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Nonholonomic mechanics, which relies on the method of Lagrange multipliers in the d’Alembert–Lagrange formulation of the classical equations of motion, is the standard description of the dynamics of systems subject to nonholonomic constraints. Vakonomic mechanics has been proposed as another possible approach to the dynamics of nonholonomic systems. Partial equivalence theorems have been found that provide conditions under which nonholonomic motions are also vakonomic motions. The aim of the present work is to show that these results do not cover at least one important physical system, whose motion brought about by vakonomic mechanics is completely inequivalent to the one derived from nonholonomic mechanics. For the rolling coin on an inclined plane, it is proved that no nontrivial solution to the equations of motion of nonholonomic mechanics can be obtained in the framework of vakonomic mechanics. This completes previous investigations that managed to show only that, for certain mechanical systems, some but not necessarily all nonholonomic motions are beyond the reach of vakonomic mechanics. Furthermore, it is argued that a simple qualitative experiment that anyone can perform at home supports the predictions of nonholonomic mechanics.

Flow of the thin layer of a nonlinearly viscous fluid subject to the gravity force down an inclined plane is considered. The capillary forces are assumed to be significant and rheology is described by the power law. The effect of a small periodic topography applied to the plane on the characteristics of steady-state flow and its stability is studied in the linear approximation; the calculations are carried out for the sinusoidal topography. It is shown that, regardless of rheology, the film on the plane with topography has the greater mean thickness as compared with the flat plane at the same flow rate, variation in the mean thickness being proportional to the second power of the topography amplitude. The variations in the increments of the normal modes and the critical Reynolds number are also proportional to the second power of the topography amplitude. The long-wave topography increases the critical Reynolds number, for dilatant fluids this effect being reached at the smaller periods than for pseudoplastic fluids.

Experiments have been carried out on determining the value of the moment of inertia of a hollow cylinder. This study aims to analyze the value of the moment of inertia with variations in the radius of the hollow cylinder using the Phyphox Application (Physical Phone Experiment). This research is experimental. The tools and material used are a 1-meter longboard, t hree hollow cylinders with different radius sizes but the same mass, smartphone, laptop , caliper, and b alance . The experiment of rolling motion on a hollow cylinder on an inclined plane is assisted by data processing in a Phyphox application. From the experiment, the result of the moment of inertia value from data processing is 4.89 x 10 -4 kg.m 2 , 9.82 x 10 -4 kg.m 2 , 12.4 x 10 -4 kg.m 2 . This research can be used as a teacher reference in teaching the topic of moments of inertia in physics learning at school. This is highly recommended for distance learning during the pandemic so that mastery of concepts is maximized. Further research is suggested that it can be tried to use other materials with other types of rigid bodies to find the moment of inertia. Further references on digital application media to assist learning are also needed considering the dynamic development of technology.

In the field of downhole drilling robot, dragging force, size and safety of the robot are the main indexes to evaluate the adaptability of downhole drilling robot. In fact, the characteristics of the gripping mechanism are the most important factors affecting the dragging force, safety and miniaturization of the drilling robot. Based on this status, a novel drilling robot with gripping mechanism of co-directional double inclined plane (GMCDIP) is proposed. The GMCDIP is divided into two processes which are composed of griping process of inclined plane (GPIP) and griping process of connecting rod (GPCR). Mechanical models of GPIP and GPCR are established, respectively. Based on the mechanical models, the boundaries of self-locking of the GPIP and GPCR are found. When the friction coefficient between friction block and wall of wellbore is 0.35 and friction coefficient between friction block and gripping block is 0.15. The inclination angle of the gripping block is 10.66°. And, the angle of the collect rod is 19.10°. To ensure that the drilling robot will not be stuck, the inclination angle of the gripping block is taken when gripping force = 30–50% of dragging force. Furthermore, an experimental device and a drilling robot are designed and processed. Experimental data verify the correctness of the novel design method and mechanical model of GMCDIP. The significance of this paper is to solve the problem of large dragging force, safety and miniaturization of the drilling robot simultaneously. It will promote the further research and application of the drilling robot.

Dynamics of nonhomogeneous disk on a rough support is considered. Since P. Painleve, this system becomes a popular simple model to demonstrate singularities intrinsic to multibody dynamics. It is shown that equations of motion can be ambiguous: along with continued rolling the body can jump or slide with nonzero initial acceleration. An extended model of the system is applied to stability analysis and exhibits bistability, where rolling and jump correspond to sinks, while sliding turns to be a saddle point. In previous studies, the support was treated as a horizontal plane, and such paradox was not detected. Here the cases of an inclined plane and a half-pipe are examined. In particular, such periodic motions with possible jumps are constructed, where continuous rolling is possible too. Generally speaking, “hidden” dynamics is an essential feature of multibody systems, not taken in account in straightforward computing.

Although the drainage and spreading processes of thin liquid films on substrates have received growing attention during the last decades, the study of three-dimensional cases is limited to a few studies on flat and axisymmetric substrates. In this work, we exploit differential geometry to study the drainage and spreading of thin films on generic curved substrates. We initially investigate the drainage and spreading processes on spheroidal and paraboloidal substrates by employing an asymptotic expansion in the vicinity of the pole and a self-similar approach, finding that the thickness distribution is set by the substrate metric and tangential gravity force components. Spheroids with a large ratio between height and equatorial radius are characterized by a growing thickness moving away from the pole, and vice versa. The non-symmetric coating on a toroidal substrate shows that larger thicknesses and a faster spreading are attained on the inner region than on the outer region of the torus. An ellipsoid with three different axes is chosen as a testing ground for three-dimensional drainage and spreading. Modulations in the drainage solution are observed, with a different variation of the thickness along the two axes. By imposing the conservation of mass, an analytical solution for the average spreading front is obtained. The analytical and numerical results are in good agreement. The resulting drainage solutions show also a good agreement with experimental measurements obtained from the coating of a curing polymer on diverse substrates., 49 pages, 28 figures (including Supporting Information)

