NONLINEAR FLUID FORCES IN CYLINDRICAL SQUEEZE FILMS. PART I: SHORT AND LONG LENGTHS

Using three approximation methods, nonlinear models have been derived for short and long cylindrical squeeze films with arbitrary inner cylinder motions. Elliptical and parabolic velocity profiles are employed in the derivation in order to determine the effects of the choice of velocity profile. The only differences in the final squeeze film equations, due to the three approximation methods and the two velocity profiles, are in the four constant coefficients. Each term in the squeeze film equations is a nonlinear function of cylinder position. Comparing the present nonlinear expressions with existing models for short cylindrical squeeze films shows that the force terms are either exactly the same or have the same trends with instantaneous eccentricity values. For long cylindrical squeeze films, the present expressions have some force terms which are essentially the same as in other studies, while other force terms show variations with position which are very different from a previously published study.     

Instantaneous Squeeze-Film Force Between a Heat Exchanger Tube and a Support Plate for Arbitrary Tube Motion

The instantaneous squeeze-film force between a heat exchanger tube and a support plate is studied. Based on a two-dimensional rectangular plate model, a short-sleeve squeeze-film model for arbitrary tube motion is developed. The instantaneous squeeze-film force is expressed in normal and tangential directions. The normal squeeze-film force consists of four nonlinear terms, the viscous, unsteady inertia, convective inertia and centripetal inertia terms. Three nonlinear terms, the viscous, unsteady inertia and Coriolis inertia terms, make up the tangential squeeze-film force. An experimental apparatus was developed in order to evaluate the theoretical models against measurements of a finite length squeeze film. A modified model based on the experimental data is obtained where the viscous terms for both directions are multiplied by the instantaneous Reynolds number. All the inertia terms are multiplied by constant coefficients. The modified model is in good agreement with most experimental cases for unsymmetrical linear motion, approximate circular motion and elliptical motion. The form of the modified model is suitable for predicting instantaneous squeeze-film forces in the simulation of heat exchanger tube vibration. Further work using different sized components and fluid properties is required in order to finalize coefficient values.     

SQUEEZE FILM FLOW WITH NONLINEAR BOUNDARY SLIP

A nonlinear boundary slip model consisting of an initial slip length and a critical shear rate was used to study the nonlinear boundary slip of squeeze fluid film confined between two approaching spheres. It is found that the initial slip length controls the slip behavior at small shear rate, but the critical shear rate controls the boundary slip at high shear rate. The boundary slip at the squeeze fluid film of spherical surfaces is a strongly nonlinear function of the radius coordinate. At the center or far from the center of the squeeze film, the slip length equals the initial slip length due to the small shear rate. However, in the high shear rate regime the slip length increases very much. The hydrodynamic force of the spherical squeeze film decreases with increasing the initial slip length and decreasing the critical shear rate. The effect of initial slip length on the hydrodynamic force seems less than that of the critical shear rate. When the critical shear rate is very small the hydrodynamic force increases very slowly with a decrease in minimum film thickness. The theoretical predictions agree well with the experiment measurements.     

MODELING THE UNSTEADY LIFT AND DRAG ON A FINITE-LENGTH CIRCULAR CYLINDER IN CROSS-FLOW

Semi-empirical models for unsteady lift and drag are developed to predict the spectral features of the unsteady forces on a finite-length, right circular cylinder in cross-flow. In general, the models consist of two parts; the spatial variation of r.m.s wall pressure on the cylinder, and the correlation lengths which describe the spatial extent of the correlation of the unsteady wall pressures. Experiments were conducted in a low noise wind tunnel as a function of cylinder diameter Reynolds number (19 200<Re<32 000) and the Strouhal number (0·05< St<3·33), to measure the statistics of the unsteady wall pressures on a model cylinder. These results are incorporated into the theoretical models, and predictions of the spectral characteristics of the lift and drag are made. The r.m.s. wall pressures on the cylindrical surface are found to have the largest amplitude near the cylinder end-cap, and on the rearward portion of the cylinder body. The high levels in these locations are attributed to the separated flow region over the end-cap. The circumferential and axial length-scales decrease exponentially with Strouhal number. Both length-scales exhibit maxima near the Strouhal shedding frequency of St=0·21. The axial length-scales are found to depend on the measurement reference location due to the three-dimensional flow and separated flow region near the end-cap. The unsteady lift and drag predictions using the models developed in this work agree well with previously measured unsteady force data measured on inertial hydrophones exposed to flow. The broadband unsteady lift is found to be greater than the broadband unsteady drag by nominally 3dB.     

