Effects Of Internal Resonances Research Articles

A continuum model for the tensegrity Maxwell chain

<abstract><p>A recent study has presented a Maxwell mass–spring model for a chain formed by two different types of tensegrity prisms alternating with lumped masses. Such a model shows tensegrity theta prisms arranged in parallel with minimal regular prisms acting as resonant substructures. It features a tunable frequency bandgap response, due to the possibility of adjusting the width of the bandgap regions by playing with internal resonance effects in addition to mass and spring contrasts. This paper expands such research by presenting a continuum modeling of the tensegrity Maxwell chain, which is useful to conduct analytic studies and to develop finite element models of the plane wave dynamics of the investigated system. In correspondence to the high wave-length limit, i.e., in the low wave number regime, it is shown that the dispersion relations of the discrete and continuum models provide similar results. Analytic solutions to the wave dynamics of physical systems are presented, which validate the predictions of the bandgap response offered by the dispersion relation of the continuum model.</p></abstract>

Effects of internal resonance on the nonlinear acoustic radiation induced by axial and bending vibration of hyperelastic cantilever beams

This paper provides a numerical nonlinear vibro-acoustic study of a slender hyperelastic cantilever beam with 2:1 internal resonance condition. The model of beam includes geometric and material nonlinearities, while the model of acoustics includes the Arbitrary Lagrangian-Eulerian frame and the perfectly matched layers. The finite element method and the time integration method are used to discretize the structure and acoustic field. A loosely coupled serially staggered procedure is used to compute the variable values of structure and acoustic field. The results show that the transition from the axial mode to the bending mode of beam can be excited by the axial mode excitation. This leads to the alteration of acoustic directivity pattern and frequency components. Besides, due to the weak or strong coupling effects of axial and bending modes of beam, the symmetric or asymmetric acoustic directivity patterns can be witnessed in different frequency components.

Qualitative effect of internal resonance on the dynamics of two-dimensional resonator

Nonlinear modal interactions and associated internal resonance phenomena have recently been used to demonstrate improved oscillator performance and enhanced sensing capabilities. Here, we show tunable modal interaction in a molybdenum disulfide (MoS2) resonator. We achieve the tunability of coupling between these initially uncoupled modes by using electrostatic gate voltages. This tunable coupling enables us to make the modes commensurate and observe energy exchange between the modes. We attribute the strong energy exchange between the vibrational modes to 1:2 internal resonance. This interaction strongly impacts the dynamics of the modal response of such resonators. We observe peak splitting, a signature of energy exchange between the modes even when the modal response is in the linear regime. We model our device to explain the observed effect of excitation, detuning of modal frequencies, and intermodal coupling strength on the resonator dynamics. MoS2 resonators explored in this work are ideal for understanding the rich dynamics offered through the intermodal coupling.

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

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

Internal resonance of structure with hysteretic base-isolation and its application for seismic mitigation

The internal resonances of a five-storey concrete frame structure with hysteretic type base-isolation under harmonic excitation are studied. The participation of different vibration modes in the non-linear responses is systematically studied taking advantages of the extended incremental harmonic balance (EIHB) method. The energy transfer between different modes is clearly noted when internal resonance is triggered. The energy transfer from lower mode to higher mode or in the reverse direction may depend on the configuration of excitations and the system parameters. The strain energy and floor acceleration in the superstructure is proposed as indices to evaluate the effect of internal resonance. Results show that when energy is channeled from higher mode to lower mode in resonance, the structure may suffer larger deformation with more strain energy, but lower acceleration. The favorable internal resonance of the structure can be used for the protection of structural safety or the serviceability requirement of human comfortness by proper design of the structure.

Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams

Although the transverse vibration of the cantilever structure is of great interest, the internal resonance of the vertical cantilever structure is always ignored. In this paper, nonlinear free vibration and 1/3 super-harmonic resonance of a hanging cantilever beam are firstly presented with 3:1 internal resonance caused by gravity. By employing the temporal multi-scale method, mode responses in these two vibrations are obtained. The harmonic balanced method is used for solving the excitation component and the whole deflection in super-harmonic resonance. For nonlinear free vibration, beat phenomenon is found and the effect of mode interaction is determined by gravity, damping and initial perturbation. For super-harmonic resonance, the amplitude of the first two modes can be quite large and exceed the excitation component if damping is not strong. Besides, the softening-type frequency curves are predominated by the inertia nonlinearity. Compared with the excitation component and the whole deflection, the mode responses are more easily affected by gravity. If the value of gravity parameter is a little bit lower than the one in the condition of strict internal resonance, the energy transmission, multiple solutions and bifurcations and three complex types of frequency curves can be found in mode responses. Hysteresis and saturation phenomena in mode responses can be discovered as well. Results from analytical methods are almost identical to those from numerical ways. In summary, the internal resonance of slender hanging cantilever structures should be aroused more attention, for hanging cantilevers are common in practical engineering and considerable mode responses can be induced by large initial perturbation in weakly damped free vibration or by low excitation frequency and large excitation amplitude in super-harmonic resonance. In addition, the complex dynamics can be worthwhile to judge the occurrence of internal resonance and its impact on structures. Since gravity has tensile effects on hanging cantilevers, these phenomena may also occur in beams with axial tension.

