Viscoelastic Beam Research Articles

On the static and dynamic stability of thin beam conveying fluid

In this paper, numerical investigation of the statical and dynamical stability of aligned and misaligned viscoelastic cantilevered beam is performed with a terminal nozzle in the presence of gravity in two cases: (1) effect of fluid velocity on the flutter boundary of beam conveying fluid and (2) effect of gravity on the buckling boundary of beam conveying fluid. The beam is assumed to have a large width-to-thickness ratio, so the out-of-plane bending rigidity is far higher than the in-plane bending and torsional rigidities. Gravity vector is considered in the vertical direction. Thus, deflection of the beam because of the gravity effect couples the in-plane bending and torsional equations. The beam is modeled by Euler–Bernoulli beam theory, with the flow-induced inertia, Coriolis and centrifugal forces along the beam considered as a distributed load along the beam. Furthermore, the end nozzle is regarded as a lumped mass and modeled as a follower axial force. The extended Hamilton’s principle and the Galerkin method are utilized to derive the bending–torsional equations of motion. The coupled equations of motion are solved as eigenvalue problems. Also, several cases are examined to study the impact of gravity, beam inclination angle, mass ratio, nozzle aspect ratio, bending-to-bending rigidity ratio and bending-to-torsional rigidity ratio on flutter and buckling margin of the system.

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.

Analysis of the Primary and Secondary Resonances of Viscoelastic Beams Made of Zener Material

The problem of geometrically nonlinear, steady-state vibrations of beams made of viscoelastic (VE) materials is considered in this paper. The Euler–Bernoulli and the von Kármán theories are used to describe the dynamic behavior of beams. The VE material of the beams is modeled using the Zener model. Two harmonics are present in the assumed steady-state solution of the problem at hand, which enables an analysis of both the primary and secondary resonances. The virtual work equation and the harmonic balance method are used to derive the amplitude equations in the explicit form. The response curves are determined using the continuation method and treating the frequency of excitation as the main parameter. The results of several examples, which illustrate the dynamic behavior of the considered beams, are presented and discussed.

Nonlinear vibration of axially moving beams with internal resonance, speed-dependent tension, and tension-dependent speed

In this paper, a new nonlinear model of an axially accelerating viscoelastic beam is established. The effects of a speed-dependent tension (a time-dependent tension due to perturbation of the speed) and a tension-dependent speed (a time-dependent speed due to perturbation of the tension) are highlighted. However, there was one common characteristic in previous studies about parametric vibration of axially moving beams with time-dependent tensions and time-dependent speeds: Axial tension and axial speed are independent of each other. Another highlight is that the inhomogeneous boundary conditions arising from Kelvin viscoelastic constitutive relation are taken into account. The technique of the modified solvability conditions is employed to resolve the existing solvability conditions failure because of the inhomogeneous boundary conditions. The influences of material’s viscoelastic coefficients, axial speed fluctuation amplitude, axial tension fluctuation amplitude, and dependent and independent models on the steady-state vibration responses are demonstrated by some numerical examples. Furthermore, the approximate analytical results are verified by using the differential quadrature method.

Nonlinear dynamics of a viscoelastic traveling beam with time-dependent axial velocity and variable axial tension

In this paper, stability, various bifurcations and chaotic behavior of a traveling Euler beam are studied. Kelvin–Voigt viscoelastic model is adopted for the beam material. The beam has time dependency in velocity as well as tension in axial direction which is considered for the first time for traveling viscoelastic beam. The multi-frequency parametric resonance is assumed to be comprised of simultaneous principal parametric resonance of first mode due to the time dependency of speed, while the principal parametric resonance of second mode is due to the variable tension in the axial direction. The two-frequency parametric resonance is considered along with 3:1 internal resonance. Such multi-resonant case has not been considered so far in any available literature. The higher-order integro-partial differential equation of motion is attacked with direct method of multiple scales. Solvability condition is incorporated to get two complex variable modulation equations for amplitude and phase. From these two equations, a set of normalized reduced equations are derived through Cartesian transformation which are subsequently utilized to get numerical solutions for the nonlinear system. Continuation algorithm is used to depict the effect of system parameters such as fluctuating tension component, fluctuating velocity component, internal frequency detuning parameter, different damping on the frequency response curves, their stability and bifurcation analysis. This research work reveals interesting results which are not found so far in the existent literature.

Stabilisation of a viscoelastic beam conveying fluid

In this paper, vibration reduction of a viscoelastic flexible beam conveying fluid with variable velocity is studied. A control is designed at the top boundary of the beam based on original infinite dimensionality PDEs model and Lyapunov's direct method to reduce the beam's vibrations. The uniform stability under external disturbance and exponential stability without external disturbance are proved based on the proposed boundary control.

