Direct Method Of Multiple Scales Research Articles

Vibration suppression of pipe conveying fluid using a nonlinear absorber in longitudinal direction

The article outlines the design of a nonlinear absorber that features cubic stiffness. The absorber was created for a pipe conveying fluid and was based on the internal resonance mechanism. To create the nonlinear absorber, a sliding mass oscillator was attached along to the pipe conveying fluid. Nonlinear equations were derived using Hamilton's principle. The flowing fluid was considered both as a constant value and a fluctuating term. To investigate the nonlinear behavior of the coupled pipe-absorber system, the direct multiple scales method was used. By selecting appropriate values of mass and stiffness for the absorber, the natural frequency of the absorber was tuned in three cases: 1–1, 2–1, and 3–1 internal resonance ratios between the natural frequency of the absorber and the first natural frequency of the pipe. The goal of the mechanism was to transfer energy from the pipe to the absorber, and the design was successful in achieving this. A saturation phenomenon was detected in the force modulation response at the three-to-one internal resonance. The results showed a reduction in amplitudes in both frequency and force response curves under the external primary resonance and pulsating flowing fluid in the supercritical regime. The nonlinear absorber has a good performance in a wide range of flow velocity and excitation. The accuracy of the results was verified using Galerkin and Finite difference methods.

Primary and subharmonic resonance features of a graphene nanoplatelet strengthened beam in a thermal ambient directly attacking nonlinear integro-partial differential governing equations

The nonlinear aspects of advanced structures may be utilized for design purposes and making use of advantages and avoiding disadvantages of such structures. On the other hand, developing information from secondary resonance examinations of advanced structures is worthwhile for engineering objectives such as mass detection, energy harvesting and damage detection of systems likewise structures. Moreover, a precise solution methodology discards any possibility of qualitative errors and incorrect prejudice. Accordingly, for the first time, the nonlinear primary resonance alongside the subharmonic resonance of an advanced beam structure strengthened by means of graphene nanoplatelets in a thermal ambient is addressed in details employing the direct method of multiple scales. With the aim of defeating to the disadvantageous influences of the discretization procedure on the nonlinear dynamics of the graphene nanoplatelet strengthened beam, the nonlinear integro-partial differential equation of motion in accompanied by the in-homogeneous boundary conditions are attacked directly resorting to the method of multiple scales. It is shown that how an FGX distribution pattern of graphene nanoplatelets for a beam subjected to a subharmonic resonance excitation relative to an FGO distribution pattern effectively extends the range of excitation frequency that leads to a bistable solution. This deduction is very proficient for harvesting energy scopes. A similar finding is observed for a CC beam with respect to a CS beam. It is seen that when an FGX beam is exposed to a subharmonic resonance in a wider range of force amplitude experiences a non-trivial equilibrium state relative to an FGO beam. However, the strengthening of the beam with graphene nanoplatelets decreases the lower bound of the excitation frequency required for the activation of subharmonic resonance more significant for an FGX distribution pattern. Moreover, the primary resonance frequency response shows different trend prior to and after the situation of the external force frequency to the first natural frequency ratio equal to one for an FGX and an FGO graphene nanoplatelet strengthened beams. The presented results open a new view of the future of engineering designs for various engineering scopes such as energy harvesting, mass detection and damage detection purposes employing advanced structures.

Nonlinear Dynamics and Bifurcation Analysis of a Hinged-Clamped Beam Under Parametric and Internal Resonances

The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.

Dynamic stability of an axially transporting beam with two-frequency parametric excitation and internal resonance

Abstracts In this paper, the dynamic stability of axially transporting viscoelastic beams with two-frequency parametric excitation and 1:3 internal resonance is investigated for the first time. The governing equation and corresponding inhomogeneous boundary conditions are developed by the Newton's second law. The viscoelastic characteristic of the transporting Euler-Bernoulli beam obeys the Kelvin-Voigt model. The axial tension is considered to vary longitudinally. Direct method of multiple scales is employed to obtain the solvability conditions in principal parametric resonances. The stability boundary conditions are obtained by the Routh-Hurwitz criterion. Numerical examples are shown to illustrate the effects of relevant parameters on the stability boundaries. Unusual and interesting phenomena of stability boundaries occur in two-frequency parametric excitation and 1:3 internal resonance. The accuracies of the approximate analytical results are verified by comparing with the numerical results, which obtained by a differential quadrature method.

