Multiple Frequency Excitations Research Articles

Nonlinear vibration and stability analysis of a dual-disk rotor-bearing system under multiple frequency excitations

Nonlinear vibration and stability analysis of a dual-disk rotor-bearing system under multiple frequency excitations

Route to mixed-mode oscillations via step-shaped sharp transition of equilibria in a nonlinear gyroscope oscillator

Sharp transitions in relation to the variation of system parameters are frequently encountered in many multiple-timescale systems, and they have been found to be an important factor related to the generation of mixed-mode oscillations (MMOs). The present paper aims to report a novel type of sharp transition, referred to as step-shaped sharp transition, in a nonlinear gyroscope oscillator with multiple-frequency excitations, and investigate the resulting MMOs. We show that step-shaped sharp quantitative changes in relation to the variation of system parameters can be observed in the equilibrium branch, which yields the step-shaped sharp transition. In particular, with the increase of the frequency ratio between the parametric and external excitations, more step-shaped sharp transitions appear in the equilibrium branches, which evolve into the ones displaying different structures. Based on this, the rectangular-pulse-shaped explosion of equilibria is created. Furthermore, these sharp transitions can form active areas for the MMOs, leading to the alternations between large-amplitude and small-amplitude oscillations, and finally the route to MMOs is created. Our findings have significant implications for understanding the fast-slow dynamics of the nonlinear gyroscope oscillator, contributing to the exploration of new routes to MMOs. Thus, the results could provide theoretical support for the potential application of gyroscopes.

Analysis of Dynamic Characteristics of Attached High Rise Risers

The exhaust chimney of third-generation nuclear power units is a typical attached high-rise riser structure. In this paper, the simplified mechanical model and dynamic model of China’s third-generation nuclear power Hualong-1 VNA system, including multiple nonlinear factors, are established for the first time. The DTM (differential transformation method) was first applied to solve the natural vibration characteristics of a multi-point constrained variable cross-section riser structure, and the effects of variable cross-section, variable mass, variable axial force, and different elastic constraint parameters on the natural vibration characteristics of the system were studied. The dynamic behavior of the VNA system under the combined action of internal flow velocity, vortex excitation, and foundation excitation was studied. The results show that the outer diameter function of the VNA system pipeline should be designed as a quadratic function or a near quadratic multi-segment constant value function. The “limiting” effect of constraining large stiffness can force low-order vibration modes with high constraint stiffness to jump to high-order vibration modes with low constraint stiffness. The elastic constraint arrangement scheme with near center symmetry can make the system vibration mode present a half stable and half-curved form. A new optimization design scheme has been proposed regarding the layout and stiffness parameters of the VNA system guide bracket. This can enable the VNA system pipeline to avoid severe oscillations near the response extreme values caused by multiple frequency excitations of seismic loads under design and accident conditions and ensure the service life of the equipment.

Determination of all hydrodynamic added properties resulting from fluid structure interactions using singular value decomposition

The hydrodynamic added effects such as inertia, damping and stiffness resulting from a fluid flow surrounding a structure strongly affect its dynamic response. Thus, accurate determination of such effects is of prime importance in the design of modern mechanical systems exposing fluid–structure interactions. The main goal of this study is to develop a unified methodology for the simultaneous identification of all hydrodynamic added effects caused by fluid–structure interactions. The proposed model is based on the least squares solution of an over-determined system of equations using singular value decomposition. The singular value decomposition determine the added inertia and stiffness without any assumption on their respective magnitude and variation, necessary for other methods. The method is firstly validated using the analytical solutions for single degree of freedom free vibration and forced oscillations under single and multiple frequency excitations for a variety of cases. It is found that the proposed method can predict all presumed added properties for transient forced oscillations under single and multiple external forces. Next, the method is applied to two challenging fluid structure interaction test studies; an oscillating hydrofoil in a turbulent flow and a Kaplan turbine runner subjected to single and multiple frequency perturbations. In both cases, the results of the proposed method are found to be consistent with results of methods in the literature and confirmed the underlying hypothesis of their analysis. The results confirm the ability of the newly developed methodology in evaluation all fluid added effects in complex engineering turbulent flows subject to one to several excitations, simultaneously.

