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Cutting force prediction is very important for optimizing machining parameters ahead of the costly physical test. Ball-end milling operation is widely used for machining sculptured surface. Mechanistic approach can precisely predict elemental cutting force at each cutting element and integrate them into the cutter tooth with high fidelity to predict the cutting force for ball-end milling operation. However, the intersection between the cutting tool and workpiece could be complicated due to the trochoid motion trajectory of cutting edge and constantly changing workpiece geometry, making it difficult to determine the cutter-workpiece engagement (CWE) and undeformed chip thickness (UCT). In this present research, a modified cutting force prediction model was developed with considering the true trajectory of cutting edge and in-process workpiece geometry in ball-end milling operation. First, a triangular mesh model of the in-process workpiece surface was developed, and its mesh points were continuously updated by the intersection between the vertical reference line of the selected mesh point and the motion trajectory of cutting edge. Secondly, the UCT was calculated directly using the linear distance between a selected point on the cutting edge and the intersection between the radial reference line of the selected point and the triangular mesh of the in-process workpiece surface. Meanwhile, the CWE was expressed as a step function of UCT. Thirdly, a modified mechanistic approach was established by incorporation into the developed UCT and CWE models. The cutting force of ball-end milling operation was predicted with mechanistic approaches. Finally, ball-end milling experiments of AISI P20 steel were carried out for calibrating cutting force coefficients and validating cutting force model. The relative error between the predicted and measured cutting force is less than 15%, which indicates the predicted cutting force is in good agreement with measured cutting force. The works presented in this paper are one important step for optimizing machining parameters and compensating cutting force induced form error, which could improve the surface accuracy and machining efficiency.

In modern day high-performance space optical systems, precision surface grinding of quartz glass freeform surface has been widely utilized in the initial stage of component machining. Investigations regarding material removal mechanisms present an essential link to ensuring both machining accuracy and quality. Currently, removal mechanisms of quartz glass investigations are mainly focused on straight line scratch and parallel scratch experiments. The material removal mechanism of high-speed cross scratch on machined surface is rarely considered. In order to investigate the removal mechanism of the high-speed cross scratch material, a new scratch test method based on trochoid feed was proposed. The cross-point effect of quartz glass was investigated through high-speed scratch experiment with variable cutting depth. The experimental results show that the cross-point area formed by high-speed cross-scratches is more damaged and enters brittle removal earlier than other areas. In the area of brittle removal, the damage distribution of high-speed cross scratches at the crossover point is approximately symmetrical. The cutting forces at the inlet and outlet of the intersection area are different, resulting in a different fracture morphology. The reflection fracture of stress wave caused by impact load is the key factor leading to the cross-point effect of the intersection area.

The advanced computers with CAM software and the fast computing speeds behind them, made it possible to use an unknown roughing machining process that can only be known as trochoid milling. My goal...

A trochoid cam gear (TCG) is a kind of precise transmission device with high performance, such as non-backlash, high precision, low noise, etc. Generally, the tooth profile modification technique is essential to influence the dynamic performance of TCG, which has not been studied. In the present, the equations of tooth profile modification are constructed to analysis the nonlinear dynamics of the TCG. Firstly, the tooth profile modification equation is deduced according to the requirement of meshing impact reduction. Secondly, considering the time-varying principal curvature radius of tooth profile and the nonlinear relationship between forces and deformations, a translation-torsion nonlinear dynamic model is established to further construct the nonlinear model of TCG. Finally, the dynamic characteristics of the TCG with three key parameters (the damping c, the radius dr of roller and the short amplitude coefficient K) are investigated. The tooth profile modification technique proposed herein can improve the system stability. Moreover, by increasing c and K, or decreasing dr, the performance of TCG can be improved to maintain stable motion state.

Laser surface transformation hardening becomes one of the most modern processes used to improve fatigue and wear properties of steel surfaces. In this process, the material properties and the heating parameters are the factors that present the most significant effects on the hardened surface attributes. The control of these factors using predictive modeling approaches to achieve desired surface properties leads to conclusive results. However, when the dimensions of the surface to be treated are larger than the cross-section of the laser beam, various laser-scanning patterns are involved. This paper presents an experimental investigation of laser surface hardening of AISI 4340 steel using different laser scanning patterns. This investigation is based on a structured experimental design using the Taguchi method and improved statistical analysis tools. Experiments are carried out using a 3 kW Nd: YAG laser source in order to evaluate the effects of the heating parameters and patterns design parameters on the physical and geometrical characteristics of the hardened surface. Laser power, scanning speed and scanning patterns (linear, sinusoidal, triangular and trochoid) are the factors used to evaluate the hardened depth and the hardened width variations and to identify the possible relationship between these factors and the hardened zone attributes. Various statistical tools such as ANOVA, correlations analysis and response surfaces are applied in order to examine the effects of the experimental factors on the hardened surface characteristics. The results reveal that the scanning patterns do not modify the nature of the laser parameters’ effects on the hardened depth and the hardened width. But they can accentuate or reduce these effects depending on the type of the considered pattern. The results show also that the sinusoidal and the triangular patterns are relevant when a maximum hardened width with an acceptable hardened depth is desired.