Previous analyses of the flow of low-Reynolds-number, viscous gravity currents down inclined planes are investigated further and extended. Particular emphasis is on the motion of the fluid front and tail, which previous analyses treated somewhat cavalierly. We obtain reliable, approximate, analytic solutions in these regions, the accuracies of which are satisfactorily tested against our numerical evaluations. The solutions show that the flow has several time scales determined by the inclination angle, $\alpha$ . At short times, the influence of initial and boundary conditions is important and the flow is governed by both the pressure gradient and the direct action of gravity due to inclination. Later on, the areas where the boundary conditions are important shrink. This fact explains why previous solutions, being inaccurate near the front and the tail, described experimental data with high accuracy. At larger times, of the order of $\alpha ^{-5/2}$ , the influence of the pressure gradient may be neglected and the fluid profile converges to the square-root shape predicted in previous works. Important extensions of our approach are also outlined.

We demonstrate a novel instability found within unconfined viscous bands/rims, or free-surface flows involving a longitudinal viscosity contrast. Such instabilities may be described as viscous banding instabilities, non-porous viscous fingering instabilities or unconfined viscous fingering instabilities of free-surface flows involving the intrusion of a less viscous fluid into a band of more viscous fluid. A consequence of this work is that viscous fingering instabilities, widely known to occur in porous media following the seminal work of Saffman & Taylor (Proc. R. Soc. Lond. A, vol. 245, 1958, pp. 312–329), also occur in non-porous environments. Although the mechanism of the viscous banding instability is characteristically different from that of the Saffman–Taylor instability, there are important similarities between the two. The main similarity is that a viscosity contrast leads to instability. A distinguishing feature is that confinement, such as the rigid walls of a Hele-Shaw cell, is not necessary for viscous banding instabilities to occur. More precisely, Saffman–Taylor instabilities are driven by a jump in dynamic pressure gradient, whereas viscous banding instabilities, or non-porous viscous fingering instabilities, are driven by a jump in hydrostatic pressure gradient, directly related to a slope discontinuity across the intrusion front. We examine the onset of instability within viscous bands down an inclined plane, determine conditions under which viscous banding instabilities occur and map out a range of behaviours in parameter space in terms of two dimensionless parameters: the viscosity ratio and the volume of fluid ahead of the intrusion front.

This work relates to the investigation of indentation induced damage resistance in glass fiber reinforced polymer composite laminates under normal and inclined planes. Uni-directional [0] and cross-ply [0/90] glass fiber reinforced polymer composite specimens were subjected to 0° and 20° indentation loading with acoustic emission monitoring. The specimens were loaded at the centre using a hemispherical steel indenter with 12.7 mm diameter. Mechanical responses such as force–displacement behavior, absorbed energy and resulting damage area were used for the quantification of the indentation damage. Acoustic responses such as normalized cumulative counts, energy, duration and peak frequency were considered for monitoring damage progression during 0° and 20° indentation loading. The residual compressive strength of the glass fiber reinforced polymer specimens following indentation was measured by testing them under in-plane loading, once again with acoustic emission monitoring. The correlation between mechanical and acoustic strain energy was used for the evaluation of the damage severity of the laminates. The result revealed indentation damage resistance in cross-ply laminates as 65.08% and 68.01% higher than uni-directional laminates for 0°, whereas for 20°, there was a reduction in the damage resistance in cross-ply laminates to 43.27% and 57.39%. These were higher than uni-directional laminates under the displacements of 2 mm and 3 mm, respectively. The conclusion from these results was that transverse shear load at 20° inclination of laminate leads to reduction in residual compressive strength. Moreover, uni-directional glass/epoxy laminates have better residual compressive strength than cross-ply laminates for both 2 mm and 3 mm indentation displacements.

In this paper, we present a comparison of classical theory with video analysis techniques to teach kinetic and potential energy of a device with a boomerang effect as an observable and measurable concept. The device can store energy through an elastic band when it rolls downan inclined plane and can release the energy when it rolls on a horizontal surface; hence, the name Boomerang. In theoretical terms, the details of energy charge and discharge processes are analysed with Newton’s laws and Lagrangian method. The experimental results were recorded with cell phone cameras and processed with an open-source video analysis software, called ’Tracker’. The comparison shows relevant concepts about kinetic and potential energy, which can help the student to overcome some of the typical student misconceptions.

Purpose The purpose of this study is to examine the simultaneous effects of Hafnium particles and partially submerged metallic particles for the flow of bi-phase coupled stress fluid over an inclined flat plane. Design/methodology/approach An unflinching free stream flow that stretches far from the surface of the plane with the possibility of containing some partially submerged metallic particles is considered. Innovative model has been proposed and designed using Runge–Kutta–Fehlberg method. Findings The findings show that the drag force resists the couple stress fluid, whereas the Newtonian flow is supported by increasing the velocity. For both types of flows, movement of the particle is retarded gradually against the drag force coefficient. Originality/value To the best of the authors’ knowledge, this model is reported for the first time.

We consider a 2D thin film flow of a power-law fluid on an inclined plane due to gravity. The plane oscillates harmonically keeping the inclination angle constant. The longitudinal oscillations produce additional shear. We calculate the effect of these oscillations on the mean flow rate for fixed film thickness assuming that free surface is flat. Using the integral boundary layer approach, we investigate the stability of the waveless flow due to low-frequency disturbances. We show that the oscillations of the plane stabilize the flow.

