Change In Friction Coefficient Research Articles

Effect of hexagonal boron nitride (h-BN) addition on friction behavior of low-steel composite brake pad material for railway applications

In this study, low-steel composite friction materials containing different amount of h-BN were developed for railway vehicles. Addition of h-BN has no considerable effect on mechanical properties of the matrix structure. However, h-BN addition significantly affects friction and wear properties of the matrix material. While h-BN increases coefficient of friction of samples at low disk surface temperatures, it has a reverse effect after the temperature reaches to a critical level of 250–300 °C. Also, addition of 1.5 wt% h-BN to the matrix brings about more stable changes in friction coefficient during the braking periods under the extreme conditions as compared to the h-BN-free sample. Furthermore, low h-BN addition into the matrix results in better lubricity properties at relatively high temperatures.

Evaluation of Wear of Disc Brake Friction Linings and the Variability of the Friction Coefficient on the Basis of Vibroacoustic Signals

The article presents the results of friction and vibroacoustic tests of a railway disc brake carried out on a brake stand. The vibration signal generated by the friction linings provides information on their wear and offers evaluation of the braking process, i.e., changes in the average friction coefficient. The algorithm presents simple regression linear and non-linear models for the thickness of the friction linings and the average coefficient of friction based on the effective value of vibration acceleration. The vibration acceleration signals were analyzed in the amplitude and frequency domains. In both cases, satisfactory values of the dynamics of changes above 6 dB were obtained. In the case of spectral analysis using a mid-band filter, more accurate models of the friction lining thickness and the average coefficient of friction were obtained. However, the spectral analysis does not allow the estimation of the lining thickness and the friction coefficient at low braking speeds, i.e., 50 and 80 km/h. The analysis of amplitudes leads to the determination of models in the entire braking speed range from 50 to 200 km/h, despite the lower accuracy compared to the model, based on the spectral analysis. The vibroacoustic literature presents methods of diagnosis of the wear of various machine elements such as bearings or friction linings, based on amplitude or frequency analysis of vibrations. These signal analysis methods have their limitations with regard to their scope of use and the accuracy of diagnosis. There are no cases of simultaneous use of different methods of analysis. This article presents the simultaneous application of the amplitude and frequency methods in the analysis of vibroacoustic signals generated by brake linings. Moreover, algorithms for assessing the wear of friction linings and the average coefficient of friction were presented. The algorithm enables determination of the time at which the friction linings should be replaced with new ones. The final algorithm analyzes the vibration acceleration signals using both amplitude analysis for low braking speeds, as well as spectral analysis for medium and high braking speeds.

INVESTIGATION ON DRY SLIDING WEAR BEHAVIOR OF TITANIUM DIOXIDE-REINFORCED MAGNESIUM MATRIX COMPOSITE

In this work, the dry sliding wear behaviors of pure monolithic magnesium and magnesium–titanium dioxide (Mg–TiO2) composites were studied using pin-on-disc tribometer against an oil-hardened nonshrinking die steel (OHNS) counter-disc with a normal load of 0.5–2[Formula: see text]kg and a sliding velocity of 1.5–2.5[Formula: see text][Formula: see text] with the sliding distance and wear track diameter of 1500[Formula: see text]m and 90[Formula: see text]mm, respectively. The pin samples were characterized for their microstructural, nanomechanical and tribological properties such as wear rate, coefficient of friction and wear fractographs. Scanning electron microscopy (SEM) was used to analyze the worn-out surfaces of each pin sample in order to identify the different types of wear and wear mechanisms and the chemical constituents of each element were quantified by energy-dispersive spectroscopy. The influence of TiO2 reinforcements on the nanomechanical behavior was studied by nanoindentation technique. As compared with pure Mg, the nanoindentation strengths of Mg–1.5TiO2, Mg–2.5TiO2 and Mg–5TiO2 composites were found to increase by 11.9%, 22.2% and 35.8%, respectively, which was due to the addition of TiO2 particles and also due to the good bonding at the interface of TiO2 and magnesium particles. From the wear test results, a significant change in wear rate was observed with the change in normal load than that of sliding speed, whereas a significant change in coefficient of friction was noticed with the changes in both normal load and sliding velocity. The dominant wear mechanisms involved under the testing conditions were identified through plotting the contour maps and SEM fractographs. Also, from the fractographs it was noticed that delamination and plowing effect have been the significant wear mechanisms observed during low wear rate of samples, whereas melting, delamination and oxidation wear have been observed during high wear rate of pure Mg and its composites.