A new route to pulse-shaped explosion of limit cycles and its induced amplitude-modulated bursting

This paper aims to report a new route to pulse-shaped explosion (PSE) related to the limit cycle attractor in a mechanical system with multiple-frequency excitations. We show that the compound bursting connected by PSE of equilibria can be observed in this system. Then, with the decrease of the amplitude of parametric excitation, the compound bursting pattern begins to change to other types, accompanied by the generation of PSE of limit cycles. Typically, as a novel sharp transition behavior underlying the occurrence of bursting oscillations, the PSE can be created by the disappearance of critical escape transitions. In present study, based on the two-parameter bifurcation analysis, we find that the PSE can also be created in the branch of limit cycles with the disappearance of fold points of homoclinic bifurcation curves, which is quite different from the ones reported in previous works and can be regarded as a new route to the PSE of limit cycles. Based on this, the compound bursting induced by the PSE of limit cycles is investigated. Besides, it should be pointed out that the limit cycle exhibiting PSE can be regarded as an amplitude-modulated limit cycle attractor, and thus we obtain a newly released bursting rhythm, namely the amplitude-modulated bursting, based on the PSE of limit cycles. These findings have significant implications for the understanding of PSE and the diversity of bursting dynamics induced by it.

Experimental and numerical study of nonlinear modal characteristics of Faraday waves

Faraday wave means the standing wave produced in a vertically-excited tank. The wave characteristics and trigger conditions of various modes especially the high-order modes are rarely reported. In this paper, an in-house-code NEWTANK and a hexapod motion sloshing test platform were employed to study mode 1 to mode 5 Faraday waves. The two-dimensional (2D) nonlinear mode 1 and mode 2 Faraday waves were obtained through both experiments and simulations. Besides, the 2D strongly breaking mode 2, 2D steep mode 3 and fully three-dimensional (3D) mode 4 Faraday waves were investigated numerically. The free surface displacement and snapshots of Faraday waves in connection with the non-linearity and breaking phenomenon were presented and analyzed. The fast Fourier transform (FFT) technique was adopted to identify the frequency response. The transfer of transient nonlinear energy was also identified by using the Wavelet analysis technique. Finally, a fully 3D multiple modal Faraday wave was triggered under multiple frequency excitations.

Analysis of multiple frequency excitations of a cracked beam using nonlinear vibrations

In this study, under harmonic multiple frequency excitations, the dynamic response of a cracked cantilever beam is investigated. The breathing crack model is assumed to show the nonlinear behaviour of a transverse crack. The first mode of vibration and the single degree freedom lumped system is considered to simplify the case study. Because of applying the multiple frequency excitations, the analysis is applied in a combinational resonance. Multiple time scales method is employed to solve the motion equation of the crack, and the nonlinear vibrational responses are obtained. Then, by changing the crack parameters and frequency of the excitations, the different dynamic responses of the crack are demonstrated. The proposed model shows that the crack parameters analysis in nonlinear vibration of multiple excitations could be an appropriate method to recognise the crack and the depth of the damage. Results indicate that the beam analysis under multiple frequency excitations is more sensitive than the single frequency excitation to illustrate the impacts of the crack parameters on its vibrational nonlinearity responses.

