Accumulative Pitch Error Research Articles

The study of cutting force, temperature, and geometric error in cutting a large screw

For cutting a large screw, the cutting force and temperature have direct impact on machined surface quality, such as accumulated pitch error in geometric error. Excited by cutting force, the axial deflections directly affecting accumulated pitch error were studied based on the beam theory. By experiments, the axial cutting force was predicted with a 96% confident interval, and the tool number and cutting depth were found to be significant terms. Tested under increasing cutting speeds, the temperature rise of workpiece and accumulated pitch error both decreased obviously. Thus the higher cutting speed could improve the cutting precision of a full-length large screw.

Analysis of forming error during thread and spline synchronous rolling process based on motion characteristic

The thread meshing and the spline/gear meshing are intercoupling during thread and spline synchronous rolling (TSSR) process. The subtle difference between different meshing types will lead to profile error of workpiece. Exploring the influence of motion characteristic on the forming error during TSSR process is important for control of pitch error, design of profile compensation of rolling die, and improvement of rolling quality. Thus, in this paper, the accumulative pitch error for spline section of workpiece during TSSR process was defined and modeled according to the motion characteristic in the synchronous rolling process. A quantitative calculating program of the accumulative pitch error has been compiled by using MATLAB code based on the motion models during thread rolling process and spline rolling process. Then, the influences of calculation step, rotating parameter of rolling die at initial contact state, and initial values for numerical solution of motion models of thread section on the calculation accuracy were investigated. The influences of processing parameters and geometrical parameters on the accumulative pitch error were studied by means of orthogonal experiment design method.

Analysis of tooth profile and accumulative pitch errors of end-toothed disc and its tolerance development

Tooth profile errors of the end-toothed disc describe the tooth profile deviations which mainly contain tooth trace error and tooth profile half-angle error while accumulative pitch error describes the tooth position deviation of the end-toothed disc. In traditional analysis, neither the deviation of the benchmark of tooth trace error nor that of tooth profile half-angle error is considered, while the deviations exist in engineering. Hence, tooth width error is proposed in order to describe the deviations of the benchmarks and the indexing accuracy error model is also established according to these four errors in this paper. Meanwhile, Tolerance Proportional Distribution Method is presented to establish the tolerance proportional model of end-toothed disc. Therefore, according to the indexing accuracy error model and tolerance proportional model, the pass rate model of end-toothed disc is established and a new approach for tolerance development is put forward based on this pass rate model. Finally, a simulation program is developed for random sequence generation, which replaces the processing errors measurement, and meshing process simulation, and verifies the correctness of the approach for tolerance development.

Global sensitivity analysis of transmission accuracy for RV-type cycloid-pin drive

Global sensitivity analysis of transmission accuracy for RV-type pin-cycloid drive is studied by Sobol' method in comprehensive consideration of nonlinear factors and uncertainty factors (such as manufacturing errors, assembly errors, bearing clearances, etc.). The purpose is to study the effect of various errors on transmission accuracy, then find out main errors so as to regulate them, thus not only improve transmission accuracy, but also reduce manufacturing cost. The results show that the effect on transmission accuracy is larger of the runout error of eccentricity cam of crank shaft, the tooth groove error and accumulative pitch error of cycloid gear, the bearing clearance between cycloid gear and crank shaft, and the tooth groove error and accumulative pitch error of pin gear, large of the runout error of cycloid gear hole at crank shaft and the runout error of carrier hole at crank shaft, small of the assembly error of carrier, the bearing clearance between carrier and frame, etc. The presented method provides a basis for global sensitivity analysis of transmission accuracy for the RV-type pin-cycloid drive. It is also beneficial to global sensitivity analysis of transmission accuracy of other similar gear drives. The results of this study provide an important guideline for the design and manufacture of high accuracy and complex gear drive.

Frequency spectrum and vibration analysis of high speed gear-rotor system with tooth root crack considering transmission error excitation

A finite element node dynamic model of gear-rotor-bearing system with different lengths of crack by taking the time-varying mesh stiffness, backlash, transmission error excitation, flexible shaft and supporting bearing into account is proposed. The time-varying mesh stiffness of gear-pair with cracked tooth is obtained by applying the improved potential energy method. Due to the periodic features of the dynamic responses of the system when the tooth is cracked, the short term component (tooth profile error and tooth pitch error) and long term component (accumulative pitch error) transmission error excitations are introduced, the RMS values, kurtosis values, and frequency spectrum diagrams of the dynamic response with respect to input speed considering different forms of transmission error excitation are gained. The influences of transmission error excitation and crack length on the dynamic responses are investigated. The effectiveness of the RMS value, kurtosis value and frequency spectrum in the judgment of the crack length is analyzed.

