Internal Resonance Conditions Research Articles

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.

Large amplitude free vibration of a viscoelastic beam carrying a lumped mass–spring–damper

Numerous engineering structures can be modeled as flexible beams carrying appendages. Physical time-dependent phenomena in such structures appear by considering nonlinear models. Accordingly, the free vibration of a nonlinear Rayleigh beam carrying a linear mass–spring–damper is investigated in this research. The nonlinearity is due to the axial stress changes, and the viscoelastic characteristic of the beam is described by the Kelvin–Voigt model. The governing nonlinear equations of motion are derived by using Hamilton’s principle and solved by the multiple scales method. The presence of the linear damper in the mass–spring–damper system causes a contradiction in the time response of the mass obtained from the classical procedure of the multiple scales method. Therefore, the problem is solved by some manipulations in the process of the multiple scales method. Then, the complex-valued resonance frequencies and mode-shapes are derived by applying the method of power series with the aid of Green function concept. Considering the solvability condition, the nonlinear time-dependent resonance frequencies and the vibration response are obtained. The stability of the solution is studied by using the forward and pullback attractor ideas. Finally, the vibration of the system involving 3:1 internal resonance is investigated, and the stability boundaries of the trivial and non-trivial steady-state solutions are derived based on Lyapunov’s first method. It is shown that the main parameters of the MSD can provide 3:1 internal resonance condition.

Forced vibration of axially moving beam with internal resonance in the supercritical regime

Local and global resonances under the condition of 3:1 internal resonance of a super-critically axially moving beam, subjected to a harmonic exciting force, are investigated in the present work. The governing equation is derived from the generalized Hamilton's principle and discreted into a multiple-degrees-of-freedom system by the Galerkin's method. In the super-critical regime, the axially moving beam becomes a bistable system with two symmetrical non-trivial equilibrium configurations. Based on the transformation around one of them, natural frequencies and the condition of internal resonance are obtained. By employing the method of multiple scales, resonances for first-two modes and harmonics under the condition of internal resonance are discussed analytically. Total displacement at the middle of the beam is composed by them and confirmed by direct numerical method. Internal resonance is found to have a big effect on the phase angle of and the amplitude. Coupling ship between the first-two modes is verified to be produced by the cubic nonlinearity and the 3:1 commensurability together. The effect of moving speed acting on the internal resonance is discussed and an energy transmission region is found. Different with the internal resonance in the sub-critical regime, most of the transferred energy is absorbed by the quadratic nonlinearity in the super-critical regime. The critical excitation of the local response is predicted by the analytical method and certified by simulations. The global response for the primary resonance has two stable focal points. However, the global response for the secondary resonance only has one stable focal point for the non-trivial equilibrium configuration is counteracted.

Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber

A semi-active vibration absorption method is put forward to attenuate nonlinear vibration of a flexible arm based on the internal resonance. To maintain the 2:1 internal resonance condition and the desirable damping characteristic, a virtual vibration absorber is suggested. It is mathematically equivalent to a vibration absorber but its frequency and damping coefficients can be readily adjusted by simple control algorithms, thereby replacing those hard-to-implement mechanical designs. Through theoretical analyses and numerical simulations, it is proven that the internal resonance can be successfully established for the flexible arm, and the vibrational energy of flexible arm can be transferred to and dissipated by the virtual vibration absorber. Finally, experimental results are presented to validate the theoretical predictions. Since the proposed method absorbs rather than suppresses vibrational energy of the primary system, it is more convenient to reduce strong vibration than conventional active vibration suppression methods based on smart material actuators with limited energy output. Furthermore, since it aims to establish an internal vibrational energy transfer channel from the primary system to the vibration absorber rather than directly respond to external excitations, it is especially applicable for attenuating nonlinear vibration excited by unpredictable excitations.