This letter introduces a novel control framework for a unicycle model to generate an extensive set of motion patterns. With a set of design parameters, the proposed control law leads to Andronov–Hopf bifurcation in speed-scaled angular velocity space of the agent. Using the normal form theory, the generated patterns are shown to be identical to the parametric form of trochoid curves. Considering a comprehensive classification of the generated patterns, the limit cycle characteristics are used to deduce corresponding ranges of the design parameters.

Now progressing cavity pumps (PCPs) and downhole mud-motors (DHMs) with cycloidal profiles are widely used in oil and gas industry. This study presents the method to design the cross section of PCP or DHM with hypotrochoidal multilobe. The conditions for conjugate envelope of hypotrochoidal rotor were derived by using the differential geometric approach. On the basis of this, we made the formulae for the envelope and its offset curve and worked to calculate the area efficiency of cross section and curvature of hypotrochoid. The field of trochoid ratio and offset distance were selected to get the rational offset curve. These results enable to design the cross section of PCP or DHM with multilobe of hypotrochoid which has the continuous and limited curvature. These results also can be used for multilobe PCP or DHM with solid stator.

Gerotor pumps are well known by a compact design, simple structure and low noise level, which makes them suitable for use in the automotive industry, and especially in hydraulic systems for engine lubrication. One of the main disadvantages of gerotor pumps is the inability to adjust to wear, which significantly reduces the pump efficiency. In order to mitigate the negative effect of the inevitable wear process, this paper presents a methodology for determining the optimal combination of trochoid gears design parameters for a defined aspect. An appropriate mathematical model has been developed to analyze the effect of changes in gear design parameters in relation to maximum contact stresses, pressure changes in gerotor pump chambers and wear rate proportional factor (WRPF). Verification of the developed models was performed by realizing physical pairs of gears and laboratory experiments with simulation of pump operating conditions. The results and conclusions presented in this paper, with an emphasis on the actual work processes, bring very important perspectives for the gerotor pumps design with improved performance.

For camera systems mounted on a rotating platform, it would be difficult to install the image sensor concentrically with the rotation axis. Free rotors are unmanned aerial vehicles that rotate about its centre of mass for flight. Small changes in mass distribution is sufficient to displace the free rotor’s rotation axis from the camera’s principal axis. This displacement can be modeled as a trochoid equation and the rotation centre of the trochoid for one arch recovered. The algorithm optimally compensates for the image eccentricity in three steps. Firstly, derotating the images about its principal axis. Secondly, computing the optimal values to translate the images in the image stream. Lastly, to shift the images using an affine transformation matrix. The results show that the algorithm is sufficiently accurate to compensate a time-varying rotation and time-varying translation image stream. It is also possible to visually observe the gyroscopic precession of the free rotor.

A trochoid cam gear (TCG) is a kind of precise transmission device with high performance, such as non-backlash, high-precision, low noise, etc. However, the nonlinear dynamics has not been studied. In the present paper, the nonlinear dynamic differential equation of the TCG is constructed. First, the trochoid equation and the tooth profile equation are deduced to satisfy the conditions of the continuous transmission. Second, a pure torsional nonlinear dynamic model is established to further construct the nonlinear model of the TCG according to a Lagrange equation. Finally, the dynamic characteristics of the TCG are investigated. For the short amplitude coefficient K, break of quasiperiodic torus is the main reason for the chaotic vibration. By increasing K, the performance of TCG can be improved to maintain stable motion.