Research on Gear Total Accumulative Pitch Error Measure Method

In the light of the shortcoming that gears traditional measurement methods not only consume time and labor, but also have a great contrived factor, a new methods of total accumulative pitch errors measurement based on the coordination measurement machine is put forward, which can scan tooth profile coordinates automatically and quickly within the effective work scope of tooth profile, conduct the data processing directly and obtain the deviation through the involutes deviation to achieve online measurements. The theoretical analysis and experimental results show that the measurement method has many advantages, such as simple, non-contact and efficient.

Rotational Transmission Error of K-H-V-Type Planetary Gears With Cycloid Gears. 2nd Report, Effects of Manufacturing and Assembly Errors on Rotational Transmission Error.

Since the K-H-V-type planetary gears with cycloid gears can be made compact and light, and have a large reduction ratio, high efficiency, and high torsional stiffness, they have been used in industrial robots and machines requiring precise positioning. The rotational transmission error of the K-H-V-type planetary gears with cycloid gears should be as small as possible in order to apply the planetary gears in various fields. In this paper, K-H-V-type planetary gears with cycloid gears were transformed into a two-dimensional model with twenty degrees of freedom, and the effects of machining and assembly errors of elements on the rotational transmission error were determined analytically, and some analytical results obtained were checked experimentally, giving the following results. The major errors affecting the rotational transmission error are the position error of holes for the crank cam in the cycloid gear ; runout and accumulative pitch error of the cycloid gear ; error in the crank cam ; the accumulative pitch error of the internal gear. Directions of position error of the hole, runout and accumulative pitch errors of one cycloid gear should be identified with those of the other cycloid gear respectively, so as to decrease the rotational transmission error.

Rotational Transmission Error of K-H-V-Type Planetary Gears With Cycloid Gear. 3rd Report, Mutual Effects of Errors of the Elements on the Rotational Transmission Error.

The rotational transmission error of K-H-V-type planetary gears with cycloid gears should be as small as possible in order to apply these planetary gears to the industrial robot and a machine requiring precise positioning. In this paper, after the K-H-V-type planetary gears with cycloid gears were transformed into a two-dimensional model with twenty degrees of freedom, both the mutual effects of errors of elements and the effect of plural cycloid gears on the rotational transmission error were determined theoretically, and moreover the theoretical results were checked experimentally in some cases, with the following conclusions. (1) The rotational transmission error cannot be determined by superposition of each rotational transmission error produced by individual error. (2) The tooth profile error where the tip has a convex error of cycloid gear should be avoided to decrease the rotational transmission error. (3) The rotational transmission error whose frequency coincides with the carrier rotational frequency is produced by errors of cycloid gears or carrier and assembly error or accumulative pitch error of internal gear. (4) The rotational transmission error increases when the backlash increases. (5) Plural cycloid gears have a beneficial effect on decreasing the rotational transmission error.

Rough forming of ring gear by indentation of a rack-shaped punch. Deformation of ring-shaped specimens and theoretical calculation of indentation pitches.

In our previous paper, it was shown that the ring-shaped spur gear can be formed by successive indentation of a rack-shaped punch. A ring-shaped specimen is deformed by the flow of the indented material in the circumferential direction during working in this method, causing changes in the pitches of the formed grooves. The specimen is observed experimentally to be deformed with a similar pattern throughout working. A deformation model of the specimens is proposed using an elastic-plastic finite element method to simulate the actual deformation behavior. Spur gears with a maximum accumulated pitch error of 0.3 mm are produced by indentation of a rack-shaped punch with pitches determined theoretically based on this deformation model.

Rough forming of ring gear by indentation of a rack-shaped punch. (2nd Report, Deformation of ring-shaped specimens and theoretical calculation of indentation pitches).

In the previous paper it was shown that the ring-shaped spur gear can be formed by successive indentation of a rack-shaped punch. A ring-shaped specimen is deformed by the flow of the indented material in the circumferential direction working in this method, causing changes pitches of the formed grooves. The specimen is observed experimentally to be deformed with a similar pattern throughout working. A deformation model of the specimens is proposed using an elastic-plastic finite element method to simulate well the actual deformation behavior. Spur gears with a maximum accumulated pitch error of 0.3 mm are produced by indentation of a rack-shaped punch with pitches determined theoretically based on this deformation model.

