Presence Of Internal Resonance Research Articles

Internal resonance analysis of bio-inspired X-shaped structure with nonlinear vibration absorber

In this paper, a vibration isolation system based on bio-inspired structure with a nonlinear absorber is proposed. The nonlinear vibration properties and complex dynamical characteristics are explored elaborately in the presence of internal resonance. Lagrange’s principle is employed to derive the governing equations, which can describe the coupled nonlinear vibration system. The method of multiple scales is utilized to obtain the steady-state responses of the coupled system. It is found that the frequency-amplitude relationships reveal the typical nonlinear phenomena including jumping and multi-value. The influences of parameters on the dynamical characteristics are discussed in detail. Numerical integrations are directly implemented on the vibration equations to verify the aforementioned analytical results. And furthermore, bifurcations and chaotic motions are detected in the bifurcation diagrams via the Poincare section techniques. The proposed system and the corresponding researches would be of significance for the design and practical applications of the passive vibration control.

Vibration energy harvesting under concurrent base and flow excitations with internal resonance

In this study, internal resonance is investigated to further explore the potential of energy harvesting under concurrent base and flow excitations. The effects of system parameters on the performance of energy harvester with three-to-one internal resonance are analyzed analytically. At first, a lumped-parameter model for the energy harvester, which consists of a two-degree-of-freedom airfoil with the piezoelectric coupling introduced to the plunging motion, is established by using a nonlinear quasi-steady aerodynamic model. Subsequently, the method of multiple scales is implemented to derive the approximate analytic solution of the energy harvesting system under three-to-one internal resonance. Then, the bifurcation characteristics of the energy harvester with respect to various system parameters are analyzed. Finally, the numerical solutions are presented to validate the accuracy of the approximate analytic solutions. The study shows that the harvested voltage and power of the energy harvester can be significantly improved in the presence of internal resonance. In addition, the analytic solutions of internal resonance and the bifurcation analysis can provide an essential reference for design of such a kind of energy harvester.

A Semi-Analytical Method for Calculation of Strongly Nonlinear Normal Modes of Mechanical Systems

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

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 Dynamics of a Travelling Beam Subjected to Multi-Frequency Parametric Excitation

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

Active Control of a Non-Linear Ship model with External and Parametric Excitation

The response of a ship model with non-linearly coupled pitch and roll modes under modulated external and parametric solved and studied. The active control is applied to reduce the vibration of the system . The method of multiple scale perturbation technique is applied to obtain the periodic response equation near the primary resonance in the presence of internal resonance of the system. The objective of this research is focused on the stability of this periodic solution, dynamical properties and chaotic response. The stability of the obtained numerical solution is studied using both frequency response equation and phase-plane methods. The effects of some parameters on the vibrating system are investigated and reported in this paper.

Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance

Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance is investigated for the first time. For appropriate choice of system parameters, the natural frequency of the second mode is approximately twice that of the first providing the condition for 2 : 1 internal resonance. The ordinary differential equations of the two mode shapes are established using the Galerkin method. The problem is replaced by two coupled second-order differential equations with quadratic and cubic nonlinearities. The multiple scales method is applied to derive the modulation-phase equations. Steady-state solutions of the system as well as their stability are examined. The frequency-amplitude curves exhibit the steady-state response in the directly excited and indirectly excited modes due to modal interaction. The double-jump, the saturation phenomenon, and the nonperiodic region phenomena are observed illustrating the influence of internal resonance. The validity range of the analytical approximations is assessed by comparing the analytical approximate results with a numerical solution by the Runge-Kutta method. The unstable regions in the internal resonance are explored via numerical simulations.

