Nonlinear Vibrations Of Beams Research Articles

Analytical and numerical study of the influence of different support types in the nonlinear vibrations of beams

This paper aims at studying the influence of the different support types on the free nonlinear vibration frequency of a beam, both by analytical and numerical approaches and its dependence on the vibration amplitude. First, an analytical study of two cases of simply supported beams is carried out by using the nonlinear normal modes (NNM) and the multiple scale methods to obtain the analytical expressions of the nonlinear frequencies as a function of the vibration amplitude. Such results have been compared and discussed with other authors’ results. In addition, a nonlinear finite element model of these two cases showed that the analytical and numerical results are in good agreement. Additionally, several numerical studies have been carried out for different support types. A total of seven different boundary conditions have been numerically analysed and the corresponding frequency-amplitude relations have been obtained and compared. In addition, the effect of the axial inertial forces has been enhanced by adding concentrated masses at one end of the beam. It was found for instance that the hinged-hinged beam shows a hardening behaviour, which depends on the beam slenderness, or that the hinged-simply supported beam results in the largest softening behaviour. The fundamental causes of nonlinearity are, on one hand the coupling of the midline deformation and bending and, on the other hand, the coupling of axial inertial forces and bending. The significance of the first of these effects depends on the beam material and geometrical properties.

Nonlinear free vibration of damped and undamped bi-directional functionally graded beams using a cubic-quintic nonlinear model

In this paper, a novel higher order cubic-quintic nonlinear model is proposed for the nonlinear free vibration of damped and undamped bi-directional functionally graded (2D FG) beams. To the best of the researchers’ knowledge, no study has focused on damping characteristics of 2D FG beams. Thus, the present paper extends the previous studies in this field. It is assumed that the material properties of the beam change in the axial and lateral directions simultaneously according to exponential and power-law functions, respectively. A new neutral surface is defined to remove the stretching and bending couplings effect. The variational iteration method (VIM) and the Hamiltonian approach (HA) are applied to obtain closed-form analytical solutions for the nonlinear vibration of damped and undamped 2D FG beams, respectively. Based on the results, an increase in the material grading indices can similarly decrease the damping coefficient for an underdamped 2D FG beam, and more time is required for dying out the oscillations. Further, compared to the homogeneous beams, a significant difference is observed between the cubic and cubic-quintic models indicating the importance of applying higher order nonlinear models for the nonlinear analysis of 2D FG beams.

Analysis of the component forces of the nonlinear magnetic force in energy harvester

In the piezoelectric vibration energy harvester, permanent magnets are often used to generate nonlinear applied magnetics force to improve the energy utilization rate of the system, the modeling analysis and accurate calculation of the force between magnets in the system is a difficult problem in the study of nonlinear bistable vibration energy harvesting. During the deformation of the cantilever beam, the direct force of the permanent magnet block can be divided into horizontal and vertical component forces. In most existing literatures analyzing such problems, the magnetizing current method magnetic force calculation model of double-stabilized electric beam mainly considers the influence of vertical magnetic force on the system of cantilever beam, and it is considered that the horizontal component of magnetic force has little influence on the vibration response of cantilever beam, but there is no detailed proof and elaboration of this problem. In this paper, the magnetic force, the effect of magnetic force on natural frequency and magnetic potential energy are calculated and simulated from three aspects. Through the comparison of results, it is proved that the effect of the horizontal magnetic force on the whole nonlinear piezoelectric beam vibration energy harvester can be ignored.

