investigated. The frequency response diagrams of the cantilever beam for the fundamental harmonic vibration, super-harmonic condition $$({\omega }\approx \frac{1}{2}{\omega }_0,\,{\omega }\approx \frac{1}{3}{\omega }_0\hbox { and }{\omega }\approx \frac{1}{5}{\omega }_0 )$$ and also sub-harmonic condition $$({\omega }\approx 2\omega _0,\,{\omega }\approx 3\omega _0\hbox { and }{\omega }\approx 5\omega _0 )$$ are obtained. The stable and unstable segments of each frequency-response curves are then numerically determined. It is shown that the frequency response of the cantilever beam is strongly affected by damping and excitation level. Particularly, for the sub-harmonic resonance of the system, instability can be observed by increasing the amplitude of excitation. As for a detailed parameter sensitivity study, the influences of different parameters on the frequency response of the cantilever beam have been examined. By varying the values of the parameters of the model, it is found that transition from the periodic solution to chaos occurred for the mechanical system.

Nonlinear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations

In a recent work, a discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions has been developed and validated via comparison with known results corresponding to nonlinear vibration of clamped beams carrying a concentrated mass. It is extended here to continuous beams carrying two or three concentrated masses at various locations and subjected to large vibration amplitudes. The discrete model used is an N -dof ( N -Degrees of Freedom) system made of N masses placed at the ends of solid bars connected by springs, presenting the beam flexural rigidity. The large transverse displacements of the bar ends induce a variation in their lengths giving rise to axial forces modelled by longitudinal springs causing nonlinearity. The calculations made allowed application of the semi-analytical model developed previously for nonlinear structural vibration involving three tensors, namely the mass tensor m ij , the linear rigidity tensor k ij and the nonlinearity tensor b ijkl presenting the effect of the change in the bar lengths. The addition of three concentrated masses studied here induces a change in the mass matrix. By application of Hamilton’s principle and spectral analysis in the modal basis, the nonlinear vibration problem is reduced to a nonlinear algebraic system, using an explicit method, developed previously for non-linear structural vibration. This study shows that concentrated masses may be used for practical purposes to shift the resonant frequency; if the three masses locations are appropriately chosen.