Case Of Primary Resonance Research Articles

Multimode Dynamics and Out-of-Plane Drift in Suspended Cable Using the Kinematically Condensed Model

The large amplitude vibration and modal interactions of shallow suspended cable with three-to-three-to-one internal resonances are investigated. The quasistatic assumption and direct approach are used to obtain the condensed suspended cable model and the corresponding modulation equations for the case of primary resonance of the third symmetric in-plane or out-of-plane mode. The equilibrium, periodic, and chaotic solutions of the modulation equations are studied. Moreover, the nonplanar motion and symmetric character of out-of-plane vibration of the shallow suspended cables are investigated by means of numerical simulations. Finally, the role played by the quasistatic assumption, internal resonance, and static configuration in disrupting the symmetry of the out-of-plane vibration is discussed.

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlinear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of multiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequencies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency–amplitude and frequency–response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.

Nonlinear vibrations of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic responses of an inclined pinned-pinned Euler-Bernoulli beam with a constant cross section and finite length subjected to a concentrated vertical force traveling with constant velocity is investigated by using the mode summation method. Frequency analysis of the PDE's governing equations of motion for steady-state response is studied by applying multiple scales method. The nonlinear dynamic deflections of the beam are obtained by solving two coupled nonlinear PDE's governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-point of the beam are obtained for various load velocity ratios and the numerical results are compared with those obtained from traditional linear solution. It is found that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also stability analysis of the steady-state response for the modes equations having quadratic nonlinearity is carried out and it is observed from the amplitude response curves that for the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon are predicted for the longitudinal excitation.

A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case

Nonlinear vibrations of a general model of continuous system is considered. The model consists of arbitrary linear and cubic operators. The equation of motion is solved by the method of multiple scales (a perturbation method). The primary resonances of external excitation is analysed. The amplitude and phase modulation equations are presented. Approximate analytical solution is derived. Steady-state solutions and their stability are discussed. Finally, the solution algorithm is applied to two different engineering problems. One of the application is the transverse vibration of an axially moving Euler–Bernoulli beam and the other is a viscoelastic beam.

Nonlinear behavior of Van der Pol oscillators under parametric and harmonic excitations

In this paper, the vibration of nonlinear coupled Van der Pol oscillators under external and parametric excitations has been studied and solved. The method of the multiple scale perturbation technique has been applied to study the frequency response equations near the primary resonance case of this system. The study is focused on the stability of the periodic solution and the nonlinear behavior of this system. Analytical results are verified using numerical simulations and the effects of some parameters on the vibrating system are investigated and reported.

On methods for continuous systems with quadratic, cubic and quantic nonlinearities

Methods for study of weakly nonlinear continuous systems are discussed. The method of multiple scales is used to analyze the nonlinear response of a relief valve under combined static and dynamic loadings. We determine a second-order approximation to the response of the system for the case of primary resonance. Second, we derive a second-order nonlinear ordinary differential equation that describes the time evolution of a single-mode, the so-called single-mode discretization. Then, we use the multiple scales method to determine second-order approximate solutions of this equation, thereby obtaining the equations describe the modulations of the amplitude and phase of the response. We show that the results of the second approach are erroneous.

Dynamic behavior of an AMB supported rotor subject to harmonic excitation

This paper presents a study of the non-linear response of a simple rigid disk-rotor, supported by active magnetic bearings (AMB), without gyroscopic effects. The case of primary resonance is examined under multi-excitation forces. The rotating shaft is described by a coupled second order non-linear ordinary differential equations. Approximate solutions are sought applying the method of multiple scales. Numerical simulations are carried out to illustrate the steady-state response and the stability of the solutions for various parameters using the frequency response function method. It is shown that the system parameters have different effects on the non-linear response of the rotor. For steady-state response, however, multiple-valued solutions and jump phenomenon occur. Results are compared to previously published work.

Global bifurcations and chaos in a harmonically excited and undamped circular plate

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.