Mechanism analysis and improved model for stick-slip friction behavior considering stress distribution variation of interface

Studying the evolution of interface contact state, revealing the “black box” behavior in interface friction and establishing a more accurate friction model are of great significance to improve the prediction accuracy of mechanical system performance. Based on the principle of total reflection, a visual analysis technology of interface contact behavior is proposed. Considering the dynamic variation of stress distribution in interface contact, we analyze the nonlinear characteristics of contact parameters in different stages of stick-slip process using the above-mentioned experimental technology. Then, we find that the tangential stiffness of the interface is not a fixed value during the stick-slip process and the stress distribution variation is one of the important factors affecting the tangential stiffness of interface. Based on the previous experimental results, we present an improved stick-slip friction model, considering the change of tangential stiffness and friction coefficient caused by the stress distribution variation. This improved model can characterize the variation characteristics of contact parameters in different stages of stick-slip process, whose simulation results are in good agreement with the experimental data. This research may be valuable for improving the prediction accuracy of mechanical system performance.

Molecular Dynamics Study on Nano-Friction and Wear Mechanism of Nickel-Based Polycrystalline Superalloy Coating

In this work, molecular dynamics simulations are employed to study the nanotribological process of nickel-based polycrystalline superalloy coating. A series of simulations were carried out using the method of repeated friction to explore the influence of frictional force, friction coefficient, grinding groove morphology, wear scar depth, debris flow direction, subsurface damage degree and evolution of defects during the nano-friction process. In addition, the change mechanism of different grain sizes on wear scar depth, frictional force, friction coefficient, and internal damage in the repeated friction process is also explored. The results show that the frictional force is related to the direction of the dislocation slip, and that the friction coefficient change is related to the number of repeated frictions. Moreover, it is observed that the grinding ball has a shunting effect on the formed wear debris atoms, and the shunt point is located at the maximum horizontal radius. We reveal that the grain boundary structure has a strengthening effect. When the grinding ball rubs to the grain boundary, the nucleation of dislocation defects inside the workpiece is obviously hindered by it. Simultaneously, we also find that the closer the subsurface is to the bottom of the grinding ball, the greater the degree of damage to the workpiece by friction. Furthermore, with the grain size decreases that the material begins to soften, resulting in a decrease of frictional force, friction coefficient, and smaller defects are formed inside the workpiece. The research of this work can better clarify the microscopic mechanism of the polycrystalline friction process.

Influence of the running-in process on the working ability of contact surfaces in lubricated sliding conditions

The influence of the running-in process operating parameters on tribological properties of the block-on-disc samples in lubricated sliding conditions is analyzed and discussed in detail. Different running-in regimes are achieved by varying the normal load and sliding speed. After the running-in period, during which the operating parameters are varied, all samples are placed in a working regime under the same set of operating conditions. At the end of the running-in period, as well as at the end of the working period, an analysis of the changes in the surface roughness, microhardness, wear rate, and coefficient of friction is performed. Less desirable properties in terms of wear rate and steady-state coefficient of friction are noticed for the samples that were run-in with the operating conditions which were the same as the working regime operating conditions. In the defined test conditions, it is shown that the intensity of normal load applied during the running-in process has a dominant influence on the amount of wear and coefficient of friction value. It was also shown that the running-in process can significantly improve the roughness of the initially rough contact surfaces. The results of experimental testing indicate that the variation of the operating parameters during the running-in process can be used to improve the working ability of the sliding contact surfaces under the mixed lubrication regime.

