Hydrodynamic Stiffness Research Articles

Nonlinear Energy Transfer of a Spar-Floater System Using the Inerter Pendulum Vibration Absorber

Abstract The inerter pendulum vibration absorber (IPVA) is integrated between a spar and an annulus floater using a ball-screw mechanism to study its wave energy conversion potential. Hydrodynamic stiffness, added mass, and radiation damping effects on the spar-floater system are characterized using the boundary element method. It is found that a 1:2 internal resonance via a period-doubling bifurcation in the system is responsible for nonlinear energy transfer between the spar-floater system and the pendulum vibration absorber. This nonlinear energy transfer occurs when the primary harmonic solution of the system becomes unstable due to the 1:2 internal resonance phenomenon. The focus of this paper is to analyze this 1:2 internal resonance phenomenon near the first natural frequency of the system. The IPVA system when integrated with the spar-floater system is shown to outperform a linear coupling between the spar and the floater both in terms of the response amplitude operator (RAO) of the spar and one measure of the energy conversion potential of the system. Finally, experiments are performed on the IPVA system integrated with single-degree-of-freedom system (without any hydrodynamic effects) to observe the 1:2 internal resonance phenomenon and the nonlinear energy transfer between the primary mass and the pendulum vibration absorber. It is shown experimentally that the IPVA system outperforms a linear benchmark in terms of vibration suppression due to the energy transfer phenomenon.

The hydroelastic response of a surface-piercing hydrofoil in multiphase flows. Part 2. Modal parameters and generalized fluid forces

The fluid–structure interactions (FSI) of compliant lifting surfaces is complicated by free-surface and multiphase flows such as cavitation and ventilation. This paper describes the dynamic FSI response of a flexible surface-piercing hydrofoil in dry, wetted, ventilating and cavitating conditions. Experimental modal analysis is used to quantify the resonant frequencies and damping ratios of the fluid–structure system in each flow regime. The generalized hydrodynamic stiffness, fluid damping and fluid added mass are also determined as ratios to the corresponding structural modal forces. Added mass increases with increasing partial immersion of the hydrofoil and decreases in the presence of gaseous cavities. In particular, modal frequencies were observed to increase significantly in fully ventilated flow compared to fully wetted flow. The modal frequencies varied non-monotonically with speed in fully wetted flow. Gaseous cavities reduced the modal added mass and reduced the fluid disturbing force. Modal damping increases non-monotonically with increasing immersion depth. Forward speed causes the fluid damping force to increase with an approximately quadratic functional behaviour, consistent with a series expansion of the Morison equation, although damping identification became increasingly difficult at high flow speeds. The results indicate that fluid damping is greater than the associated structural damping in a quiescent liquid, and increasingly so with increasing immersion, suggesting viscous dissipation as a dominant mechanism. A preliminary investigation of modal vibration as a means of controlling the size and stability of ventilated cavities indicates that low-order modes encourage the formation of ventilation, while higher-order modes encourage the washout and elimination of ventilation.

The hydroelastic response of a surface-piercing hydrofoil in multi-phase flows. Part 1. Passive hydroelasticity

Compliant lift-generating surfaces have widespread applications as marine propellers, hydrofoils and control surfaces, and the fluid–structure interactions (FSI) of such systems have important effects upon their performance and stability. Multi-phase flows like cavitation and ventilation alter the hydrodynamic and hydroelastic behaviours of lifting surfaces in ways that are not yet completely understood. This paper describes experiments on one rigid and two flexible variants of a vertical surface-piercing hydrofoil in wetted, ventilating and cavitating conditions. Tests were conducted in a towing tank and a free-surface cavitation channel. This work, which is Part 1 of a two-part series, examines the passive, or flow-induced, fluid–structure interactions of the hydrofoils. Four characteristic flow regimes are described: fully wetted, partially ventilated, partially cavitating and fully ventilated. Hydroelastic coupling is shown to increase the hydrodynamic lift and yawing moments across all four flow regimes by augmenting the effective angle of attack. The effective angle of attack, which was derived using a beam model to account for the effect of spanwise twisting deflections, effectively collapses the hydrodynamic load coefficients for the three hydrofoils. A generalized cavitation parameter, using the effective angle of attack, is used to collapse the lift and moment coefficients for all trials at a single immersed aspect ratio, smoothly bridging the four distinct flow regimes. None of the hydrofoils approached the static divergence condition, which occurs when the hydrodynamic stiffness negates the structural stiffness, but theory and experiments both show that ventilation increases the divergence speed by reducing the hydrodynamic twisting moment about the elastic axis. Coherent vortex shedding from the blunt trailing edge of the hydrofoil causes vortex-induced vibration at an approximately constant Strouhal number of 0.275 (based on the trailing edge thickness), and leads to amplified response at lock-in, when the vortex-shedding frequency approaches one of the resonant modal frequencies of the coupled fluid–structure system.

