Trochoidal Curves Research Articles

Generating high-power Lissajous structured modes and trochoidal vortex beams by an off-axis end-pumped Nd:YVO4 laser with astigmatic transformation.

High-power structured beams with the transverse morphologies as the Lissajous figures are generated by an Nd:YVO4 laser under two-dimensional off-axis pumping. By fine-tuning the cavity length in the neighborhood around the condition of longitudinal-transverse coupling, different cases of accidental degeneracies from the intracavity astigmatism are achieved to lead the output emission to be various Lissajous modes with different transverse frequency ratios. The generated Lissajous modes reveals good power performance with slope efficiency up to 47% and optical-to-optical conversion efficiency to be higher than 37.5% at a pump power of 16 W. Moreover, by applying beam transformation via a single-lens astigmatic mode converter, the generated Lissajous modes are further converted into structured vortex beams with transverse patterns localized on the trochoidal curves. The transformed trochoidal vortex beams are confirmed to preserve well-defined mode structures even when the average output power has been scaled up to be higher than 4 W.

Fatigue Fracture Behaviour of Asymmetric Spur Gear Tooth Under Cyclic Loading

When the research on the gears are investigated, it is seen that there are different studies on the design and analysis. In the majority of research studies, the tooth root curve is designed as a trochoid curve as a results of production method. According to literature, application of the circular fillet method instead of the trochoid curve in the tooth root would increase the strength of the tooth root.Fatigue life will also be positively affected with increase of tooth strength. Because the less stress will develop in the tooth root, the fatigue life of the gear tooth will increase. In this study, we have designed and manufactured involute profile asymmetric spur gears using the circular fillet method in tooth root region.Involute profile asymmetric gears tooth have lower contact stress and superior tooth root strength than symmetric gears tooth. In this work, fatigue damages on asymmetric gear tooth caused by cyclic loads and effects of material hardness on fatigue life of gear tooth were investigated.In the extent of the study, a new single-tooth bending fatigue test apparatus (STBFT) was developed to investigate the fatigue performance of the gears. Low-cycle and high-cycle tests were done to detect the fatigue performance of the asymmetric gears.

Geometric Pattern Formation on a Plane Under a General Graph Topology

This paper deals with the problem of geometric pattern formation by a group of autonomous agents. The patterns are formed by the trajectories of the agents modelled with single integrator kinematics. To achieve this, a distributed control law, which is an extension of the consensus protocol, is presented. The control law allows the freedom to generate different circular formations or trochoidal curves, for any underlying graph topology, which represents the interaction among the agents. The existence of spanning tree is the only condition that the graph needs to satisfy. The patterns is achieved by manipulating the location of the eigenvalues of the overall system in order to achieve the desired pattern. The patterns are annular which can either be closed or can cover the entire space. The patterns are not only aesthetically appealing but can also be used in civilian applications like search, exploration and surveillance. We validate the control law with various examples that present appealing patterns.

Spherical gearing with intermediate ball elements: parameter ranges with a high contact ratio

The paper presents analytical research of the geometry and kinematical parameters of spherical gearing with ball intermediate elements. The main attention is paid to the influence of the offset coefficient on the tooth geometry generation, the contact ratio and the motion transmission angle. Intermediate ball element racetracks on the gear are trochoidal curves on a spherical surface. Two areas for the offset coefficient values providing a high value of the contact ratio – basic trochoid (without offset) and prolate trochoid with abutting racetracks of adjacent ball elements ― were revealed. Analysis of the investigated parameters showed that for power transmission, it is preferable to use spherical gearing without an offset, and for kinematic transmission, it is possible to use profiles with a large offset. The present study allows making a rational choice of geometrical parameters depending on the transmission predestination.

The Mechanism of Surface Trajectories and Milling Forces in Micro-Milling Processing

In practical industrial processing, in order to make the tiny functional devices assembly accuracy as high as possible, which raises an very high requirement for surface quality of micro workpiece. Some of the important factors in controlling the surface quality with the level of the trajectories and forces of the blade across on the workpiece have a great relationship. This paper presents generalized mathematical models of the trochoid and forces combined with the helix cutter geometry used in the industry. And then from both of these aspects analytical laws in-depth of the trochoidal curves and micro-cutting forces variation. It will be of great reference value for the study of micro-milling mechanism.