Development of Online Adaptive Traction Control for Electric Robotic Tractors

Estimation and control of wheel slip is a critical consideration in preventing loss of traction, minimizing power consumptions, and reducing soil disturbance. An approach to wheel slip estimation and control, which is robust to sensor noises and modeling imperfection, has been investigated in this study. The proposed method uses a simplified form of wheels longitudinal dynamic and the measurement of wheel and vehicle speeds to estimate and control the optimum slip. The longitudinal wheel forces were estimated using a robust sliding mode observer. A straightforward and simple interpolation method, which involves the use of Burckhardt tire model, instantaneous values of wheel slip, and the estimate of longitudinal force, was used to determine the optimum slip ratio that guarantees maximum friction coefficient between the wheel and the road surface. An integral sliding mode control strategy was also developed to force the wheel slip to track the desired optimum value. The algorithm was tested in Matlab/Simulink environment and later implemented on an autonomous electric vehicle test platform developed by the Nanjing agricultural university. Results from simulation and field tests on surfaces with different friction coefficients (μ) have proved that the algorithm can detect an abrupt change in terrain friction coefficient; it can also estimate and track the optimum slip. More so, the result has shown that the algorithm is robust to bounded variations on the weight on the wheels and rolling resistance. During simulation and field test, the system reduced the slip from non-optimal values of about 0.8 to optimal values of less than 0.2. The algorithm achieved a reduction in slip ratio by reducing the torque delivery to the wheel, which invariably leads to a reduction in wheel velocity.

Influence of realistic rheological properties on the style of mantle convection: roles of dynamic friction and depth-dependence of rheological properties

SUMMARY In order to generate plate tectonics, the near surface layer should not be too strong, but the causes for not-so-strong near surface layer remains unclear. We conduct mantle convection modelling in the spherical geometry to investigate the influence of the strength of the near surface layer. We explore a range of friction coefficients including the static high friction coefficient (∼0.6) as well as the reduced friction coefficients by fast fault motion in earthquakes. When the friction coefficient is low enough (<0.03), the surface layer is yielded by the convective stress, and the style of mantle convection appears the mobile-lid mode (plate tectonics style of convection). This style is relevant for the Earth where fault motion is unstable because of the low surface temperature. In contrast, for a high friction coefficient, the surface layer is too strong, generating the stagnant-lid mode. This case corresponds to Venus where fault motion is stable because of high surface temperature. Our calculations show that, in plate tectonic style of convection, the mantle convection is likely to be more vigorous, inducing the high convective stress that helps the operation of plate tectonics. In contrast, when stagnant-lid mode of convection appears, the convective vigor is likely to be low, inducing the low convective stress. Therefore, in each case, the interplay between the surface strength and convective stress tends to maintain the same mode of convection in a self-consistent way. We also investigate the relationship between mantle temperature and heat flux for two different modes of convection upon a change in friction coefficient. We found that the heat flow associated with mobile lid convection caused by low friction is less sensitive to the mantle temperature compared to a conventional mantle convection model, where the heat flow is highly sensitive to mantle temperature. This provides a possible mechanism to solve the thermal runaway paradox.

Experimental investigation of the effect of wave turbulators on heat transfer in pipes

This experimental study deals with the heat transfer and friction effects of sinusoidal part turbulators for single-phase flows occurring in a circular shaped pipe. Turbulators with three different radius values are placed in the pipe to make the flow turbulent. In this way, changes in Nusselt number and friction coefficient are examined. As a result of the experiments made with Reynolds numbers in the range of 6614-20710, the increase rates of the Nusselt numbers of turbulators with 20 mm, 110 mm, and 220 mm radius compared to the empty pipe were obtained as 153.49%, 85.36%, and 52.09%, respectively. As a result of the decrease in the radius, there was an increase in the Nusselt number and the friction factor. Parallel to the Nusselt number, the highest friction factor was obtained in the smallest radius turbulator. It was found that the thermal enhancement factors of 110 mm and 220 mm radius turbulators increased by 179.54% and 132.95%, respectively, compared to the 20 mm radius turbulator. Similarly, it was determined that the thermal enhancement factor of the 110 mm radius turbulator increased by 20% compared to the 220 mm radius turbulator.

Исследование изменения коэффициента гидравлического трения материалов напорных труб при различных температурах