Geometrically nonlinear vibration of anisotropic composite beams using isogeometric third-order shear deformation theory

Due to broad usage of anisotropic composite beams in modern engineering structures, the main goal of the present work is to examine their geometrically nonlinear vibration. To this end, a third-order shear deformation theory with a nonlinear von-Kármán strain field is used for anisotropic beams and combined with the advantages of the isogeometric framework. The layup properties are assumed to be anisotropic in the depth direction and a transient tip follower force is considered. The governing nonlinear equations of vibration are integrated by means of the Newmark approach and solved with the Newton–Raphson method. Flutter loads as well as natural frequencies are obtained by eigenvalue analysis. The effects of various important factors such as non-prismatic shape, orientation of the composite fibers, critical follower force and bifurcation point are studied, using both h- and p-refinements. The results show that the nonlinear vibration and flutter characteristics of the anisotropic composite beams are completely different from those for orthotropic and isotropic ones. Thick beams with anisotropic layups are more sensitive to the shear parameter than conventional ones and deform primarily in shear mode rather than in bending. Anisotropic beams reveal a higher flutter instability force than other cases for a given shear parameter value. Another important phenomenon is that the stress distribution in anisotropic layups shows irregular patterns both in depth and time. Anisotropic layups with ply angles between 15° and 45° seem to present enhanced nonlinear performance with respect to other layup choices. A coarse mesh of quartic C3 B-splines is observed to provide high accuracy for nonlinear deflections in anisotropic cases even for rather low shear parameters.

On size-dependent nonlinear free vibration of carbon nanotube-reinforced beams based on the nonlocal elasticity theory: Perturbation technique

Based on the first-order shear deformation (FSD) model and nonlocal elasticity theory, the simultaneous effects of shear and small scale on the nonlinear vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams are investigated for the first time. To this end, the governing equations of bending and stretching with von Kármán geometric nonlinearity are decoupled into one fourth-order partial differential equation in terms of transverse deflection. A closed-form solution of the nonlinear natural frequency, which can be used in conceptual design and optimization algorithms of FG- CNTRC beams with different boundary conditions, is developed using a hybrid method of Galerkin and perturbation technique. First, the decoupled equation is reduced to a nonlinear ordinary one with respect to time by implementing the Galerkin method. Next, multiple scales perturbation technique is used to replace this nonlinear ordinary differential equation with a series of linear equations which can be solved analytically. Finally, numerical results are presented to compare with the existing ones in the literature as well as to study the effects of CNTs distribution, boundary conditions and nonlocal parameter on the frequency ratio. It is seen that the influence of CNTs distribution becomes more significant when the nonlocal parameter increases. Also, the difference between results obtained from the classical and nonlocal elasticity theories increases as the amplitude of vibration increases. Therefore, it is concluded that the classical elasticity theory is inadequate to predict the nonlinear vibration behavior of FG-CNTRC beams with large deformation.

Effect of soil mass motion on nonlinear forced vibrations of beams on elastic foundation

Based on the nonlinear equation of motion of the beam on the Winkler foundation with the consideration of finite-depth soil mass motion, the nonlinear forced vibrations of the beam were investigated. Applying the Galerkin method and the method of multiple scales, the frequency-response equation and the second-order approximate solution of the primary resonance of the beam were obtained. Then, by means of the numerical calculation and parameter analysis, the effects of parameters about the soil mass motion on the frequency-response curves of the beam on the Winkler foundation were explored, such as foundation depth, soil mass, and Winkler foundation stiffness. The results show that the effect of the soil mass motion on the nonlinear forced vibration of the beam on the Winkler foundation is significant. When the effect of soil mass motion is introduced into the equation of motion of the beam, the range and the softening behavior of primary harmonic resonance of the system are reduced.

Bifurcation of nonlinear normal modes of a cantilever beam under harmonic excitation

Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two degrees-of-freedom nonlinear model for the vibration in the transverse direction is obtained by the discretization of the governing equation using Galerkins method based on the eigenmodes in each direction. The method of multiple scales is used to derive two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of the dominant mode for the case of 1:1 internal resonance. The bifurcation diagrams are computed considering the frequency of excitation and the magnitude of the excitation as the control parameters. The stability of the fixed point is determined by examining the eigenvalues of the Jacobian matrix. The results show that a saddle-node-type bifurcation of the solution can occur under certain parameter conditions.

Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams

Although the transverse vibration of the cantilever structure is of great interest, the internal resonance of the vertical cantilever structure is always ignored. In this paper, nonlinear free vibration and 1/3 super-harmonic resonance of a hanging cantilever beam are firstly presented with 3:1 internal resonance caused by gravity. By employing the temporal multi-scale method, mode responses in these two vibrations are obtained. The harmonic balanced method is used for solving the excitation component and the whole deflection in super-harmonic resonance. For nonlinear free vibration, beat phenomenon is found and the effect of mode interaction is determined by gravity, damping and initial perturbation. For super-harmonic resonance, the amplitude of the first two modes can be quite large and exceed the excitation component if damping is not strong. Besides, the softening-type frequency curves are predominated by the inertia nonlinearity. Compared with the excitation component and the whole deflection, the mode responses are more easily affected by gravity. If the value of gravity parameter is a little bit lower than the one in the condition of strict internal resonance, the energy transmission, multiple solutions and bifurcations and three complex types of frequency curves can be found in mode responses. Hysteresis and saturation phenomena in mode responses can be discovered as well. Results from analytical methods are almost identical to those from numerical ways. In summary, the internal resonance of slender hanging cantilever structures should be aroused more attention, for hanging cantilevers are common in practical engineering and considerable mode responses can be induced by large initial perturbation in weakly damped free vibration or by low excitation frequency and large excitation amplitude in super-harmonic resonance. In addition, the complex dynamics can be worthwhile to judge the occurrence of internal resonance and its impact on structures. Since gravity has tensile effects on hanging cantilevers, these phenomena may also occur in beams with axial tension.

Nonlinear vibrations of beams and plates with fractional derivative elements subject to combined harmonic and random excitations

This paper proposes an efficient approach for estimating reliably the second order statistics of the response of continua excited by combinations of harmonic and random loads. The problem is relevant in several engineering applications, where, for instance, the harmonic load is influenced by significant noise that cannot be neglected when computing the response statistics. The considered problems pertain to the vibration of beams and of plates endowed with fractional derivative elements. In both cases, it is shown that by representing the system response by the linear modes of vibration, systems of nonlinear fractional ordinary differential equations describing the time-dependent variation of the modes amplitudes are obtained. These equations are coupled and are treated by combining the harmonic balance and statistical linearization techniques, leading to the determination of the second-order statistics of the response. Relevant Monte Carlo data demonstrate the reliability of the proposed solution approach. The specific numerical examples considered pertain to simply supported beams, and plates with simply supported stress-free edges conditions.

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic parameters

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports.

Nonlinear Free Vibration of Hyperelastic Beams Based on Neo-Hookean Model

The investigation of hyperelastic responses of soft materials and structures is essential for understanding of the mechanical behaviors and for the design of soft systems. In this paper, by considering both the material and geometrical nonlinearities, a new neo-Hookean model for the hyperelastic beam is developed with focus on its nonlinear free vibration with large strain deformations. The neo-Hookean model is employed to capture the large strain deformation of the hyperelastic beam. The governing equations of the hyperelastic beam are derived by using Hamilton’s principle. To avoid expensive calculations for solving the nonlinear governing equations, a simplified Taylor-series expansion model is proposed. The effects of two key system parameters, i.e. the initial displacement amplitude and the slenderness ratio, on the nonlinear free vibrations of the hyperelastic beam are numerically analyzed. The bifurcation diagrams, displacement time traces, phase portraits and power spectral diagrams are presented for the nonlinear free vibrations of the hyperelastic beam. For small initial displacement amplitudes, it is found that the hyperelastic beam will undergo limit cycle oscillations, depending on the initial amplitude employed. For initial displacement amplitudes large enough, interestingly, the free vibration of the hyperelastic beam will become quasi-periodic or chaotic, which were rarely reported for the free vibration of linearly elastic beams. Also observed is the traveling wave feature of oscillating shapes of the hyperelastic beam, indicating that higher-order modes of the beam are excited even for free vibrations. All these new features in the nonlinear free vibrations of hyperelastic beams indicate that the material and geometric nonlinearities play a great role in the dynamic analysis of hyperelastic beams.