Parametric multi-axis tool path planning for CNC machining: the case of spur gears

Four and five axis machining strategies are difficult to standardise in the general case. General purpose CAM systems are an important tool allowing the experienced machinist to experiment with machining strategies and assess quality of the resulting tool paths. However, in the case of machine elements, such as gears, shape definition by parametric equations is most inviting for development of fit-for-purpose machining strategies and corresponding parametric tool paths without the use of CAM software. Such an approach is materialised in this paper for spur gears, drawing on discretisation of involute and trochoidal curves constituting the tooth profile, so as to satisfy scallop height constraints. The roughing stage is performed with a flat end mill, implementing slot milling along a meander type tool path. The finishing stage is performed with a bull-nose end mill, implementing peripheral milling circumferentially around the full gear profile. The algorithms were tested on a three-axis machining centre equipped with an additional 4th rotary axis to manufacture a plastic spur gear.

Analysis of Milling Forces in Slot Milling of Aluminum Alloy with Three Flutes Cutter

A new finite element simulation model based on two trochoidal tool-paths was constructed, and was used to simulate a full immersed milling with three flutes cutter. In this FEM model, the special method used here is to construct two trochoidal curves at the beginning with an offset value which is just equal to the feed value per tooth. These two linked trochoidal curves represent tool paths generated by two flutes respectively so as to construct the workpiece model. Cutting forces in the feed direction and in the normal direction were analyzed in detail. A verifying experiment shows this simulation is credible in general. This method can also be applied into situations of milling with more flutes such as face milling.

Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations

The geometry of classical dynamics in coupled oscillators with SU(2) transformations is explored and found to be relevant to a family of continuous-transformation orbits between Lissajous and trochoidal curves. The quantum wave-packet coherent states are derived analytically to correspond exactly to the transformation geometry of classical dynamics. By using the quantum wave-packet coherent states derived herein, stationary coherent states are constructed and are shown to possess spatial patterns identical to the transformation geometry between Lissajous and trochoidal orbits.

Modeling of Current Distribution on Smooth and Columnar Platinum Structures

Studying the growth and stability of anisotropic or isotropic disordered surfaces in electrodeposition is of importance in catalytic electrochemistry. In some cases, the metallic nature of the electrode defines the topography and roughness, which are also controlled by the experimental time and applied external potential. Because of the experimental restrictions in conventional electrochemical techniques and ex situ electron microscopies, a theoretical model of the surface geometry could aid in understanding the electrodeposition process and current distributions. In spite of applying a complex theory such as dynamic scaling method or perturbation theories, the resolution of mixed mass-/charge-transfer equations (tertiary distribution) for the electrodeposition process would give reliable information. One of the main problems with this type of distribution is the mathematics when solving the spatial n-dimensional differential equations. Use of a primary current distribution is proposed here to simplify the differential equations; however it limits wide application of the first assumption. Distributions of concentration profile, current density, and electrode potential are presented here as a function of the distance normal to the surface for the cases of smooth and rough platinum growth. In the particular case of columnar surfaces, cycloid curves are used to model the electrode, from which the concentration profile is presented in a parameterized form after solving a first-type curvilinear integral. The concentration contour results in a combination of a trigonometric inverse function and a linear distribution leading to a negative concavity curve. The calculation of the current density and electrode potential contours also show trigonometric shapes exhibiting forbidden imaginary values only at the minimal values of the trochoid curve.

Spatial transformation of coherent optical waves with orbital morphologies

We theoretically verify that converting the Hermite-Gaussian modes into the Laguerre-Gaussian can lead to the spatial morphologies of the two-dimensional (2D) coherent states to be transformed from Lissajous figures to trochoidal curves. With this transformational relationship, we experimentally generate various structured light beams by exploiting a cylindrical-lens mode converter to transform the optical Lissajous modes. The present investigation manifests a notable method to generate optical coherent waves with various orbital spatial morphologies.

Generation Simulation of Involute Spur Gears Machined by Pinion-Type Shaper Cutters

A pinion-type shaper cutter is considered as the generating tool for the generation of the gear, and a mathematical model of spur gears with asymmetric involute teeth is given according to the gearing theory. The equations of the profile of the cutter, the principle of coordinate transformation, the theory of differential geometry, and the theory of gearing are applied for describing generating and generated surfaces. Trochoidal envelope traced by cutter during the generating process is also investigated. Trochoidal curves of the cutter depends on the type of tip rounding. Computer graphs of involute spur gears are presented based on the given model. In addition, generation simulation is illustrated. The simulated motion path of the cutter can be used to determine chip geometry for further analysis. The results of this research should be helpful in the design and manufacture of spur gears.