Снижение затрат электроэнергии на механическую транспортировку воды по напорным трубопроводам за счет уменьшения гидравлического сопротивления внутренних стенок труб носит актуальный характер. Это обстоятельство предопределяет необходимость проведения соответствующих гидравлических испытаний альтернативных материалов, которые могут быть использованы при проектировании и строительстве трубопроводных систем. Целью исследований является выявление характера изменения величины гидравлического трения в зависимости от температурных условий транспортируемой воды и окружающей среды в определенных диапазонах с последующей возможностью управления процессом транспортировки при минимуме затрат на электроэнергию. Достижение поставленных целей осуществляется путем опытного и расчетно-аналитического подходов к определению динамики изменения гидравлических показателей трубопроводов из различных материалов. Представлен комплексный анализ экспериментальных и расчетных данных, полученных с помощью запатентованных автоматизированных комплексов, по определению коэффициентов гидравлического трения для полиэтиленовой трубы и трубы из непластифицированного поливинилхлорида и динамики изменения этих коэффициентов в зависимости от температуры транспортируемых вод и температурных условий прокладки трубопроводов. Reducing the cost of electricity for the mechanical transportation of water through pressure pipelines by reducing the hydraulic resistance of the inner walls of the pipes is of current importance. This circumstance predetermines the need for appropriate hydraulic testing of alternative materials that can be used in the design and construction of pipeline systems. The aim of the study is to identify the nature of the change in the value of hydraulic friction depending on the temperature conditions of the transported water and the environment in certain ranges, with the subsequent possible control of the transportation process with a minimum of energy costs. The set goals are achieved by experimental computational and analytical approaches to determining the dynamics of changes in the hydraulic parameters of pipelines made of various materials. A comprehensive analysis of the experimental and calculated data obtained using patented automated systems for determining the coefficients of hydraulic friction for a polyethylene pipe and a pipe made of unplasticized polyvinylchloride, and the dynamics of changes in these coefficients depending on the temperature of the transported water and the temperature conditions of pipeline laying is presented.

Modeling and verification of life prediction of a V-shaped linear ultrasonic motor.

In this paper, a wear model is established for a V-shaped linear ultrasonic motor. The influence of the change in the dynamic friction coefficient and the roughness parameters of the contact surface of the stator/slider on the output performance of the motor is considered in this model. The wear mechanism of the motor and the variation rules of the morphology of the contact surface during the operation of the motor are studied, as well as the relation of the wear height of the driving tip with the preload force and the relation of the preload force with the resonance frequency of the stator. The accuracy of the model is verified by numerical simulation and experimental methods. A test platform for friction and wear experiments is constructed, and the wear characteristics and the output performance of the motor are tested. The obtained results indicated that the model can accurately predict the dynamic changes during the wear process and the service life and output performance of the motor.

Comparison of the Tribological Properties of the Thermal Diffusion Zinc Coating to the Classic and Heat Treated Hot-Dip Zinc Coatings.

The presented studies are focused on the wear resistance and friction coefficient changes of the thermal diffusion (TD) zinc coating deposited on steel. The aim of research was to evaluate the variation in coating properties during dry friction as a result of the method of preparation of the basis metal. The measured properties were compared to those obtained after classic hot-dip (HD) zinc galvanizing—heat treated and untreated. Thermal diffusion zinc coatings were deposited in industrial conditions (according to EN ISO 17668:2016-04) on disc-shaped samples. The results obtained during the tribological tests (T11 pin-on-disc tester) were analysed on the basis of microscopic observations (with the use of optical and scanning microscopy), EDS (point and linear) analysis and microhardness measurements. The obtained results were similar to effects observed after heat treatment of HD zinc coating. The conducted analysis proved that the method of initial steel surface preparation results in changes in the coating’s hardness, friction coefficient and wear resistance.

Research of physically modified antifriction-wear-resistant composite polymer materials and coatings on their basis for machine-building purpose

The article presents the results of studies of the tribotechnical properties of composite thermosetting polymeric materials based on an epoxy compound and ultrasonicated oligomeric fillers operating under conditions of contact interaction with a pulp using the example of raw cotton. Regularities of changes in tribotechnical properties (coefficient of friction, intensity of wear) of composite thermosetting epoxy polymeric materials associated with their filling with organomineral fillers and ultrasonic treatment, in contact with raw cotton have been revealed. The temperature and the magnitude of the electrostatic charge arising in the friction zone of rubbing polymer- cotton and composite-cotton pairs on the type and content of organic-mineral fillers were studied.

Cartilage Assessment Requires a Surface Characterization Protocol: Roughness, Friction, and Function.

