Traveling Wave Response Research Articles

Numerical investigation of VIV responses of the flexible riser system modelled as tensioned cable subjected to shear flow

The crossflow vortex-induced vibration (VIV) responses of the tensioned cable subjected to the sheared flow velocity profile are investigated numerically using the wake oscillator model. A comparative study is carried out on the VIV response characteristics between the tensioned cable and beam with an aspect ratio (L/D) of 3000. The study was conducted on tensioned cable having different L/D of 2000 and 3000 achieved by varying both diameter and length of the cable. The effect of top tension on the RMS crossflow displacement responses is investigated. The RMS crossflow displacement response is not sensitive to the boundary conditions for the flexible cylinder modeled as a tensioned cable. In contrast, the RMS crossflow displacement response of the lower L/D flexible cylinder modeled as a beam is sensitive to boundary conditions. The standing wave response pattern is observed for the beam with L/D of 500. The standing and traveling wave response patterns are combined for the tensioned cable with L/D = 500. Unlike finite tensioned beams, the higher mode responses are obtained even in the lower aspect ratio for tensioned cables.

Experimental Investigation on the Mechanism of Impeller Synchronous and Nonsynchronous Vibrations in an Industrial Centrifugal Compressor

Abstract Understanding of the fluid–structure interaction phenomena in centrifugal compressors is essential for the impeller structural integrity design. To enhance the understanding of blade vibration in a realistic flow environment, a single-stage centrifugal compressor representative of industrial architecture has been investigated experimentally. For deterministic synchronous vibrations, the response amplitudes at different operating lines are measured and compared. The fundamental relations between flow excitation and rotor modal resonance are established by pairwise strain gages from experimental perspective. Compared with the third mode, the impeller first bending mode shows traveling wave response characteristic only within the frequency band where two adjacent blades have the same resonant frequency. Besides, the impeller encounters unexpected nonsynchronous vibrations when operating near the flow instability boundary. Speed ramp tests show that the stall cell propagating speed increases along with the rotor rotational speed without changing the cell count. Response amplification is further measured when throttling the compressor into stall. Experimental findings point toward vaned diffuser rotating stall and the corresponding propagating pressure waves in circumference leads to the impeller vibration. The aerodynamic asymmetry of stall cells increases the possibility of aeroelastic coupling with the blade modes. These results contribute to an in-depth understanding of the aeroelastic phenomena in industrial centrifugal compressors. The observed nonsynchronous vibration is important for the aeroelastic design and also of great interest for numerical predictions near stall.

Nonlinear vibrations and stability of rotating cylindrical shells conveying annular fluid medium

This study aims to investigate the nonlinear dynamic characteristics of annular cylinders coupled with a fluid medium to identify the effective parameters on stability margins at different rotation speeds. To achieve this, stability and nonlinear vibrations of rotating cylindrical shells containing incompressible annular fluid are considered. The fluid medium is bounded by a rigid external cylinder. Sanders–Koiter kinematic assumptions are utilized to determine the geometrically nonlinear structural equations of motion. These equations are then used to investigate finite amplitude vibrations of various fluid-loaded shells at different states. A penalty approach is introduced by using boundary springs with arbitrary stiffness to simulate any desired set of end conditions. Both driven and companion modes of vibration are included in the solution algorithm to capture the traveling wave response. Fluid equations are determined by using the linearized Navier–Stokes equations. The two sets of governing equations are coupled through the flow impermeability condition on the internal shell’s interface and the zero fluid particle velocity condition on the external boundary. Nonlinear numerical solutions to rotating shells with extremal fluid loading are conducted by the Runge–Kutta direct integration technique. The minimum number of driven/companion modes is determined and the results are verified with the available literature. Results show a significant exchange of energy between the driven and companion modes which consequently initiates a traveling wave response in rotating shell configurations. Moreover, the centrifugal loading and external excitation are shown to be effective on the oscillation amplitude as rotation speed changes leading to alteration of the response quality from periodic to chaotic.