Asymptotic approximation of the band structure for tetrachiral metamaterials

Tetrachiral materials are characterized by a cellular microstructure made by a periodic pattern of stiff rings and flexible ligaments. Their mechanical behaviour can be described by a planar lattice of rigid massive bodies and elastic massless beams. Therefore, the periodic cell dynamics is governed by a monoatomic structural model, conveniently reduced to the only active degrees-of-freedom. The paper presents an explicit parametric description of the Floquet-Bloch spectrum (or band structure) governing the propagation of elastic waves through the tetrachiral material. By virtue of multiparametric perturbation techniques, an analytical asymptotic approximation is achieved for the dispersion surfaces in the Brillouin zone. Since different optimization strategies tend to fail in opening low-frequency band gaps in the material spectrum, this specific design purpose is commonly pursued by introducing interring inertial resonators. The paper demonstrates that multiparametric perturbation methods can efficiently deal with the consequent enlargement of the parameter space, necessary to describe the resulting inertial metamaterial. Indeed, paying due attention to the doubling of internal resonance conditions, an accurate parametric approximations of the enriched band structure can be achieved. From the applicative perspective, the research findings furnish suited analytical tools for the optimal design of pass and stop bands.

Primary resonance of traveling viscoelastic beam under internal resonance

Under the 3:1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The undetermined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.

Nonlinear coupling of transverse modes of a fixed–fixed microbeam under direct and parametric excitation

Tuning of linear frequency and nonlinear frequency response of microelectromechanical systems is important in order to obtain high operating bandwidth. Linear frequency tuning can be achieved through various mechanisms such as heating and softening due to DC voltage. Nonlinear frequency response is influenced by nonlinear stiffness, quality factor and forcing. In this paper, we present the influence of nonlinear coupling in tuning the nonlinear frequency response of two transverse modes of a fixed–fixed microbeam under the influence of direct and parametric forces near and below the coupling regions. To do the analysis, we use nonlinear equation governing the motion along in-plane and out-of-plane directions. For a given DC and AC forcing, we obtain static and dynamic equations using the Galerkin’s method based on first-mode approximation under the two different resonant conditions. First, we consider one-to-one internal resonance condition in which the linear frequencies of two transverse modes show coupling. Second, we consider the case in which the linear frequencies of two transverse modes are uncoupled. To obtain the nonlinear frequency response under both the conditions, we solve the dynamic equation with the method of multiple scale (MMS). After validating the results obtained using MMS with the numerical simulation of modal equation, we discuss the influence of linear and nonlinear coupling on the frequency response of the in-plane and out-of-plane motion of fixed–fixed beam. We also analyzed the influence of quality factor on the frequency response of the beams near the coupling region. We found that the nonlinear response shows single curve near the coupling region with wider width for low value of quality factor, and it shows two different curves when the quality factor is high. Consequently, we can effectively tune the quality factor and forcing to obtain different types of coupled response of two modes of a fixed–fixed microbeam.

On Energy Transfer between Vibration Modes under Frequency-Varying Excitations for Energy Harvesting

This paper presents an analytical and numerical analysis of vibration energy harvesting from the dynamic interaction and energy transfer between the two vibration modes of a 2-D-O-F model of a flexible portal frame. The frequencies of these modes are set in a two-to-one internal resonance condition. Excitation is provided by eccentric rotating mass-motor captured in external resonance. Next, we consider the same flexible portal frame excited from various directions with time-variable frequency. A piezoelectric device is used for energy harvesting. Depending on how the piezoelectric energy harvester is installed (or coupled) different gains are obtained. Good performance of the harvester generator is detected. We also observed periodic, quasi-periodic or chaotic oscillations, depending on the saturation phenomenon.

Nonlinear vibration of rotating pre-deformed blade with thermal gradient

In this paper, the nonlinear dynamic behavior is investigated for a rotating pre-deformed pre-twisted blade subjected to harmonic gas pressure. The pre-deformation curve produced by thermal gradient is derived. A novel nonlinear vibration system is established by considering the influence of pre-deformation. The combination of the Lagrange principle and the assumed modes method is utilized to obtain the motion equations, and then the equations are transformed into a dimensionless form through introducing a set of dimensionless parameters. For the purpose of ensuring high precision in determination of the internal resonance condition, the equations of motion are discretized by adequate trial functions. An eigenvalue analysis is conducted on the corresponding linear system to obtain the natural frequencies and examine the possibility of the 2:1 internal resonance. The method of multiple scales is developed to solve the resulting multi-degree-of-freedom nonlinear ordinary differential equations. A numerical integration by means of Runge–Kutta is performed to establish the validity of the derived formulations. The evolution of frequency response curves with the rotating speed is observed. The influences of the thermal gradient, gas pressure and damping coefficient on the resonant dynamics of the system are investigated in detail. For the purpose of comparison, another distribution profile of pre-deformation, which is frequently used in previous literature, is also examined. It could be found that not only the amplitude but also the distribution of the initial deflection could influence the steady-state nonlinear response of the pre-deformed blade. A series of interesting nonlinear dynamic phenomena are discovered from the results.