The surface of articular cartilage is integral to smooth, low-friction joint articulation. However, the majority of cartilage literature rarely includes measurements of surface characteristics and function. This may, in part, be due to a shortage of or unfamiliarity with fast, nondestructive, and, preferably, noncontact methods that can be applied to large cartilage surfaces for evaluating cartilage surface characteristics. A comprehensive methodology for characterizing cartilage surfaces is useful in determining changes in tissue function, as for example, in cases where the quality of cartilage grafts needs to be assessed. With cartilage storage conditions being an area of ongoing and active research, this study used interferometry and tribology methods as efficient and nondestructive ways of evaluating changes in cartilage surface topography, roughness, and coefficient of friction (CoF) resulting from various storage temperatures and durations. Standard, destructive testing for bulk mechanical and biochemical properties, as well as immunohistochemistry, were also performed. For the first time, interferometry was used to show cartilage topographical anisotropy through an anterior-posterior striated pattern in the same direction as joint articulation. Another novel observation enabled by tribology was frictional anisotropy, illustrated by a 53% increase in CoF in the medial-lateral direction compared to the anterior-posterior direction. Of the storage conditions examined, 37°C, 4°C, -20°C, and -80°C for 1 day, 1 week, and 1 month, a 49% decrease in CoF was observed at 1 week in -80°C. Interestingly, prolonged storage at 37°C resulted in up to an 83% increase in the compressive aggregate modulus by 1 month, with a corresponding increase in the glycosaminoglycan (GAG) bulk content. This study illustrates the differential effects of storage conditions on cartilage: freezing tends to target surface properties, while nonfreezing storage impacts the tissue bulk. These data show that a bulk-only analysis of cartilage function is not sufficient or representative. The nondestructive surface characterization assays described here enable improvement in cartilage functionality assessment by considering both surface and bulk cartilage properties; this methodology may thus provide a new angle to explore in future cartilage research and tissue engineering endeavors.

Experimental study on the velocity-dependent frictional resistance of a rough rock fracture exposed to normal load vibrations

Changes in shear velocity can strengthen or weaken the frictional resistance of joints/faults in natural systems, but the mechanism remains unclear. We investigated the shear behavior of a rough basalt fracture in well-controlled, repeatable shear tests under constant and dynamic normal load conditions at different shear velocities. Normal load vibrations, simulating a dynamic normal load, were applied to the upper block of a fractured basalt sample. Simultaneously, a shear load was applied to the bottom block, providing a constant shear velocity. The peak shear strength increased with increasing shear velocity under constant normal load conditions. The peak shear strength decreased at a lower shear velocity under normal load vibrations. When the shear velocity exceeded the critical value, vc, the peak shear strength increased. The apparent coefficient of friction reduced under normal load vibrations. The reduction in the dynamic coefficient of friction increased with increasing shear velocity. We identified a phase shift between the peak normal load and peak shear load with peak shear load delay (D1) and a phase shift between peak normal load and the peak coefficient of friction with the peak coefficient of friction delay (D2). D1 and D2 were dependent on the quasi-static coefficient of friction and shear velocity, and both decreased with increasing shear velocity. D1 decreased with the increasing quasi-static coefficient of friction, while D2 was almost constant with changes in the quasi-static coefficient of friction. A new shear strength criterion was proposed for a rough joint under a constant shear velocity and normal load vibrations.

The Effect of Bionic 3D Printed Structure Morphology on Skin Friction

Abstract Dragonfly has remarkable flight efficiency, with unique wing structural properties such as the surface topological vein structures, corrugation, etc. The object of this paper is to identify how the polygonal patterns of the samples with bionic wing veins affected the skin friction. Four kinds of polygonal three-dimensional (3D) patterns were designed and fabricated by additive manufacturing technology, and the skin friction coefficients (Cf) of various models were measured by the wind channel experiments. The quantitative effects of models on Cf with different Reynolds numbers (Re) in laminar, transitional, and turbulent flow conditions were obtained. Results indicated that the law of whole change of the skin friction coefficient versus Re is the same for all patterns which can be expressed by an empirical formula Cf=kReα. The model with mixed square and pentagonal patterns always generates the highest skin friction in the different flow conditions, which was speculated to play an important role on the attenuation of the flow separation of the dragonfly wing.