Investigating nonlinear vibrations of higher-order hyper-elastic beams using the Hamiltonian method

This paper presents a higher-order shear deformation beam theory for modeling and nonlinear vibration analysis of hyper-elastic beams made of silicon rubber and unfilled natural rubber. Four models named neo-Hookean, Mooney–Rivlin, Ishihara, and Yeoh models are presented, and their efficacy in nonlinear dynamic modeling of hyper-elastic beams has been explored. Geometric nonlinearity of the hyper-elastic beam is considered based on von-Karman-type nonlinearity. The hyper-elastic beam is resting on a nonlinearly hardening elastic foundation. It is shown that the Ishihara model is a suitable model for nonlinear vibration analysis of hyper-elastic beams accounting for the shear deformation effect. The nonlinear governing equations based on the presented beam theory are analytically solved via the Hamiltonian method to find nonlinear vibration frequencies. It is shown that the nonlinear vibration behavior of hyper-elastic beams is influenced by rubber-material type and material parameters of the hyper-elastic model.

A total Lagrangian elasticity formulation for the nonlinear free vibration of anisotropic beams

The large amplitude free vibration response of isotropic and anisotropic beams is considered using a total Lagrangian description of the motion along with Ritz-based approximations to the dynamic statement of virtual work. Results are presented for both axial and bending modes and are compared with existing results in the literature using various formulations along with a finite element solution of the Euler–Bernoulli beam with Föppl–von Karman nonlinearity. Results from the present formulation include all nonlinear strain terms and give frequency ratios that are in excellent agreement with both Euler–Bernoulli and higher order beam theories for slender beams under small amplitudes but grow noticeably higher than those of other beam models as the beam becomes thick and the amplitude of vibration increases.

Fatigue Precracking Time Estimates for Three-Point Bending Specimens

Specimens containing sharp cracks are needed in certain types of mechanical tests, first and foremost for fracture toughness measurement of materials. Their use, however, is not just limited to this type of test. Another category of experiments deals with characterizing nonlinear vibrations of beams containing breathing cracks. To produce cracked beam specimens, fatigue cracks can be grown ahead of sharp notches under controlled loading. ASTM E399 and ASTM E1820 standards provide guidance on such procedures for preparation of fracture toughness test specimens. However, certain issues which might become important in testing of vibrations of cracked beams, such as the time required for specimen preparation is not addressed in these standards. In this article, both low cycle fatigue methods and linear elastic fracture mechanics methods are used to estimate the number of loading cycles required to have a crack of desired length at the notch tip. Calculation results are compared with experimental ones, and the effects of various factors influencing the required number of loading cycles for a certain crack size are discussed.

Nonlinear Vibration and Stability Analysis of Viscoelastic Rayleigh Beams Axially Moving on a Flexible Intermediate Support

In this study, the nonlinear vibration and stability of a simply supported axially moving Rayleigh viscoelastic beam equipped with an intermediate nonlinear support are investigated. The type of considered nonlinearity is geometric and is due to the axial stretching. The Kelvin–Voigt model is used to regard the beam internal damping. The Hamilton’s principle is employed to derive the governing equations and corresponding boundary conditions. The multiple scales method is applied to the dimensionless form of the governing equations and the nonlinear frequencies, time response of the system for two cases of the axial velocity fluctuation frequency are obtained. The stability of the system is investigated via solvability condition and Routh–Hurwitz criterion. Some case studies are accomplished to demonstrate the effect of rotary inertia, axial velocity and parameters of intermediate support on the system response, critical velocity and the system stability. Furthermore, the variation of the first two resonance frequencies with respect to mean axial velocities for different locations of the intermediate support are investigated. It is found that by moving the intermediate support from one end of the beam to its midpoint, the region in which the first mode undergoes static instability, shrinks. Moreover, although rotary inertia impressively decreases the natural frequencies, intermediate support has the dominant effect on increasing the natural frequencies.

