Case Of Principal Parametric Resonance Research Articles

Effect of spring mid-support condition on the vibrations of the axially moving string

In this study, the axially moving string with spring-loaded middle support is discussed. The supports assumed as simple support on the string both ends. The intermediate support shows the characteristics of the spring. The string velocity is accepted as harmonically varying around a mean value. The Hamiltonian principle is used to find the equations of motion. The equations of motion become nonlinear, considering the nonlinear effects caused by string extensions. The equations of motion and boundary conditions are become dimensionless by nondimensionalization. Approximate solutions were found by using multiple time scales which is one of the perturbation methods. By solving the linear problem that is obtained by the first terms of the perturbation series, the exact natural frequencies were calculated for the different locations of the mid-support, various spring coefficients, and various axial velocity values. The second-order nonlinear terms reveal the correction terms for the linear problem. Stability analysis is carried out for cases where the velocity change frequency is away from zero and two times the natural frequency. Stability boundaries are determined for the principal parametric resonance case.

Analysis of Parametric Resonances in In-Plane Vibrations of Electrostrictive Hyperelastic Plates

Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice.

Dynamic Response of Axially Moving Beams with Small Non-Idealities at Boundary Supports

The nonlinear response of an axially moving Euler-Bernoulli beam having one intermediate simple support and clamped boundaries with small non-idealities is examined. A small parameter representing real physical clearances at ends is used. The clamped support conditions mean that the beam is passing through two frictionless guides and one intermediate simple support placed between two guides. It is assumed that the beam has immovable boundaries at the outer ends. The assumption introduces nonlinearity because of stretching of neutral fibers. It is considered that the beam is axially moving along its length at a harmonically varying velocity about a constant mean value. In this paper variations at the velocities are assumed to be suitably weak. Differential equations governing the motion of the two sides of the beam are derived using variational formulation. A damping term is added into the equations. The method of multiple scales is applied to obtain approximate analytical solutions in this weakly nonlinear system. Non-ideal conditions at boundary supports of the beam are taken into consideration as small parameters at nonlinear equations. Solvability condition for the case of principal parametric resonance is derived, amplitude-phase curves of steady-state responses and stability conditions are investigated. It is found that the effect of small non-ideality at boundary supports has a role on nonlinear response: the width of unstable regions becomes larger with increase of non-ideality value.

Nonlinear Vibrations and Chaotic Dynamics of the Laminated Composite Piezoelectric Beam

This paper investigates the complicated dynamics behavior and the evolution law of the nonlinear vibrations of the simply supported laminated composite piezoelectric beam subjected to the axial load and the transverse load. Using the third-order shear deformation theory and the Hamilton's principle, the nonlinear governing equations of motion for the laminated composite piezoelectric beam are derived. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of principal parametric resonance and 1:9 internal resonance. From the averaged equations obtained, numerical simulation is performed to study nonlinear vibrations of the laminated composite piezoelectric beam. The axial load, the transverse load, and the piezoelectric parameter are selected as the controlling parameters to analyze the law of complicated nonlinear dynamics of the laminated composite piezoelectric beam. Based on the results of numerical simulation, it is found that there exists the complex nonlinear phenomenon in motions of the laminated composite piezoelectric beam. In summary, numerical studies suggest that periodic motions and chaotic motions exist in nonlinear vibrations of the laminated composite piezoelectric beam. In addition, it is observed that the axial load, the transverse load and the piezoelectric parameter have significant influence on the nonlinear dynamical behavior of the beam. We can control the response of the system from chaotic motions to periodic motions by changing these parameters.

Dynamic stability of a rotating sandwich beam with magnetorheological elastomer core

Abstract In this work the dynamic stability of a rotating three layered symmetric sandwich beam with magnetorheological elastomer (MRE) core and conductive skins subjected to axial periodic loads has been investigated using finite element method (FEM). The derived governing equation of motion is in the form of a multi-degree of freedom Mathieu–Hill's equation with complex coefficients. The instability regions of the sandwich beam for the principal parametric resonance case have been determined by using the harmonic balance method. Effects of applied magnetic field, rotating speed, setting angle, hub radius, static load and dynamic load on the dynamic characteristics and instability regions of the sandwich beam are investigated. This work will find application in the passive and active vibration reduction of rotating sandwich structure using magnetorheological elastomer core, magnetic field and periodic axial load.

Dynamic stability of magnetorheological elastomer based adaptive sandwich beam with conductive skins using FEM and the harmonic balance method

Abstract The dynamic stability of a partially treated magnetorheological elastomer (MRE) cored sandwich beam with conductive skins subjected to time varying axial load has been studied. The finite element method (FEM) and Guyan reduction method are used to derive the governing equation of motion which is similar to that of Mathieu's equation. The instability regions of the sandwich beam for the principal parametric resonance case are calculated by using the harmonic balance method. Effects of applied magnetic field, static load, dynamic load, the length and the location of the MRE patch on the stability of the sandwich beam are investigated. The results suggest that the stability of the MRE embedded sandwich beams are strongly influenced by the strength of the applied magnetic field, static load, dynamic load, the location and the length of the MRE patch.

Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions

In this study, nonlinear vibrations of an axially moving multi-supported string have been investigated. The main difference of this study from the others is in that there are non-ideal supports allowing minimal deflections between ideal supports at both ends of the string. Nonlinear equations of the motion and boundary conditions have been obtained using Hamilton’s Principle. Dependence of the equations of motion and boundary conditions on geometry and material of the string have been eliminated by non-dimensionalizing. Method of multiple scales, a perturbation technique, has been employed for solving the equations of motion. Axial velocity has been assumed a harmonically varying function about a constant value. Axially moving string has been investigated in three regions. Vibrations have been examined for three different cases of the velocity variation frequency. Stability has been analyzed and stability boundaries have been established for the principal parametric resonance case. Effects of the non-ideal support conditions on stability boundaries and vibration amplitudes have been investigated.

Stability and bifurcations of hysteretic systems subjected to principal parametric excitation

Dynamic behavior of smooth hysteretic systems subjected to harmonic parametric excitation is investigated. Wen's differential equation model for hysteresis, which can describe a large class of hysteretic systems, is used. Employing the piecewise power series expression for hysteresis proposed in the previous paper, the method of multiple scales is applied for the case of principal parametric resonance to obtain the second-order approximate solutions and their stability. It is shown that the hysteretic systems are unstable inside the principal resonance region but the responses are bounded having symmetric periodic solutions. It is also shown that the systems are stable outside the region even if viscous damping does not exist because of Hopf bifurcation. Numerical integrations are also performed to illustrate the nonlinear resonant characteristics of the systems and confirmed that trivial stationary solutions and stable symmetric periodic solutions in principal resonance region bifurcate to produce a pair of non-symmetric periodic solutions. Stability regions with regard to excitation parameters are also illustrated. The theoretical and numerical results are compared to examine the validity of the present analysis.

A time-varying stiffness rotor-active magnetic bearings system under parametric excitation

The method of multiple scales is applied to investigate the non-linear oscillations and dynamic behaviour of a rotor-active magnetic bearings (AMBs) system, with time-varying stiffness. The rotor-AMB model is a two-degree-of-freedom non-linear system with quadratic and cubic non-linearities and parametric excitation in the horizontal and vertical directions. The case of principal parametric resonance is considered and examined. The steady-state response and the stability of the system at the principal parametric resonance case for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviours including multiple-valued solutions, jump phenomenon, hardening and softening non-linearity. Different effects of the system parameters on the non-linear response of the rotor are also reported. Results are compared with available published work.

Response of parametrically excited Duffing-van der Pol oscillator with delayed feedback

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

Dynamic Behavior of an AMB/Supported Rotor Subject to Parametric Excitation

The dynamical behavior of a parametrically excited simple rigid disk-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The principal parametric resonance case is considered and studied. The motion of the rotor is modeled by a coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought applying the method of multiple scales. A reduced system of four first-order ordinary differential equations are determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency response function method. The numerical results show that the system behavior includes multiple solutions, jump phenomenon, and sensitive dependence on initial conditions. It is also shown that the system parameters have different effects on the nonlinear response of the rotor. Results are compared to previously published work.

Spatial nonlinear dynamics near principal parametric resonance for a fluid-conveying cantilever pipe

Abstract Flow-induced vibrations of a fluid-conveying cantilever pipe are examined theoretically under the condition that the fluid velocity has a small harmonic pulsatile component. More specifically, the case of principal parametric resonance is considered for the pipe free to undergo three-dimensional motions. The mean flow is considered to be near the critical flow rate at which the tube undergoes a Hopf bifurcation into self-excited oscillations. When the governing equations of motion for the tube with steady flow are reduced to those on the center manifold in the neighborhood of Hopf bifurcation, the normal form equations are O(2)-equivariant. The weak harmonic fluctuations due to pulsatile flow result in symmetry-breaking terms in the normal form. The eigenvalues of an O(2)-equivariant system undergoing a symmetry-breaking Hopf bifurcation have multiplicity two. When an additive linear term, arising from time-periodic modulations of the original dynamic system, is introduced into the normal form, the symmetry-breaking bifurcation structure for the trivial solution splits into three categories: a steady-state bifurcation giving rise to standing wave fixed-point solutions, a Hopf bifurcation giving rise to two-frequency solutions, and an O(2)-Takens–Bogdanov bifurcation. The resulting dynamics in each case are studied along with secondary and tertiary bifurcations. The dynamics of the tube system are studied as a function of the mean flow rate and the frequency of flow fluctuations. Amplitude response diagrams constructed for a specific example tube system using the continuation and bifurcation analysis software package AUTO illustrate the variety of possible behaviors.

Thermally Induced Principal Parametric Resonance in Circular Plates

We consider the problem of large‐amplitude vibrations of a simply supported circular flat plate subjected to harmonically varying temperature fields arising from an external heat flux (aeroheating for example). The plate is modeled using the von Karman equations. We used the method of multiple scales to determine an approximate solution for the case in which the frequency of the thermal variations is approximately twice the fundamental natural frequency of the plate; that is, the case of principal parametric resonance. The results show that such thermal loads produce large‐amplitude vibrations, with associated multi‐valued responses and subcritical instabilities.

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.

PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

Damping of parametrically excited single-degree-of-freedom systems

A single-degree-of-freedom parametrically excited system coupled with a Lanchester damper, a mass-dashpot device, is studied. The two equations governing the total system are solved using the method of multiple scales for the case of principal parametric resonance. Steady-state solutions are obtained and the effect of the various system parameters examined. The stability analysis for the steady-state solution is also carried out. Results show that this damper can limit the maximum response of the main system and delay the onset of the force threshold necessary to trigger a non-trivial stable response.