Amplification Factors in the Case of Beam Under Moving Force – Theoretical Analysis

AbstractThe impact of a moving load speed on the dynamic overload of beams, assuming that the track of the load has no unevenness, is examined. First the problem of a visco-elastic beam on a Winkler foundation subjected to a force moving at a constant speed will be solved. Using the Bubnov-Galerkin method, the deflections of the beam, and then the bending moments and shear forces will be determined. The solution of the problem will be obtained both for the case of a forced vibration and the case of a free vibration after the moving force has left the beam. Using these solutions, dynamic amplification factors will be determined for the deflections, bending moments, and shear forces, which are different for the two cases.The magnitude of the amplification factors increases and decreases alternately as a function of the speed. In the case of a single force on a beam, the dynamic overloads are limited, and do not exceed 60%. There is no resonance phenomenon in the beam subjected to the single moving force. The dynamic amplification factors determined in this way can be used as correction coefficients when designing engineering structures subjected to moving loads by static methods.

Stochastic P-bifurcation in a nonlinear viscoelastic beam model with fractional constitutive relation under colored noise excitation

In this paper, we study the stochastic P-bifurcation problem for an axially moving bistable viscoelastic beam with fractional derivatives of high-order nonlinear terms under colored noise excitatio...

Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions

In this research, the nonlinear static, buckling and vibration analysis of viscoelastic micro-composite beam reinforced by various distributions of boron nitrid nanotube (BNNT) with initial geometrical imperfection by modified strain gradient theory (MSGT) using finite element method (FEM) are presented. The various distributions of BNNT are considered as UD, FG-V and FG-X and also, the extended rule of mixture is used to estimate the properties of micro-composite beam. The components of stress are dependent to mechanical, electrical and thermal terms and calculated using piezoelasticity theory. Then, the kinematic equations of micro-composite beam using the displacement fields are obtained. The governing equations of motion are derived using energy method and Hamilton\'s principle based on MSGT. Then, using FEM, these equations are solved. Finally the effects of different parameters such as initial geometrical imperfection, various distributions of nanotube, damping coefficient, piezoelectric constant, slenderness ratio, Winkler spring constant, Pasternak shear constant, various boundary conditions and three material length scale parameters on the behavior of nonlinear static, buckling and vibration of micro-composite beam are investigated. The results indicate that with an increase in the geometrical imperfection parameter, the stiffness of micro-composite beam increases and thus the non-dimensional nonlinear frequency of the micro structure reduces gradually.

Nonlinear vibration of a beam with asymmetric elastic supports

Under the conditions of horizontal placement and only considering geometric nonlinearity, depending on the boundary constraints, primary resonances of an elastic beam exhibit either hardening or softening nonlinear behavior. In this paper, the conversion of softening nonlinear characteristics to hardening characteristics is studied by using the multi-scale perturbation method. Therefore, in a local sense, the condition is established for the resonance of the elastic beam exhibits only linear characteristics by finding the balance between asymmetric elastic support and geometric nonlinearity. A viscoelastic beam supported by vertical springs is proposed with nonrotatable left boundary and freely rotatable right end. In order to truncate the continuous system, natural frequencies and modes of the proposed asymmetric beam are analyzed. The steady-state responses of the beam excited by a distributed harmonic force are, respectively, obtained by an approximate analytical method and a numerical approach. Under the condition that the beam is placed horizontally, the transition from the cantilever state to the clamped–pinned state is demonstrated by constructing different asymmetry support conditions. The resonance peak of the first-order primary resonance is used to demonstrate the transition from softening nonlinear characteristics to the hardening characteristics. This research shows that the transformation from softening characteristics to hardening characteristics caused by asymmetric elastic support and geometric nonlinearity exists only in the first-order mode resonance.

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

A viscoelastic beam theory of polymer jets with application to rotary jet spinning

Complex deformation of a polymer jet appears in many manufacturing processes, such as 3D printing, electrospinning, blow spinning, and rotary jet spinning. In these applications, a polymer melt or solution is first extruded through an orifice and forms a jet. The polymer jet is then dynamically deformed until the polymer solidifies. The final product is strongly affected by the deformation of the polymer jet. And the deformation is strongly affected by the viscoelasticity of the polymer. Here we develop a beam theory to incorporate both the nonlinear viscoelasticity and the bending/twisting stiffness of a polymer jet. As a demonstration, we study the formation of a polymer fiber under strong centrifugal force, a fiber manufacturing process known as rotary jet spinning.

General decay of energy to a nonlinear viscoelastic two-dimensional beam

A viscoelastic beam in a two-dimensional space is considered with nonlinear tension. A boundary feedback is applied at the right boundary of the beam to suppress the undesirable vibration. The well-posedness of the problem is established. With the multiplier method, a uniform decay result is proven.

