Dimensionless Load Parameter Research Articles

Points of inflection of special eigenvalue functions as indicators of stiffness maxima/minima of proportionally loaded structures

The stiffness of a proportionally loaded structure may continuously increase or decrease. As a special exception, it may be constant. On the other hand, an initially stiffening (softening) structure may turn into a softening (stiffening) structure. At the load level of such a change the stiffness of the structure attains an extreme value. The task of this work is to present mathematical conditions for these load levels. Lack of them represents a void in the pertinent literature. The practical significance of the aforementioned changes is the one of indicators of the mechanical behavior to be expected after their occurrence. The second and the last author have recently presented a condition for the load level at which the stiffness of a proportionally loaded structure becomes a minimum value. It is given as d2(ℜ(χ1(λ)))/dλ2=0, representing the condition for a point of inflection of the real part of a complex eigenvalue function χ1(λ), where λ denotes a dimensionless load parameter. The underlying linear eigenvalue problem has two indefinite coefficient matrices, which is a necessary condition for complex regions of eigenvalue functions. These matrices are established with hybrid elements, available in a commercial finite element program. In the present work, d2(χ1(λ))/dλ2=0 is shown to be the condition for the load level at which the stiffness of a proportionally loaded structure attains a maximum value. The eigenvalue function concerned has no complex region. It is also shown that the displacement elements, which are the basis for their extension to the employed hybrid elements, are unable to indicate the load level at an extreme value of the stiffness.

Experimental Analysis of Rolling Torque and Thermal Inlet Shear Heating in Tapered Roller Bearings

The investigation in this article focuses on the rolling resistance torque and thermal inlet shear factor in tapered roller bearings (TRBs) through systematic experiments using a modular test setup. TRBs typically operate under Elastohydrodynamic Lubrication (EHL) conditions. At sufficiently high speeds, the majority of rolling friction is due to a significant shift of the pressure centre in the EHL contact. While at lower speeds, sliding friction in the roller-rib contact becomes dominant, which operates under mixed lubrication conditions. Limited literature exists on the impact of inlet shear heating on effective lubricant temperature (Tin_c) and rolling friction in TRBs. To fill this gap, experimental measurements of the total frictional torque under axial loading at different speeds and oil temperatures are performed. With existing models for different friction contributions described in the literature, the rolling resistance due to EHL has been determined for various operating conditions. The effects of dimension-less speed (U), material (G), and load (W) parameters have also been investigated. Under fully flooded conditions, it was observed that the influence of material (G) and load (W) parameters on rolling friction is minor, while the impact of velocity (U) is significant. In the context of rolling resistance, the heating due to shear of the lubricant in the inlet zone plays a significant role. For higher rotational velocities, the estimated rotational torque reduction resulting from inlet shear heating was found to be approximately 6–8%.

A combined experimental and analytical method to determine the EHL friction force distribution between rollers and outer raceway in a cylindrical roller bearing

Friction force is a crucial factor causing power loss and fatigue spalling of rolling element bearings. A combined experimental and analytical method is proposed to quantitatively determine the elastohydrodynamic lubrication (EHL) friction force distribution between rollers and outer raceway in a cylindrical roller bearing (CRB). An experimental system with the instrumented bearing and housing was developed for measuring radial load distribution and friction torque of bearings. A simplified model of friction force expressed by dimensionless speed, load, and material parameters was given. An inequality constrained optimization problem was established and solved by using an experimental data-driven learning algorithm for determining the uncertain parameters in the model. The effect of speed, load, and lubricant property on friction force and friction coefficient was discussed.