Nonlinear free and forced vibrations of porous functionally graded pipes conveying fluid and resting on nonlinear elastic foundation

This paper studies the nonlinear free and forced vibrations of fluid-conveying pipes that are made of porous functionally graded materials and supported on nonlinear elastic foundation. A modified power-law function is adopted to model the porous functionally graded materials, with both even and uneven porosity distributions to represent different material imperfections. Equations of motion of the pipe system are then derived through Hamilton’s principle based on Euler–Bernoulli beam theory, with foundation and von Kármán nonlinearities and damping effect being considered. The nonlinear free vibration and the primary resonance of the pipe system are investigated analytically, by employing the variational iteration method and the direct method of multiple scales, respectively. Detailed parametric studies for the free and forced vibrations are then carried out numerically, considering different parameters including power-law index, porosity characteristics, fluid velocity, pipe geometries, foundation parameters and so on.

Bifurcations and Chaotic Dynamics of an Axially Accelerating Hinged–Clamped Viscoelastic Beam

This paper investigates stability, bifurcation and dynamic behavior with chaotic response of an axially accelerating hinged–clamped viscoelastic beam under the combined effect of parametric and internal resonant conditions. The beam material obeys the Kelvin–Voigt model in which the material time derivative is used. The axial time-dependent velocity is characterized as a simple harmonic variation about the constant mean speed. The direct method of multiple scales is applied to the nonlinear integro-partial differential equation of motion to reduce it into a set of first-order equations. Here, in particular, aim is fixed to examine the system response in the close proximity of a 3:1 internal resonance between first two transverse modes. Specifically, the fixed point responses are obtained using continuation technique and the dynamic solution is obtained through direct time integration. The steady-state solution has both trivial and two-mode nontrivial solutions with hardening type of nonlinear characteristics. The system displays various bifurcations including super and subcritical pitchfork, Hopf and saddle node bifurcations. The multiple equilibrium solutions get transition to instability of both a saddle node type and a Hopf bifurcation. The damping has different effects on different modes. With decrease in external damping, the maximum amplitude of directly excited first mode is nearly unaffected, but the maximum amplitude of the indirectly excited second mode grows. This shows the energy exchange between the involved modes is encouraged due the presence of internal resonance. Internal damping has stabilizing effect on the transverse modes in the Hopf bifurcation region of the frequency and amplitude response equilibrium curves. It suppresses the nonlinear characteristics, viz. the number of Pitchfork, saddle node and Hopf bifurcation points even to zero for certain cases. Most importantly, multiple nontrivial equilibrium solution curves reduce to two in case of frequency response analysis and a single isolated closed-loop nontrivial solution in amplitude response analysis. Dynamic solution reveals the existence of not only stable periodic, mixed mode and quasiperiodic solutions, but also unstable chaotic oscillations depending on the system parameters and initial conditions in the neighborhoods of Hopf bifurcation regions. The system displays a wide array of interesting dynamic results equipped with various nonlinear features which are not present in the available literature on nonlinear traveling systems.

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.

Nonlinear free vibration analysis of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes

In this paper, the linear and nonlinear free vibrations of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes are presented. The formulation is based on the assumptions of Timoshenko beam theory in addition to consideration of the nonlinear von Karman strain–displacement relationship. The effective material properties of carbon nanotube reinforced composites are determined employing the Mori–Tanaka micromechanics model and the extended mixture rule. For the carbon nanotube reinforced composite beams, uniform distribution and four types of functionally graded distribution patterns of single-walled carbon nanotube reinforcements are considered. A differential transform method is applied on the nondimensionalized equations of motion to release the flapping modeshapes and the associated natural frequencies. The direct method of multiple scales is implemented to derive the effective nonlinearity and the corresponding nonlinear natural frequency. The accuracy of the present outcomes is validated by the comparison with the results given in the literature. The numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fractions, distribution types of the carbon nanotubes, and rotation speed on linear and nonlinear free vibration characteristics of carbon nanotube reinforced composite beam. The results demonstrate the important role of carbon nanotube distribution profile on linear and nonlinear free vibration features.

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.