An investigation into electromagnetic force models: differences in global and local effects demonstrated by selected problems

This study investigates the implications of various electromagnetic force models in macroscopic situations. There is an ongoing academic discussion which model is “correct,” i.e., generally applicable. Often, gedankenexperiments with light waves or photons are used in order to motivate certain models. In this work, three problems with bodies at the macroscopic scale are used for computing theoretical model-dependent predictions. Two aspects are considered, total forces between bodies and local deformations. By comparing with experimental data, insight is gained regarding the applicability of the models. First, the total force between two cylindrical magnets is computed. Then a spherical magnetostriction problem is considered to show different deformation predictions. As a third example focusing on local deformations, a droplet of silicone oil in castor oil is considered, placed in a homogeneous electric field. By using experimental data, some conclusions are drawn and further work is motivated.     

A critical overview of internal and external cylinder contact force models

Most of the analytical models found in the literature, to study the contact between cylindrical bodies, are based on the Hertz pressure distribution. The major shortcomings associated with these cylindrical models concern their nonlinearity. Firstly, the indentation is expressed as an implicit function of the contact force, thus a numerical iterative technique is required to evaluate the contact force for a given indentation. In a dynamic analysis code, it is implied that at each integration time step, the iterative process for the solution of the nonlinear equations has to be solved. Secondly, the current cylindrical contact models include logarithmic functions, which impose mathematical and physical limitations on their application, particularly for conformal contact conditions with lower clearance values. The validity domain of each contact model is identified in this work with relation to the clearance value and material properties of the contacting cylinders. A comparative assessment of the performance of each model is performed calculating the relative difference of each one in relation to Johnson’s model. The results show that, in general, different models exhibit distinct behavior for both the internal and external contact between cylinders. The load limit of each model and the restrictions on its application is identified using two simple examples of mechanical engineering practice in which internal contacting cylinders are involved and analyzed to include: journal bearings and roller chain drives.     

Validation of turbulence models in a simulated air-conditioning unit

Details are given of a study to obtain experimental data in an idealized environment for the purpose of evaluating the corresponding computational predictions and which supplement parallel measurements made in actual packaged air-conditioning units. The system consisted of a purpose-built low-speed wind tunnel with a working section constructed to reproduce particular features of the real units. In the experiment, both the mean velocity profiles and turbulence properties of the flow are obtained from triple-hot-wire anemometry measurements. A numerical model, based on finite volume methodology, was used to obtain the solution of the Reynolds-averaged Navier–Stokes equations for incompressible isothermal flow. The Reynolds stress terms in the equations are calculated using the standard k–ϵ model and second-moment closure (Reynolds stress) models. The accuracy of the two models was evaluated against the experimental measurements made 10 mm downstream of a baffle. The results show that the standard k–ϵ model gave the better agreement except in regions of strong recirculation. © 1997 John Wiley & Sons, Ltd.     

Parameter identification of systems with multiple disproportional local nonlinearities

Identification of nonlinear systems, especially with multiple local nonlinearities exhibiting disproportional ratios of the degree of nonlinearity and present at a single or multiple spatial locations, is a highly challenging inverse problem. Identification of such complex nonlinear systems cannot be handled easily by the existing conventional restoring force or describing function methods. Further, noise-corrupted measured time history responses make the parameter identification process much more difficult. Keeping this in view, we propose a new meta support vector machine (meta-SVM) model to precisely identify the type, spatial location(s) and also the nonlinear parameters present in disproportionate levels using the noisy measurements. Apart from the conventional SVM model, we also explore the effectiveness of the non-batch processing models like incremental learning for lesser computational cost and increased efficiency. Both incremental and conventional support vector regression models are explored to precisely identify the nonlinear parameters. A numerically simulated multi-degree of freedom spring-mass system with limited multiple local nonlinearities at a few selected spatial locations is considered to illustrate the proposed meta-SVM model for nonlinear parametric identification. However, the extension of the proposed meta-SVM model is rather straightforward to include all types of nonlinearities and cases with the simultaneous existence of multiple numbers of same or different nonlinearities (i.e. combined nonlinearities) at single or multiple locations. It is also clearly established from the numerical simulation studies that the proposed incremental meta-SVM model paves way for online real-time identification of nonlinear parameters which is not yet been addressed in the existing literature.

     

Steady-state behaviour of flexible rotordynamic systems with oil journal bearings

This paper deals with the long term behaviour of flexible rotor systems, which are supported by nonlinear bearings. A system consisting of a rotor and a shaft which is supported by one oil journal bearing is investigated numerically. The shaft is modelled using finite elements and reduced using a component mode synthesis method. The bearings are modelled using the finite-length bearing theory. Branches of periodic solutions are calculated for three models of the system with an unbalance at the rotor. Also self-excited oscillations are calculated for the three models if no mass unbalance is present. The results show that a mass unbalance can stabilize rotor oscillations.     