Model of cochlear microphonic explores the tuning and magnitude of hair cell transduction current

The mammalian cochlea relies on the active forcing of sensory outer hair cells (OHCs) to amplify traveling wave responses along the basilar membrane. These forces are the result of electromotility, wherein current through the OHCs leads to conformational changes in the cells that provide stresses on surrounding structures. OHC transducer current can be detected via the voltage in the scala tympani (the cochlear microphonic, CM), and the CM can be used as an indicator of healthy cochlear operation. The CM represents a summation of OHC currents (the inner hair cell contribution is known to be small) and to use CM to probe the properties of OHC transduction requires a model that simulates that summation. We developed a finite element model for that purpose. The pattern of current generators (the model input) was initially based on basilar membrane displacement, with the current size based on in vitro data. The model was able to reproduce the amplitude of experimental CM results reasonably well when the input tuning was enhanced slightly (peak increased by ∼6 dB), which can be regarded as additional hair bundle tuning, and with a current/input value of 200–260 pA/nm, which is ∼4 times greater than the largest in vitro measures.

A numerical study on the vortex-induced vibration of flexible cylinders covered with differently placed buoyancy modules

Buoyancy modules are essential parts in the offshore oil and gas drilling and production practices. They provide weight compensation for the riser and help avoid excessive top tension exerted by the floating platform. However, the deployment of buoyancy modules with larger diameter than the bare riser may alter the geometric shape of the riser system and greatly influence the hydrodynamic performance. Engineering designs with buoyancy modules are mostly based on experience and rigid cylinder forced vibration experimental data. Since it is difficult if not impossible to perform real sea experiments, numerical simulation becomes important in aiding the buoyancy module design and setup. In the present work, a fluid–structure interaction code based on the computational fluid dynamic model is firstly validated with the riser experimental data from MARINTEK (the Norwegian Marine Technology Research Institute) and then used to simulate the riser responses with different continuous buoyancy module setups. The Reynolds number based on the incoming velocity and the bluff body diameter ranges from 4×103 to 1.68×104. The simulation results suggest that the shape alteration from the buoyancy deployment will change the vortex-induced vibration(VIV) response mode along with the vibration frequency. The cylinder fatigue damage, travelling wave response and the fluid–structure energy transfer characters are discussed under different buoyancy module setups.

Nonmonotonic traveling waves in an unsaturated medium induced by a solid force and fluid mass sources

The traveling wave responses in an unsaturated medium induced by a solid force and fluid mass sources are derived in 3D Green's functions using the Dirac Delta function decompositions. The wave behaviors are described in pore fluid pressures and solid frame displacement along with fluctuations of capillary pressure in response to time variant fluid injections and solid force loadings. These solutions are numerically calculated and analyzed for unconsolidated sands as a function of water saturation as representative examples. Our results reveal and validate four wave modes, a fast compressional wave (P1), two slow compressional waves (P2 and P3), and a shear wave. The P1 wave and shear wave can be effectively observed. The slow waves with diffusive characters bear low amplitudes and low velocities. The traveling waves in the pore fluid pressures and solid displacement are sensitive to the different sources and the water saturation. The solid force induces the total stress perturbation while the fluid mass sources directly make the volume strains in the fluid phases. This fluid-solid coupling difference causes the distinct traveling wave signature contrasts in the pore-fluid and solid-frame response amplitudes and tendencies with water retention. For the variation of water saturation, the traveling wave shows a nonmonotonic feature. The fluctuation of capillary pressure exhibits a concave upward relationship, which can further interpret the changes of wave amplitude with water saturation. A solid force may enhance the fluctuation. However, for high water saturation, the interactive fluid-solid effect by the solid force may break the fluid contact due to the extremely compressible air.

Nonlinear Dynamical Responses of Rotary Cylindrical Shells with Internal Resonance

The nonlinear forced vibration response of a thin, elastic, rotary cylindrical shell to a harmonic excitation is investigated in this study. Nonlinearities due to the large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structural damping. Different from the conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing the Airy stress function is utilized to study the nonlinear dynamics of thin shells. The system is discretized using the Galerkin method, while a model involving two degrees of freedom and allowing for the traveling wave response of the shell is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the two-degree-of-freedom system. In addition, the stability of steady-state solutions is analyzed in detail. Finally, results are given for exploring the effects of different parameters on the nonlinear dynamic response with internal resonance.

Numerical investigation of vortex-induced vibration of a riser with internal flow

Vortex-induced vibration (VIV) of a riser considering both internal and external flows is investigated numerically. The model is firstly verified based on the comparison of the experimental data and numerical results. Then the internal flow is taken into account while VIV of a riser subjected to external flow in the subcritical regime of the Reynolds number is studied. Correspondingly, typical VIV characteristics such as the dominating mode and frequency in the in-line (IL) and cross-flow (CF) directions are analyzed, as well as the Root Mean Square (RMS) of the amplitudes, standing and traveling waves for the IL and CF responses. The results indicate that the IL and CF dominating modes are obviously influenced by the internal flow while the dominating frequency and RMS in both IL and CF directions are slightly changed during the investigation. The dominating modes for both IL and CF responses increase with the internal flow velocity increasing. Moreover, conspicuous traveling wave response is detected for both IL and CF responses while the internal flow interacts with the external flow for VIV.