The stability behavior of a visco-elastic (Upper Convected Maxwell) liquid film flowing down an inclined plane is examined using the standard normal mode linear stability analysis in creeping flow limit. The inclined plane is coated with a deformable solid layer, and the gas-liquid interface is covered with a monolayer of an insoluble surfactant. There exist three modes of instabilities for this composite fluid-solid system: (i) the free surface gas-liquid (GL) mode, (ii) the surfactant or Marangoni mode, and (iii) the liquid-solid (LS) mode. Previous studies have shown that the Marangoni mode remains stable for film flowing down a rigid inclined plane. We show that when the rigid wall is replaced by a deformable wall, the Marangoni mode becomes unstable when a non-dimensional deformability parameter, defined as G = μ V / E s R , increases above a threshold value (μ, V, Es, and R are fluid viscosity, free-surface velocity, shear modulus of soft solid layer, and fluid thickness, respectively). It is well known that the GL mode remains stable for Newtonian liquid film at zero Reynolds number, but becomes unstable for a visco-elastic film flowing past a rigid inclined plane even in creeping flow limit purely because of fluid elasticity. Previous studies have shown that the presence of soft solid layer can suppress this free surface GL mode instability. The presence of surfactant also has a stabilizing effect on this GL mode instability in rigid limit. Here, we evaluate whether a soft solid coating can be used to obtain a stable flow configuration for a surfactant-covered visco-elastic film as well particularly (i) in view of the above mentioned Marangoni mode destabilization observed solely due to wall deformability, and (ii) when the surfactant stabilizing contribution remains insufficient to suppress the free-surface instability. We demonstrate that there exists a window in terms of parameter G where the free surface GL mode instability is suppressed without exciting any other mode. We also identify the parameter regimes where it is not possible to achieve stable film flow, however, in these cases, we show that the Marangoni mode can be selectively destabilized by appropriately selecting the thickness and shear modulus of solid layer.

High depth-to-width ratio friction stir welding is an attractive method for the joining demands of aluminum profiles, which is sparked with its extremely low heat input and high mechanical performance. In this study, the joint formation mechanism was studied by a numerical model of plastic flow combined with experimental approaches. A fluid-solid-interaction algorithm was proposed to establish the coupling model, and the material to be welded was treated as non-Newtonian fluid. The thread structure and the milling facets on tool pin promoted drastic turbulence of material. The thread structure converged the plasticized material by its inclined plane, and then drove the attached material to refill the welds. The milling facets brought about the periodic dynamic material flow. The thread structure and the milling facets increased the strain rate greatly under the extremely low heat input, which avoided the welding defects. The condition of the peak temperature of 648 K and the strain rate of 151 s−1 attributed to the lowest coarsening degree of precipitate. The tensile strength of the joint reached 265 MPa, equivalent to 86% of base material. The amelioration via the material flow model inhibits the welding defects and optimizes the parameter intervals, providing references to extracting process-structure-property linkages for friction stir welding.

The water film due to rain falling on a radome surface causes severe losses in radio wave transmission. Hydrophobic coatings have been applied as a remedy to reduce the film thickness and to minimize the losses. However, quantitative accounts of the wave scattering are mostly based on empirical estimates of the film thickness. We describe a fluid-mechanical theory for the film under steady rain falling on a textured surface formed by a square array of pillars. Assuming the water surface on top of the pillars to be in the Cassie–Baxter state, the analysis is carried out by making use of the sharp contrast of length scales between the film thickness and the radome radius. The textured surface is viewed as a periodic array of cells around pillars. The macro-scale flow is simple and linear but the micro-scale flow in a typical lattice period is fully nonlinear. These two problems are coupled and are solved iteratively to obtain the slip length and the spatial variation of the film thickness. Numerical results are presented to show the effect of solid fraction on local flow field, the slip length and the non-uniform reduction of the film thickness. To examine the influence of the macro-scale geometry on film formation, the theory is also modified for a hydrophobic roof top formed by two inclined planes.

We consider a thin viscous liquid film flow down an inclined plane with sinusoidal topography. We use the integral boundary layer approach to derive governing equations. By use of Floquet theory, we study linear stability of steady-state flow. For small values of the topography amplitude, perturbation analysis gives an analytical expression for corrections to the growth rate of a normal mode disturbance. This correction is proportional to the second power of the topography amplitude. The topography can both stabilize and destabilize the flow, depending on the corrugation period and the plane inclination angle. We study the topology of neutral curves and the critical Reynolds number dependence on topography parameters.

When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwardset al.(J. Fluid. Mech., vol. 823, 2017, pp. 278–315,J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.

We investigate the stability of a thin Newtonian fluid spreading on a horizontal cylinder under the action of gravity. The capillary ridge forming at the advancing front is known to be unstable with respect to spanwise perturbations, resulting in the formation of fingers. In contrast to the classic case of a flow over an inclined plane, the gravity components along a cylindrical substrate vary in space and the draining flow is time-dependent, making a modal stability analysis inappropriate. A linear optimal transient growth analysis is instead performed to find the optimal spanwise wavenumber. We not only consider the optimal perturbations of the initial film thickness, as commonly done in the literature, but also the optimal topographical perturbations of the substrate, which are of significant practical relevance. We found that, in both cases, the optimal gains are obtained when the perturbation structures are the least affected by the time horizon. The optimal spanwise wavenumber is found to be dependent on the front location, due to the dependence of the characteristic length of the capillary ridge on its polar location.

Many studies use an inclined ice plane to determine the effect of different materials and processing conditions on sliding over an ice surface. Experiments measure the sliding time at different stages on the track to determine the sliding speed and the coefficient-of-friction. Here, the sliding body vibrations and, the oscillations in axial directions are addressed. This paper analyzes the sample’s acceleration by attaching a portable High Sensitivity 1.25g USB Accelerometer and compares the data with 3D CAM modeling results to identify the causes for interrupted sliding.