Conditions for minimum stiffness of proportionally loaded structures

During proportional loading of structures a significant re-arrangement of stresses may occur. It may result e.g. in the transition from a softening to a stiffening structure. This may even be the case for constant material stiffness. Since the terms stiffening structure and softening structure are lacking universally accepted definitions, it is not astonishing that the search for conditions for minimum stiffness of proportionally loaded multi degree of freedom structures has so far attracted very little attention. Presentation of such a condition, in the framework of the Finite Element Method (FEM), is the task of this publication. It is based on a linear eigenvalue problem that renders complex eigenvalues and eigenvectors possible. The point of inflection in the real part of the complex section of a specific eigenvalue function is shown to mark the load-level of minimum stiffness of a proportionally loaded structure. Complex eigenvalues require the indefiniteness of both real symmetric coefficient matrices of the underlying linear eigenvalue problem. In the given case, the first one depends on a dimensionless load parameter, whereas the second one is obtained by specialization of the first one for the onset of loading. These matrices are established with the help of a special hybrid finite element, available in a commercial FE program. The proposed condition for minimum stiffness of proportionally loaded structures is verified numerically. The results obtained in a parameter study of the investigated structure are found to agree well with the ones following from an alternative condition for the transition from a softening to a stiffening structure. This condition is based on a mechanically objective arc length in the form of a representative displacement of the structure.

Экспериментальное и аналитическое исследование геометрически-нелинейного изгиба консоли под действием распределенной нагрузки гравитационного типа

This paper describes an approximate analytical solution for the geometrically nonlinear bending of a thin elastic cantilever beam under a uniformly distributed gravity load. The solution is based on the linearized Euler-Bernoulli equation of mechanics of materials. Traditionally, such a linear approach is used for small (geometrically linear) deflections. The authors have modified the original equation with an arc-length preservation condition. The modified solution allows one to obtain bending shapes, deflection, and axial displacement in the range of loads corresponding to geometrically nonlinear bending of a beam (large deflections). An experimental study is conducted to verify the proposed solution. A thin steel band bent by gravity is used as a sample. Changes in the length of the bent sample part allow one to obtain various dimensionless load parameters. The deflections and axial displacements averaged on experimental statistics are determined. Bending shapes are obtained by the least square method of 5th order. Experimental and theoretical data are shown to be in good agreement. This fact confirms that the approximate analytical solution can be applied to solve large deflection problems in a wider range of loads than normally considered in the original linear theory.

A multi-scale stiffness fractal model of joint interfaces

Stiffness characterization of mechanical interfaces is quite crucial for the analysis of several tribological behaviors. The stiffness of different machine tools varies greatly, particularly for computer numerical control machine. Therefore, this research aims at providing an assessment of influence factors for stiffness of joint interfaces theoretically. Based on fractal roughness parameters independent of scale and contact mechanics theory, the contact area of joint interface is studied, and the multi-scale normal contact stiffness model and multi-scale tangential contact stiffness model are proposed. Meanwhile, the problem of the deformation of any contact asperity is considered as three separate regimes. The laws of area-displacement and force-displacement under elastic-plastic regime are established. The transition which is in the deformation mechanism of asperity from elastic to plastic is consistent with classical contact mechanics. The analysis of numerical calculation results indicates the approximate linear relation among dimensionless normal load and key parameters. Moreover, these key parameters have been divided into two main categories for the multiscale model of joint interfaces, one is fractal parameters such as fractal dimension D and fractal roughness parameter G, and the other is interfacial parameters. In addition, tangential load and friction factor are two important factors to the tangential stiffness.

线接触弹流润滑状态下的油膜压力极值研究

用Newton-Raphson法求解线接触弹流润滑问题基本方程，研究了线接触弹流润滑状态下中心压力、二次峰压力及其位置的影响因素(载荷、速度及材料参数)。在大量数值解结果基础上，通过多元回归分析，给出了中心压力、二次峰压力及其位置关于载荷、速度、材料参数的拟合公式，并对拟合公式的显著性和精确性进行了检验。结果表明：在很宽的参数范围内，本文所提出的拟合公式具有高度的显著性，变量之间具有密切的相关性及交互性；同时，拟合公式具有很高的计算精度，完全可以满足工程应用的需要。 Based on Newton-Raphson Method for line contact EHL problem, the influence factors (dimensionless load, speed and materials parameters) of center pressure, pressure spike and the location of pressure spike of the line contact EHL state were investigated. A large amount of numerical solution of center pressure, pressure spike and the location of pressure spike corresponding to load, speed and material parameter were obtained, and by making multiple regression analysis of these data, this paper proposed fitting formulas of center pressure, pressure spike and the location of pressure spike on operating parameters (dimensionless load, speed, and material parameter), and made significance test and accuracy test of the formulas. The result shows that in a rather wide range of operating parameters, the fitting formulas proposed in this paper are highly significant, and the variables are closely related and interactive; and the accuracy of the formulas are satisfactory, which can completely meet the need of engineering application.