Internal resonance in parametric vibrations of axially accelerating viscoelastic plates

In this paper, instability boundaries of axially accelerating plates with internal resonance are investigated for the first time. The relation between the acceleration and the longitudinally varying tensions are introduced. The governing equation and the corresponding boundary conditions are derived from the generalized Hamilton principle. The effects of internal resonances and the nonhomogeneous boundary conditions on the instability boundaries are highlighted. By the method of multiple scales, the modified solvability conditions in principal parametric and internal resonances are established. The Routh-Hurwitz criterion is introduced to determine the instability boundaries. The effects of the viscoelastic coefficient and the viscous damping coefficient on the instability boundaries are examined. Abnormal instability boundaries are detected when the internal resonance is introduced. The phenomenon of local zigzag and V-shape boundaries are explained from the viewpoint of modal interactions. The numerical calculations of the differential quadrature schemes about the first four complex frequencies, the first four complex modes, and the stability boundaries are used to confirm the results of the analytical method.

Effects of internal resonances in the pitch glide of Chinese gongs.

The framework of nonlinear normal modes gives a remarkable insight into the dynamics of nonlinear vibratory systems exhibiting distributed nonlinearities. In the case of Chinese opera gongs, geometrical nonlinearities lead to a pitch glide of several vibration modes in playing situation. This study investigates the relationship between the nonlinear normal modes formalism and the ascendant pitch glide of the fundamental mode of a xiaoluo gong. In particular, the limits of a single nonlinear mode modeling for describing the pitch glide in playing situation are examined. For this purpose, the amplitude-frequency relationship (backbone curve) and the frequency-time dependency (pitch glide) of the fundamental nonlinear mode is measured with two excitation types, in free vibration regime: first, only the fundamental nonlinear mode is excited by an experimental appropriation method resorting to a phase-locked loop; second, all the nonlinear modes of the instrument are excited with a mallet impact (playing situation). The results show that a single nonlinear mode modeling fails at describing the pitch glide of the instrument when played because of the presence of 1:2 internal resonances implying the nonlinear fundamental mode and other nonlinear modes. Simulations of two nonlinear modes in 1:2 internal resonance confirm qualitatively the experimental results.

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement.

Internal resonance in forced vibration of coupled cantilevers subjected to magnetic interaction

Forced vibration is investigated for two elastically connected cantilevers, under harmonic base excitation. One of the cantilevers is with a tip magnet repelled by a magnet fixed on the base. The cantilevers are uniform viscoelastic beams constituted by the Kelvin model. The system is formulated as a set of two linear partial differential equations with nonlinear boundary conditions. The method of multiple scales is developed to analyze the effects of internal resonances on the steady-state responses to external excitations in the nonlinear boundary problem of the partial differential equations. In the presence of 2:1 internal resonance, both the first and the second primary resonances are examined in detail. The analytical frequency–amplitude response relationships are derived from the solvability conditions. It is found that the frequency–amplitude response curves reveal typical nonlinear phenomena such as jumping and hysteresis in both primary resonances as well as saturation in the second primary resonance. The frequency–amplitude response curves may be converted from hardening-type single-jumping to double-jumpings, and further to softening-type single-jumping by adjusting the distance between two magnets. It is also found that the unstable parts of the frequency–amplitude response curves correspond to quasi-periodic motions. The finite difference scheme is proposed to discretize both the temporal and the spatial variables, and thus the numerical solutions can be calculated. The analytical results are supported by the numerical solutions.

Nonlinear vibrations and mode interactions for a continuous rod with microstructure

Natural longitudinal vibrations of a periodically heterogeneous rod embedded into a nonlinear elastic matrix are considered. The governing macroscopic dynamical equation was obtained by the higher-order asymptotic homogenization method. An asymptotic solution is developed by the method of multiple time scales. The effects of internal resonances and modes coupling are predicted. The specific objective of the paper is to analyse how the presence of the microstructure influences on the processes of mode interactions. It is shown that depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell different scenarios of the modes coupling can be realized.

Scattering and absorption of a complex source point beam by a grounded lossy dielectric slab with a superstrate

Considered is the scattering and absorption of an E-polarized two-dimensional (2D) beam generated by a complex-source-point (CSP) feed in the presence of a lossy grounded dielectric slab with a lossless superstrate. The solution is found analytically in the Fourier-transform domain. Numerical results demonstrate the ways of obtaining enhanced absorption in the slab by a proper selection of the slab electrical thickness and the parameters of the superstrate. The effects of internal resonances in the slab are analyzed and discussed.

Analysis of Axial Vibration in the Laminated Rubber-Metal Spring

The laminated rubber-metal spring is well known in application as the vibration isolator for earthquake protection. The spring is therefore designed to be able to sustain the vibration waves from horizontal direction. This paper discusses the possibility of the laminated spring to be employed for other applications where the excitation mainly comes from axial direction, such as to isolate vibration transmission from heavy engine. The model is first developed for a simple finite rod to simulate the effect of internal resonances at high frequency when the wavelength is much smaller than the length of the rod. The effect of metal plates inserted in the rubber is then modelled using the lumped parameter system. The results are presented in terms of the vibration transmissibility.

