Analysis of the Pitch Deviation in Involute Splined Connections

Transcription

1 Machine Dynamics Research 2013, Vol. 37, No 1, Analysis of the Pitch Deviation in Involute Splined Connections Abstract Jacek Kroczak, Marian Dudziak Poznan University of Technology 1 jacek.kroczak@put.poznan.pl The paper presents the measurement results of the pitch deviation for the selected involute splined connections applied in aircraft engines. The authors have stated that there is a lack of exact information on a load distribution in involute splined connections with this type of manufacturing error. Hence, the analysis of the influence of the pitch deviation in a two-dimensional model on the load distribution and stress state is presented. This model gives us an information on the load distribution coefficient in a cross-section of the involute splined connection which depends on torque, tooth pair stiffness and manufacturing accuracy of a part, ie. the pitch deviation. Keywords: involute spline, pitch deviation, load distribution coefficient. 1. Introduction Splined connections are used to align two elements and transfer the torque. In the mechanical engineering the following modifications of splines are applied: straight-sided spline, involute spline and serrations. Involute splined connections are commonly used in aircraft and automotive industry, because in comparison with other types of splined connections these ones are characterized by: higher strength of teeth, lower stress concentrations, radial alignment of mating elements and self-acting coaxial align of the hub and the shaft under loading [Klebanov et al., 2008]. These connections can be found in many mechanical transmission systems, e.g. gas turbine engine mainshafts, intermediate shafts of tractor gearbox and automobile drive shafts. The geometry of involute splines is very similar to gear wheels, but their teeth are shorter in height. Therefore, an involute spline can be cut and measured with the same machines as for gear teeth. Standard involute splines utilize the 1 Chair of Basics of Machine Design

2 66 J. Kroczak, M. Dudziak nominal pressure angle values of 30, 37.5 or 45 degrees. However, in special applications, the pressure angles of 14.5, 20 or 25 degrees are sometimes chosen, which are commonly used in gears. 2. Design and standarization of involute splines Involute splines are specified by the international standard ISO 4156 or by the national standards, e.g. ANSI B92.2M or DIN According to the ISO standard, there are four classes of machining tolerance and six fit classes. Fittings mentioned above concern side fits only. After machining and heat treatment of involute splines, we can not obtain a perfect external or internal involute spline. There will always be some form deviations. These deviations affect the maximum material condition and hence the fit of connection. The form deviations are complex and occur on each flank of tooth and tooth space. The most important spline deviations are: pitch deviation, profile deviation and lead deviation. Therefore, the tolerance zone of the tooth thickness and space width has been enlarged for these deviations [Kroczak, Dudziak, 2011]. This additional zone is defined as the deviation allowance λ. This deviation is the accumulation of the total pitch deviation, total profile deviation and total lead deviation and has an effect on the effective fit of an involute spline [Cedoz, Chaplin, 1994]. The deviation allowance is calculated as follows: F p + Fα + Fβ λ = (1) where: F p total pitch deviation, F α total profile deviation, F β total lead deviation. Spline tolerance class IT Total pitch deviaton F p [µm] mz π / mz π / mz π / mz π / Table 1. The formulas for allowable total form deviations Total profile deviation F α [µm] 1.6( m mz) ( m mz) ( m mz) ( m mz) + 40 Total lead deviation F β [µm] 0.8 L + L L L + 10 where: m module, z number of teeth, L the spline length [mm] The formulas for allowable total form deviations are given in Table 1. These relationships are restricted to the assumption that the length of spline engagement is equal to one half of the pitch diameter. 6.3