Assessment of solutions from the consistently linearized eigenproblem by means of finite difference approximations

The consistently linearized eigenproblem (CLE) plays an important role in stability analysis of structures. Solution of the CLE requires computation of the tangent stiffness matrix K∼T and of its first derivative with respect to a dimensionless load parameter λ, denoted as K∼̇T. In this paper, three approaches of computation of K∼̇T are discussed. They are based on (a) an analytical expression for the derivative of the element tangent stiffness matrix K∼Te, (b) a load-based finite difference approximation (LBFDA), and (c) a displacement-based finite difference approximation (DBFDA). The convergence rate, the accuracy, and the computing time of the LBFDA and the DBFDA are compared, using the analytical solution as the benchmark result. The numerical investigation consists of the analysis of a circular arch subjected to a vertical point load at the vertex, and of a thrust-line arch under a uniformly distributed load. The main conclusion drawn from this work is that the DBFDA is superior to the LBFDA.

Displacement‐based finite difference approximations of derivatives of the tangent stiffness matrix with respect to the load parameter

AbstractThe vehicle to investigate to which extent energy‐based categorization of buckling can be linked up with spherical geometry is the so‐called consistently linearized eigenproblem. This investigation requires computation of the first and the second derivative of the tangent stiffness matrix $\tilde{\bf{K}}_T$ with respect to a dimensionless load parameter λ in the frame of the Finite Element Method (FEM). A finite‐difference approximation of the first derivative of $\tilde{\bf{K}}_T$ , redefined as a directional derivative, has proved to meet the requirements of computational efficiency and sufficient accuracy. It represents a displacement‐based finite‐difference approximation, abbreviated as DBFDA. The present work is devoted to the computation of a DBFDA of the second derivative of ˜ KT with respect to λ. For the special case of a two‐dimensional co‐rotational beam element, an analytical solution of this derivative is presented. A circular arch, subjected to a vertical point load on its apex, serves as an example for numerically assessing the usefulness of the computed DBFDAs of the first and the second derivative of $\tilde{\bf{K}}_T$ with respect to λ. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bifurcation buckling from a membrane stress state

SUMMARY A membrane stress state may be considered as a general stress state in which mechanical constraints have been imposed. Therefore, it would appear that there are mathematical subsidiary conditions for bifurcation buckling from such a stress state. In this work, such conditions are derived in the frame of the Finite Element Method. The basic condition follows from disintegration of the second derivative of the mathematical formulation of the so-called consistently linearized eigenproblem with respect to a dimensionless load parameter. It is used for deriving another condition, characterized by the vanishing of a particular bilinear form. Linear stability analysis and bifurcation buckling from linear prebuckling paths are two special cases for which this condition is satisfied. Sensitivity analysis of bifurcation buckling of a two-hinged arch, subjected to a uniformly distributed static load, by varying the geometric form of its axis serves the purpose of non-trivial verification of the derived condition for the special case of a thrust-line arch. Copyright © 2011 John Wiley & Sons, Ltd.