Rapid shallow granular flows over inclined planes are often seen in nature in the form of avalanches, landslides and pyroclastic flows. In these situations the flow develops an inversely graded (large at the top) particle-size distribution perpendicular to the plane. As the surface velocity of such flows is larger than the mean velocity, the larger material is transported to the flow front. This causes size segregation in the downstream direction, resulting in a flow front composed of large particles. Since the large particles are often more frictional than the small, the mobility of the flow front is reduced, resulting in a so-called bulbous head. This study focuses on the formation and evolution of this bulbous head, which we show to emerge in both a depth-averaged continuum framework and discrete particle simulations. Furthermore, our numerical solutions of the continuum model converge to a travelling wave solution, which allows for a very efficient computation of the long-time behaviour of the flow. We use small-scale periodic discrete particle simulations to calibrate (close) our continuum framework, and validate the simple one-dimensional (1-D) model with full-scale 3-D discrete particle simulations. The comparison shows that there are conditions under which the model works surprisingly well given the strong approximations made; for example, instantaneous vertical segregation.

In this letter a deep learning-based model is proposed for online inspection of silicone rubber outdoor insulators. The inclined plane tracking and erosion test is used as per ASTM D2303 in order to simulate standard erosion on silicone rubber insulation composites. Photos taken for the tested composites are used as training and testing inputs for a convolutional neural network topology in the proposed deep learning model, thereby classifying the degree of erosion damage into light, moderate and severe. The remarkable classification accuracy obtained shows the potential of utilizing the proposed framework for online monitoring of outdoor silicone rubber insulators in the transmission and distribution grid.

International audience; We study experimentally the primary instability of a visco-plastic film flow down an inclined plane. The experimental setup is a channel with a varying slope angle, in which a permanent flow of a Herschel-Bulkley fluid (carbopol or kaolin) is established. Controlled perturbations are imposed at the entrance of the channel to generate surface waves, and their downstream evolution is observed with a laser sensor system measuring the local fluid thickness. Growth rates and cutoff frequencies are obtained after processing the thickness signal, and experimental critical Reynolds and Bingham numbers are deduced. We find that the experimental stability map obtained is well described by the pseudo-plug model of Balmforth & Liu (2004), a model obtained after neglecting the film thickness compared to its length. This is not the case for dispersion effects (growth rates, cutoff frequencies, phase speeds), for which a more accurate model is needed.

The net erosion-deposition rate of a snow avalanche is fundamental to the dynamics of the flow and in determining its growth or decay. Small-scale experiments are performed in this paper that are analogous to a finite mass release on a marginally stable surface layer of snow. An avalanche is initiated by releasing a given volume of yellow sand onto a stationary erodible layer of the same material, but coloured red, which coats the rough bed of an inclined plane. Depending on the erodible layer depth and the slope angle, the resulting avalanche is found to either decay, grow, propagate steadily or rapidly shed grains to produce secondary avalanches. The use of different coloured sand with identical frictional properties shows that a particle exchange occurs during erosion from the substrate and deposition by the avalanche, which eventually results in a flow that is comprised entirely of particles from the stationary layer rather than the initial release. It is notoriously difficult to model the erosion and deposition processes in granular flows such as these. Nonetheless, it is shown in this paper that a two-dimensional shallow-water-like avalanche model, together with a basal friction rule that is responsible for handling the transition between stationary and moving regions of the flow, is able to qualitatively reproduce of all the observed avalanche behaviours. The model is solved numerically together with equations governing the three-dimensional motion of individual particles, which allows the positions of specific grains to be tracked. The results illustrate how a continuous exchange of particles with the substrate layer is fundamentally important to the propagation of such avalanches. An investigation into long distance propagation distance behaviour reveals that avalanches can reach a steady state, the size and speed of which are independent of the initially released volume but are set by the slope angle and basal layer thickness. In certain conditions, including an abundance of erodible material available, avalanches can sometimes grow to steady states that are significantly more massive than the flows from which they are originally formed. This paper therefore demonstrates the dominant effect that erosion and deposition may have, and the importance of including these effects correctly in operational forecast models of snow avalanches and geophysical mass flows.

This paper reports the dependences that have been derived to determine the effective width of a free flange in a dangerous cross-section of the wide-flange hull girder with the breaking of the wall/edges/axis at elastic-plastic deformation depending on the applied load for a perfectly plastic material without strengthening. Currently, there are no systematic dependences to determine the effective width of the free flange of girders of this type, except for certain cases. The technique is suitable for use for both purely elastic and elastic-plastic deformation. To calculate the stressed-strained state (SSS), a finite-element method (FEM) was used to solve the three-dimensional problem from the elasticity and plasticity theory. It has been shown that the node is exposed to simple loading. The reported results were derived within the framework of the deformation theory of plasticity. The largest ratio of external load to the boundary of fluidity is 0.9. The estimation scheme takes into consideration the most unfavorable working conditions of the examined node when the safest data are acquired. The dependences were built for the theoretical coefficient of concentration in a dangerous cross-section, which is used in the proposed procedure to determine the moment of transition from the elastic stage of deformation to elastic-plastic. When determining the effective width, the complex work of the flange and its deplanation was taken into consideration by defining the SSS components in the median plane. The feasibility of the idea of designing the SSS components on the inclined plane of a free flange has been proven. In this case, there is practically a (quasi) flat stressed state, suitable for the application of classical methods to determine the effective width. The proposed technique simplifies the calculations of the strength of the examined girder.