A thermo-viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials

A temperature-dependent viscodamage model is proposed and coupled to the temperature-dependent Schapery’s nonlinear viscoelasticity and the temperature-dependent Perzyna’s viscoplasticity constitutive model presented in Abu Al-Rub et al., 2009, Huang et al., in press in order to model the nonlinear constitutive behavior of asphalt mixes. The thermo-viscodamage model is formulated to be a function of temperature, total effective strain, and the damage driving force which is expressed in terms of the stress invariants of the effective stress in the undamaged configuration. This expression for the damage force allows for the distinction between the influence of compression and extension loading conditions on damage nucleation and growth. A systematic procedure for obtaining the thermo-viscodamage model parameters using creep test data at different stress levels and different temperatures is presented. The recursive-iterative and radial return algorithms are used for the numerical implementation of the nonlinear viscoelasticity and viscoplasticity models, respectively, whereas the viscodamage model is implemented using the effective (undamaged) configuration concept. Numerical algorithms are implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The model is then calibrated and verified by comparing the model predictions with experimental data that include creep-recovery, creep, and uniaxial constant strain rate tests over a range of temperatures, stress levels, and strain rates. It is shown that the presented constitutive model is capable of predicting the nonlinear behavior of asphaltic mixes under different loading conditions.     

MODELLING THE UNSTEADY AXIAL FORCES ON A FINITE-LENGTH CIRCULAR CYLINDER IN CROSS-FLOW

A semi-empirical model for unsteady axial forces is developed to predict the spectral features of the force generated by the flow over the end-caps on a finite-length, right circular cylinder in cross-flow. In general, the model consists of two parts: the spatial variation of r.m.s. wall pressure on the cylinder end-caps, and the correlation lengths and areas, which describe the spatial extent of the correlation of the unsteady wall pressures. Experiments were conducted in a low-noise wind tunnel as a function of cylinder diameter Reynolds number (19 200<Re<32 000) and the Strouhal number (0·05<St<3·33) to measure the statistics of the unsteady wall pressures on a model cylinder. These results are incorporated into the theoretical models, and prediction of the spectral characteristics of the axial force are made. The r.m.s. wall pressures on the end-caps are found to have the largest amplitude at circumferential locations (from the forward stagnation point) in the 90–120° range. The high levels at these locations are attributed to reattachment of the separated flow over the end-cap. The radial and circumferential correlation areas have a maximum value at St=0·21. Due to the 3-D flow over the end-caps, the radial correlation areas are found to depend on the circumferential measurement reference location, and the circumferential correlation lengths are found to depend on the radial measurement location. The unsteady axial force predictions using the model show a very broad spectral character.     

Design of nonlinear springs for attaining a linear response in gap-closing electrostatic actuators

Gap-closing electrostatic actuators are inherently nonlinear and their dynamic range is often limited by the pull-in instability. To overcome this, we propose a nonlinear spring that counteracts the nonlinear effects of electrostatic attraction. The nonlinear spring is designed to extend the stable range of the actuator and to enforce a linear electromechanical response. We present a method for designing elastic springs with monotonically increasing stiffness. The mechanism we propose is effective shortening of a straight clamed-guided beam flexure, by wrapping it over a cam. We consider two specific cases. The first case assumes the wrapped section of the beam flexure fully conforms to the cam shape. The second case assumes that there is a single contact point between the beam flexure and the cam. To validate the concept we have designed and measured the response of a nonlinear spring with a prescribed force–displacement law. Experimental measurements of a macro-scale spring are in good agreement with the model predictions.     

Determination of surface heat transfer coefficients from measured temperature data for spherical and cylindrical bodies during cooling

In this paper which is a combination of the methodological and experimental aspects, models were developed for determining surface heat transfer coefficients for spherical and cylindrical bodies from their center temperature measurements during forced-cooling. Experiments involved the cooling of the individual spherical and cylindrical products as test samples in the air flow. The cooling parameters in terms of the cooling coefficients and lag factors were also determined to use in the present models. The results show that the surface heat transfer coefficients of the individual spherical and cylindrical products increased with an increase in the flow velocities from 1 to 2 m/s. It can be concluded that the present models have the capabilities of determining the surface heat transfer coefficients for spherical and cylindrical bodies with a single transient experiment.     

考虑进油压力的滑动轴承非线性油膜力数据库

通过对ReyTlolds方程的非线性变换，提出了考虑进油压力边界条件时径向滑动轴承非线性油膜力数据库的建库方法，扩展了油膜力数据库计算方法的应用，通过引入2个有限数据域的新变量对转子轴心速度项和进油压力边界条件进行有限化处理，得到了更符合实际工况的连续性油膜力数据库及计算模型，同有限元法对比分析了非线性油膜力数据库的适用性．结果表明，非线性油膜力数据库模型的精度较高，所建立的非线性油膜力计算模型可用于对转子系统瞬态运动进行简便和快捷的分析．     

Drag measurements at Mach 5 using a stress wave force balance

The stress wave force balance, which has been used for measurements of drag on short models in hypersonic impulse facilities, is investigated here for its suitability for drag measurements on a longer, axisymmetric model. The sensitivity of the balance to loading distribution is investigated and results are reported for experiments on a 5° semi-angle cone, 425 mm in length and of 1.71 kg mass. Experimental drag measurements are shown to be in good agreement with theoretical levels. An investigation into the period over which the stress wave force balance can be used is addressed and, for the present model, the balance is shown to be suitable for measurements in flows of durations of one to several milliseconds with an estimated accuracy of ±10%.     

