Epihypocurves and epihypocyclic surfaces with arbitrary base curve

If a circle rolls around another motionless circle then a point bind with the rolling circle forms a curve. It is called epicycloid, if a circle is rolling outside the motionless circle; it is called hypocycloid if the circle is rolling inside the motionless circle. The point bind to the rolling circle forms a space curve if the rolling circle has the constant incline to the plane of the motionless circle. The cycloid curve is formed when the circle is rolling along a straight line. The geometry of the curves formed by the point bind to the circle rolling along some base curve is investigated at this study. The geometry of the surfaces formed when the circle there is rolling along some curve and rotates around the tangent to the curve is considered as well. Since when the circle rotates in the normal plane of the base curve, a point rigidly connected to the rotating circle arises the circle, then an epihypocycloidal cyclic surface is formed. The vector equations of the epihypocycloid curve and epihypocycloid cycle surfaces with any base curve are established. The figures of the epihypocycloids with base curves of ellipse and sinus are got on the base of the equations obtained. These figures demonstrate the opportunities of form finding of the surfaces arised by the cycle rolling along different base curves. Unlike epihypocycloidal curves and surfaces with a base circle, the shape of epihypocycloidal curves and surfaces with a base curve other than a circle depends on the initial rolling point of the circle on the base curve.

Improving the Oil-Free Scroll Vacuum Pump Efficiency

Improving the quality of manufactured products involves reducing any contaminants introduced into the vacuum chamber from the pumping means. Scroll vacuum pumps, which are constantly developing, are the most promising for oil-free pumping. Relying on the mathematical model developed, we examined the influence of the main parameters of the scroll on the scroll pump performance, in particular, the influence of the radius of the base circle on the pumping speed and the power consumption of the scroll pump at fixed and variable radii of the pump housing bore. Maintaining the overall dimensions of the pump with an increase in the radius of the base circle proves to lead to an exponential increase in the limiting residual pressure and a decrease in energy efficiency due to a decrease in the number of scroll wraps and, as a consequence, an increase in backflows. For the pump under consideration, when the base circle radius is more than 3.5 mm, the limiting residual pressure exceeds 10 Pa, and the vacuum scroll pump can no longer be a full-fledged oil-free alternative to oil-sealed vacuum pumps. With an increase in the radius of the base circle, while maintaining the number of scroll wraps, the radius of curvature of the outer wraps will increase; this results in the backflow decrease, leading to a decrease, albeit insignificant, in the limiting residual pressure. The energy efficiency of the scroll pump decreases with decreasing pump inlet pressure. The above dependencies make it possible to choose the optimal geometry of the scrolls, based on the specific conditions for which the vacuum scroll pump is designed

Accurate calculations of working chamber volume constructed from involute of variable radii circle with double arcs modification in a scroll type compressor

A mathematical modeling of working chamber volume, constructed from the involute of variable radii circle with double arcs modification, is established in this study. The formulations of working chamber volume which is consisted of the involute of variable radii circle and modified by double arcs at center profiles are accurately deduced, and the orbiting angle of scroll at every moment has a corresponding working chamber volume. The effect of parameters on the working chamber volume is investigated. The different values of initial radius of base circle, starting involute angle, corrected increment, polytropic index and modified angle will change the working chamber volume. Furthermore, the effect of parameters on suction volume and volume ratio has been considered in this study, and the design schemes to obtain the highest volume radio are also studied. These results can be used to provide general implications in calculation of working chamber volume constructed from other type involutes and modifications.

Construction and Analysis of Variable Wall Thickness Profile of Double Scroll Teeth of Scroll Compressor

Based on the changing trend of the geometric parameters of the scroll profile of the involute of circle with variable radii, a new type of double scroll teeth profile of scroll compressor composed of involute of circle with variable radii is constructed. The construction process of the baseline is discussed, and the baseline equations are deduced. According to the principle of normal equidistance, the geometric model of variable wall thickness double scroll teeth is established, and the suction volume, tooth thickness, and scroll area of double scroll teeth are compared with those of the double scroll teeth composed of ordinary involute of circle. The results show that: compared with ordinary double scroll teeth with constant thickness, the scroll diameter when two scrolls mesh is reduced and the revolutions of orbiting scroll required in the working process are reduced and it is easy to select better scroll profile through adjusting the changing coefficient of the base circle radius k, which provides a reference for scroll machinery using environment-friendly working fluids.

Determination of the mean base circle radius of gears by optical multi-distance measurements

The required reliability of wind turbine gearboxes increases the requirements for large gear measurements. Extensive measurements to reliably assess the geometry of large gears in the single micrometer range are necessary. Due to an individually fixed measuring volume, standard methods like coordinate and gear measuring instruments reach their limits for large gears with diameters > 1 m. Therefore, a scalable optical measurement approach consisting of a single sensor in combination with a rotary table for multi-distance measurements with subsequent model-based evaluation of shape parameters of gears is presented. The scalable measurement approach is to be extended to a multisensory system in further work. As a fundamental shape parameter the mean base circle radius using the example of spur gears is determined. The base circle radius is used due to the geometric relationship to further shape parameters for example to the profile slope deviation. The theoretically achievable measurement uncertainty of the mean base circle radius due to sensor noise is estimated to less than 5 µm (k=2) for a small and a large gear, which verifies the scalability of the sensor system. In order to show a general proof of principle, two series of optical measurements on a gear with a diameter of 0.105 m are performed and referenced with a tactile measurement. As a result, random errors of 1.2 µm for k=2 are determined. The remaining systematic deviations to the reference value amount to 4.3 and 1.6 µm, respectively. Hence, the total measurement uncertainty is currently limited by systematic effects, and the defined aim of a total uncertainty of less than 5 µm (k=2) is narrowly missed by 1.5 µm. The random errors of 1.2 µm (k=2) show, however, that an adequate measurement precision is achieved and that the multi-distance measurement approach has the potential to reach the aimed measurement uncertainty with appropriate strategies to compensate for the systematic influences. The experimental and theoretical results prove the principle applicability of the proposed single sensor multi-distance approach for the precise inspection of gears.