The ability to robustly and efficiently control the dynamics of nonlinear systems lies at the heart of many current technological challenges, ranging from drug delivery systems to ensuring flight safety. Most such scenarios are too complex to tackle directly and reduced-order modelling is used in order to create viable representations of the target systems. The simplified setting allows for the development of rigorous control theoretical approaches, but the propagation of their effects back up the hierarchy and into real-world systems remains a significant challenge. Using the canonical setup of a liquid film falling down an inclined plane under the action of active feedback controls in the form of blowing and suction, we develop a multi-level modelling framework containing both analytical models and direct numerical simulations acting as an in silico experimental platform. Constructing strategies at the inexpensive lower levels in the hierarchy, we find that offline control transfer is not viable, however analytically-informed feedback strategies show excellent potential, even far beyond the anticipated range of applicability of the models. The detailed effects of the controls in terms of stability and treatment of nonlinearity are examined in detail in order to gain understanding of the information transfer inside the flows, which can aid transition towards other control-rich frameworks and applications., Comment: 19 pages, 8 figures

We explore flow of a completely wetting fluid in a funnel, with particular focus on contact line instabilities at the fluid front. While the flow in a funnel may be related to a number of other flow configurations as limiting cases, understanding its stability is complicated due to the presence of additional azimuthal curvature, as well as due to convergent flow effects imposed by the geometry. The convergent nature of the flow leads to thickening of the film, therefore influencing its stability properties. In this work, we analyse these stability properties by combining physical experiments, asymptotic modelling, self-similar type of analysis and numerical simulations. We show that an appropriate long-wave-based model, supported by the input from experiments, simulations and linear stability analysis that originates from the flow down an incline plane, provides a basic insight allowing an understanding of the development of contact line instability and emerging length scales.

A new type of rigid-flexible coupling three degrees of freedom (3-DOF) micro-positioning platform with high positioning accuracy and high bearing capacity is developed, which consists of flexible drive mechanism and rigid platform. The flexible drive mechanism consists of three sets of symmetrical parallel round flexible hinge structures, each with a wedge structure in the middle of the symmetrical parallel flexible hinge. The rigid platform has an inclined plane with the same angle as the wedge, while the wedge structure is used to achieve the self-locking effect. The flexibility matrix method and ANSYS are used to analyze the statics of the flexible drive mechanism. The first four natural frequencies of the platform are obtained by dynamic simulation analysis. A symmetrical rigid flexible coupling micro-positioning platform experimental system is developed. Output characteristics, positioning accuracy, relationship between frequency and amplitude, and bearing performance of the micro-positioning platform are tested. These experimental results obviously show that the micro-positioning platform has good motion characteristics, high positioning accuracy, large movement distance, and large load bearing capacity performance.

We analyse the near-rolling motion of nonwetting droplets down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau (Phys. Fluids, vol. 11, 1999, pp. 2449), we focus upon the limit of small Bond numbers, where the drop shape is nearly spherical and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. In that region, where the fluid interface appears flat, we obtain an analytical approximation for the flow field. By evaluating the dissipation associated with that flow we obtain a closed-form approximation for the drop speed. This approximation reveals that the missing prefactor in the Mahadevan–Pomeau scaling law is (3π/16)p 3/2 ≈ 0.72 — in good agreement with experiments. An unexpected feature of the flow field is that it happens to satisfy the no-slip and shear-free conditions simultaneously over both the solid flat spot and the mobile fluid interface in its vicinity. Furthermore, we show that close to the near-circular contact line the velocity field lies primarily in the plane locally normal to the contact line; it is analogous there to the local solution in the comparable problem of a two-dimensional rolling drop. This analogy breaks down near the two points where the contact line propagates parallel to itself, the local flow being there genuinely three dimensional. These observations illuminate a unique ‘peeling’ mechanism by which a rolling droplet avoids the familiar non-integrable stress singularity at a moving contact line.

Exact solutions for steady fully developed dry granular and saturated granular–liquid mixture flows down rough inclined planes are presented with an emphasis on nonlocal effects. The partic...

A small target is an obstacle situated along the trajectory path of a moving drop, and when the drop hits it, a consequent noticeable reduction in the drop velocity occurs. In the present study we investigate the evolution of the velocity and the shape of a free falling drop impacting the upper disc-shape surface of an up-right cylindrical rod. Drops of water are created by a drop generator and some color is added to enhance visualization. The drops’ impact and propagation were traced, using two high-speed cameras. The cameras were positioned to capture the side and top views of the drop impact. It is shown that the drops crawl downwards along the cylinder in a deccelerating manner while wetting and coating the rod surface. We use a small target with a divided impact surface to enable a simple visual comparison between the simultaneous impact outcomes on normal and on inclined surfaces. We show that while the combined target divides the drop into two portions, each portion continues to move independently from its counterpart in the same manner as on the undivided target. The experimental observations show that each portion is then split into two segments; one segment hits the target and the other misses it. Thereafter the two segments coalesce and crawl together downward. Off-center hitting has also been studied and analyzed by using an extended version of a simple energy conservation equation which was modified to take into account the target plane inclination.

In this work is presented a discussion about the Galileo’s inclined plane showing aspects from its evolution and scientific development together with a didactical proposal to teaching move concepts from learning sciences. After an introduction of physicalmathematics concepts, it is showing the construction of an experimental aparatus with Mindstorms NXT LEGO© pieces and light sensors to measure time steps in mechanical experiments. The experiment consists of measuring the time of rolling a ball along a plane with different degrees of inclination. The getting and analysis of space and time data allow an assessment of the movement. The experimental setup allows a direct comparison between Newtonian mechanics considering the existence of a pure rolling (without sliding) showing the friction force as a precursor of the rotation of the sphere with the equation of uniform motion (without bearing). The comparison suggests that there is no pure rolling. Taking into account the easy handling of the equipment involved and the method used, it is believed that he can help spread the use of technology in teaching Physics.

In this investigation, an answer is given to the question of whether Newton’s law of motion is of integer or non-integer, i.e., fractional, order differential form. The answer is given by seeking Newton’s law of motion in the form of a fractional differential operator. Then, applying an identification procedure using separately virtual Galileo’s experimental data on the inclined plane and Kepler’s laws of planetary motion, the fractional differential operator is established yielding the equation of motion. Both identifications yield the law of motion in the form of a fractional differential equation, which is converted into a second-order differential equation, verifying thus that for a body with constant mass Newton’s law of motion is indeed of integer differential form.