Nonlinear dynamic response of elliptical cylindrical shell underHarmonic excitation

The nonlinear dynamic response of elliptical cylindrical shell under transverse harmonic excitation is experimentally investigated. The elliptical cylindrical shell depicts softening nonlinearity for the excitation range considered. The circumferential variation of acceleration at different time instants of a cycle depicts standing, travelling waves over part of the circumference and travelling waves along the full circumference with/without variable amplitude depending upon the forcing frequency. For symmetric excitation, the travelling wave response is observed for the forcing frequency ωF=81, 85–89, 97.5–99.6, 112, 123.2, 149–155, 178–200 Hz. The participation of different modes involving 1:1, 1:2, 1:3 external, 1:1, 1:2, 2:3 internal resonances and the presence of harmonics including multiples of 1/2, 1/3 are reported.

Distorted wave response of ultrasonic annular stator incorporating non-uniform geometry

The traveling wave ultrasonic stator is normally fabricated with teeth. The tooth geometry improves the driving speed, but it creates natural frequency splitting and mode contamination, especially a distorted traveling wave. A dynamic model of a stepped-plate periodic stator is developed to examine the distortion. The stator is treated as an annular supported by a thin mid plate, and the support stiffness is formulated by using equivalent energy principle. The effects of the tooth and mid plate on the natural frequency and vibration mode are examined by using the perturbation method. The rules governing the frequency splitting, frequency perturbation as well as mode contamination are also identified. The traveling wave response and elliptical trace on stator surface are obtained by using the mode superposition method and they are proved to be distorted due to the tooth geometry. The response at the repeated doublets becomes coupled forward and backward traveling waves, but that at the split doublets becomes coupled forward traveling, standing and backward traveling waves. The results indicate that the tooth mass instead of the stiffness decreases the vibration amplitude and driving speed of the dominant wave, but their effects are different at the repeated and split doublets. Inspection of the model implies that the distortion can be suppressed by using a suitable combination of the wavenumber, tooth count, tooth height and occupying fraction. Numerical calculations are carried out to demonstrate the tooth geometry effect on the transient waveform, driving speed and elliptical trace. The optimization of the tooth geometry that can help achieve a purer traveling wave is discussed.

Travelling wave and non-stationary response in nonlinear vibrations of water-filled circular cylindrical shells: Experiments and simulations

The nonlinear vibrations of a water-filled circular cylindrical shell subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated experimentally and numerically by using a seamless aluminium sample. The experimental boundary conditions are close to simply supported edges. The presence of exact one-to-one internal resonance, giving rise to a travelling wave response around the shell circumference and non-stationary vibrations, is experimentally observed and the nonlinear response is numerically reproduced. The travelling wave is measured by means of state-of-the-art laser Doppler vibrometers applied to multiple points on the structure simultaneously. Chaos is detected in the frequency region where the travelling wave response is present. The reduced-order model is based on the Novozhilov nonlinear shell theory retaining in-plane inertia and the nonlinear equations of motion are numerically studied (i) by using a code based on arclength continuation method that allows bifurcation analysis in case of stationary vibrations, (ii) by a continuation code based on direct integration and Poincaré maps that evaluates also the maximum Lyapunov exponent in case of non-stationary vibrations. The comparison of experimental and numerical results is particularly satisfactory.

Motion complexity in a non-classically damped system with closely spaced modes: From standing to traveling waves

This article addresses the phenomenon of motion complexity in a periodically oscillating system, i.e. the occurrence of non-trivial phase lags among the system's coordinates. Specifically, the steady-state forced response of a linear, weakly damped, self-adjoint system is studied, for which the extent of motion complexity is typically expected to be small. Yet, it is shown that under the condition of closely spaced modes, weak non-classical damping may lead to a transition from standing waves to traveling waves. A system of two oscillators weakly coupled via a linear spring-damper element is considered. The emergence of these motions is related to the distribution of the applied forces and the effect of adding nonlinearity in the form of a cubic spring to the system is investigated. Moreover, it is demonstrated that under certain conditions, the traveling wave response is critical, i.e. it is associated with a resonance or anti-resonance.