Stress distribution and fatigue life of nonlinear vibration of an axially moving beam

In this paper, the effects of the nonlinear vibration on stress distribution and fatigue life of the axially moving beam are studied. The parametric excitation of the flexible material is created by the pulsating moving speed. Three-to-one internal resonance condition is satisfied. The three-parameter model is adopted in the viscoelastic constitutive relation. The nonlinear vibration of the axially moving beam with parametric and internal resonance are studied by using the direct multiple scales method (MSM) with numerical simulation confirmation. Based on the approximate analytical solution, the distribution of tensile stress and bending stress on the axially moving beam is presented by adopting a V-belt as the prototype. Based on the maximum stable cyclic stress, the limit cycle response of the V-belt is utilized to evaluate the effect of the resonance on the fatigue life. Also, the influences of the internal resonance on the steady-state responses and the fatigue life of the V-belt are revealed. Numerical examples illustrate that large unwanted resonances occur and the second-order mode receives vibration energy from to the first-order mode. The numerical results demonstrate that the nonlinear vibration significantly reduces the fatigue life of the V-belt. The fatigue life analysis method in this paper can be applied to the excited vibration of other axially moving systems and even static continuum.

Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance

Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincare sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.

On Parametrically Excited Vibration and Stability of Beams with Varying Rotating Speed

In this paper, principal parametric resonance of rotating Euler–Bernoulli beams with varying rotating speed is investigated. The model contains the geometric nonlinearity due to the von Karman strain–displacement relationship and the centrifugal forces due to the rotation. The rotating speed of the beam is considered as a mean value which is perturbed by a small harmonic variation. In this case, when the frequency of the periodically perturbed value is twice the one of the axial mode frequencies, the principal parametric resonance occurs. The direct method of multiple scales is implemented to study on the dynamic instability produced by the principal parametric resonance phenomenon. A closed-form relation which determines the stability region boundary under the condition of the principal parametric resonance is derived. Numerical simulation based on the fourth-order Runge–Kutta method is established to validate the results obtained by the method of multiple scales. The numerical analysis is applied on the discretized equations of motion obtained by the Galerkin approach. After validation of the results, a comprehensive study is adjusted for demonstrating the damping coefficient and the mode number influences on the critical parametric excitation amplitude and the parametric stability region boundary. A discussion is also provided to illustrate the advantages and disadvantages of the fourth-order Runge–Kutta method in comparison with the method of multiple scales.

Stability, Bifurcation and Chaos of a Traveling Viscoelastic Beam Tuned to 3:1 Internal Resonance and Subjected to Parametric Excitation

Analytical-numerical approach has been adopted to investigate the stability, bifurcation and dynamic behavior (including chaotic behavior) of axially moving viscoelastic beam subjected to parametric excitation resulting from speed variation in the presence of 3:1 internal resonance between the first two modes of vibration. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent coefficients. The direct method of multiple scales is employed to analyze the joint influence of the combination of parametric resonance and internal resonance with the focus on steady state responses. Equilibrium solutions along with their stability and bifurcations are determined by continuation algorithm while direct time integration is used for dynamic behavior for various system parameters. The results are compared with the previous works depicting the principal parametric resonances of the first and second modes. Significant comparative analysis results are reported in the stability and bifurcation of frequency response analysis. The dynamic responses show a range of behavior viz. stable periodic, mixed mode, quasiperiodic and unstable chaotic motion of the system. Numerical results illustrate various typical and interesting nonlinear phenomena of the traveling system which are not found in the existent literature.

Super-harmonic resonance and multi-frequency responses of a super-critical translating beam

With 3:1 internal resonance, super-harmonic resonances of a super-critical axially moving beam is studied for the first time. The response produced by 1/3 super-harmonic resonance is numerically determined and it is found to be very intense. Therefore, the direct multi-scale method is applied to fully investigate steady-state response of the super-harmonic resonance. In the super-critical speed range, the governing equation contains quadratic and cubic nonlinearity at the same time. Accordingly, the perturbation solution by applying the multiple scale method contains three time scales. The analytical solutions conclude that not only the primary resonance is contained, but also several harmonic responses are involved. Moreover, the perturbation analysis shows more harmonic responses should be involved. Furthermore, the 3:1 internal resonance investigated here indicates that energy transmits from the first mode to the second one. The amplitude-frequency response shows that nonlinearity of the super-critically translating beam is soft. Besides, the analytical process with internal resonance proves that the direct method of multiple scales can be applied to investigate the multi-mode responses.

Nonlinear Dynamics of Traveling Continua with Low Flexural Stiffness under Parametric and Internal Resonances

This study focuses on the trivial state stability boundary, steady state periodic response and chaotic behavior in the transverse motion of an axially moving viscoelastic continuum subject to parametric excitation and internal resonance. This parametric excitation comes from harmonic fluctuations of the traveling speed. The motion is restricted by viscous damping as well as material damping. The derived nonlinear integro-partial-differential equation of motion is solved through analytic-numerical approach. Direct method of multiple scales is used to solve the nonlinear equation of motion. A continuation algorithm is used to find the steady state response. Furthermore, Runge-Kutta method is applied to find the dynamic behavior of the system. The stable periodic response and the unstable chaotic motions are identified using different tools including the time traces, Phase portraits, Poincare map, fast Fourier transforms. Evolution of maximum Lyapunov exponent is used to locate the range of system parameters for which chaotic behavior exists.

Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.

Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam

In this paper, analytical and numerical approach is applied to find the steady-state and dynamic behaviors of an axially accelerating viscoelastic beam subject to two-frequency parametric excitation in presence of internal resonance. Direct method of multiple scales is employed to solve the cubic nonlinear integro-partial differential equation. As a result, the governing equation of motion is reduced to a set of nonlinear first-order partial differential equations. These equations are solved through continuation algorithm approach to find the frequency and amplitude response curves and their stability and bifurcation. The system reveals the presence of Hopf, saddle node, and pitchfork bifurcations. The dynamic bevavior of the system is estimated through phase portraits, time traces, Poincare maps, and FFT power spectra obtained via direct time integration. The evolution of maximum Lyapunov exponent reveals the system parameter where the dynamic response changes from stable periodic to unstable chaotic motion of the system.

Nonlinear modal interaction in rotating composite Timoshenko beams

Nonlinear free vibration analysis of rotating composite Timoshenko beams featuring internal resonance is studied in this paper. Three nonlinear coupled equations of motion for flapping, shear and axial motions, are based on the assumptions of Timoshenko theory and the nonlinear von Karman strain–displacement relationships. Due to the small magnitude of low-order flapping/shear frequencies ratio, usually a big gap exists between the aforementioned frequencies especially in isotropic beams, while this gap reduces for composite beams which exhibit more characteristics of Timoshenko beams. Different material, geometrical and operational parameters effects on the frequencies of rotating beams are investigated. For the first time the possibility of internal resonance occurrence between low-order flapping and shear modes is proved. The direct multiple scales method is implemented for construction of the flapping nonlinear normal modes. Results for the flapping nonlinear normal modes are validated via comparison with the results of the fourth-order Runge–Kutta method. Depend on the amplitude ratios besides the nearness of the frequencies of the interacting modes, four bifurcation regions are defined through the nonlinear normal mode stability analysis; (a) one stable coupled mode, (b) two stable and one unstable coupled modes, (c) three stable coupled modes and (d) one stable coupled mode.

Nonlinear free vibration analysis of rotating composite Timoshenko beams

Linear and nonlinear free vibrations of rotating composite Timoshenko beams are investigated in this paper. The formulation is based on the assumptions of Timoshenko beam theory and the nonlinear von-Karman strain–displacement relationships. The updated equations of motion about the prestressed configuration are obtained from the three-coupled partial differential equations of motion. The differential transform method is implemented to find the prestressed configuration due to centrifugal forces. The Galerkin discretization approach is applied to the linearized updated equations of motion to determine the linear normal modes and the associated natural frequencies of the rotating composite Timoshenko beam. The rotation speed and number of layers variation effects on the flapping natural frequencies are studied and the necessity of employing the Timoshenko beam model for the composite beams is illustrated. The direct multiple scales method is applied to the three-coupled nonlinear updated partial differential equations for the nonlinear vibration study especially on the flapping backbone curves. The number of layers variation effects is investigated on the three lowest flapping backbone curves. The present results are validated with the previous literature results.

Nonlinear Dynamic Analysis of a Rotor Shaft System With Viscoelastically Supported Bearings

This research investigates the dynamic analysis of a single-rotor shaft system with nonlinear elastic bearings at the ends mounted on viscoelastic suspension. Timoshenko shaft model is utilized to incorporate the flexibility of the shaft; the rotor is considered to be rigid and located at the mid-span of the shaft. A nonlinear bearing pedestal model is assumed which has a cubic nonlinear spring and linear damping characteristics. The viscoelastic supports are modeled using Kelvin-Voigt model. Free and forced vibration is investigated based on the direct multiple scales method of one-to-one frequency-to-amplitude relationship using third order perturbation expansion. The results of the nonlinear analysis show that a limiting value of the internal damping coefficient of the shaft exists where the trend of the frequency-response curve switches. Also, the primary resonance peak shifts to higher frequencies with the increase of the bearing nonlinear elastic characteristics, but with a flattened curve and hence lower peak values. A jump phenomenon takes place for high values of the bearing nonlinear elastic characteristics.