Influence of Gear Error on Rotational Vibration of Power Transmission Spur Gear : 3rd Report, Accumulative Pitch Error

In this study a power transmission spur gear is recommended as a profile corrected spur gear. Supposing it is allowable that the given value deviates about ten-odd percent from actual acceleration, it is synthesized how an accumulative pitch error influences the rotational vibration of a profile corrected spur gear, using the simulator developed by authors. The influence degrees are revealed in an influence chart in which they are determined in relation to the rotational speed and the contact ratio. The permitted amounts of errors for each kind of error are shown in a table when the permissive vibration of a pair is decided. The classification of errors in the dynamic performance has been made clear.

A method of gear lapping for high pitch accuracy: Example of a Wildhaber worm wheel

A method of finishing a highly accurate external gear with lapping, from both the experimental and theoretical points of view, is discussed. First, the external gear is meshed with the lapping tool, which acts as an internal gear having the same number of teeth and the same tooth-profile as the work. The lap is reciprocated in the direction of the gear axis and is then turned at a certain angle, so as to change the meshing teeth, and lapping is repeated. In a similar manner, the lapping action is repeated, changing the meshing teeth, and the accumulative pitch error is decreased. The aim is to make a highly accurate worm wheel (plane-tooth-profile) as basic research for the study of an indexing table which is assembled with Wildhaber worm gears. The experiment was tried on a worm wheel having an outside diameter 292 mm and 72 teeth. The maximum accumulative pitch error was decreased from 15 sec (10.5 μm) to 3.15 sec (2.2 μm). Thus we obtained a highly accurate worm wheel for practical use

A study on precision thread grinding (prevention of thermal deformation of workpiece during grinding

The authors have investigated the causes of pitch errors in leadscrews during grinding and have found that the thermal deformation (elongation or shrinkage) of the workpiece during grinding is one of the most dominant causes of the errors. In case of using grinding fluid (spindle oil etc.), the temperature of the workpiece rises gradually as grinding operations are repeated and the accumulated pitch error often amounts to several tens of μm. The aim of this study is to find some means to prevent such thermal deformation of the leadscrews during grinding. The method suggested here is to splash the oil, the temperature of which is maintained to be the same as that of the room, on the workpiece all over the length. The results show that this method is very effective to prevent the thermal deformation of the workpiece. Furthermore, the theoretical analysis has been done for the temperature distribution of the workpiece and also for the limiting fluid flow rate to prevent thermal deformation. The theoretical analysis agrees with the experimental results, and it is considered that this method is applicable to various grinding conditions.

歯車の成形研削に関する研究(第3報)

A form grinding machine of spur gears, equipped with the dressing attachment drived by means of NC controllers, which is described as "form dressing an involute profile based on a circular arc movement" in the 2nd report written by authors, is developed. Here, an economical D. D. A. type NC controller, depending on a method of automatic interconnection among operational elements, is used. It enables, what you call "the on-line form grinding machine of gears", correcting the part of a punched tape, which allows easy and rapid grinding gears in obedience to their data: module, tooth number and pressure angle, further tooth profile modification. For example, standard and modified profiles, of which data is module; 2, number of tooth; 48, and pressure angle; 20 degrees, are ground by this machine, and results of profile and tooth trace inspection show the deviation from the ideal and the experimental are separately within 2 μm. Further, an automatic indexing machine using a rotary magnetic scale, which set a position demand counter to the total number corresponding to the angle required, then the dividing angle is a multiple of 10 minutes, is designed. The indexing accuracy of 17 seconds of arc, is demonstrated and its repeatability is high. Using the grind wheel of a finer grain size, the accumulated pitch error due to the abrasive wear appears with distinct increase. Then, the wear quantity being detected by means of air micrometer, the result of compensating the pitch error shows that it decreases from 51 seconds to 24 seconds.