On the efficacy of friction damping in the presence of nonlinear modal interactions

This work addresses friction-induced modal interactions in jointed structures, and their effects on the passive mitigation of vibrations by means of friction damping. Under the condition of (nearly) commensurable natural frequencies, the nonlinear character of friction can cause so-called nonlinear modal interactions. If harmonic forcing near the natural frequency of a specific mode is applied, for instance, another mode may be excited due to nonlinear energy transfer and thus contribute considerably to the vibration response. We investigate how this phenomenon affects the performance of friction damping. To this end, we study the steady-state, periodic forced vibrations of a system of two beams connected via a local mechanical friction joint. The system can be tuned to continuously adjust the ratio between the first two natural frequencies in the range around the 1:3 internal resonance, in order to trigger or suppress the emergence of modal interactions. Due to the re-distribution of the vibration energy, the vibration level can in fact be reduced in certain situations. However, in other situations, the multi-harmonic character of the vibration has detrimental effects on the effective damping provided by the friction joint. The resulting response level can be significantly larger than in the absence of modal interactions. Moreover, it is shown that the vibration behavior is highly sensitive in the neighborhood of internal resonances. It is thus concluded that the condition of internal resonance should be avoided in the design of friction-damped systems.

Second harmonic generation in composites: Theoretical and numerical analyses

Second harmonic generation in a transversely isotropic plate and a symmetric composite laminate is analyzed from a theoretical perspective. The strain energy function for a nonlinear elastic transversely isotropic material is expressed in terms of the five invariants of the Green-Lagrange strain tensor. Internal resonance conditions for the generation of cumulative second harmonics indicate that a cumulative second harmonic exists when the primary-secondary mode pair satisfies the phase matching and non-zero power flux criteria. In particular, for transversely isotropic plates, when the primary mode propagates along the material principal direction, only symmetric second harmonic Lamb-like wave modes can be cumulative. Also, when the primary wave propagates along other directions, only symmetric second harmonic modes can be generated. Additionally, for the case of symmetric composite laminates, only symmetric modes can be generated as cumulative second harmonics regardless of the propagation direction of the primary mode. To validate the above theoretical predictions, finite element simulations were conducted for mode pairs that are: (i) phase matched but have zero power flux, (ii) not phase matched but have non-zero power flux, and (iii) internally resonant i.e., satisfying both phase matching and non-zero power flux criterion. The results obtained from the simulations corroborate the theoretical findings for both transversely isotropic plates and symmetric composite laminates.

Kinematic and dynamic modeling and approximate analysis of a roller chain drive

A simple roller chain drive consisting of two sprockets connected by tight chain spans is investigated. First, a kinematic model is presented which include both spans and sprockets. An approach for calculating the chain wrapping length is presented, which also allows for the exact calculation of sprocket center positions for a given chain length. The kinematic analysis demonstrates that the total length of the chain wrapped around the sprockets generally varies during one tooth period. Analytical predictions for the wrapping length are compared to multibody simulation results and show very good agreement. It is thereby demonstrated that chain drives with tight chain spans must include compliant components to function. Second, a dynamic model is presented which includes the two spans and the driven sprocket. Assuming the presence of a stationary operating state, the presented dynamic model allows for analytical studies of the coupled motion of the chain spans and driven sprocket. Parametric excitation of the spans come from sprocket angular displacements, and the driven sprocket acts as a boundary which can be compliant in the axial direction. External transverse excitation of the spans comes from polygonal action, and is treated through kinematic forcing at the moving string boundaries. Perturbation analysis of the model is carried out using the method of multiple scales. Results show a multitude of internal and external resonance conditions, and some examples are presented of both decoupled and coupled motion. Together, the kinematic and dynamic model are aimed toward providing a framework for conducting and understanding both numerical, and experimental investigations of roller chain drive dynamics.