A Time Variational Method for the Approximate Solution of Nonlinear Systems Undergoing Multiple-Frequency Excitations

Abstract The main objective of this paper is to use the time variational method (TVM) for the nonlinear response analysis of mechanical systems subjected to multiple-frequency excitations. The system response, which is composed of fractional multiples of frequencies, is expressed in terms of a fundamental frequency that is the greatest common divisor of the approximated frequency components. Unlike the multiharmonic balance method (MHBM), the formulation of the proposed method is very simple in analyzing the systems with more than two excitation frequencies. In addition, the proposed method avoids the alternate transformation between frequency and time domains during the calculation of the nonlinear force and the Jacobian matrix. In this work, the performance of the proposed method is compared with that of numerical integration and the MHBM using three nonlinear mechanical models undergoing multiple-frequency excitations. It is observed that the proposed method produces approximate results during the quasi-periodic response analysis since the formulation includes an approximation of the incommensurate frequencies to commensurate ones. However, the approximation error is very small and the method reduces a significant amount of computational efforts compared to the other methods. In addition, the TVM is a recommended option when the number of state variables involved in the nonlinear function is high as it calculates the nonlinear force vector and the Jacobian matrix directly from the displacement vector. Moreover, the proposed method is far much faster than numerical integration in capturing the steady-state, quasi-periodic responses of the nonlinear mechanical systems.

Non-smooth bursting analysis of a Filippov-type system with multiple-frequency excitations

The main purpose of this paper is to explore the patterns of the bursting oscillations and the non-smooth dynamical behaviours in a Filippov-type system which possesses parametric and external periodic excitations. We take a coupled system consisting of Duffing and Van der Pol oscillators as an example. Owing to the existence of an order gap between the exciting frequency and the natural one, we can regard a single periodic excitation as a slow-varying parameter, and the other periodic excitations can be transformed as functions of the slow-varying parameter when the exciting frequency is far less than the natural one. By analysing the subsystems, we derive equilibrium branches and related bifurcations with the variation of the slow-varying parameter. Even though the equilibrium branches with two different frequencies of the parametric excitation have a similar structure, the tortuousness of the equilibrium branches is diverse, and the number of extreme points is changed from 6 to 10. Overlying the equilibrium branches with the transformed phase portrait and employing the evolutionary process of the limit cycle induced by the Hopf bifurcation, the critical conditions of the homoclinic bifurcation and multisliding bifurcation are derived. Numerical simulation verifies the results well.

Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations

The purpose of this paper is to report a novel route to mixed-mode oscillations (MMOs), i.e., the mechanism which we call speed escape of attractors, based on a Rayleigh equation with multiple-frequency excitations. The fast subsystem exhibits two critical points, which bound the region of a periodic attractor or an equilibrium point attractor and outside which are divergent regions. We show that both the periodic attractor and equilibrium point attractor can rapidly move far away from the original place when the control parameter reaches the critical values. This helps us reveal the novel mechanism to MMOs, and two different types of MMOs, i.e., MMOs of point–point type and MMOs of cycle–cycle type, are thus obtained. Besides, the effects of excitations on the MMOs are explored. We show that the amplitudes and frequencies of excitations may have important influences on MMOs. Our results enrich the possible routes to MMOs as well as the underlying mechanisms of MMOs.

Synthesis of Experimental and Theoretical Analysis of Pneumatic Hammer Instability in an Aerostatic Bearing

The present work is focused on the pneumatic hammer instability in an aerostatic bearing with shallow recesses and orifices of four different diameters. Operating conditions were zero rotation speed, zero load, and different supply pressures. The diameters of the tested orifices were large compared to the usual practice and correspond to a combined inherent and orifice restriction. The theoretical analysis was based on the computational fluid dynamics (CFD) evaluation of the ratio between the recess and the feeding pressure and on the “bulk flow” calculation of the rotordynamic coefficients of the aerostatic bearing. Calculations showed an increase of the direct stiffness with decreasing the orifice diameter and increasing the supply pressure and, on the other hand, a decrease toward negative values of the direct damping with decreasing the orifice diameter. These negative values of the direct damping coefficient indicate pneumatic hammer instabilities. In parallel, experiments were performed on a floating bearing test rig. Direct stiffness and damping coefficients were identified from multiple frequency excitations applied by a single shaker. Experiments were performed only for the three largest orifices and confirmed the decrease of the direct damping with the orifice diameter and the supply pressure. The identification of the rotordynamic coefficients was not possible for the smallest available orifice because the aerostatic bearing showed self-sustained vibrations for all feeding pressures. These self-sustained vibrations are considered the signature of the pneumatic hammer instability. The paper demonstrates that aerostatic bearings with shallow recesses and free of pneumatic hammer instabilities can be designed by adopting orifice restrictors of large size diameter.