Studies have found that milling with an inclined plane when using ball-end milling cutters can reduce heating of the cutting tool. It has also been found that micro-textures on the rake face can play a role in the reduction of friction and wear. Taking these things together, milling titanium alloy with micro-textured ball-end milling cutters with workpieces at an inclined angle can have a significant influence on tool wear and the surface quality of machined workpieces. We have therefore looked at the factors affecting the quality of machined surfaces using an orthogonal test and, in particular, the effect of different micro-texture parameters and different workpiece inclination angles. Through means of an extreme difference analysis, the primary and secondary relationship between micro-texture parameters, workpiece inclination angles, and surface roughness is assessed. The analysis focuses in particular on the distance between micro-pits, micro-pit diameter, the workpiece inclination angle, the distance between the first row of micro-pits and the cutter edge, and the depth of the micro pits. On the basis of the analysis, a model that can predict surface roughness is established that provides a means of optimizing micro-textured cutter design and machining inclination angles when milling titanium alloy.

When a droplet is placed on an inclined plane, it can slide down the plane by gravitational force. This phenomenon can occur in many industrial applications such as surface coating and printing, as well as in evaluating the characteristics of a surface. The slippery droplet can be analyzed from the balance between the gravitational force and the sum of the capillary and resistance forces. Researchers have suggested various equations of force balance with several hypotheses. To increase the reliability of the force balance correlation, it is important to obtain an accurate inside velocity profile and the three-dimensional (3D) shape of a droplet. In this paper, an analysis system of the slippery droplet was developed which includes optical correction based on a ray tracing method for velocity field measurement and an ellipse fitting method for droplet shape reconstruction. These correction methods were verified by a numerical simulation and applied to a slippery droplet on an inclined plate coated with Poly methyl methacrylate (PMMA). The wall shear force calculated from the velocity field and that calculated from the force balance were in good agreement.

As a high efficient machining technology, five-axis flat end milling is applied to free-form surface more widely. Cutter workpiece engagement (CWE) is an important issue for studying the milling mechanics. In this paper, milling force prediction model for five-axis flat end milling of free-form surface is presented based on a new analytical CWE model. Free-form surface is discretized into a series of tiny inclined plane. Three-axis flat end milling of inclined plane being study object, considering that the depth of cut is stacked by a series of thin cutting layers, the parametric formulas of CWE boundary curves can be obtained by calculating the intersection between cutter geometry and each thin cutting layer. The CWE of five-axis flat end milling of inclined plane can be transformed from three-axis flat end milling of inclined plane. The effective boundary points of in-cut cutting edge are derived by intersecting the analytical boundary curves of CWE with the parameter formula of cutting edge. Then combined with elemental cutting theory, this paper establishes force model of five-axis flat end milling of inclined plane as well as free-form surface. The experiments and simulations of CWE of five-axis milling show that the results of analytical method are in reasonable agreement with those of experiments and only a slight error with those of solid modeling method. Further, the influence rules of curvature radius of tool path on the accuracy of analytical CWE model are explored. Arranged milling experiments show that predicted milling forces are in reasonable agreement with measured forces in trend and amplitude, validating the effectiveness of force model for five-axis flat end milling of free-form surface.

A mathematical modeling of geometric and kinematic relations has been made in respect of an operational zone for one of the standard machine tool sections which is used for simultaneous double-sided abrasive processing of highly-accurate lenses with a small rigidity (with a thin centre) under free lapping conditions. An analytical expression has been obtained for calculation of a sliding velocity in an arbitrarily selected point either on a surface to be processed or on a processing surface. As the proposed technology for simultaneous double-sided processing presupposes oscillatory motion of only processing tools then in order to eliminate a joint opening (a local contact fault between lapping surfaces of a tool and a work-piece) length of a drive piece must be not less than a specified value. In this case a convex tool is rigidly connected with a drive piece and it makes a reversing rotary motion (an oscillatory motion) around a centre of the processed spherical surface and a hinged joint of the centre with an output element of the technological equipment actuation mechanism is realized by transition of the drive piece ball end with a spherical seat in the output unit. In order to reveal analytical dependence of tool drive piece length on radius value of the processed spherical lens surface and friction coefficient in the contact zone of the tool and a work-piece the paper has considered a flow pattern of force while processing concave surfaces of lenses having small radius of curvature in case when the tool is positioned at the top. The friction coefficient included in the obtained expression has been determined for grinding while using suspensions of М40, М28, М10 micro-powders in a cast-iron grinding instrument and polishing while using polyrhythm suspension in a pitch and urethane-foam polisher. A method of the inclined plane has been used in this case and following the method a work-piece of optical glass has been initially lapped to the tool with the help of the appropriate abrasive suspension and the required angle has been determined at the moment when the work-piece started its sliding movement along the tool surface.

Milling force, closely related to machining efficiency and machining quality, is the most important technical parameter in computer numerical control machining. Aiming at five-axis flat-end milling, this paper proposes a milling force prediction model with arbitrary feed direction and cutter workpiece engagement (CWE). CWE, defining the interaction region between cutter and workpiece, is the basis for studying statics and dynamics of cutting. The efficiency of traditional solid modeling method is low and not suitable for large complex sculptured surface. Z-map method is also difficult to apply in five-axis flat-end milling. In response to this issue, analytic CWE model for five-axis flat-end milling of inclined plane is established by calculating the intersection of spatial surfaces; then, an algorithm is proposed to investigate the in-cut cutting edge. Different from the conventional slot cutting and half slot cutting processing, non-slot cutting processing with changing arbitrarily of cutter axis direction is used to calibrate the coefficients which are more suitable for five-axis machining. Combined with the instantaneous chip thickness considering cutter run-out, the cutting force coefficients which have quartic polynomial relationship with instantaneous chip thickness and the cutter run-out parameter identification model are established. The validity of this analytic model has been proved by experimental tests. The results also show that the measured milling force and predicted milling force are in good agreement in the aspect of the amplitude and trend in the different cutting depth, path interval, feed per tooth, and slope inclination angle, which verifies the efficiency of the coefficient identification model.