内部構造物を有する矩形容器内液面の非線形揺動特性

The nonlinear liquid motion in a partially filled rectangular tank with a core barrel is investigated. Basic equations are derived by employing the variational principle. Equations of motion governing nonlinear surface oscillations are derived by applying the Galerkin’s method to basic equations. Admissible functions are assumed to be represented by combining the modes of vibration. Because of complicated shape tank like a rectangular tank with a core barrel, the modes of vibration and the natural frequencies are not obtained by analytical method. Therefore, they are calculated by using the finite elements method. Occurrence condition of internal resonance were considered. In the case of a complicated shape tank, since the nonlinear coupling between the modes become vague, it is difficult to predict the internal resonant modes. By considering the coefficients in the derived nonlinear equations of motion of nonlinear coupling modes, an index to predict the mode of which amplitude may become large by internal resonance is proposed. Experiments were carried out at the condition when the nonlinear liquid sloshing occurs. The occurrence of the internal resonance was confirmed by the experimental results. It is shown that the amplitude of the predicted internal resonant mode become large in both analytical and experimental results. Effectiveness of the analysis and validity of the proposed index are verified.

Study on galloping behavior of iced eight bundle conductor transmission lines

Wind tunnel test was carried out to obtain the aerodynamic coefficients of an eight bundle conductor accreted with crescent-shaped ice. A user-defined cable element with torsional degree of freedom is developed in ABAQUS software to capture the torsional deformation of the iced conductors during galloping. By means of the user-defined cable element, different damping ratios in in-plane, out-of-plane and torsional directions of the conductors can be defined and the aerodynamic forces varying with their motion status can be exerted on the conductors conveniently when ABAQUS is used to simulate galloping of transmission lines. A wind tunnel test to model galloping of an iced eight bundle conductor segment was carried out, and the validity of the numerical simulation method is demonstrated by the agreement of the galloping orbit of the bundle conductor segment model recorded in the test and that by the numerical simulation. Furthermore, galloping behavior, including dynamic responses, galloping orbits, frequencies, vibration modes and amplitudes, of typical iced eight bundle conductor transmission lines in the cases of different span lengths, initial tensions in sub-conductors, wind velocities, angles of wind attack and damping ratios is studied, and the galloping behavior of the lines with internal resonance conditions is discussed. The obtained results may provide a fundamental tool for the development of anti-galloping techniques of eight bundle conductor transmission lines.

Internal resonance of nonlinear sloshing in rectangular liquid tanks subjected to obliquely horizontal excitation

Nonlinear sloshing in rectangular tanks subjected to obliquely horizontal, harmonic excitation is investigated when the internal resonance condition 1:1 is satisfied between the natural frequencies of predominate modes (1, 0) and (0, 2). Galerkin’s method is employed to derive the nonlinear modal equations of motion for sloshing, considering nine sloshing modes. Then, van der Pol’s method is applied in order to obtain the expressions of the frequency response curves for amplitudes and phase angles of the predominate modes. The frequency response curves are calculated and reveal that (0, 2) mode may occur even though it is not directly excited because it is nonlinearly coupled with (1, 0) mode due to the autoparametric terms. In the numerical simulations, it is found that planar motions of (1, 0) mode, clockwise and counter-clockwise swirl motions, and translational motions may appear. Furthermore, Hopf bifurcation occurs, and amplitude modulated motions (AMMs), including chaotic motions, may appear depending on the value of the excitation frequency. Three-dimensional distribution charts of the maximum liquid surface elevation are calculated to show the risk of liquid overspill. The influence of the difference between the horizontal excitation direction and the tank side on the frequency response curves is also examined. Bifurcation sets are calculated to clarify this influence. Experimental data confirmed the validity of the theoretical results.