Study on ringing effect in ultrasonic transducer

This paper presents theoretical experimental and evidence of the effect of internal resonance at silicon substrate in micro fabrication ultrasonic transducer. This ringing is clearly observed in immersion transducers with 4.2 MHz and harmonics. A mathematical analytical model of the attenuation yield by ringing effective and simulation is introduced. Experimental results are further compared to simulations carried out in time-domain and qualitative agreement is provided. This paper summarizes the theoretical measured, performance backing layer. Experiment and simulation proved that it is possible to design a custom backing material with acoustic impedance that decreases the ringing mode. Key words: Capacitive micro-machined ultrasonic transducers (CMUT), ultrasound sensor, backing layer, attenuation.

Nonlinear Analysis of Forced Responses of an Axially Moving Beam by Incremental Harmonic Balance Method

This article analyzes nonlinear vibration of an axially moving beam subject to periodic lateral forces. The governing equation of the beam vibration is developed using Hamilton's Principle. The stable and unstable periodic solutions are obtained by employing the multivariable Floquet theory and incremental harmonic balance (IHB) method. In the solution procedure, Hsu's method is applied for computing the transition matrix at the end of one period. The effects of internal resonance on the beam responses are discussed. The periodic solutions obtained from the IHB method are in good agreement with the results obtained from numerical integration.

Saturation in asymmetric structures under internal resonance

In this paper, a novel model for a single-storey irregular-plan building, subjected to earthquake and harmonic translational excitation, is presented to consider non-linear inertial coupling terms, which are ignored in conventional linear models. Furthermore, the effect of internal resonance is investigated. The governing equations of motion for a model of this type of structure are achieved in a non-inertial rotational system of reference, attached to the centre of mass of the rigid diaphragm of its floor. Non-linearities are caused by non-linear inertial coupled lateral and torsional responses. Using the method of multiple scales, the non-linear equations are reduced to approximate linear equations, and subsequently the dynamic stability of the steady-state responses near a two-to-one ratio of internal resonant conditions is studied. The difference between the responses of the proposed non-linear model and the conventional linear one is shown in time history and frequency domain analyses. By inspecting the effects of amplitude and frequency detuning parameters in the responses, some non-linear phenomena, such as saturation, hysteretic and jumping, are observed.

Dynamic Characteristics of Counter-current Flow Processes

In industry counter-current flow processes are common. Although these processes have been widely studied in literature, relatively little has been published on their dynamic behaviour. Two very common counter-current flow processes are heat exchangers and distillation columns. Ma's study based on dynamic models of heat exchanger's dynamic behaviour [1] reports an internal resonance effect, also earlier reported by Profos in 1943 [4] and Friedly in 1972 [3]. Here the study is extended to lumped models, first for heat exchangers and thereafter for an extremely simplified distillation column. Not unexpectedly, the dynamic properties change gradually as the number of lumps increases towards the distributed systems and for high frequencies similar internal resonance effects evolve with the envelopes showing a very low-order behaviour, which though somewhat surprisingly is independent of the number of lumps. Finally we show that the eigenvalues of the normed system matrix lie on a circle in the complex plane.

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.

Recent Advances in Liquid Sloshing Dynamics

A liquid free surface in partially filled containers can experience a wide spectrum of motions such as planar, non-planar, rotational, quasi-periodic, chaotic, and disintegration. Civil engineers and seismologists have been studying liquid sloshing effects on large dams, oil tanks and elevated water towers under ground motion. Since the early 1960’s, the problem of liquid sloshing dynamics has been of major concern to aerospace engineers studying the influence of liquid propellant sloshing on the flight performance of jet vehicles. Since then, new areas of research activities have emerged. The modern theory of nonlinear dynamics has indeed promoted further studies and uncovered complex nonlinear phenomena. These include rotary sloshing, Faraday waves, nonlinear liquid sloshing interaction with elastic structures, internal resonance effects, stochastic sloshing dynamics, hydrodynamic sloshing impact dynamics, g-jitter under microgravity field, cross-waves, and spatial resonance. The dynamic stability of liquid gas tankers and ship cargo tankers, and liquid hydrodynamic impact loading are problems of current interest to the designers of such systems. This article will address the means of passive control of liquid sloshing and the use of liquid sloshing forces to control vibratory structures. Other important contributions include the development of digital computer codes to solve complex problems that were difficult to handle in the past. The purpose of this article is to review the research work developed in different applications. It will highlight the major achievements and results reported in the literature. Some early work will be cited very briefly in order to provide an updated bibliography of liquid sloshing dynamics. This review article contains 1,319 references.

Vibrations of an Asymmetrical Shaft with Gravity and Nonlinear Spring Characteristics. Effect of Internal Resonance.

Vibrations of an Asymmetrical Shaft with Gravity and Nonlinear Spring Characteristics. Effect of Internal Resonance.