Nonlinear Vibrations Of Beams Research Articles

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

Nonlinear Random Vibrations of Beams with In-Span Supports via Statistical Linearization with Constrained Modes

AbstractA statistical linearization technique is developed for determining second-order response statistics of beams with in-span elastic concentrated supports. The nonlinearities considered relate...

High damping and nonlinear vibration of sandwich beams with entangled cross-linked fibres as core material

This article investigates the use of a recently developed fibrous core material to increase vibration damping in sandwich beams. The entangled cross-linked fibre (ECF) material is made of short carbon fibres cross-linked with epoxy resin. Dry friction between fibres provides energy dissipation when the material is deformed. Previous measurements on the material are post-processed to provide a simplified viscoelastic description of the material, for an easier interpretation of subsequent structural testings. Two sandwich beams are compared with reference honeycomb beams: a sandwich beam with an ECF core, and a hybrid beam with a honeycomb core and an ECF insert. Steady-state tests are performed on both types of beams to obtain their frequency responses for different excitation levels, and the corresponding apparent loss factors are computed. The beam with a full ECF core shows an apparent loss factor more than ten timess higher than the reference honeycomb beam. The hybrid sandwich beam provides an apparent loss factor four times higher than the reference honeycomb beam. All beams exhibit nonlinear softening responses consistent with a dry friction phenomenon in the material: the resonance frequencies decrease with increasing excitation amplitude, and damping increases then decreases again at very high amplitudes while remaining largely superior to that of the honeycomb beams. Transient impact testings are also presented for a qualitative comparison of the ECF and reference beams, and the ECF beams lead to shorter decay times compared to the reference beams.

A vibro-impact acoustic black hole for passive damping of flexural beam vibrations

Nonlinear flexural vibrations of slender beams holding both an Acoustic Black Hole termination and a contact non-linearity are numerically studied. The Acoustic Black Hole (ABH) effect is a passive vibration mitigation technique, which has shown attractive properties above a given cut-on frequency. In this contribution, a vibro-impact acoustic black hole (VI-ABH) is introduced, the contact nonlinearity being used as a mean to transfer energy from low to high frequencies. A numerical model of a VI-ABH is derived from a Euler-Bernoulli beam. The contact law is handled with a penalization approach, the visco-elastic layer with a Ross-Kerwin-Ungard model and the problem is solved with a modal approach combined with an energy-conserving time integration scheme. Numerical results show that the VI-ABH brings about important modifications, and changes the nature of more traditional black holes, by redistributing all the vibrational energy. It can lead to a strong decrease of the resonance magnitude at low frequencies. Under steady state noise excitation, parametric studies are realised in the cases of a single contact, a grid of contacts and bilateral contacts layouts, in order to find some optimal designs. Transient dynamics is also studied through the analysis of displacement signal envelope and energy decay time. All the numerical results constantly show that the combination of the ABH effect and an energy transfer provided by contact nonlinearity leads to very attractive mitigation template including low frequencies.

Nonlinear Vibration of Sandwich Beams with Functionally Graded Negative Poisson’s Ratio Honeycomb Core

This paper investigates the nonlinear flexural vibration of sandwich beams with functionally graded (FG) negative Poisson’s ratio (NPR) honeycomb core in thermal environments. The novel constructions of sandwich beams with three FG configurations of re-entrant honeycomb cores through the beam thickness direction are proposed. The temperature-dependent material properties of both face sheets and core of the sandwich beams are considered. 3D full-scale finite element analyses are conducted to investigate the nonlinear vibration, and the variation of effective Poisson’s ratio (EPR) of the sandwich beams in the large deflection region. Numerical simulations are carried out for the sandwich beam with FG-NPR honeycomb core in different thermal environmental conditions, from which results for the same sandwich beam with uniform distributed NPR honeycomb core are obtained as a basis for comparison. The effects of FG configurations, temperature changes, boundary conditions, and facesheet-to-core thickness ratios on the nonlinear vibration ratio curves and EPR–deflection curves of sandwich beams are discussed in detail.

Exact mathematical solution for nonlinear free transverse vibration of beams

In the present paper, an exact mathematical solution has been obtained for nonlinear free transverse vibration of beams, for the first time. The nonlinear governing partial differential equation in un-deformed coordinates system has been converted in two coupled partial differential equations in deformed coordinates system. A mathematical explanation is obtained for nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of beam. It is shown that as the th mode of transverse vibration excited, the mode 2 th of in-plane vibration will be developed. The result of present work is compared with those obtained from Galerkin method and the observed agreement confirms the exact mathematical solution. It is shown that the governing equation is linear in the time domain. As a parameter, the amplitude to length ratio has been proposed to show when the nonlinear terms become dominant in the behavior of structure.