3 Analysis of the Pitch Deviation in Involute Splined Connections 67 The strength of involute splined connection is checked on the basis of calculations of contact pressures between the external and internal spline teeth. According to the handbooks for machine design the analytical formula for maximum contact pressures is the following: p F z h l p (2) max allowable = δ where: F = 2T/D circumferential force, T torque, D = 0.5mz, z number of teeth, m - module, h height of the active surface of the tooth, l engaged length of the teeth, δ accuracy factor, p allowable allowable stress. Formula 2 assumes that only a fraction of teeth transfer the load due to the manufacturing accuracy. The accuracy factor δ in the denominator is not directly connected with the specific class of tolerance. Its value is arbitrarily taken between 0.5 for low manufacturing accuracy and 0.8 for high manufacturing accuracy. Some handbooks for engineers impose to take into account the value of threequarters. So, there is a need to do research on the influence of the individual form and postion deviations on the load distribution for different tolerance classes. 3. Pitch deviations measuring Knowledge of measuring methods and modern measuring devices gives to a designer the possibility of taking into account the real shape of the structure of connection elements in transverse and longitudinal sections [Podolski et al., 2010]. Traditional methods and measuring devices allow for a simple measurement of particular dimensions of involute splines and their deviations. In order to perform the measurement of all types of deviations which characterize the tested element, one should use many different measuring devices. This situation increases the time of production control process. So, in many production plants coordinate measuring machines are used. These types of measuring machines allow to measure all types of deviations at one attachement of the part. Involute splines are manufactured using identical machining methods as gear wheels with involute profile. Therefore, involute splines have the same types of form and postion errors as gear wheels. In order to check the range of value variations of the pitch deviations in a permissible tolerance zone and characterisitc of these deviations, the authors performed the measurements of pitch deviations for external and internal involute splines. The connection applied in aircraft engine between the driving shaft and the drive gear wheel of regulator for a rotational speed was investigated. Single pitch deviation fpt which is the difference between two adjacent teeth pitch values was measured. The value of this deviation can be with plus or

4 68 J. Kroczak, M. Dudziak negative sign. It was measured using the coordinate measuring machine Klingelnberg P40 with measuring uncertainty of +/-1µm for the pitch deviation. The examplary raport with the measurement results of the single pitch deviations is presented in Fig ,00 Deviation value [um] 5,00 0,00-5,00-10, ,00 Tooth number Fp fpt Fp' Er Fig. 1. Example of the measurement results of the single pitch deviation f pt, calculation of the cumulative pitch deviation F p, the actual pitch deviation F p ' and the concentricity deviation E r for the analysed internal involute spline Pitch deviation measurements allow to determine the accuracy of the tooth locations in a cross-section of the involute spline. Two parameters are used to classify the pitch accuracy of a part, i.e. the maximum single pitch deviation and the total (cumulative) pitch deviation. The last type of deviation is the total amplitude of the cumulative pitch deviations plot and its allowable values are defined by the standards (see Table 1). The measurement results of the pitch deviations are distorted by the concentricity deviation Er which is the result of the eccentricity of external and internal involute splines. In this situation we have to separate the concentricity deviation from the cumulative pitch deviation in relation to rotation axis of involute spline. In order to do this, we can perform the harmonic analysis of the curve of the obtained cumulative pitch deviations. The obtained results of the harmonic analysis and separation of the pitch deviation from the concentricity deviation are called the actual pitch deviations Fp' and are presented in Fig. 1. The values of the actual pitch deviations were checked if these random variables derive from a normal distribution. To perform the normality test, the test of Ryan-Joiner was applied. This test is similar to the Shapiro-Wilk test normality. We prepared and analysed histograms and probability plots of the measured pitch deviations for external and internal involute splines. On the basis of the p-values given by the test normality, which were equal or

5 Analysis of the Pitch Deviation in Involute Splined Connections 69 greater than 0.05, we have stated that the data being tested is derived from the normal distribution. Besides the authors verified the percentage participation of the actual pitch deviations in the permissible tolerance zone. It was stated that the value of the actual pitch deviaton is within % of the tolerance zone for external involute splines manufactured with hobbing and the value of this deviation is within % for internal involute splines manufactured with broaching. 4. Load distribution in a cross-section of involute splined connection with the pitch deviations Form and position errors of involute splines have an essential influence on the value and distribution of contact pressures, clearance between mating tooth flanks, resistance to motion for sliding connections and reliability of stationary connections. These errors cause faster volumetric wear of parts and variation of contact stiffness of the connection [Kroczak et al., 2007]. In order to analyse a load distribution coeffiecient in a cross-section of involute splined connection with the pitch deviations, the authors prepared a full two-dimensional FEM model of the connection with two types of fits, ie. H/h and H/f, and with different tolerance classes and total and single pitch deviations (Fig.2). Fig. 2. Two-dimensional FEM model of the involute splined connection The parameters of the geometry and the material of the model were defined as follows: pitch diameter - defined as a reference parameter and was equal 20mm, so the number of teeth - 10 and 20, module - 1 and 2, pressure angle - 30, Young's