Global resonance optimization analysis of nonlinear mechanical systems: Application to the uncertainty quantification problems in rotor dynamics

An efficient method to obtain the worst quasi-periodic vibration response of nonlinear dynamical systems with uncertainties is presented. Based on the multi-dimensional harmonic balance method, a constrained, nonlinear optimization problem with the nonlinear equality constraints is derived. The MultiStart optimization algorithm is then used to optimize the vibration response within the specified range of physical parameters. In order to illustrate the efficiency and ability of the proposed method, several numerical examples are illustrated. The proposed method is then applied to a rotor system with multiple frequency excitations (unbalance and support) under several physical parameters uncertainties. Numerical examples show that the proposed approach is valid and effective for analyzing strongly nonlinear vibration problems with different types of nonlinearities in the presence of uncertainties.

Experimental investigation of a vibration isolation system using negative stiffness structure

The purpose of this study is to investigate and propose a vibration isolation system with a negative stiffness structure (NSS) for driver seats in low frequency vibration conditions. In this paper, the nonlinear stiffness characteristic of the system is presented. The relationship between the configurative parameters and the stiffness of the elastic element, for which the dimensionless nonlinear stiffness of the proposed system ranges from zero to one and the range of the displacement of the isolation equipment (mass) is the maximum without exceeding a desired value of stiffness, will be obtained. An experimental apparatus is subsequently configured to examine the characteristic of the vibration transmissibility of the system according to various values of the configurative parameters. Next, the time responses to the multiple frequency and random base excitations are also investigated. In addition, the dynamic responses of the system with and without the negative stiffness structure are considered. The results show that the proposed system has a wide frequency range of isolation, especially since the resonance phenomenon almost does not occur.

Identification and Prediction of Force Coefficients in a Five-Pad and Four-Pad Tilting Pad Bearing for Load-on-Pad and Load-Between-Pad Configurations

This paper presents the identification of the rotordynamic force coefficients for direct lubrication five-pad and four-pad tilting pad bearings. The bearing is 110 mm in diameter with a L/D of 0.4 pad axial length (44 mm). The experiments include load-on-pad and load-between-pad configurations, with 0.5 and 0.6 pivot offsets, for rotor speeds ranging from 7500 rpm to 15,000 rpm. The bearing force coefficients are identified from multiple frequency excitations (20–300 Hz) exerted on the bearing housing by a pair of hydraulic shakers and are presented as a function of the excitation frequency and rotor speed for a 300 kPa unit load. The experimental results also include temperatures at the trailing edge of three pads. The direct force coefficients, identified from curve-fits of the complex dynamic stiffness, are frequency independent if considering an added mass term much smaller than the test device modal mass. The force coefficients from the four-pad bearing load-between-pad configuration show similar coefficients in the loaded and orthogonal directions. On the other hand, as expected, the five-pad bearing load-on-pad shows larger coefficients (∼25%) in the loaded direction. The maximum pad temperature recorded for the 0.5 pivot offset configurations is up to 20°C higher than those associated to the 0.6 offset configuration. Results from a predictive code are within 50% of the experimental results for the direct stiffness coefficients and within 30% for the direct damping coefficients.

Biomechanical mechanism for transitions in phase and frequency of arm and leg swing during walking.