A robust Bézier based solution for nonlinear vibration and post-buckling of random checkerboard graphene nano-platelets reinforced composite beams

In the present study, an accurate Bézier based multi-step method is developed and implemented to find the nonlinear vibration and post-buckling configurations of Euler-Bernoulli composite beams reinforced with graphene nano-platelets (GnP). The GnP is assumed to be randomly and uniformly dispersed in the composite mix-proportion, with a random checkerboard configuration. Therefore, a probabilistic model together with an efficient simulation technique is proposed to find the effective moduli of a matrix reinforced GnP. It is worth noting that the presented micro-mechanics model found by the employed Monte-Carlo simulation matches exactly the experimental data and predicts the composite elastic constants more accurate than that found from other common methods, including the Halpin-Tsai theory. Also, for mathematical simplification, the composite beam in-plane inertia is neglected. The presented multi-step method is based on Burnstein polynomial basis functions while shows interesting potential to provide robust solutions for various initial and boundary value problems. It is found that adding a relatively low content of GnP would drastically increase the composite elastic constants, particularly in the transverse direction to fiber. In addition, the numerical results are compared with those provided by exact analytical solutions, where the stability of results suggests the effectiveness of the presented methodology.

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

Nonlinear vibration analysis of different types of functionally graded beams using nonlocal strain gradient theory and a two-step perturbation method

In the framework of nonlocal strain gradient theory, the nonlinear vibration of beams subjected to different types of functionally graded distribution is studied in this paper. They are beams with bottom-up functionally graded distribution, beams with inside-out functionally graded distribution and beams with mirror symmetrical functionally graded distribution. The effective material properties of FGM beams are defined based on the proposed assumptions and approximate models. Different displacement functions that can satisfy the stress boundary conditions are used to analyze respective types of FGM beam. The governing equations of nonlinear vibration, including a material length-scale parameter and a nonlocal parameter are derived via the principle of Hamilton, then solved by a two-step perturbation method. With the aid of the obtained analytical solutions, the effects of various parameters on nonlinear vibration problem are studied in detail, including temperature, nonlocal parameter, strain gradient parameter, scale parameter ratio, slenderness ratio, volume index and different kinds of functionally graded distribution. Two new approaches are suggested in this study to change linear and nonlinear frequencies of the beam.

Nonlinear Free Transverse Vibration Analysis of Beams Using Variational Iteration Method

In this study, Variational Iteration Method is employed so as to investigate the linear and non-linear transverse vibration of Euler-Bernoulli beams. This method is a very powerful approach with a high convergence speed providing an analytical and semi-analytical solution to the linear equations and is able to be extended to present semi-analytical solution to the non-linear ones. In this method, firstly, Lagrange`s multiplier and Initial Function should be chosen. The suitable choice of these two elements would effectively affect the convergence speed. In this attempt, in addition to presenting a discussion on how to choose these two functions appropriately, the calculated frequencies in the non-linear state are compared with the available results in the literature, and the accuracy and convergence speed are studied, as well.

Nonlinear damping in nonlinear vibrations of rectangular plates: Derivation from viscoelasticity and experimental validation

Even if still little known, the most significant nonlinear effect during nonlinear vibrations of continuous systems is the increase of damping with the vibration amplitude. The literature on nonlinear vibrations of beams, shells and plates is huge, but almost entirely dedicated to model the nonlinear stiffness and completely neglecting any damping nonlinearity. Experiments presented in this study show a damping increase of six times with the vibration amplitude. Based on this evidence, the nonlinear damanaping of rectangular plates is derived assuming the material to be viscoelastic, and the constitutive relationship to be governed by the standard linear solid model. The material model is then introduced into a geometrically nonlinear plate theory, carefully considering that the retardation time is a function of the vibration mode shape, exactly as its natural frequency. Then, the equations of motion describing the nonlinear vibrations of rectangular plates are derived by Lagrange equations. Numerical results, obtained by continuation and collocation method, are very successfully compared to experimental results on nonlinear vibrations of a rectangular stainless steel plate, validating the proposed approach. Geometric imperfections, in-plane inertia and multi-harmonic vibration response are included in the plate model.

Comments on “Nonlinear vibration of viscoelastic beams described using fractional order derivatives”

This discussion seeks to address some of the inconsistencies and concerns regarding fractional derivatives when applied to vibration analysis of viscoelastically damped beams [1]. Detailed numerical assessments of the major fractional derivatives of concern have been carried out with their relationships numerically examined. Numerical results have shown that two of the key equations of [1] are mathematically inappropriate and their applications could lead to serious modelling errors. A simplified yet representative nonlinear system with fractional viscoelastic damping and cubic stiffness nonlinearity is used to illustrate the existence of periodic steady-state solutions, as well as amplitude frequency characteristics of such fractional nonlinear systems.