Gear Shape Parameter Measurement Using a Model-Based Scanning Multi-Distance Measurement Approach

To reduce wind turbine failures by defective drive trains, deviations in the geometry of large gears (diameter ≳ 1 m) must be extensively determined with single-digit micrometer uncertainties. Fixed measuring volumes limit standard measuring methods like coordinate and gear measuring instruments for large gear measurements. Therefore, a model-based scanning multi-distance measurement approach for gear shape parameters is presented. The measurement approach has a scalable design and consists of a confocal-chromatic sensor, rotary table as a scanning unit and model-based signal processing. A preliminary study on a midsize spur gear demonstrates the general feasibility of the model-based scanning multi-distance measurement approach. As a result, the mean base circle radius as the fundamental gear shape parameter is determined with an uncertainty of <5 μm. The calibration and adjustment of the sensor arrangement were performed with a known calibration gear. Scalability is not experimentally validated in this article. However, simulations verify the scalability of the measurement approach in a first step. For gears with 1 m in diameter and varying tooth flank geometries, the estimated achievable uncertainty of the mean base circle radius is still <5 μm. Therefore, the model-based scanning multi-distance measurement approach is a promising alternative for gear inspection.

A transient contact position prediction method for worn tooth profile under dry running conditions

Purpose The purpose of this paper is to establish a transient contact position prediction method of gears at the meshing point based on the equivalent contact model. Design/methodology/approach In this method, the contacting surface profiles are constantly updated by changing the pressure angle and the chord tooth thickness, which has a direct connection with the equivalent base circle radius. According to the equivalent base circle radius, the equivalent pressure angle at the pitch circle and equivalent pitch point can be calculated. The equivalent contacting surface profile is determined by the equivalent pressure angle at the pitch circle; for each meshing point, there is one equivalent pressure angle at the pitch circle. Therefore, each meshing point can be regarded as a point on the equivalent contacting surface profile. Findings The model is applicable to find out the contact position after a series of meshing cycles through the law of pressure angle change and intentionally kept as simple as possible with the aim to be used in further study of gear flanks at the point of the actual contact. Practical implications The results of the experiment are applied to the equivalent contact model to describe the transient contact position and assess the model accuracy. Originality/value The determination of the contact position of the worn tooth profile provides the action points of the force for the study of the contact fatigue.

Nonlinear dynamic analysis of a microsegment gear system with a time-varying base circle

A systematic dynamic analysis of a microsegment gear system with a time-varying base circle, time-varying mesh stiffness, and gear backlash is carried out in this paper. By discretizing the meshing process, a six degree-of-freedom nonlinear dynamic model of a microsegment gear pair is established. To study the dynamic response of the microsegment gear and involute gear under various operating conditions, the numerical integration method is adopted. The dynamic transmission error (DTE) of the two gears is analysed in terms of time history charts, phase diagrams, fast Fourier transformation spectra, and Poincaré maps. The effects of support damping and support stiffness on radial vibration are also investigated. Results reveal that, compared with the involute gear system, the microsegment gear system is more stable at the high-speed condition and has a smaller amplitude of DTE under medium-speed and heavy-load, high-speed, and heavy-load conditions. The support damping and support stiffness have great effects on the resonant peak in the radial direction of the microsegment gear. Both the proposed model and numerical results are expected to provide a useful source of reference for the dynamic design of the microsegment gear system.

Solution and Verification of Cutter Position for Machining Split Equal-Base Circle Bevel Gear

The purpose of this study is to achieve machining of a split equal-base circle bevel gear. Based on the equal-base circle bevel gear theory, a cutting coordinate system for the separate piece is developed, and the tooth surface processing path of the separate piece is analysed and planned. According to the working principle of the equal-base circle bevel gear, by analysing the cutter position and posture, the calculation method for the angle between the wheel blank coordinate system and the fixed space coordinate system is derived when machining the separate piece for different spiral angles as well as various concavity and convexity properties. The explicit function expressions for the cutter centre coordinates and axis vectors for machining the separate piece are obtained. Using MATLAB, the tool position model is verified by means of calculation, and the tooth cutting simulation of the separate piece is carried out using VERICUT software. By machining experiment and tooth surface measurement analysis, the feasibility of machining the separate piece and the correctness of the tool position mathematical model are verified.

THEORETICAL AND FINITE ELEMENT ANALYSIS TO DETERMINE CONTACT PRESSURE, VONMISSES STRESSES AND WEAR IN CAM AND FOLLOWER FOR VARIOUS CAM ROTATIONAL ANGLES IN IC ENGINE

The contact between the cam and follower that exists in the valve strain system of IC engine influences wear. The dynamic analysis of cam and follower system in carried to find the normal compressive force for various cam rotational angles. Based on this compressive force on the cam, the hertz contact stresses and surface wear are calculated theoretically. Finite element analysis was carried out in the three critical portions of the cam such as cam nose region, cam tangent region and cam base circle region to compare the results. The results showed that cam rotational angle directly affects the contact pressure. The max contact pressure occurs in the nose end of the cam. The results showed that principle stress and wear also increases with cam rotational angle