Resonance behavior of a rotor-active magnetic bearing with time-varying stiffness

Abstract The non-linear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The rotor-AMB system is subjected to a periodically time-varying stiffness. The simultaneous primary resonance case is considered and examined. The vibration of the rotor is modeled by a coupled second-order non-linear ordinary differential equations with quadratic and cubic non-linearities. Their approximate solutions are sought applying the method of multiple scales. The steady-state response and the stability of the system at the simultaneous primary resonance case for various parameters are studied numerically, applying the frequency response function method. It is found that different shapes of chaotic motion exist, which are determined using phase-plane method. It is also shown that the system parameters have different effects on the non-linear response of the rotor. For steady-state response, however, multiple-valued solutions, jump phenomenon, hardening and softening non-linearity occur. Results are compared to previously published work.

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.

Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt

Abstract In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equations of motion for viscoelastic moving belt with the external damping and parametric excitation are given. The four-dimensional averaged equation under the case of primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving belt. From the averaged equations obtained here, the theory of normal form is used to give the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on normal form, the energy-phrase method is employed to analyze the global bifurcations and chaotic dynamics in parametrically excited viscoelastic moving belt. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type multi-pulse homoclinic orbits in the averaged equation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense in parametrically excited viscoelastic moving belt. The chaotic motions of viscoelastic moving belts are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits is observed from three-dimensional phase space.

훅조인트로 연결된 축계의 비선형 비틀림 진동 : 조합공진의 경우

Torsional oscillations of a system incorporating a Hooke's joint are investigated by studying a simple similar nonlinear 2-degree-of-freedom model, which has linear and quadratic nonlinear parametric excitations. The simple system is identified to have the possibilities of primary, sub harmonic and combination resonances. The case of simultaneous primary and combination resonances is selected for perturbation analysis to have the reduced amplitude-equations of motion. The same procedure is applied to the system incorporating a Hooke's joint.

Coexistence of Stable Solutions in the Nonlinear Ship Motion

The dynamic stability and complicated motions of a vessel in regular sea are investigated when the frequency in the pitch is nearly twice the frequency in the roll. We use the Method of Multiple Scales to determine four first-order nonlinear ordinary differential equations that govern the modulation of the amplitudes and phases when the forcing frequency is near either the pitch or the roll natural frequency. These equations are used to determine the steady-state solutions and their stability. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found in the presence of quadratic nonlinearities. The linear solution is symmetric. The pitch mode has multi-valued solutions and the roll mode has a single-valued solution for the case of primary resonance of the pitch mode. Both pitch and roll modes have multivalued solutions in small intervals for the case of primary resonance of the roll mode.

Analytical approximation of the primary resonance response of a periodically excited piecewise non-linear–linear oscillator

An analytical approximate solution is constructed for the primary resonance response of a periodically excited non-linear oscillator, which is characterized by a combination of a weakly non-linear and a linear differential equation. Without eliminating the secular terms, a valid asymptotic expansion solution for the weakly non-linear equation is analytically determined for the case of primary resonances. Then, a symmetric periodic solution for the overall system is obtained by imposing continuity and matching conditions. The stability characteristic of the symmetric periodic solution is investigated by examining the asymptotic behaviour of perturbations to the steady state solution. The validity of the developed analysis is highlighted by comparing the first order approximate solutions with the results of numerical integration of the original equations.

Multiple time scale analysis of hysteretic systems subjected to harmonic excitation

Dynamic behavior of smooth hysteretic systems subjected to harmonic excitation is analyzed. Wen's differential equation model for hysteresis, which can be applied to a large class of hysteretic systems, is used. A piecewise power series expression for hysteretic restoring force is derived from Wen's model assuming that steady state force–displacement curve draws a single loop and that the non-linearity of the restoring force is weak. The method of multiple scales is applied to the equation of motion by using the piecewise power series expression for the cases of primary and secondary resonance to derive the approximate solutions and the differential equations governing the amplitude and phase of the solutions. Phase plane trajectories, resonance curves and stability limit of the solutions are obtained and compared with the results of numerical integration in order to examine the validity of the present analysis.