Hydrodynamic and hydrostatic modelling of hydraulic journal bearings considering small displacement condition

This paper proposes modified coefficients for the dynamic model of hydraulic journal bearing system that integrates the hydrodynamic and hydrostatic properties. In recent years, design of hydraulic bearing for machine tool attracts worldwide attention, because hydraulic bearings are able to provide higher capacity and accuracy with lower friction, compared to conventional bearing systems. In order to achieve active control of the flow pressure and enhance the operation accuracy, the dynamic model of hydraulic bearings need to be developed. Modified coefficients of hydrostatic stiffness, hydrodynamic stiffness, and squeeze damping of the dynamic model are presented in this work, which are derived referring to small displacement analysis from literature. The proposed modified coefficients and model, which consider the pressure variations, relevant geometry size, and fluid properties of the journal bearings, are able to characterise the hydrodynamic and hydrostatic properties with better precision, thus offering the following pragmatic contribution: (1) on-line prediction of the eccentricity and the position of the shaft in the face of external force that results in vibration; (2) development of active control system to regulate the supply flow pressure and to minimize the eccentricity of the shaft. Theoretical derivation and simulation results with different vibration cases are discussed to verify the proposed techniques.

Experimental Investigation of Thermal and Hydrodynamic Effects on Radially Grooved Thrust Washer Bearings

In this study, an experimental investigation on the effects of grooves on thrust washer bearings is investigated. Eight equally sized grooves are machined about 100 μ m deep into one side of a flat-faced steel washer. This thrust washer bearing is located between a helical gear and its carrier and is tested on a test rig capable of measuring frictional torque and the temperature of the bearing at different speeds. It is found that the grooved washers had lower bearing temperatures and failed at significantly higher loads than the control washer with no grooves. For a test procedure with varying operating conditions, the coefficient of friction is also significantly lower for the grooved washers. However, the grooved washers had about the same coefficient of friction as the control washers at each step when the speeds are very high. The results from various tests suggest that the increased amount of lubricant passing through the grooved surface of the washer removes heat from the washer bearing by convection. This decrease in stored heat conducted from friction deters thermoelastic instabilities and the reduction of hydrodynamic stiffness due to the decrease in viscosity. Enhanced hydrodynamic load-carrying capacity is also evident in the grooved washers test results. Review led by Jane Wang

An adaptive finite element strategy for analysis of air lubrication in the headdisk interface of a hard disk drive

In this paper a finite element solution of the modified Reynolds equation using isoparametric, bilinear quadrilateral elements with an adaptive meshing strategy is presented. The modified hydrodynamic stiffness method (Smith, 1995) was used to obtain a coupled solution of the air bearing equation with the slider equilibrium equations. The vertex label based adaptive meshing algorithm of Cheng et al. (1999) was also implemented. The problem is initially solved with a regular quadrilateral FE mesh. The mesh adaptation (hrefinement) is based on the relative pressure gradients in the initial solution, and on the geometry of the slider. The refinement is implemented on an existing element, if preset criteria on the pressure gradient and/or slider geometry are exceeded. The method is described in detail. Examples are presented. Two types of sliders have been used, namely the 50% taper flat slider, and the negative air bearing slider.

THE NONLINEAR HYDROELASTIC BEHAVIOUR OF FLEXIBLE WALLS

A computational model is developed to study the hydroelastic response of simple panels and compliant walls to a uniform flow. Numerical experiments are presented which simulate the history of hydroelastically unstable disturbances as they evolve from amplitudes that may be treated by linear theory to amplitudes for which non-linearity in both the wall and flow cannot be neglected. The method is first applied to simple unsupported flexible panels. Unstable deformations of these panels are seen to be dominated by the fundamental mode. When some panel damping is incorporated, the panel ultimately settles into a static buckled state; however, this long-time response may be preceded by sustained nonlinear oscillations. The amplitudes and frequencies of these oscillations are characterized as a function of wall and flow properties. The method is then used to study a compliant wall comprising a spring-backed flexible plate. For low levels of wall damping, linearly unstable waves evolve into a complex limit-cycle flutter-type response. For high levels of damping small-amplitude unstable disturbances evolve into saturated nonlinear divergence waves that have sharp peaks and shallow troughs. These have much slower downstream wave travel than small-amplitude growing divergence waves. Features of the simulated waves and their dependence on the freestream flow show good qualitative agreement with experimentally measured nonlinear divergence waves. The characteristic waveform of the nonlinear divergence waves is shown to be attributable to the hydrodynamic stiffness pressure field generated by large-amplitude disturbances.