Automatic polishing process of plastic injection molds on a 5-axis milling center

The plastic injection mold manufacturing process includes polishing operations when surface roughness is critical or mirror effect is required to produce transparent parts. This polishing operation is mainly carried out manually by skilled workers of subcontractor companies. In this paper, we propose an automatic polishing technique on a 5-axis milling center in order to use the same means of production from machining to polishing and reduce the costs. We develop special algorithms to compute 5-axis cutter locations on free-form cavities in order to imitate the skills of the workers. These are based on both filling curves and trochoidal curves. The polishing force is ensured by the compliance of the passive tool itself and set-up by calibration between displacement and force based on a force sensor. The compliance of the tool helps to avoid kinematical error effects on the part during 5-axis tool movements. The effectiveness of the method in terms of the surface roughness quality and the simplicity of implementation is shown through experiments on a 5-axis machining center with a rotary and tilt table.

ベールオープナーの開繊機構 （第１報） タフト重量，歯先軌跡の理論的解析と繊維層モデルによる実験的確認

The purpose of bale opener is to open the bale to fine and uniform tufts.Therefore, the way how to open the bale influences strongly the mean value and the distribution of tuft weight.In the present paper, the difference between theoretical and practical tuft weight is investigated, and one theoretical and two experimental models to simulate real bale opening process are presented.Finally the effectivity of a locus analysis and two models to explain the opening mechanism are examined.The results can be summarized as follows : 1. Mean tuft weight investigated at a opening process in commercial mills depends on a production rate of tuft and increases proportionally with the production rate.2. There is a big difference between theoretical mean tuft weight calculated from opening conditions including production rate and practical mean tuft weight collected from actual opening process in the mills.It means that the opener spike plucks only one tuft approximately at its 15∼30 times strikes.3. Ideal opening curve of opener's spike tip for bales is theoretically indicated as a trochoid curve and it can be classified into down-cut opening (D.C.O.) curve and up-cut opening (U.C.O.) curve.4. It is suggested by the two experimental models that the size of tufts opened with D.C.O.are smaller and more uniform because of the smoother surface of bale formed by the strike of D.C.O.than those of U.C.O..5. The mechanism to cause the miss swing can be qualitatively explained by experimental observations by the two experimental models.

直動形トロコイド歯車の基礎解析(第1報)

It is important to eliminate the backlash and to reduce inertia for high accuracy linear motion system. For this reason, the trochoid cam rack has been developed. The profile undercut must be avoided to satisfy the required performance. So this rack profile is ground using trochoid curve loci. In this report, the motion principle has been analyzed, and it becomes clear that there is the optimal value of rack profile offset factor which affects on pressure angle and radius of curvature. Also, roller diameter and tooth height affects on those characteristics. The optimal cam rack without profile undercut is designed using the selected each design factor. The prototype cam rack system using optimal design value is made and tested. It is proved that the system has high positioning accuracy without backlash.

Über zykloidale radlinien und planetenbewegungen

In his fundamental article “Hohere Radlinien” W. WUNDERLICH has shown, that for planetary motions the polodes, the paths of points and the envelopes of straight lines are trochoidal curves without exception. The last ones are called cycloidal trochoids and among all trochoidal curves they have the characteristic property, that the motion of the canonical frame is a planetary motion (Theorem 1). This well known theorem, for which a new proof is given, is fundamental for the following new statements. If parallel rays are reflected on a given cycloidal trochoid h of degree n, they envelope a cycloidal trochoid k of degree 2n (Theorem 2). Figure 1 shows a nephroid h with the given rays parallel to the common tangent in the cusps of h. Figure 2 shows a STEINER-cycloid h, the given rays have a general direction. In Section 4 first is shown, that any given trochoid can be used as moving or fixed polode of an one-parametric family of planetary motions (Theorem 3). Further it is proven, that either no one of the two polodes of a planetary motion is cycloidal or both of them (Theorem 4). Therefore one can distinguish between “not cycloidal planetary motions” and “cycloidal planetary motions.” It is surprising, that among all planetary motions the cycloidal ones can be characterized by a constant proportion between the curvatures of the polodes in the respective poles (Theorem 5). The ordinary planetary motions with the polecircles represent a trivial special case.

On the Relation between the Run-Out Errors and the Motion of the Center of Sun Gear In a Stoeckicht Planetary Gear

The authors considered a single-stage Stoeckicht planetary gear (Type 2K-H) as a model which took the tooth stiffness and the run-out errors of each gear and so on. And the authors statically analyzed the relation between the run-out errors and the motion f the center of the sun gear, and further compared the calculated results with the measured ones. The conclusions are as follows : (i) The motion of the center of sun gear draws a trochoid curve. (ii) In the torque of higher-speed shaft, there is a variation whose frequencies coincide with the number of the relative revolutions of each gear for carrier, and also there may be a variation whose frequencies coincide with the number of the revolutions which can be obtained by adding or subtracting these numbers of the relative revolutions.

On Adjustable Cycloidal and Trochoidal Curves

On Adjustable Cycloidal and Trochoidal Curves