Stability and primary simultaneous resonance of harmonically excited non-linear spring pendulum system

The considered system is a harmonically excited, non-linear spring-pendulum, which simulates the ship roll motion. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the fourth order approximation. All possible resonance cases were extracted at this approximation order. This work is limited only to the primary resonance cases. The stability of the system is investigated using both frequency response equations and phase-plane method. The effects of the different parameters on system behavior are studied numerically. Variation of some equations parameters leads to the bending of the frequency response curves and hence to the jump phenomenon. It is quite clear that some of the simultaneous primary resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported.

VIBRATIONS OF CONTINUOUS SYSTEMS WITH A GENERAL OPERATOR NOTATION SUITABLE FOR PERTURBATIVE CALCULATIONS

The operator notation previously developed to analyze vibrations of continuous systems has been further generalized to model a system with an arbitrary number of coupled differential equations. Linear parts of the equations are expressed with an arbitrary linear differential and/or integral operators, and non-linear parts are expressed with arbitrary quadratic and cubic operators. Equations of motion are solved in their general form using the method of multiple scales, a perturbation technique. The case of primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations is considered. The algorithm developed is applied to a non-linear cable vibration problem having small sag-to-span ratios.

Multiple resonances in suspended cables: direct versus reduced-order models

We apply two analytical approaches to construct asymptotic models for the non-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and a two-to-one internal resonance with the first symmetric out-of-plane mode. First, we apply the method of multiple scales directly to the governing two integral-partial-differential equations and associated boundary conditions. Reconstitution of the solvability conditions at second and third orders leads to a system of four coupled non-linear complex-valued equations describing the modulation of the amplitudes and phases of the interacting modes. The homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problem are needed to make the reconstituted modulation equations derivable from a Lagrangian. However, this procedure leads to an indeterminacy, indicating a likely inconsistency with this specific application of the method of multiple scales. Then, we apply the method to a four-degree-of-freedom Galerkin discretized model obtained using the pertinent excited eigenmodes. Again, the homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problems are required to make the reconstituted modulation equations derivable from a Lagrangian. Frequency–response curves obtained using the two generated asymptotic models, for a specific choice of the arbitrary constant appearing in both models, show different qualitative as well as quantitative predictions for some classes of motions. The effects of an inconsistent reconstitution in the direct approach are also investigated.

NON-LINEAR VIBRATIONS OF A SLIGHTLY CURVED BEAM RESTING ON A NON-LINEAR ELASTIC FOUNDATION

In this study, non-linear vibrations of slightly curved beams are investigated. The curvature is taken as an arbitrary function of the spatial variable. The initial displacement is not due to buckling of the beam, but is due to the geometry of the beam itself. The ends of the curved beam are on immovable simple supports and the beam is resting on a non-linear elastic foundation. The immovable end supports result in the extension of the beam during the vibration and hence introduces further non-linear terms to the equations of motion. The integro-differential equations of motion are solved analytically by means of direct application of the method of multiple scales (a perturbation method). The amplitude and phase modulation equations are derived for the case of primary resonances. Both free and forced vibrations with damping are investigated. Effect of non-linear elastic foundation as well as the effect of curvature on the vibrations of the beam are examined. It is found that the effect of curvature is of softening type. For sufficiently high values of the coefficients, the elastic foundation may suppress the softening behaviour resulting in a hardening behaviour of the non-linearity.

Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam

An experimental validation of the suitability of reduction methods for studying nonlinear vibrations of distributed-parameter systems is attempted. Nonlinear planar vibrations of a clamped-clamped buckled beam about its first post-buckling configuration are analyzed. The case of primary resonance of the nth mode of the beam, when no internal resonances involving this mode are active, is investigated. Approximate solutions are obtained by applying the method of multiple scales to a single-mode model discretized via the Galerkin procedure and by directly attacking the governing integral-partial-differential equation and boundary conditions with the method of multiple scales. Frequency-response curves for the case of primary resonance of the first mode are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For high buckling levels above the first crossover point of the beam, the computed frequency-response curves are qualitatively as well as quantitatively different. The experimentally obtained frequency-response curves for the directly excited first mode are in agreement with those obtained with the direct approach and in disagreement with those obtained with the single-mode discretization approach.