Typical damage to gears is pitting and breakage of teeth, so it was traditionally designed based on these two damage and tooth surface shape design was done to avoid edge contact. However, in recent years, a design method that minimizes the tooth surface modification amount to maximize the load capacity and a tooth surface modification method that can secure a contact region even in a low load region are adopted. However, damage due to trochoid interference occurs outside the geometric meshing range and can’t be calculated by the strength calculation formula (ex. gear strength standard). And, with such a tooth surface modification, since tooth edge contact is accompanied in a high load region, it is necessary to consider edge contact stress analysis of the tooth due to trochoid interference. For this reason, this software (CT-FEM Opera iii [1]) adopts the finite element method (FEM) as a method capable of edge contact analysis and analyzes it. With the occurrence of edge contact, a large stress is generated in that portion and the flash temperature becomes high temperature, so temperature analysis of this portion is important. In the stress analysis of the teeth, it is necessary to analyze considering the tooth surface modification, the tooth profile error, the pitch error, and the posture error of the axis in consideration of tooth deformation and load sharing. At the time of engagement of the teeth, the curvature of the tooth profile, the sliding speed, the tooth surface roughness and the lubricant greatly change the friction coefficient and the oil film thickness depending on the tooth surface position. Therefore, tooth surface stress, edge contact stress, and flash temperature are also greatly affected. In this paper, the damaged picture of the experiment gear and the edge contact stress analysis result are shown, and they are in good agreement. And the analysis results of flash temperature, friction coefficient distribution, oil film thickness, power loss, transmission error, and tooth bending stress are shown.

The highly dynamic production puts the versatility of production systems in the center of industrial interests. Due to their flexibility and low energy consumption industrial robots are a big part of this development. They have been used as handling systems for decades and are already state of the art in this regard. This is not the case for other surrounding tasks such as milling. These systems have in comparison to conventional machine tools a relatively low stiffness c, which allows only low cutting forces Fc and therefore low feed rates fz and speeds vc. This leads to higher processing times and therefore to a higher energy consumption. A highly dynamic milling strategy, which results in significantly lower cutting forces Fc than in conventional milling and which at the same time has high cutting rates is trochoid milling. This paper shows the basic usability of this milling strategy on standard industrial robots. Furthermore, the challenges of the near future will be shown.

Trochoidal milling is an alternative tool path strategy which has been shown to increase productivity, improve tooling life, and reduce resultant cutting forces. While the advantages of trochoidal milling over conventional slot or shoulder milling were previously reported, the complexity of trochoid tool path makes developing an analytical force model highly complicated. In this work, a numerical algorithm to construct the uncut chip thickness model in trochoidal milling is introduced, which is based on the geometrical relation of self-intersection and cross-intersection points of the tool path curve. In addition, an extensive series of experiments is carried out in slot and trochoidal-milling operations in order to investigate the dependency of cutting pressure coefficients on tool path parameters and to develop a model to relate the two. Furthermore, the performance of the developed model for uncut chip thickness and the cutting pressure coefficients are evaluated in predicting cutting forces; it is shown that the model predicts the maximum cutting force in the feed direction and between all the testing sets with 8% total average error, while 17% total average error is observed in predicting maximum cutting force in the lateral direction. This demonstrates the potential of the proposed approach in offline simulation of the cutting forces which is a critical step in selecting proper tooling and process parameters to increase productivity of the cut.

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.

Carbon fibre-reinforced plastic (CFRP) has good specific mechanical properties, it is, therefore, an often used structural material in the high-tech industries. Due to the anisotropy and inhomogeneity of CFRP, the machining properties are highly depending on the cutting directions. Based on previous experimental works, a new trochoid tool patch was developed in order to minimise cutting force and uncut fibres when milling unidirectional CFRP. The main objective of the present work is to test and analyse the effect of the special trochoid milling technology on the quality parameters (surface roughness and characteristics of uncut fibres) of machined unidirectional CFRP specimens. The machining experiments were analysed using digital image processing (DIP) method. The influence of the chosen independent variables (cutting tool and tool patch) were analysed at fixed process parameters (cutting speed, feed rate, depth of cut). It was found that the quality of the machined edge is better when the new trochoid milling technology is applied.

Automation of polishing planar surfaces on 3-axis machining centers is treated from the point of view of defining suitable tool paths that provide full and even coverage of the surface both along and across straight line guide curves, which are routinely produced by any CAM system. A fast but approximate and a slower but precise trochoidal path calculation model are presented. The parameters of the trochoid and the side step are analytically calculated for any given polishing ring diameter in order to safeguard full coverage as well as to avoid over-polishing. Thus, a region of feasible trochoid parameter values is offered to the machinist. Verification of polishing towards full surface coverage is ensured by a simulation tool. Validation of the approach through an actual polishing example is presented.