Squeeze flow of a second-order fluid between two parallel disks or two spheres

The normal viscous force of squeeze flow between two arbitrary rigid spheres with an interstitial second-order fluid was studied for modeling wet granular materials using the discrete element method. Based on the Reynolds‘ lubrication theory, the small parameter method was introduced to approximately analyze velocity field and stress distribution between the two disks. Then a similar procedure was carried out for analyzing the normal interaction between two nearly touching, arbitrary rigid spheres to obtain the pressure distribution and the resulting squeeze force. It has been proved that the solutions can be reduced to the case of a Newtonian fluid when the non-Newtonian terms are neelected.     

Modelling for heat and mass transfer parameters in deep-frying of products

In this paper, the development of new models to determine heat and mass transfer parameters (HMTPs) in terms of the thermal diffusivity, heat transfer coefficient, moisture diffusivity and moisture transfer coefficient for slab, cylindrical, and spherical products being deep-fried was presented. In the model development, two cases of the Biot number, i.e., 0<Bi<100, and Bi≥100 in the transient heat and mass transfer were considered. In order to verify the models, frying experiments were performed using the cylindrically shaped potatoes as test samples, and these samples were fried in a deep fryer at 180°C. The lag factor and frying coefficient for a frying process, which affect heat and mass transfer parameters, were first-defined and considered most important process parameters. By using the temperature and moisture measurements, the frying coefficient and lag factor were determined and incorporated into the models. The HMTPs were determined in a simple and effective manner. In this respect, we can conclude that the present models are useful tools for determining the HMTPs for the products during frying and will be beneficial to the practical applications.     

A magnetoelastic theoretical model for soft ferromagnetic shell in magnetic field

With the aid of a generalized variational method, in this paper, a theoretical model for soft ferromagnetic shells is derived to describe their magnetoelastic behavior in an applied magnetic field. Having made a quantitative comparison between the numerical predictions given by several theoretical models and the experimental results on strains of a cylindrical shell, we find that the predictions got by our model are in good agreement with the experimental data. It is also found that the Moon’s model is a special case of the model derived in this paper when the relative magnetic permeability μ_r>10⁴, which confirms that it is reasonable for the Moon’s model to calculate strains of the soft ferromagnetic shells. Having displayed the distribution of the equivalent magnetic force in the length of the cylindrical shell and its circumferential bending strains with different elastic end constraints, we give an explanation for the discrepancy between Moon’s analytical results and his experimental ones.     

有限长柱形药包土中爆腔特征尺寸的计算方法

为计算柱形药包土中爆腔尺寸，提出了一种有限长柱形药包在土中爆炸的特征尺寸近似计算方法，该方法利用球形药包爆腔膨胀准静态模型叠加的方式，给出了长径比较大情况下柱形药包爆腔特征尺寸及塑性区半径。与数值模拟对比表明，该方法的误差随长径比的增大而减小，当球形药包数量N=n、长径比在10及以上时，误差在12.2%以内，表明该方法能够较准确地预测有限长柱形药包爆腔的特征尺寸。     

A THEORETICAL MODEL FOR NONLINEAR PLANAR MOTIONS OF ROTORS UNDER FLUID CONFINEMENT

Following previous papers by Axisa, Antunes and co-workers, the authors address a theoretical model for immersed rotors, under moderate confinement, using simplified flow equations on the gap-averaged fluctuating quantities. However, in contrast to our previous efforts, the nonlinear terms of the flow equations are here fully accounted. Because such nonlinear analysis is quite involved, this paper will focus on the simpler case of planar motions, in order to emphasize the main aspects of our approach. A direct integration of the continuity and momentum equations leads to extremely lengthy formulations. Here, in order to solve the flow equations, we perform an exact integration of the continuity equation and an approximate solution of the momentum equation, based on a Fourier representation of the azimuthal pressure gradient. Then, an exact formulation for the dynamic flow force can be obtained. Our solution is discussed in connection with physical phenomena. Numerical simulations of the nonlinear rotor-flow coupled system are presented, showing that the linearized and the fully nonlinear models produces similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative dynamics stemming from these models are quite distinct. Experimental results indicate that the nonlinear flow model leads to better predictions of the rotor dynamics when the eccentricity is significant, when approaching instability and for linearly unstable regimes.