Diagnosis of and remedy for imperfection sensitivity of arch bridges

AbstractUnless the hangers of arch bridges are sufficiently stiff, such bridges are imperfection sensitive [1]. Increasing the stiffness of the hangers, such structures eventually become imperfection insensitive. The mathematical definition of imperfection insensitivity follows from a series expansion of the dimensionless load parameter Δλ(κ, η), relative to the stability limit λ = λS, given as [2] Δλ(κ, η) = λ1(κ)η + λ2(κ)η2 + λ3(κ)η3 + O(η4), where λ1, λ2, … are coefficients depending on the stiffness of the hangers representing the design parameter κ and η is a path parameter describing the postbuckling path.A necessary condition for imperfection insensitivity is [3] λ1(κ) = 0 ∀κ. If, for a specific value κ of κ, also λ2(κ=κ) > 0, then the structure is imperfection insensitive for κ=κ.It will be shown numerically that the increase of the stiffness of the hangers is the remedy addressed in the title of the paper. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the influence of the prebuckling behavior of elastic structures on their initial postbuckling behavior

AbstractDepending on whether or not bending and/or torsion are part of the prebuckling deformation state of elastic structures, the initial postbuckling behavior is either completely general or characterized by specific restrictions [1]. In this context, the term ‘completely general’ means (a) no restriction to monotonic improvement or deterioration of the initial postbuckling behavior in consequence of a monotonic change of a design parameter and (b) the impossibility of ‘zero‐stiffness postbuckling’ characterized by a postbuckling path at a constant load level.The mathematical reason for restrictions of the postbuckling behavior is a constraint condition for the eigenvector ν1 appearing in the classical static stability condition where $\tilde {\bf K}_T$ denotes the tangent stiffness matrix [2]. This condition contains the first and the second derivative of $\tilde {\bf K}_T$ with respect to the dimensionless load parameter λ. The theoretical findings will be verified numerically. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modes of Bed-Load Movement on a Smooth Rigid Surface

Sediment grains transported as supply-limited bed load on a rigid surface move either discretely or collectively as bed forms, with significantly different effective grain speeds and active storage volumes. The adopted mode has implications for sediment sorting and heavy mineral placer formation, dispersal of grain-associated pollutants, and accumulation and flushing of sediment deposits in unlined canals and sewers. The threshold condition between the two modes has been established for a smooth surface from flume experiments with different sediment types, flow conditions, and sediment supply rates. This is expressed in terms of a relationship between the sediment movability number, a dimensionless bed load parameter, and a grain shear Reynolds number.

Development of Theoretical Contact Width Formulas and a Numerical Model for Curved Rough Surfaces

The apparent contact area of curved rough surfaces can be larger than that predicted by the Hertz theory due to asperity interaction outside the Hertzian region. In the present study, simple theoretical formulas for the contact semi-width and radius for Gaussian and truncated Gaussian height distributions were derived, and a numerical contact model was developed based on a general power-law relationship between the local apparent pressure and real-to-apparent contact ratio. Numerical results of the contact semi-width agree well with the prediction of the formula. The apparent contact region becomes increasingly larger than the Hertzian region as a dimensionless roughness parameter increases or as a dimensionless load parameter decreases. The ratio of the contact semi-width to the Hertzian semi-width and the apparent pressure distribution are completely determined by a dimensionless contact parameter and the dimensionless roughness parameter, which are both independent of the instrument resolution, thus providing a long awaited solution to the problem of instrument dependency in a traditional theory. An application to fractal-regular surfaces indicates that the influence of the fractal dimension on the contact behavior is due to its effects on both the area-load coefficient and the load exponent.

Refined Solar Sail Force Model with Mission Application

The aim of this paper is to propose a refined mathematical model for describing the acceleration experienced by a solar sail. Unlike the conventional model characterized by constant coefficients, the force coefficients of the sail are now assumed to depend on the light incidence angle, the sail surface roughness, and the sun–sail distance. The new model is elaborated with the support of experimental data that show how the main variable affecting the force coefficients is the light incidence angle. To emphasize the differences between the refined force model with respect to the conventional one, a comparison is established through the analysis of a circle-to-circle interplanetary rendezvous problem between coplanar orbits. The problem is solved using an indirect approach and the optimal steering law is approximated in polynomial form. A number of optimal trajectories toward Mars and Venus are simulated and the results obtained are discussed as a function of the dimensionless sail loading parameter and the sail surface roughness.