Vibration suppression of atomic-force microscopy cantilevers covered by a piezoelectric layer with tensile force

In this paper, vibration suppression of a micro-beam covered by a piezoelectric layer is studied. The micro-beam is modeled with the specific attention to its application in AFM. The AFM micro-beam is a cantilever one which is stimulated close to its natural frequency by applying a harmonic voltage to the piezoelectric layer. The beam is an Euler-Bernoulli beam which abbeys Kelvin-Voigt model. Using such model supplies the comparison between elastic and viscoelastic beams; and one of the most important properties of viscoelastic materials, damping effect can readily be investigated. The pump provides an axial load with the result that it suppresses the vibrations. First, the vibration equations are extracted using Lagrangian and extended Hamiltonian method in vertical, longitudinal, as well as torsional directions and are discretized by exploiting the Galerkin mode summation approach. The discretized time-domain equations are solved by the aid of the Runge-Kutta method. The viscoelastic beam is compared with the elastic one, and the effects of damping ratio on vibration responses are presented. Additionally, the effects of micro-pump load, excitation voltage, and initial twist angle are investigated on the amplitude of vibration and natural frequency of system. It is observed that viscoelasticity of beam and axial load of the pump reduce vibrations and provide uniform time-domain responses without beatings.

Forced Vibration and Self-Heating of a Flexible Viscoelastic Beam with Piezoelectric Sensor and Actuator with Account of Shear Strain

The problem of the forced resonant vibrations and dissipative self-heating of a hinged flexible beam with piezoelectric sensors and actuators is formulated taking into account the in-plane shear strain. The effect of geometrical nonlinearity, in-plane shear strain, and heat transfer conditions on the beam surfaces on the amplitude- and temperature–frequency characteristics of the beam as well as on the thermal failure of the system is studied. The possibility of active damping of the flexural vibration mode with piezoactuators and sensors is examined.

High Resolution Wavenumber Analysis (HRWA) for the mechanical characterisation of viscoelastic beams

The High-Resolution Wavenumber Analysis (HRWA) is presented. It identifies complex wavenumbers and amplitudes of waves composing the harmonic response of a beam. Based on the frequency dependence of these wavenumbers, experimental dispersion equations of various beam mechanisms (e.g bending, torsion) can be retrieved. The HRWA method is compared to the Mc Daniel and the Inverse Wave Correlation (IWC) methods. It overcomes some drawbacks of these methods: the wavenumber resolution is enhanced. Also, the wavenumber search problem is expressed as a linear problem, making the method computationally efficient. A number of wavenumbers can be identified automatically, thanks to a statistical criterion. First, the noise sensitivity of each method is investigated in the basis of synthesised measurements. For this criterion, the HRWA and Mc Daniel method performances are close and much better than IWC. Moreover, the HRWA is five to twenty times faster to compute than other methods, depending on the mesh size. Second, an experimental case is presented where bending and torsion waves are identified, yielding an apparent viscoelastic Young and shear moduli on a wide-frequency range.

Stability approach to the fractional variational iteration method used for the dynamic analysis of viscoelastic beams

Non-integer derivatives are frequently used to describe the constitutive behavior of viscoelastic materials. The dynamic analysis of a simply supported viscoelastic beam for a fractional Zener model is performed with the help of a modified variational iteration method. The structure is subjected to two loading scenarios: a uniformly distributed transverse load and a periodic concentrated force at the center of the beam. Using the properties of convolution product in Banach spaces L p ( R n ) and the fixed point theory, the stability temporal intervals of a nonlinear operator for a given fractional order ν are determined. The graphical representations present in the numerical examples show how the existence of fractional derivative in the selected rheological model influences the dynamic response of the structure compared to the classical Zener model. The results of this study prove that the presented algorithm is an efficient tool for solving the fractional integro-differential equations. • An algorithm based on the modified VIM and Laplace transformation techniques is presented. • Approach of the variable change in fractional integral calculus. • Stability study of a fractional nonlinear operator defined by VIM. • Using the fixed point theory are determined the stability intervals. • Numerical examples show the influence of fractional derivative.

Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

On Parametric Instability Boundaries of Axially Moving Beams with Internal Resonance

In this paper, the instability boundaries of an axially moving viscoelastic beam due to parametric resonance are revisited for the internal resonance case. The relation between the time-dependent tension and the time-dependent axial speed is constructed, which provides a new model in the study of axially moving material with pulsation parameters. The instability boundaries caused by the combination of parametric and internal resonances are studied using the method of multiple time scales. Some strange instability boundaries are detected when the internal resonance is considered. The phenomenon of local zigzag boundary contour is explained from the viewpoint of modal interactions.

Effect of Dynamic Soft Elasticity on Vibration of Embedded Nematic Elastomer Timoshenko Beams

This paper addresses the transverse vibration of a nematic elastomer (NE) beam embedded in soft viscoelastic surroundings with the aim to clarify a new dissipation mechanism caused by dynamic soft elasticity of this soft material. Based on the viscoelasticity theory of NEs in low-frequency limit and the Timoshenko beam theory, the governing equation of motion is derived by using the Hamilton principle and energy method, and is solved by the complex modal analysis method. The dependence of vibration property on the intrinsic parameters of NEs (director rotation time, rubber relaxation time, anisotropic parameter) and foundation (spring, shear and damping constants) are discussed in detail. The results show that dynamic soft elasticity leads to anomalous anisotropy of energy transfer and attenuation. The relative stiffer foundation would restraint the rubber dissipation of viscoelastic beams, but has less influence on the director rotation dissipation, which is particular for NE beams. This study would provide a useful guidance in the dynamic design of NE apparatus embedded in soft viscous media.