Linear stability of liquid and gas counterflows in an inclined channel is considered. The full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode (corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase. The wave length and the phase velocity of neutral disturbances of both modes are calculated as functions of the Reynolds number for the liquid. It is shown that these dependences for the surface mode are significantly affected by the gas velocity.

The main objective of this study was to obtain new experimental data for conditions not previously tested for discharging two-phase flow through two 6.35 mm diameter branches with centerlines falling in an inclined plane. The present results are relevant to many industrial applications including headers and manifolds, multichannel heat exchangers and small breaks in horizontal pipes. In the experimental investigation, the critical heights for the onsets of liquid and gas entrainment (OLE and OGE, respectively) were obtained, analyzed and correlated for two different branch spacings and two different angles between the branches. For each combination of branch spacing and angle between the branches, a wide range of Froude numbers was used. Two-phase mass flow rate and quality results were also obtained and analyzed for a range of interface heights for 16 different combinations of branch spacing, inclination angle, test section pressure and pressure drop across each branch. New empirical correlations were developed to predict the dimensionless mass flow rate and quality. The new correlations show good agreement with the present data and with previous correlations.

The paper deals with searching for the phenomenological model of a fibre. A multibody model of a simple weight-fibre-pulley-drive mechanical system is created. This mechanical system is focused on the investigation of a fibre behaviour and experimental laboratory measurements are performed on it. The carbon fibre, which is driven by one drive, is led over a pulley. On its other end there is a prism-shaped steel weight, which moves in a prismatic linkage on an inclined plane. Drive excitation signals can be of different shapes with the possibility of variation of a signal rate. Time histories of the weight position and of the force acting in the fibre are measured. At simulating with the multibody model of this system there is investigated the influence of dependences of the fibre spring-damper coefficients on the velocity of the weight motion in the computational model, on the coincidence of the simulation results and the experimental measurement results are evaluated. The aim of the simulations is to create a phenomenological model of the fibre, which will be utilizable in fibre modelling in the case of more complicated mechanical or mechatronic systems.

© 2017 Elsevier Ltd Increasing the inclination of a rigid surface that is impacted by a collimated granular flow reduces the fraction of granular matter momentum transferred to the surface. Recent studies have shown that the momentum reduction depends upon a frictional interaction between the granular flow and the impacted surface. High coefficient of friction surfaces suffer significantly more momentum transfer than predicted by resolution of the incident momentum onto the inclined plane. This discovery has raised the possibility that inclined surfaces with very low friction coefficients might reduce the impulsive transferred by the impact of high velocity granular matter. Here the use of a lubricated sliding plate is investigated as a means for reducing interfacial friction and impulse transfer to an inclined surface. The study uses a combination of experimental testing and particle-based simulations to investigate impulse transfer to rigid aluminum surfaces inclined either perpendicular or at 53° to synthetic sand that was impulsively accelerated to a velocity of 350–500 m/s. The study shows that impact of this sand with lubricated plates attached to an inclined surface rapidly accelerates them to a velocity of about 55–70 m/s, and reduces the impulse transferred to the inclined surface below. The reduction of impulse by this approach is comparable to that achieved by changing the inclination of the surface., This research was funded by the Defense Advanced Research Projects Agency (DARPA) under grant number W91CRB-11-1-0005 (Program manager, Dr. J. Goldwasser).

This paper studied the aerodynamic characteristics of cylinder and pantograph based on large eddy simulation and the acoustic boundary element method, compared sound pressure levels of different observation points and adopted measures to reduce radiation noises. Pressure distributions on the upper and lower surfaces of the cylinder were symmetrical, and radiation noises were basically the same too. Sound pressure level in the front of the cylinder was more than that on the upper and lower surfaces of the cylinder. Sound pressure level at the back of the cylinder was the maximum. With the increase of the computational time, shedding vortexes gradually developed backwards, and the influence area was bigger. The radius of radiation sound field of the cylinder was obviously reduced after applying porous materials. Additionally, contours for noise were highly symmetrical in the whole analyzed frequency band. At 2000 Hz, the cylinder presented the obvious characteristics of dipole noise source. The aerodynamic drag and lift of pantographs through numerical computation were similar to experimental results and increased with the increase of flow velocity, which showed that the computational model for the aerodynamic characteristics of pantographs in this paper was effective. Pressure was large in the area close to the pantograph head, in the connection position between upper and lower pull rods and near the base. Long dragged airflow was formed at the pantograph head, upper and lower pull rods and at the back of the base, which had a serious influence on the distribution of aerodynamic noises of pantographs. Obvious shedding vortexes were formed in the area far away from the pantograph rather than near the pantograph. The density and strength of shedding vortexes were not large. Sound pressure levels of observation points which had the same distance from the center of pantograph base were basically the same in change trend, value and peak frequency. However, the sound pressure level of observation points around the pantograph head was obviously greater than that of other observation points when the analyzed frequency was over 1000 Hz. Radiation noises in the connection position between upper and lower pull rods were greater than that of other observation points. With the increase of the analyzed frequency, the dispersive-ness of aerodynamic noises of pantographs was increasingly obvious. The radiation noise of pantographs was mainly spread along the inclined plane of 45°. When a layer of porous materials was covered on the surface of pantographs, the influence area of radiation noises was obviously reduced. In addition, the sound pressure level of radiation noises was improved.

Motion analysis and control of a pendulum-driven spherical robot (PDSR) on an inclined plane with a variable slope is investigated. Firstly, the mathematical model of a PDSR on a variable-slope inclined plane is deduced applying a Lagrangian formulation. Afterwards, in the presence of an unknown external disturbance, the terminal sliding mode control (TSMC) technique is employed to stabilize the robot on the inclined plane, while the plane is still moving. In other words, the terminal sliding mode disturbance observer is used to estimate the unknown disturbance. Based on the disturbance estimation, the TSMC scheme is established to control the single-input and single-output nonlinear system with control singularity and an unknown nonsymmetric control input saturation. In fact, a compound disturbance is defined and estimated, which includes the external disturbance, the control singularity and the unknown input saturation. Simulations are then conducted to validate the proposed approach for motion control of a PDSR on a variable-slope inclined plane with an unknown external disturbance and nonsymmetric input limits.