With the ever increasing pressure to reduce processing time and cost, researchers in machining have begun to develop a body of work centered around increasing the throughput of machining operations. While standard toolpaths exist, such as raster and zig-zag, alternative toolpaths have been developed to achieve beneficial kinematics and dynamics for the cutting tool to better achieve high-speed machining conditions. One such toolpath, trochoidal milling, has been identified to decrease machining process time and increase overall tool life. Understanding the undeformed chip thickness produced utilizing trochoidal milling is critical to developing advances in the field. This paper presents a novel approach to modelling the chip thickness of the process for low to medium range cutting speeds. It has been found that the tool path cannot be described as a purely circular path, instead requiring the model of a true trochoid, which is presented in this work. Utilizing efficient, numerical method, the instantaneous chip thickness is solved for and validated experimentally with cutting force measurement, using a semi-mechanistic force model, where the experimental cutting forces find good agreement with the simulated results.

Due to the complexity of ruled surface blade in five-axis flank milling and the limited interpolation capacity of Computer Numerical Control system, the continuous tool location trajectory can only be approximated as a combination of a series of micro linear segments in flank milling process. As a result, the smooth ruled surface is decomposed into many micro planes. In this paper, the generation principle of surface morphology for the micro plane is studied without considering the tool vibration, the workpiece vibration, and shape and position errors of the tool in the milling. In the surface morphology simulation of each micro plane, a fixed plane I which is stationary and perpendicular to the micro plane is firstly defined, then the intersection distribution between each cutter tooth and the fixed plane I is studied when the cutter has no speed in the direction that is vertical to the tool axis, and then the intersection distribution can turn to the groove lines of surface morphology when the speed vertical to the cutter axis exists. For the other part of morphology besides groove lines, tool’s trochoid motion along the workpiece is approximated as arc motion, so the ridges and grooves are taken shape combined with the entry angle and the exit angle of the tool. The simulation result proves the correctness of the model.

Nowadays one trend of automotive transmission system is that gerotor pump is widely applied for power train lubrication due to its compact size and low cost. The most of traditional gerotor pumps are designed with trochoid curve. In order to obtain higher volumetric efficiency of the gerotor pump applied in transmission of All Wheel Drive (AWD) vehicles, a new gear profile so-called K-floid is designed. The K-floid profile of the inner rotor is designed by combination of three segments: hypotrochoidal curve, epitrochoidal curve, and common tangent line. The outer rotor profile coordinate is then obtained by means of rotation of the inner rotor. On the other hand, design of a dual-rotor gerotor pump is also presented in the paper so as to reduce flow rate pulsation. CFD simulation is preformed to validate design effect and results indicate that the designed dual-rotor gerotor pump with K-floid can greatly decrease flow rate pulsation and also improve volumetric efficiency compared with the conventional single rotor pump of trochoidal profile. A prototype pump is also manufactured and tested to prove its performance.

In this paper, I shall discuss Leibniz’s considerations on the problem of exactness, as they can be reconstructed from published and unpublished letters, notes, drafts and sketches he did while Paris between 1673 and 1676. Leibniz’s critical target was Descartes’ Geometrie (1637). Descartes had managed to include into geometry all algebraic curves on the ground that they can be constructed by one continuous motions. Few rogue elements, called “mechanical curves” did not comply with this criterion of exactness, and therefore were not allowed in solving geometrical problems: these were, for example, the Archimedean spiral, the quadratrix and the cycloid. From the beginning of his mathematical studies, in 1673, and throughout his career, Leibniz set the task to reform the Cartesian criterion of exactness. Leibniz objected that Descartes’ criterion is not tenable, because there exist curves constructible by one continuous motion which cannot be associated to any algebraic equation. In this paper, I shall discuss several examples of such geometrical, non-algebraic curves: evolutes and involutes, the parabolic trochoid (the curve generated by the rolling of a parabola along one of its tangents) and the cycloid. The key to reform the Cartesian demarcation was, for Leibniz, to admit within geometry curve constructions obtained by strings that could be bent into arcs. In this way, the previously mentioned curves resulted geometrical insofar as they could be generated by one and continuous motion.