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

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

Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

Periodic and nonperiodic modes of postbuckling and nonlinear vibration of beams attached to nonlinear foundations

• Periodic and Nonperiodic Modes on Postbuckling and Nonlinear have been studied. • Closed from solutions for cosine and sine imperfection have been derived. • Effect of non-periodic imperfection has been investigated. • Numerical differential-integral quadrature method is compared by analytical method. • Numerical solution is adopted to investigate the effect of nonlinear elastic foundation. This manuscript presents the influence of periodic (sine and cosine) and nonperiodic imperfections modes on buckling, postbuckling and dynamics of beam rested on nonlinear elastic foundations. The geometric von Kármán nonlinear strains induced by mid-plane stretching is considered. So, the generated equation of motion of imperfect beam is obtained as nonlinear integro-partial-differential equation. Elastic cubic nonlinear and linear springs and shearing layer are proposed to model elastic medium around the imperfected beam. Numerical differential-integral quadrature method (DIQM) in combination with Newton's method are used to find buckling loads and modes of imperfect beam that described by nonlinear integro-differential equation. In linear vibration analysis, the problem is discretized by DIQM and solved as a linear eigenvalue problem. The closed form solutions for static response and linear vibration problem of imperfect beam are obtained. Excellent agreement is observed between proposed numerical and analytical solutions. The numerical results indicate that imperfection modes and its amplitude have a weighty influence on the buckling load, postbuckling configurations and natural frequencies. Finally, numerical results for clamped-clamped (C-C) and simply supported (SS-SS) beams with local (nonperiodic) imperfection modes are presented.

Asymmetric viscoelastic nonlinear vibrations of imperfect AFG beams

The nonlinear asymmetric viscoelastic vibration of imperfect axially functionally graded (AFG) beams is investigated within this study; this is for the first time. As the mechanical properties vary in the longitudinal direction, asymmetry in the vibration response appears. Viscosity (or internal friction between beam elements) dissipates the vibration energy; this is incorporated via the Kelvin-Voigt scheme. A stretching type nonlinearity due to clamped-clamped boundary conditions gives rise to geometric nonlinearities. A geometric imperfection, due to improper manufacturing, is incorporated. The Kelvin-Voigt type of viscosity relates the strain field to the stress one. The energy loss is dynamically balanced by the kinetic and potential energies using Hamilton’s principle. Two nonlinear coupled equations for transverse/axial motions are obtained, and then are solved using a parameter-continuation method. It is shown that how the gradient index, viscosity, and imperfection affect the asymmetric vibrations.

Nonlinear vibration and modal analysis of FG nanocomposite sandwich beams reinforced by aggregated CNTs

In the present work, by considering the aggregation effect of single‐walled carbon nanotubes (SWCNT), the nonlinear vibration of functionally graded (FG) nanocomposite sandwich Timoshenko beams resting on Pasternak foundation are presented. The material properties of the FG nanocomposite sandwich beam are estimated using the Eshelby–Mori–Tanaka approach and differential quadrature method (DQM) is used to obtain natural frequency. The nonlinear governing equations and boundary conditions are derived using the Hamilton principle and von Kármán geometric nonlinearity. The higher order nonlinear governing equations and boundary conditions are calculated using the Hamilton principle. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the beams. It is shown that the mechanical properties and therefore vibration of functionally graded carbon nanotube reinforced (FG‐CNTR) sandwich beams are severely affected by CNTs aggregation. A detailed parametric study is carried out to investigate the influences of Winkler foundation modulus, shear elastic foundation modulus, length to span ratio, thicknesses of face sheets on the nonlinear vibration of the structure. POLYM. ENG. SCI., 59:1362–1370 2019. © 2019 Society of Plastics Engineers