高いピッチ精度を目的とする歯車ラップ仕上法

In this paper, a method of finishing the highly accurate external gear by lapping is reported. First, the external gear is meshed with the lapping tool which is the internal gear having the same number of teeth and the same tooth-profile with the work. The lap is reciprocated in the direction of the gear axis. Then the lap is turned at a certain angle so as to change the meshing teeth, and the lapping is done again. In a similar manner, the lapping action is repeated changing the meshing teeth, and the accumulative pitch error is decreased.It is calculated statistically how the accumulative pitch error decreases in the case of changing the meshing teeth at random, then an experiment is tried on a spur gear. Next it is calculated how the accumulative pitch error is decreased when the work is placed at a certain angle with the lap, then two experiments are tried on a plane-tooth gear. It is concluded from the experiments that the experimental values have the same tendency with the theoretical results, and the best accuracy of the accumulative pitch error obtained in these experiments is 4.1 seconds (1. 3 μ).

A Study on Precision Thread Grinding with Numerically Controlled Compensation

For leadscrews used on machine tools or length measuring instruments, requirement of grinding after hardening is increasing. Therefore, ground leadscrews of a higher pitch accuracy are required. Taking into account that one of the causes of pitch error in thread grinding is thermal deformation in the workpiece screw and the grinding machine, it is desirable that the pitch error compensation be performed according to the grinding condition of the workpiece screw. The purpose of this study is to improve the pitch accuracy of ground leadscrews by using the compensation of a numerical control system. The method is as follows : Initially thread grinding without compensation is performed. On the basis of the pitch error obtained in this case, a tape for compensation is punched, and a second thread grinding is performed with the operation of the numerically controlled compensation device. As the result of an experiment, in a ground leadscrew of 1m in thread length the accumulated pitch error in the whole length was reduced to about 5 μ by using this device.

歯車の個別誤差と総合誤差との関係について

In this paper, the relations between the individual errors of a gear and the angular position error of the gear are studied.The angular position error of a gear can be expressed with a Fourier expansion formula, and the equation which expresses the relation between the pitch error and the angular position error is obtained. From the study following results are obtained.1. In evaluating the angular position error from the individual errors, the essential errors are accumulative pitch error and the tooth profile error. The former is to be used to evaluate the lower harmonics components of the Fourier expansion of the angular position error and the latter is to be used to evaluate the higher harmonics components of it.2. It is not advisable to obtain the maximum value of the accumulative pitch error from the measured values of the individual pitch error. As the measurement error in the individual pitch error measurements are also accumulated and maximum value has largely changed. The value is rather to be obtained from the measured value of z/2 pitch error measurement or from the value of run-out error measurement.

A Study on Precision Thread Grinding : Concerning Thermal Deformation of Workpiece

One of the most important problems in precision thread grinding is thermal deformation of a workpiece caused by grinding heat. This deformation has direct influence on its pitch accuracy. The temperature rise, distribution and longitudinal thermal expansion of a workpiece of 1 m in length are investigated on three processes, that is, (a) dry grinding, (b) grinding with spindle oil and (c) grinding with water soluble oil. The main results are as follows : 1. In the case of (a), the temperature rise in a workpiece is remarkable, so this process is undesirable in practice. 2. In the case of (b), the grinding heat is little generated, but the temperature of a workpiece rises gradually as grinding process is repeated. Accordingly the workpiece expands and this expansion tends to cause negative pitch error. 3. In the case of (c), the temperature of a workpiece falls down owing to the evaporation of grinding fluids, and its temperature distribution is stable irrespective of the grinding strokes. For example, the accumulated pitch error owing to thermal contraction is +35μ/1 m in the experiment. 4. In the cases of (a) and (b), the measured temperature distributions on workpieces conform to the results of theoretical calculations based on the heat conduction with moving heat source.

自動記録式歯車ピッチ測定器の試作

A prototype apparatus is designed for measuring the accumulated pitch error of gears automatically.This apparatus has a set of master worm gears similar to that used in a gear hobbing machine. The angle transmission errors of the master worm gears are corrected by two cams which rotate synchronously with both of the worm and worm wheel.Before manufacturing the prototype, operations of the master worm gears and the bearing of the master worm wheel are examined experimentally and it is proved that they are suitable for this prototype apparatus.Practical measurements of a spur and a helical gear each having a pitch diameter of about 120 mm are carried out and the following results are obtained.1) Inconsistencies of the accumulated pitch error curves among the successively obtained four records are at most 1.3μm and 1.8μm respectively with the spur and the helical gear.2) The errors of the master worm gears are not entirely eliminated leaving angle transmission errors of 7.0 seconds and 7.7 seconds which are respectively thought to be directly resulted from the profile errors of the two cams. When the cams are made more nearly perfect, these errors will be reduced.

歯車ピッチ誤差の迅速測定法

歯車ピッチ誤差の迅速測定法