Design of Spur Gears Using Profile Modification 739 Downloaded by [Manipal University], [sachidananda H K] at 21:24 26 May 2015 The normal force along the different points along the length of path of contact was calculated using Eq. : F t FNX D ;  cos tan ¡ 1 where F t is the tangential force (N). The individual load along the different points on the length of the path of contact was calculated using Eq. : T 1 X R b1 QX D FNX b ;  where b is the face width (mm). The maximum contact stress along different points on the length of the path of contact was calculated using Eq. : where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QX .s max / X D con ;  ðRTR rffiffiffiffiffiffiffiffiffiffiffiffi 0:35E con D 2 Þ X  if the pinion and gear are of the same material. A computer program was developed based on the above methodology using C language to compute the contact stresses. Experimental Investigation Figure 2 shows the digital photograph of back-to-back cyclic loading experimental test rig. It is one of the standard test rigs used to test gears for finite life fatigue (Nabih, et al. (1)). In the present research work, a selected tooth-sum of 96, 100, and 104 gears was tested for a total of 10 6 cycles. Fig. 2—FZG testing machine used for experimentation. 1, Gear train under test (heat treated); 2, shaft; 3, torque meter; 4, vernier coupling; 5, support plate; 6, transmission gear (untreated). Subsurface cracks, multiple cracks on the surface, scuffing, and pitting are the surface defects that are commonly developed during fatigue loading of gear teeth. In this work, in order to obtain a clear picture of surface damage, a morphological study of the tested gear tooth was conducted using scanning electron microscopy (SEM; JEOL-JSM-6380LA, Japan). The samples were prepared by wire electrodischarge machining at appropriate points on the gear tooth and the sample surface was coated with gold using a sputtering unit. SEM microphotographs were taken at appropriate voltage. RESULTS AND DISCUSSION Contact Stress The s max point shifts along the path of contact and it depends on profile shift and position of pitch point C. It is shown in Fig. 1a that T 1 C > T 1 B; hence, the ratio T1C T >1 and s 1B max for the pinion appears at point B. Otherwise, if T 1 C T 1 B, the ratio T1C T 1B 1 and s max for the pinion appears at point C. Similarly, if T 2 C > T 2 D, and if T 2 C T 2 D, the s max for the gear appears at points D and C, respectively (Moldovan, et al. (24)). The length of T 1 T 2 for 100 tooth-sum gears altered by §4% for 20 and 25 pressure angles was computed using Eq.  and tabulated as shown in Table 1. Based on the these tabulated values, the ratio of contact lengths versus profile shift for 20 and 25 pressure angles was computed using the developed C program and the plots were drawn as shown in Fig. 3. The ratio of T1C T2C T 1B and T at s 2D max point for 20 and 25 pressure angles was identified by the output and tabulated as shown in Table 2. The computed values of contact ratio, addendum, dedendum, and whole depth using C program were compared with the appendix F and appendix D in Gitin (20) and it was found that the values are in line. From the ratio of contact lengths versus profile shift plots (refer to Fig. 3), the following analysis was made. The ratio T1C T 1B decreases and the ratio T2C T 2D increases with an increase in profile shift on the pinion. The range of profile shift (X 1 ) for a 20 pressure angle varies between ¡0.6 to 1 and ¡1.6 to 0.6 for negative and positive alteration tooth-sum, respectively. Similarly, the range of profile shift for 25 pressure angle varies between ¡0.6 to 1.8 and ¡1.8 to 0.7 for negative alteration and positive alteration tooth-sum, respectively. From Table 2, it is seen that for a tooth-sum of 96 teeth the ratio T1C T 1B is 1.014 for the profile shift X 1 D 1.6 and the ratio T1C T is 0.996 for profile shift X 1B 1 D 1.7 for a 20 pressure angle. This indicates that s max shifts from point B to point C. Similarly, for the same tooth-sum the ratio of T2C T 2D is 0.984 for the profile shift X 1 D 0.3 and 1.001 for the profile shift X 1 D 0.4, where the s max shifts from point C to point D. These are the critical points where s max shifts from the inner contact point (B) of a single-pair mesh to the pitch point (C) and vice versa. In general, from Table 2, for negative alteration (96, 97, 98, and 99) tooth-sum and for standard tooth-sum (100), it is seen that the ratio of T1C T2C T 1B and T 2D changes from point B to point C and from point C to point D for positive values of profile shift. Similarly, for positive alteration (101, 102, 103, and 104) in tooth-sum
740 H. K. SACHIDANANDA ET AL. TABLE 1—LENGTH OF PATH OF CONTACT (MM) FOR TOOTH-SUM OF 100 ALTERED BY §4TEETH Altered tooth-sum 96 97 98 99 100 101 102 103 104 Length of T 1 T 2 (’ D 20 ) 43.15 41.13 38.98 36.68 34.2 31.5 28.51 25.14 21.2 Length of T 1 T 2 (’ D 25 ) 49.22 47.66 45.95 44.15 42.26 40.26 38.14 35.86 33.41 Downloaded by [Manipal University], [sachidananda H K] at 21:24 26 May 2015 the ratio of T1C T 1B T2C and T 2D changes from point B to point C and point C to point D for negative values of profile shift. The magnitude and point of s max for altered tooth-sum gearing has been computed by C program and tabulated in Table 3. The following salient features were observed from Table 3. It is seen from the results at a profile shift of 1.6 the s max is in a single-pair mesh (point B). For f D 20 and X 1 D 1.6 it is seen that value of s max for a tooth-sum of 96 at point C is 189.48 MPa. In this case, the length of T 1 C is greater than T 1 B (21.57 mm > 21.27 mm) and T 1 C is less than T 1 E (27.17 mm). Hence, the s max is at pitch point C, which lies in a single-pair mesh between point B and point D. For the same tooth-sum, the value of s max for the profile shift X 1 D 1.7 at point C is 133.98 MPa, which is less compared to previous value of profile shift and is due to smaller length of T 1 C (21.57 mm), which is less than T 1 B (21.65 mm). Therefore, the magnitude of s max is reduced as the pitch point C lies in between point A and point B (two-pair mesh). In addition, for f D 25 and X 1 D 1.6, a similar trend was observed. It is observed that the magnitude of s max at points B and C is lower at a 25 pressure angle compared to a 20 pressure angle for profile shifts of 1.6 and 1.7. The length of the path of contact T 1 T 2 is 49.22 mm in the case of a 25 pressure angle is higher compared to the length of the path of contact at a 20 pressure angle (43.15 mm). This leads to an increased radius of curvature and results in reduced contact stress. In the case of f D 20 , X 1 D 0.1, and tooth-sum of 100, the s max at pitch point C is 208.53 MPa, and the length of T 1 C is greater than T 1 B (17.10 mm >16.84 mm) and less than T 1 D (18.29 mm). The pitch point C lies in between point B and point D; hence, C lies in a single-pair mesh. In this case, the contact ratio is 1.75. Similarly, for f D 25 for the same tooth-sum, the s max at point Fig. 3—Plots of ratio of contact lengths versus profile shift: (a) (T 1 C)/(T 1 B) versus X 1 and (b) (T 2 C)/(T 2 D) versus X 1 for a 20 pressure angle; (c) (T 1 C)/(T 1 B) versus X 1 and (d) (T 2 C)/(T 2 D) versus X 1 for a 25 pressure angle.