6 70 J. Kroczak, M. Dudziak modulus MPa, Poisson ratio The external and internal teeth were flat root fillet. The torque Mo was equal first 80Nm and afterwards 800Nm. The model was meshed with element type CPE4R, i.e. a 4-node, bilinear plane strain, quadrilateral element with reduced integration and hourglass control. Thickness effects were modelled utilizing plane strain with a thickness of 1mm. Element size was defined by a local number of elements on the tooth height and root fillet. The contact was modelled as hard with type and discretization method of contact - surface to surface; constraint enforcement method - penalty [Podolski, Kroczak, 2010]. On the basis of the numerical calculations of the contact pressures, we elaborated the load distribution coeffiecients Kp=pmax/pmean for the different types of fits and tolerances. The obtained results are presented in Fig. 3 and 4. 2,5 2 Kp 1,5 1 m1 z20, Mo=80Nm m1 z20, Mo=800Nm m2 z10, Mo=80Nm m2 z10, Mo=800Nm 0, IT (H/h) Fig. 3. Load distribution coefficient K p for involute splined connection with sliding fit H/h and different tolerance classes IT 2,5 2 Kp 1,5 1 m1 z20 dla Mo=80Nm m1 z20 dla Mo=800Nm m2 z10 dla Mo=80Nm m2 z10 dla Mo=800Nm 0, IT (H/f) Fig. 4. Load distribution coefficient K p for involute splined connection with normal running fit H/f and different tolerance classes IT

7 Analysis of the Pitch Deviation in Involute Splined Connections 71 On the basis of the analysis of Fig. 3 and 4, we can see that the load distribution coefficient is within range of for low torque, and for high torque. We can also conclude that increasing the torque, decreasing the load distribution coefficient. This situation shows us that the values of the pitch deviations for the individual tooth pairs are reduced and more tooth pairs come in contact, i.e. are mating. The increase of the number of teeth (in this case: two times) does not result in a significant variation of the load distribution coefficient for the case of low torque, while this situation is changed for high torques. 5. Conclusions The two-dimensional model of the involute splined conenction does not take into account all parameters which have an influence on a load distribution. However, this planar model gives us an information on the values and distribution of contact pressures in a connection. The calculated load distribution coefficients depend on the value of torque, teeth number and manufacturing accuracy of a part, i.e. the pitch deviation. The obtained values of the load distribution coefficients are related with the defined fits and tolerance classes. References Cedoz, R.W., Chaplin, M.R., 1994, Design Guide for Involute Splines, SAE, Warrendale, PA. Klebanov, B., Barlam, D., Nystrom, F., 2008, Machine Elements: Life&Design, CRC Press, Boca Raton, FL. Kroczak, J., Dudziak, M., Kołodziej, A., 2007, Analiza konstrukcyjna połączenia wielowypustowego, XXIII Sympozjon PKM, Rzeszów-Przemyśl, Tom II, pp Kroczak, J., Dudziak, M., 2011, Tolerance analysis of involute splines, Lecture Notes in Engineering and Computer Science, vol. 2192, issue 1, pp Podolski, T., Dudziak, M., Kroczak, J., 2010, Aspekty projektowania pasowań w połączeniach osiowosymetrycznych, Modelowanie Inżynierskie nr 39, Gliwice, pp Podolski, T., Kroczak, J., 2010, Application of FEM in analysis of spigot joint contact problems, Progress in Industrial Mathematics at ECMI2008, Springer-Verlag Berlin Heidelberg, pp American Standard ANSI B92.2M-1980 Involute Splines, Metric Module. German Standard DIN 5480 Zahnwellen-Verbindungen mit Evolventenflanken. International Standard ISO 4156 Straight cylindrical involute splines Metric module, side fit.