Method for Measuring the Losses and Loading of a Quartz Crystal Microbalance

A novel, experimentally simple, and highly sensitive method for measuring the loading of a quartz crystal resonator was developed. The method is based on the use of double-sideband suppressed-carrier modulated high-frequency signal, which is swept through the resonance range of the resonator. Induced current in the resonator is passed through a capacitor, and the voltage over the capacitor is demodulated on an analog multiplier. The phase and amplitude information is carried to the frequency-doubled modulation signal and measured on a conventional low-frequency two-phase lock-in amplifier. A complex dimensionless loading parameter is obtained from the experimental data by nonlinear model fitting. The real and complex parts of this loading parameter have a simple relationship with other parameters commonly used for characterizing the resonator loading. The performance of the method was demonstrated by measuring a series of different glycerol-water mixtures ranging from 0 up to 100% glycerol. The results were close to the shear acoustic impedance of these mixtures measured and calculated from their viscosities and densities.

Stability and shrinkage of a cavity in stressed grain

A cavity in a stressed grain may change its shape through surface diffusion and volume through lattice diffusion. A suitable thermodynamic potential is established, from which a dimensionless loading parameter that controls the cavity evolution is introduced. Above a critical value of this parameter the cavity collapses to a crack; below that the cavity will stay stable. Then, the shrinkage rate, thereby the healing time, of a spherical cavity is derived as a function of external applied pressure and internal gas pressure. The effects of the applied pressure and the internal pressure on the rate process of the cavity are discussed.

Comparison of various adhesion contact theories and the influence of dimensionless load parameter

Three models, JKR (Johnson, Kendall and Roberts), DMT (Derjaguin, Muller, and Toporov) and MD (Maugis-Dugdale), are compared with the Hertz model in dealing with nano-contact problems. It has been shown that both the dimensionless load parameter, P = P /(πΔγR), and the transition parameter have significant influences on the contact area at micro/nano-scale and should not be ignored in nano-indentation tests.

Diffusive shrinkage of a void within a grian of a stressed polycrystal

A method to predict the shrinkage rate of a void centered at an equiaxial grain under hydrostatic pressure is developed. The rate process is controlled by lattice diffusion under the combined influence of void surface energy, grain boundary interface energy and elastic energy stored in the solid. A dimensionless loading parameter that controls the void evolving is introduced. Below a certain value of this parameter the void will be rounded and is stable. Then the shrinkage behavior of the voids is described by the parameters controlling the lattice diffusion. It is found that only gas-free voids could be totally eliminated through lattice diffusion; for a gas-filled void, the internal pressure produced by the gas firstly retards the rate of the shrinkage and eventually stops the shrinkage process as it becomes high enough.

The state of stress induced by the plane frictionless cylindrical contact. II. The general case (elastic dissimilarity)

In Part I of the paper, the authors have studied the contact problem between a pin and an infinite plate containing a conforming hole, in the absence of friction and in the case of elastic similarity, obtaining a closed form result which generalizes the identical materials analysis of Persson (On the stress distribution of cylindrical elastic bodies in contact, Ph.D. dissertation, 1964). Here, in Part II, the general case of contacting materials is first studied numerically, finding that the effect of elastic dissimilarity (i.e. the second Dundurs' constant not being zero) is negligible for the dimensionless pressure distribution, the maximum influence being less than 2%. Vice versa, the influence on the relation between the contact area arc semi-width, ε, and the dimensionless loading parameter E 1 ∗ΔR/Q is indeed significant; however, considering as an approximate pressure distribution the one of the elastically similar case, an extremely good approximation is obtained for the general relation ε vs. E 1 ∗ΔR/Q which can now take into account of both Dundurs' elastic parameters. In particular, the limiting value for ε lim, towards which the contact tends under very high loads both under initial clearance or interference (or for any load for the perfect fit limiting case) is given as a function of both Dundurs' elastic parameters, α, β as well as the load when complete contact is lost in an interference contact, ε compl. Hence, a complete assessment of the strength of the contact can be obtained directly from the results of Part I of the paper, given that for a certain contact area extension, the correct value of load is used.