As humans increase walking speed, there are concurrent transitions in the frequency ratio between arm and leg movements from 2:1 to 1:1 and in the phase relationship between the movements of the two arms from in-phase to out-of-phase. Superharmonic resonance of a pendulum with monofrequency excitation had been proposed as a potential model for this phenomenon. In this study, an alternative model of paired pendulums with multiple-frequency excitations is explored. It was predicted that the occurrence of the concurrent transitions was a function of (1) changes in the magnitude ratio of shoulder accelerations at step and stride frequencies that accompany changes in walking speed and (2) proximity of these frequencies to the natural resonance frequencies of the arms modeled as a pair of passive pendulums. Model predictions were compared with data collected from 14 healthy young subjects who were instructed to walk on a treadmill. Walking speeds were manipulated between 0.18 and 1.52 m/s in steps of 0.22 m/s. Kinematic data for the arms and shoulders were collected using a 3D motion analysis system, and simulations were conducted in which the movements of a double-pendulum system excited by the accelerations at the suspension point were analyzed to determine the extent to which the arms acted as passive pendulums. It was confirmed that the acceleration waveforms at the shoulder are composed primarily of stride and step frequency components. Between the shoulders, the stride frequency components were out-of-phase, while the step frequency components were in-phase. The amplitude ratio of the acceleration waveform components at the step and stride frequencies changed as a function of walking speed and were associated with the occurrence of the transitions. Simulation results using these summed components as excitatory inputs to the double-pendulum system were in agreement with actual transitions in 80% of the cases. The potential role of state-dependent active muscle contraction at shoulder joints on the occurrence of the transitions was discussed. Due to the tendency of arm movements to stay in the vicinity of their primary resonance frequency, these active muscle forces were hypothesized to function as escapements that created limit cycle oscillations at the shoulder's resonant frequency.

The temporal and spatial effects of a magnetorheological elastomer in squeeze mode

The behavior of a magnetorheological (MR) elastomer in an adaptive tuned vibration absorber (ATVA) subjected to transient magnetic fields was examined. ATVAs suppress vibration under broadband, variable, and multiple frequency excitations, and are effective because they can rapidly change one or more parameters to retune in very short periods of time. The device examined here features a MR elastomer whose stiffness and static displacement length are both sensitive to magnetic fields; this means that the static displacement transient behavior directly corresponded to the stiffness transient behavior. A magnetic field applied at step intervals was applied to the ATVA, and the static displacement length change and flux density levels through the ATVA were measured as a function of time. For a flux density level change of 0.12 Tesla, a static displacement length change of about 0.4 mm, which amounted to a 4% change in the overall spring length, occurred in 65 milliseconds. The cause of the long transient response is believed to be due to the magnetic–mechanical coupling.

Optimum design for feedforward active structural acoustic control of complex structures

An efficient design formulation for feedforward active structural acoustic control systems for complex structures and disturbances is presented. The approach consists in a multilevel optimization procedure. The upper level part is carried out in the modal domain where the optimum modal control forces and modal error sensor components which minimize the total radiated power are obtained. These optimum model parameters are then used in the lower level optimization to obtain the physical characteristics of the transducers to be implemented. This separation between the modal domain and the physical domain during the design process allows the designer to investigate different control system configurations in the upper level part with minimum computational effort. The formulation is capable to handle the design of multiple‐inputs, multiple‐outputs control systems and multiple frequency excitations. The proposed approach is demonstrated for the case of a simply supported plate excited by a periodic point force. ...

Conditions on the existence of localized excitations in nonlinear discrete systems.

We use recent results that localized excitations in nonlinear Hamiltonian lattices can be viewed and described as multiple-frequency excitations. Their dynamics in phase space takes place on tori of corresponding dimension. For a one-dimensional Hamiltonian lattice with nearest neighbour interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping $M_{l+1}=F(M_l,M_{l-1})$, where $M_l$ for every $l$ (lattice site) is a function defined on an infinite discrete space of the same dimension as the torus. We consider this mapping in the 'tails' of the localized excitation, i.e. for $l \rightarrow \pm \infty$. For a generic Hamiltonian lattice the thus linearized mapping is analyzed. We find conditions of existence of periodic (one-frequency) localized excitations as well as of multiple frequency excitations. The symmetries of the solutions are obtained. As a result we find that the existence of localized excitations can be a generic property of nonlinear Hamiltonian lattices in contrast to nonlinear Hamiltonian fields.