Experimental Study on Nonlinear Vibration of Cantilever Cylindrical Shell

Nonlinear dynamic behavior of a cantilever cylindrical shell is studied experimentally using non-contact type of electromagnetic shaker, acceleration measurements, and imaging of vibrating shell using two high speed 3D cameras. The response history and frequency spectrum are measured at different locations along the circumference of the free end of the shell for different excitation amplitudes and mode of vibration. The frequency response obtained from acceleration measurements is found to be in good agreement with the simulated response in ANSYS. The response histories from acceleration data and digital image correlation are compared for one case and found to match very well. The traveling wave phenomenon along with participation of higher harmonics and softening nonlinearity are observed. The traveling wave response is observed for certain range of forcing frequency for the shell excited in modes with number of circumferential waves n = 3 and 5. It is shown that the traveling wave response can be detected based on the phase angle difference of response at different circumferential locations.

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.

Lateralization of Travelling Wave Response in the Hearing Organ of Bushcrickets

Travelling waves are the physical basis of frequency discrimination in many vertebrate and invertebrate taxa, including mammals, birds, and some insects. In bushcrickets (Tettigoniidae), the crista acustica is the hearing organ that has been shown to use sound-induced travelling waves. Up to now, data on mechanical characteristics of sound-induced travelling waves were only available along the longitudinal (proximal-distal) direction. In this study, we use laser Doppler vibrometry to investigate in-vivo radial (anterior-posterior) features of travelling waves in the tropical bushcricket Mecopoda elongata. Our results demonstrate that the maximum of sound-induced travelling wave amplitude response is always shifted towards the anterior part of the crista acustica. This lateralization of the travelling wave response induces a tilt in the motion of the crista acustica, which presumably optimizes sensory transduction by exerting a shear motion on the sensory cilia in this hearing organ.

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.

Nonlinear vibration response and bifurcation of circular cylindrical shells under traveling concentrated harmonic excitation

The nonlinear vibration of a cantilever cylindrical shell under a concentrated harmonic excitation moving in a concentric circular path is proposed. Nonlinearities due to large-amplitude shell motion are considered, with account taken of the effect of viscous structure damping. The system is discretized by Galerkin's method. The method of averaging is developed to study the nonlinear traveling wave responses of the multi-degrees-of-freedom system. The bifurcation phenomenon of the model is investigated by means of the averaged system in detail. The results reveal the change process and nonlinear dynamic characteristics of the periodic solutions of averaged equations.

Mode-splitting and quasi-degeneracies in circular plate vibration problems: The example of free vibrations of the stator of a traveling wave ultrasonic motor

In systems with rotational symmetry, bending modes occur in doubly-degenerate pairs with two independent vibration modes for each repeated natural frequency. In circular plates, the standing waves of two such degenerate bending modes can be superposed with a 1/4 period separation in time to yield a traveling wave response. This is the principle of a traveling wave ultrasonic motor (TWUM), in which a traveling bending wave in a stator drives the rotor through a friction contact. The stator contains teeth to increase the speed at the contact region, and these affect the rotational symmetry of the plate. When systems with rotational symmetry are modified either in their geometry, or by spatially varying their properties or boundary conditions, some mode-pairs split into singlet modes having distinct frequencies. In addition, coupling between some pairs of distinct unperturbed modes also causes quasi-degeneracies in the perturbed modes, which leads their frequency curves to approach and veer away in some regions of the parameter space. This paper discusses the effects of tooth geometry on the behavior of plate modes under free vibration. It investigates mode splitting and quasi-degeneracies and derives analytic expressions to predict these phenomena, using variational methods and a degenerate perturbation scheme for the solution to the plate’s discrete eigenvalue problem; these expressions are confirmed by solving the discrete eigenvalue problem of the plate with teeth.

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.

Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force

This is a study of nonlinear traveling wave response of a cantilever circular cylindrical shell subjected to a concentrated harmonic force moving in a concentric circular path at a constant velocity. Donnell's shallow-shell theory is used, so that moderately large vibrations are analyzed. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Frequency–responses for six different mode expansions are studied and compared with that for single mode to find the more contracted and accurate mode expansion investigating traveling wave vibration. The method of harmonic balance is applied to study the nonlinear dynamic response in forced oscillations of this system. Results obtained with analytical method are compared with numerical simulation, and the agreement between them bespeaks the validity of the method developed in this paper. The stability of the period solutions is also examined in detail.