Characteristics of drag due to streamwise inhomogeneous roughness

Characteristics of the skin friction due to streamwise inhomogeneous roughness are investigated by modeling it in a simplified manner using step changes in equivalent sand grain roughness height ks, denoted as P (ks = 81.25 μm), Q (ks = 325.00 μm) and R (ks = 568.75 μm). The selected ks values represent the common ship-hull roughness, i.e., from light slime to about small calcareous fouling. RANS simulations are performed to study the friction characteristics of various rough surfaces formed by a trio-combination of P, Q and R roughnesses on flat plates with streamwise lengths of 30, 60, 120 and 240 m. The three combinations of roughnesses form either homogeneous (PPP, QQQ or RRR) or inhomogeneous (PQR, PRQ, QPR, etc.) rough walled turbulent boundary layer flow. A step change in roughness height results in a sudden change of the local skin friction coefficient in the form of overshoot or undershoot, followed by a relaxation where the inhomogeneous local skin friction is slowly returning to the homogeneous local one. The sequence of roughness arrangement in a streamwise inhomogeneous roughness pattern plays a key role in the resulting overall skin friction with value increasing in the following order: PQR < PRQ < QPR < QRP < RPQ < RQP. The similarity laws hold for both homogeneous and inhomogeneous rough surfaces.

The Speed Limit Determination of Tunnel Entrance And Exit Section On Rainy Days

In order to reduce the number of traffic accidents caused by rainy days on the entrance and exit sections of tunnels, a reasonable speed limit value on the entrance and exit sections of tunnels is proposed to improve the safety of vehicles on these sections. This paper takes passenger cars as the research object, and by exploring the changes of road friction coefficient and visibility in rainy weather, the driving environment parameters under different rainfall intensity are clarified. Taking the visible distance of the driver and the critical state of vehicle sideslip as safety indexes, the speed limit value of the section outside the tunnel in rainy days is obtained, and the allowable speed of the section inside the tunnel is determined by taking the continuity of the running speed as the standard. The study shows that the speed limit value of the tunnel entrance and exit section in rainy days takes visibility as the main limit index and decreases with the increase of rainfall intensity. When the rainfall intensity is greater than 1.5mm/min, the speed limit of the tunnel section is needed.

The Impact of Heat Treatment on the Behavior of a Hot-Dip Zinc Coating Applied to Steel During Dry Friction.

The analyzed topic refers to the wear resistance and friction coefficient changes resulting from heat treatment (HT) of a hot-dip zinc coating deposited on steel. The aim of research was to evaluate the coating behavior during dry friction after HT as a result of microstructure changes and increase the coating hardness. The HT parameters should be determined by taking into consideration, on the one hand, coating wear resistance and, on the other hand, its anticorrosion properties. A hot-dip zinc coating was deposited in industrial conditions (according EN ISO 10684) on disc-shaped samples and the chosen bolts. The achieved results were assessed on the basis of tribological tests (T11 pin-on-disc tester, Schatz®Analyse device, Sindelfingen, Germany), microscopic observations (with the use of optical and scanning microscopy), EDS (point and linear) analysis, and microhardness measurements. It is proved that properly applied HT of a hot-dip zinc coating results in changes in the coating’s microstructure, hardness, friction coefficient, and wear resistance.

Numerical Investigation of the Unsteady Flows in Hydropower Plants

One of the most important hazards that threatens the stability of power plant buildings is the phenomenon of water hammer, which can occur in the Penstock pipe of a turbine due to the rapid opening and closing of a valve. Fluid Descriptive Equations in this situation, there are two hyperbolic partial nonlinear partial differential equations that are very difficult and complex to solve analytically and are possible only for very simple conditions. In this study, by examining the two numerical methods of characteristic lines and implicit finite difference with Verwy &amp; Yu schema, which are widely used in the analysis of instabilities, their disadvantages and advantages are clearly clarified and a suitable comparison basis for use. They should be provided in different conditions in hydropower plant. The results of the characteristic method in terms of maximum and minimum pressure show more and less values than the implicit finite difference method. In the characteristic method, perturbations and fast wave fronts are presented with more accuracy than the implicit finite difference method. At points near the upstream, downstream and middle boundaries, the accuracy of the characteristic method in presenting pressure and flow fluctuations is higher than the implicit finite difference method. In the characteristic method, it is recommended not to use certain time steps and try as much as possible avoid interpolation by selecting the appropriate time step. The results of examining the amount of changes in coefficient of friction in both methods show that it is not correct to take its value constant (proportional to the value obtained in stable conditions) and coefficient of friction should be calculated in proportion to changes in velocity at different times and used in the governing equation.