Reply to: Comments on “nonlinear vibration of viscoelastic beams, described using fractional order derivatives”

Reply to: Comments on “nonlinear vibration of viscoelastic beams, described using fractional order derivatives”

Modeling and mechanical analysis of multiscale fiber-reinforced graphene composites: Nonlinear bending, thermal post-buckling and large amplitude vibration

In this paper, a mathematical model was developed to predict the effective material properties of graphene nanoplatelets/fiber/polymer multiscale composites (GFPMC). The large deflection, post-buckling and free nonlinear vibration of graphene nanoplatelets-reinforced multiscale composite beams were studied through a theoretical study. The governing equations of laminated nanocomposite beams were derived from the Euler–Bernoulli beam theory with von Kármán geometric nonlinearity. Halpin–Tsai equations and fiber micromechanics were used in hierarchy to predict the bulk material properties of the multiscale composite. Graphene nanoplatelets (GNPs) were assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. A semi-analytical approach was used to calculate the large static deflection and critical buckling temperature of multiscale multifunctional nanocomposite beams. A perturbation scheme was also employed to determine the nonlinear dynamic response and the nonlinear natural frequencies of the beams with clamped–clamped, and hinged–hinged boundary conditions. The effects of weight percentage of graphene nanoplatelets, volume fraction of fibers, and boundary conditions on the static deflection, thermal buckling and post-buckling and linear and nonlinear natural frequencies of the GFPMC beams were investigated in detail. The numerical results showed that the central deflection and natural frequency were significantly improved by a small percentage of GNPs. However, addition of GNPs led to a lower critical buckling temperature.

Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations

Experiments show a strong increase in damping with the vibration amplitude during nonlinear vibrations of beams, plates and shells. This is observed for large size structures but also for micro- and nanodevices. The present study derives nonlinear damping from viscoelasticity by using a single-degree-of-freedom model obtained from standard linear solid material where geometric nonlinearity is inserted in. The solution of the problem is initially reached by a third-order harmonic balance method. Then, the equation of motion is obtained in differential form, which is extremely useful in applications. The damping model developed is nonlinear and the parameters are identified from experiments. Experimental and numerical results are compared for forced vibration responses measured for two different continuous structural elements: a free-edge plate and a shallow shell. The free-edge plate is interesting since it represents a case with no energy escape through the boundary.

Nonlinear vibration of FG beams subjected to parametric and external excitations

Abstract In this paper, the principal resonance and parametric vibration of functionally graded (FG) beams are investigated. The theoretical formulations and governing partial deferential equations of motion are derived based on the Hamilton's principle, Euler-Bernoulli beam theory, and von Karman geometric nonlinearity. The Galerkin technique and the static condensation method are employed in order to convert the nonlinear partial differential equations into a set of nonlinear ordinary differential equations. The nonlinear frequency-response of FG beams is investigated using the method of multiple scales. The magnitude and frequency of parametric excitations are analyzed by numerical integration results. The effects of different parameters on the nonlinear response are also investigated. Moreover, the period-2 and chaotic motions of FG beams are found by numerical simulation.

Analytical modeling of nonlinear flexural-extensional vibration of flexure beams with an interconnected compliant element

The constraint behavior of compliant mechanisms can be improved via strengthening the stiffness of their constitutive beams using intermediate elements. This interior element may be assumed to be perfectly rigid or one can consider its compliance as a design parameter. While modeling the static behavior of such systems is state of art, the nonlinear dynamic of such systems have been remained un-investigated. So the objective of this paper is to suggest an analytical framework for modeling nonlinear free and forced vibrations of a simple flexure beam strengthened via a compliant intermediate element. The equations of motion of the system are derived using Hamilton's principle. Based on a single mode approximation, the partial differential equations of motion are transformed into two temporal equations. Employing multiple time-scales perturbation techniques, free vibration time histories and forced vibration response due to base excitation is derived analytically. Different parametric studies are carried out to recognize the effect of the intermediate compliant element on the vibrational behavior of the flexure beam. The results of this paper are expected to develop a new approach in modeling and investigation of load-displacement behavior of compliant mechanisms.

Nonlinear discrete-time multirate adaptive control of non-linear vibrations of smart beams

The nonlinear adaptive digital control of a smart piezoelectric beam is considered. It is shown that in the case of a sampled-data context, a multirate control strategy provides an appropriate framework in order to achieve vibration regulation, ensuring the stability of the whole control system. Under parametric uncertainties in the model parameters (damping ratios, frequencies, levels of non linearities and cross coupling, control input parameters), the scheme is completed with an adaptation law deduced from hyperstability concepts. This results in the asymptotic satisfaction of the control objectives at the sampling instants. Simulation results are presented.

Reissner stationary variational principle for nonlocal strain gradient theory of elasticity

Abstract The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.