Accelerated Natural Convergence for Pivoted Slider Bearings

A solution technique is discussed which accelerates the convergence of the pivoted slider bearing problem. The technique is based on a fully populated, three-degrees-of-freedom stiffness matrix for the hydrodynamic lubricating film. This matrix is developed using a unique perturbation method which, when used in conjunction with a finite element formulation of the Reynolds equation, yields the bearing stiffnesses in a computationally efficient manner. The resulting natural convergence of this hydrodynamic stiffness approach is examined for a typical rigid disk magnetic recording slider. Presented as a Society of Tribologists and Lubrication Engineers paper at the ASME/STLE Tribology Conference in Lahaina, Hawaii, October 16–20, 1994

Viscoelastic properties of the contracting detrusor. II. Experimental approach

Mechanical properties of detrusor muscle were studied with small-amplitude oscillatory volume perturbations in isometrically contracting bladders of anesthetized dogs. Contractions were studied at oscillatory frequencies (f) of 2 and 4 Hz and at bladder volumes (Vbl) ranging from 30 to 110 ml. The magnitude of bladder hydrodynamic stiffness (magnitude of G) increased linearly with mean detrusor pressure (Pdet) while the phase angle remained relatively constant during contraction. The slope (mG) of magnitude of G-Pdet relations had a positive dependence on f and a negative dependence on Vbl. Analysis of oscillatory data, described in the companion paper, was performed using incremental lumped-parameter models consisting of a spring with incremental constant (S = dF/dL), a viscous element with incremental viscosity (b = dF/du), and a mass (m). Only the model where elastic and viscous elements were placed in series with each other and in parallel with mass was compatible with the experimental data. Both S and b increased linearly with effective force (F), defined as Pdet times the cross-sectional area of the intravesical cavity. Slopes of the S-F and b-F relationships (ms and mb) were independent of Vbl and varied only slightly with f. The importance of this finding stems from recognizing that ms and mb correspond to the exponential coefficients of nonlinear series elastic and internal viscosity elements. These parameters, when normalized by resting muscle length, represent fundamental muscle properties independent of muscle cross-sectional area, stretch, or level of activation and compare well with parameters derived from other muscle systems using techniques such as quick releases and isotonic contractions.

Viscoelastic properties of the contracting detrusor. I. Theoretical basis

This paper presents the theoretical basis for estimating the detrusor's viscoelastic properties using the small-amplitude oscillatory perturbations technique. Three possible configurations of the simplest second-order lumped-parameter model of the bladder were analyzed to derive equations of the parameters incremental resistance (R) and incremental elastance (K) in terms of the experimentally measurable magnitude and phase of hydrodynamic stiffness. In model I, single viscous, elastic, and inertial elements were assumed to to be connected in series. In model III the elastic and viscous elements were connected in series, but the inertial element was connected in parallel. With the assumption of a spherical geometry of the bladder, equations were also derived to obtain the bladder wall mechanical properties, spring incremental constant (S), and muscle incremental viscosity (b) as functions of bladder volume and the hydrodynamic properties R and K. Integration of the incremental equation describing the viscous component yields an expression that fits well the force-velocity experimental data from bladder strips reported by others. This finding suggests that muscle viscosity measured with the small-amplitude oscillations and analyzed with the proper theoretical model may be related to the force-velocity characteristics of the muscle. The equations delivered here form the basis for analyzing the experimental data described in the companion paper.

Analysis of Gas-Lubricated Foil Journal Bearings

This work is concerned with an evaluation of the performance of a gas journal bearing using a spring supported compliant foil as the bearing surface. The analysis, conducted for both single and multipad configurations, is concerned with the effects that the various structural, geometric, and operational variables have on bearing behavior. Following the solution of the relevant differential equation, tabular or graphical solutions are provided for a range of relevant geometric and operational parameters. The solutions include values of the colinear and cross-coupled spring coefficients due to both structural and hydrodynamic stiffness. Desirable design features with regard to start of bearing arc, selection of load angle, number of pads and degree of compliance are discussed.

Dynamic and Static Properties of Recessed Hydrostatic Journal Bearings by Small Displacement Analysis

Coefficients are presented for hydrostatic stiffness, hydrodynamic stiffness and squeeze damping for capillary, orifice and constant flow control. Due to the linear behaviour of such bearings these coefficients may be used for conservative design and with reasonable accuracy where the displacements are less than half the clearance. The dynamic coefficients may also be employed to predict the onset of whirl. The results are illustrated by application to rotating and non-rotating load situations. It is demonstrated that the squeeze damping constant for the concentric condition may be obtained from a steady loading test for hydrodynamic stiffness since Csq = λhd/(πN). It is argued that this relationship may be generalised for all types of bearings capable of exhibiting half-speed whirl operating at any speed.