Non-linear interactions in the flexible multi-body dynamics of cable-supported bridge cross-sections

An elastic section model is proposed to analyze some characteristic issues of the cable-supported bridge dynamics through an equivalent planar multi-body system. The quadratic non-linearities of the four-degree-of-freedom model essentially describe the geometric coupling which may strongly characterize the dynamic interactions of the bridge deck and a pair of identical suspension cables (hangers or stays). The linear modal solution shows that the flexural and torsional modes of the deck (global modes) typically co-exist with symmetric or anti-symmetric modes of the cables (local modes). The combinations of parameters which realize remarkable 2:1:1 internal resonance conditions among one of the global modes (with higher natural frequency) and two local modes (with lower and close natural frequencies) are obtained by virtue of a multiparameter perturbation method. The non-linear response of the resonant systems shows that the global deck motion – directly forced at primary resonance by an external harmonic load – can parametrically excite the local cable motion, when the deck vibration amplitude overcomes the critical value at which a period-doubling bifurcation occurs. The relevant effects of both viscous damping and internal detuning on the instability boundaries are parametrically investigated. All the internal resonance conditions as well as the critical vibration amplitudes are expressed as an explicit, though asymptotically approximate, function of the structural parameters.

Application of lumped-mass vibration absorber on the vibration reduction of a nonlinear beam-spring-mass system with internal resonances

The objective of this study was to optimize the damping effects of a lumped-mass vibration absorber (LMVA) attached to a hinged–hinged nonlinear beam held on a nonlinear elastic foundation. The LMVA was located at various points along the beam and supported by a spring from beneath. Analysis was performed on the internal resonance conditions with the aim of eliminating internal resonance and reducing the amplitude of vibrations in the beam by altering the location and mass of the LMVA as well as the spring constant. We employed the method of multiple scales (MOMS) for the analysis of frequency response in various modes. Fixed points plots were constructed and 3D maximum amplitude contour plots (3D MACPs) were compiled to identify the LMVA combinations with the optimal damping effects. The best damping results in the 3rd mode were achieved when the LMVA was placed between 1/4l and 1/2l in combinations of M^ (mass of LMVA/mass of beam) and K˜ (LMVA spring constant/elastic foundation spring constant) followed a linear relationship of K˜=10M^.

Nonlinear energy sink to control elastic strings: the internal resonance case

The use of nonlinear energy sink as a passive control device is extended here to a nonlinear elastic string, in internal resonance conditions, excited by an external harmonic force. The Multiple Scale/Harmonic Balance Method is directly applied to the partial differential equations ruling the dynamics of the system. The internal resonance condition of the string involves a rich response containing essentially both the resonant, directly excited, mode and a superharmonic one. Numerical results on a case study are presented.

Damping Effect of Pendulum Tuned Mass Damper on Vibration of Two-Dimensional Rigid Body

This study investigated the effect of a pendulum tuned mass damper (PTMD) on the vibration of a slender two-dimensional (2D) rigid body with 1:2 internal resonance. Focus is placed on the damping effect of various parameters of the PTMD on preventing the internal resonance of the system. The instruments used include fixed points plots, time response and Poincaré maps, which were compared for confirmation of accuracy. The Lagrange's equation is employed to derive the equations of motion for the system. The method of multiple scales (MOMS) is applied to analyzing this nonlinear vibration model. The internal resonance conditions of the rigid body in vibration are obtained by the eigen-analysis. Moreover, a 3D internal resonance contour plot (3D-IRCP) aided by various amplitude analysis tables is proposed for identification of the parameter combinations of the PTMD for preventing internal resonance. This approach enables the designers to evaluate the effectiveness of various parameter combinations of the PTMD prior to the design process. The present study indicates that without changing the main configuration, the vibration amplitudes in the main body can be greatly reduced under certain parameter combinations of the PTMD.

Asymptotic model-based estimation of a wake oscillator for a tethered sphere in uniform flow

We derive and investigate the nonlinear dynamics of a tethered sphere in uniform flow. A Lagrangian based model augmented by wake oscillators for the streamwise vertical and transverse directions enables model-based estimation of both structural and aeroelastic parameters via asymptotic analysis of internal resonance conditions between the transverse wake frequency and its structural counterpart. Validation of the proposed methodology is demonstrated by comparison of results with those of a benchmark experiment for a light sphere in water conducted by Govardhan and Williamson (1997) and by analysis of a spherical aerostat experiment conducted by Coulomb-Pontbriand and Nahon (2009). A numerical bifurcation analysis of the validated system reveals existence of possible quasiperiodic and non-stationary solutions that are consistent with documented instabilities of aerostats in severe environmental conditions and shed light on control mechanisms required for suppression of finite amplitude limit-cycle oscillations.