Forced vibration analysis of the milling process with structural nonlinearity, internal resonance, tool wear and process damping effects

In this paper, forced vibration analysis of an extended dynamic model of the milling process is investigated, in the presence of internal resonance. Regenerative chatter, structural nonlinearity, tool wear and process damping effects are included in the proposed model. Taking into account the average and first order expansion of Fourier series for cutting force components; their closed form expressions are derived. Moreover, in the presence of large vibration amplitudes, the loss of contact effect is included in this model. Analytical approximate response of the nonlinear system is constructed through the multiple-scales approach. Dynamics of the system is studied for two cases of primary and super-harmonic resonance, associated with the internal resonance. Under steady state motion, the effects of structural nonlinearity, cutting force coefficients, tool wear length and process damping are investigated on the frequency response functions of the system. In addition, existence of multiple solutions, jump phenomenon and energy transfer between vibration modes are presented and compared for tow cases of primary and super-harmonic resonances.

Non-linear response of buckled beams to 1:1 and 3:1 internal resonances

The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral–partial–differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency–response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.

Nonlinear Dynamics of an Euler-Bernoulli Beam with Parametric and Internal Resonances

The nonlinear transverse vibration of a simply-supported travelling Euler-Bernoulli beam subjected to principal parametric resonance in presence of internal resonance is investigated. The variable velocity of beam is assumed to be comprised of a harmonically varying component superimposed over a mean value. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times the natural frequency of first mode for a range of mean velocity of the beam, resulting in a three-to-one internal resonance. The method of multiple scales (MMS) is directly applied to the governing nonlinear integral-partial-differential equations and the associated boundary conditions. This results in a set of first order ordinary differential equations governing the modulation of amplitude and phase of the first two modes, which are analyzed numerically for principal parametric resonance of first mode. The nonlinear steady state response along with the stability and bifurcation of the beam are investigated. The dynamical behaviour of the system is observed in terms of periodic, quasiperiodic and chaotic responses, showing the influence of internal resonance.

Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Nonlinear dynamics of rotating box FGM beams using nonlinear normal modes

In this work an analysis is performed on the nonlinear planar vibrations of a functionally graded beam subjected to a combined thermal and harmonic transverse load in the presence of internal resonance. Adopting the direct perturbation MMS technique, the partial differential equations of motion of the beam are reduced to sets of first-order nonlinear modulation equations in terms of the complex modes of the beam. The assumption of steady-state values of centrifugal loads is evaluated. It has to be said that there is a lack of information about modeling of rotating box beams made of functionally graded materials (FGMs) under thermo-mechanical loads. The influence of the transverse load amplitude and the internal detuning parameter on the strength of nonlinear modal interaction is illustrated. It is also shown that the system exhibits periodic and quasiperiodic responses for a typical range of parameter values.

Internal Resonance of a Floating Roof Subjected to Nonlinear Sloshing

Internal resonance in the vibration of a floating roof coupled with nonlinear sloshing in a circular cylindrical oil storage tank is investigated. The nonlinear system exhibits internal resonance when nonlinear terms of the governing equation have a dominant frequency close to a certain modal frequency of the system. Numerical results show that when internal resonance occurs, the responses of stresses in a floating roof exhibit a long-duration period of large amplitude despite a short duration of the earthquake excitation applied to the tank. Due to the presence of internal resonance, the underestimation of the stresses associated with the use of the linear theory becomes more marked, and thus the importance of nonlinearity of sloshing in the stress estimation is accentuated. It is illustrated that the magnitudes of the stresses increase with the increase in the liquid-filling level, and that the effect of internal resonance on the stresses noted in the case of sinusoidal excitation appears under real earthquake excitation. A method for reducing the stresses is proposed.

Stability and bifurcation for a flexible beam under a large linear motion with a combination parametric resonance

Stability and bifurcation behaviors for a model of a flexible beam undergoing a large linear motion with a combination parametric resonance are studied by means of a combination of analytical and numerical methods. Three types of critical points for the bifurcation equations near the combination resonance in the presence of internal resonance are considered, which are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues in nonresonant case, respectively. The stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained in terms of the system parameters. Especially, for the third case, the explicit expressions of the critical bifurcation curves leading to incipient and secondary bifurcations are obtained with the aid of normal form theory. Bifurcations leading to Hopf bifurcations and 2-D tori and their stability conditions are also investigated. Some new dynamical behaviors are presented for this system. A time integration scheme is used to find the numerical solutions for these bifurcation cases, and numerical results agree with the analytic ones.