Aim of study: This paper presents a mathematical modeling approach to redesign the reels of chickpea harvesters for harvest efficiency.Area of study: A prototype chickpea harvester was designed and evaluated on the Dooshan farm of the University of Kurdistan, Sanandaj, Iran.Material and methods: The strategy used for reducing harvesting losses derived from the dynamic study of the reel applied to the chickpea harvester. The machine was designed such that bats of a power take-off (PTO)-powered reel, in conjunction with passive fingers, harvest pods from anchored plants and throw the pods into a hopper. The trochoid trajectory of the reel bats concerning reel kinematic index, and plant height and spacing was determined for redesigning the reel.Main results: This kinematic design allowed an estimation of the reel orientation at the time of impact. The experimentally validated model offers an accurate and low computational cost method to redesign harvester reels.Research highlights: The new chickpea harvester implemented with a four fixed-bat reel, a height of 40 cm above the ground for the reel axis, and featuring a kinematic index of 2.4 was capable of harvesting pods with harvesting efficiency of over 70%; a significant improvement in harvesting performance.

An amphibious all terrain mobile robot is required for disaster response tasks. In this paper, we propose “R-Crank”, which has a simple four wheels driving configuration with crank links, connecting to ipsilateral front and rear wheels so as to form parallel four-bar linkage systems. The crank links act as legs because their trajectory with respect to the ground become trochoid curve, which is commonly used as leg trajectories for a legged robot. These links achieve high terrain adaptability while maintaining simple structure of the wheeled robot. We developed a prototype model and carried out basic experiments. The robot achieved to climb 0.3 m of a step(76.9% of the wheel diameter), stairs and a snowy slope. In addition, we propose amphibious structure of crank links by installing multiple rigid paddles. The propulsive force generated by the crank links was measured and empirically derived the optimum parameters by using a small-sized experimental apparatus. Finally the robot equipped with the amphibious crank links achieved propulsion on water at 0.13 m/s. We also verified that the robot achieved going up and into water through stairs.

An efficient approach for milling stability and surface location error (SLE) prediction with varying time delay and cutter runout effect is presented in this paper. Firstly, based on the tooth trochoid motion, the paper proposes a varying time delay model during cutter/workpiece engagement with taking cutter runout into account, establishes a milling dynamic model under arbitrary feed direction, and then derives the state transition matrix in one cutter rotation period by using the Cotes numerical integration formula. The milling stability of the dynamics system are obtained by using Floquet theory. According to the fixed point theory, the displacement response of the dynamic system and the method for solving the SLE are both developed. Later, a series of numerical and experimental works are conducted. The numerical verification shows that the proposed method can achieve a faster convergence rate and higher calculation efficiency than other previous methods. Meanwhile, the prediction of stability and SLE are in good agreement with the experimental results, and have a high accuracy for stability prediction when cutter runout and varying time delay considered. In the end, the numerical studies show that the milling stability and SLE strongly depends on the actual milling conditions, including milling parameters, cutter runout, cutter geometric parameters, and asymmetric structural dynamic parameters, which are helpful for milling process optimization.

A well-known problem is existing in Wankel rotary engine that leads the engine to seize before it achieves operational loading conditions. Existing research literature shows one of the primary indications to cause engine seizure are the chatter marks. The present study focuses on performing bench testing to rank the combinations of chosen materials and coatings resulting in lowest friction and reduced chatter. A total combination of 11 materials and coatings were analysed and tested. It was found that titankote with PVD coating and nikasil coating with diamond particles on A-2 tool steel performed the best, resulting in no wear, scuffing or chattering, with an increased surface smoothness. The lowest coefficient of friction measured was 0.06 with HTCS-150 hard trochoid and C-Lee Cook apex seals. The maximum average face wear reduction obtained was 15 μm on titankote with PVD coated trochoid and 18 μm on ceramic apex seal.

In this research, we suggest a method to reduce the loss of the pump aiming to develop small and high-pressure pump for EHA. To reduce the loss of the pump, we focused on the internal leakage loss and the viscous friction loss. We formulated each loss and found that it is effective for loss reduction to make gaps inside the pump narrower and increase the discharge amount per 1 rad making the radius of the gear small. Based on the above, we developed the improved involute external gear pump. Compared with conventional trochoid pump, the internal leakage loss of developed pump is reduced by 22.4% at 10MPa and the viscous friction loss is reduced by 85.6% at 10MPa. Also, there is a problem that internal leakage increases when the internal pressure becomes high. This problem is caused by the deformation of the pump-casing under high pressure. We developed highly rigid ceramics-casing to reduce the increase of the internal leakage. Compared with the pump using aluminum-casing, the pump using ceramics-casing reduced 56.2% in internal leakage loss when the internal pressure increased from 3MPa to 6MPa.