In this paper, a theoretical study of the effect of surface roughness on the static characteristics of inclined plane slider-bearing lubricated with Rabinowitsch fluid is analysed. Christensen’s st...

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.

Liquid film flow along an inclined plane featuring a slit, normal to the main direction of flow, creates a second gas–liquid interface connecting the two side walls of the slit. This inner interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. The exact liquid configuration is determined by employing the Galerkin/finite element method to solve the two-dimensional Navier–Stokes equations at steady state. The interplay of inertia, viscous, gravity and capillary forces along with the substrate wettability and orientation with respect to gravity and the width of the slit determine the extent of liquid penetration and free-surface deformation. Finite wetting lengths are predicted in hydrophilic and hydrophobic substrates for inclination angles more or less than the vertical, respectively. Multiple steady solutions, connected by turning points forming a hysteresis loop, are revealed by pseudo-arclength continuation. Under these conditions, small changes in certain parameter values leads to an abrupt change in the wetting length and the deformation amplitude of the outer film surface. In hydrophilic substrates the wetting lengths exhibit a local minimum for small values of the Reynolds number and a very small range of Bond numbers; when inertia increases, they exhibit the hysteresis loop with the second limit point in a very short range of Weber numbers. Simple force balances determine the proper rescaling in each case, so that critical points in families of solutions for different liquids or contact angles collapse. The flow inside the slit is quite slow in general because of viscous dissipation and includes counter-rotating vortices often resembling those reported by Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18). In hydrophobic substrates, the wetting lengths decrease monotonically until the first limit point of the hysteresis loop, which occurs in a limited range of Bond numbers when the Kapitza number is less than 300 and in a limited range of Weber numbers otherwise. Here additional solution families are possible as well, where one or both contact points (Cassie state) coincide with the slit corners.

Nanoindentation has been used intensively during the last decades to characterize experimentally the elastic and plastic material properties of phases at the microscale. Accompanying simulations have investigated the plastic mechanisms during nanoindentation. While experiments and simulations have led to a thorough understanding of most mechanisms during nanoindentation, the plasticity on positively and negatively inclined slip planes is still not completely clear. In this work, {1 0 0}-, {1 0 1}- and {1 1 1}-grains of an austenitic stainless steel are indented to better understand the dislocation mediated plasticity through slip step analysis. We observe that slip occurs on positively and negatively inclined slip planes during nanoindentation and we propose methods to differentiate between both types of planes. We find that slip steps on positively inclined slip planes form preferentially during the early stage as compared to the formation of slip steps on negative inclination, which occurs during the later deformation stage due to the change in surface topography. By calculating the resolved shear stress in the presence and absence of pile-ups, we reveal the origin of slip on positively and negatively inclined planes as well as the reason for the sequence of occurrences. We conclude that accounting for the surface topography evolution in experiments and simulations is essential in predicting the plastic slip activation during nanoindentation. Keywords: Nanoindentation, Slip step, Plasticity, Austenitic stainless steel

This paper constitutes the latest part of an investigation into the vibration-induced instability of a fluid flowing down an inclined plane. Paper I [H. Garih et al., “Detailed analysis of the vibration induced instability of a liquid film flow,” Phys. Fluids 25(1), 014101 (2013)] involved an in-depth look at the development and characteristics of the vibration-induced instabilities via bespoke linear stability analysis via spectral methods in the case of a fluid flowing down an inclined plane. Paper II [H. Garih, J. L. Estivalezes, and G. Casalis, “On the transient phase of the Faraday instability,” Phys. Fluids 25(12), 124104 (2013)] involved solution of the problem numerically via 3-D direct numerical simulation (DNS) simulations and a study of the effect of initial conditions on the transient phase of instability development in the case of a flat horizontal receptacle with no fluid flow. Paper III [H. Garih et al., “Vibration-induced instability of a fluid film flowing down a vertically inclined plane: Experimental and theoretical comparison,” Phys. Fluids 29(10), 104103 (2017)] involved validating the theory of linear stability as applied to a fluid flowing down a vertically inclined plane via an experimental comparison. In this latest part, a two-dimensional direct numerical simulation is carried out for the case with air flow at the fluid interface. The numerical solver was compared to the experiment by evaluating the frequency spectra at specific forcing amplitudes where mode 2 and 3 instabilities become dominant. The simulation reproduced the principal features of the frequency spectra in all cases to a high degree of accuracy and demonstrates that a consideration of a two-dimensional case is sufficient to accurately resolve the onset and growth of higher mode nonlinear instabilities without the need to account for three-dimensional effects.This paper constitutes the latest part of an investigation into the vibration-induced instability of a fluid flowing down an inclined plane. Paper I [H. Garih et al., “Detailed analysis of the vibration induced instability of a liquid film flow,” Phys. Fluids 25(1), 014101 (2013)] involved an in-depth look at the development and characteristics of the vibration-induced instabilities via bespoke linear stability analysis via spectral methods in the case of a fluid flowing down an inclined plane. Paper II [H. Garih, J. L. Estivalezes, and G. Casalis, “On the transient phase of the Faraday instability,” Phys. Fluids 25(12), 124104 (2013)] involved solution of the problem numerically via 3-D direct numerical simulation (DNS) simulations and a study of the effect of initial conditions on the transient phase of instability development in the case of a flat horizontal receptacle with no fluid flow. Paper III [H. Garih et al., “Vibration-induced instability of a fluid film flowing down a vertically inclined plane...