Stability and bifurcation analysis for a model of a nonlinear coupled pitch–roll ship

Using a combination of analytical and numerical methods, the paper studies the stability and bifurcations for a model of a nonlinear coupled pitch–roll ship. The model represents a two-degree-of-freedom system with quadratic coupling subjected to a modulated sinusoidal excitation. Three types of critical points for the bifurcation response equations near the combination resonance in the presence of internal resonance are considered. These points are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues, respectively. For each case, the stability regions for the initial equilibrium solution and the critical bifurcation curves are obtained. The amplitude response curves with respect to detuning/damping parameters for critical bifurcation parameters are obtained for the first two types, and detailed analyses of the eigenvalues of the linearized system for each parametric region are given. For the third type, with the aid of normal form theory, the explicit expressions of the critical bifurcation curves leading to incipient and secondary bifurcations are obtained. Bifurcations leading to 2D, 3D tori and their stability conditions are also investigated. Some new dynamical behaviors are presented for this system.

Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances

Nonlinear dynamics of a hinged–hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance is investigated. The system has geometric cubic nonlinearity due to stretching effect out of immovable support conditions at both ends. The pipe conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. For appropriate choice of system parameters, the natural frequency of the second mode is approximately three times that of the first mode for a range of mean flow velocity, activating a three-to-one internal resonance. The analysis is carried out using the method of multiple scales by directly attacking the governing nonlinear integro-partial-differential equations and the associated boundary conditions. The set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for combination parametric resonance and principal parametric resonance. Stability, bifurcation and response behavior of the pipe are investigated. The amplitude and frequency detuning of the harmonic velocity perturbation are taken as the control parameters. The system exhibits response in the directly excited and indirectly excited modes due to modal interaction. Dynamic response of the system is presented in the form of phase plane trajectories, Poincare maps and time histories. A wide array of dynamical behavior is observed illustrating the influence of internal resonance.

Nonlinear Interactions: Analytical, Computational, and Experimental Methods

Nonlinear Interactions: Analytical, Computational, and Experimental Methods

Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 1: Equations of Motion and Numerical Results

The response-frequency relationship in the vicinity of a resonant frequency, the occurrence of travelling wave response and the presence of internal resonances are investigated for simply supported, circular cylindrical shells. Donnell’s nonlinear shallow-shell theory is used. The boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied. The problem is reduced to a system of four ordinary differential equations by means of the Galerkin method. The radial deflection of the shell is expanded by using a basis of four linear modes. The effect of internal fluid is also investigated. The equations of motion are studied by using a code based on the Collocation Method. The present model is validated by comparison of some results with others available. A water-filled shell presenting the phenomenon of 1:1:1:2 internal resonances is investigated for the first time; it shows intricate and interesting dynamics. [S0739-3717(00)01204-6]

Application of the Lie Group Transformations to Nonlinear Dynamical Systems

This paper describes the theory of Lie group operators in a form suitable for the applied dynamics community. In particular, it is adapted to analyzing the dynamic behavior of nonlinear systems in the presence of different resonance conditions. A key ingredient of the theory is the Hausdorff formula, which is found to be implicitly reproduced in most averaging techniques during the transformation process of the equations of motion. The method is applied to examine the nonlinear modal interaction in a coupled oscillator representing a double pendulum. The system equations of motion are reduced to their simplest (normal) form using operations with the linear differential operators according to Hausdorff's formula. Based on the normal form equations, different types of resonance regimes are considered. It is shown that the energy of the parametrically excited first mode can be regularly (or nonregularly) shared with the other mode due to the internal resonance condition. If the second mode is parametrically excited, its energy is localized and is not transferred to the first mode, even in the presence of internal resonance.