A Generalised Approach In Identifying Control Link Tolerances And Its Effect On Design Tolerances Of Mechanism Using Instantaneous Center

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1 A Generalised Approach In Identifying Control Link Tolerances And Its Effect On Design Tolerances Of Mechanism Using Instantaneous Center C. C. Handa and H. T. Thorat 1 KDK College of Engineering 1 Visveshvarya National Institute of Technology Nagpur Nagpur 9 ABSTRACT: Mechanism is designed for desired output or required performance against specified input. The performance is not at par with designed performance due to manufacturing inaccuracies in link lengths. The link length inaccuracies are due to inherent mechanism errors due to deflection of links, clearances in joints, manufacturing tolerances etc. In mass production of machine / mechanism it is not possible to verify functionally all its characteristics. As such the quality of mechanisms will depend upon the tolerances specified. Performance of mechanism is expressed in terms of position, location or orientation accuracy. In case when mechanism has to work as force or torque manipulator the performance can be expressed as ratio of effort and load. This ratio of effort and load is used to calculate the performance deviation of mechanism in this paper. A designer provides close possible tolerance on all the links / joints to achieve mechanisms performance variations within limits. Close or precision tolerance will increase the cost of manufacturing. Hence instead of providing close tolerance on all the links in a mechanism the paper aim at identifying link/ s named as control link/s. Close tolerance on this link will allow the designer to provide broader tolerances on remaining links in a mechanism. A computer is used to do the iterative analysis. KEYWORDS: Instantaneous centre, performance, tolerance, mechanical advantage, control link. INTRODUCTION: Machines / equipment consist of linkage mechanism. Mechanisms are designed keeping in view the desired parameter or output. Designed mechanism gives a desired output for a specified input. Designed performance is very important in working of a mechanism. Performance of these mechanisms is affected by inherent mechanical errors. The errors are due to deflection of links, clearance in joints, manufacturing tolerances etc. In bulk manufacturing it is not possible to verify functionally all the desired characteristics and if it is so, then it add to the cost of production. The quality of mechanism depends upon the tolerances specified. To assure quality and achieve desired

2 performance, designers provide closer tolerances on all links. This closer tolerance adds to cost of manufacturing. It is difficult to get desired performance for some of the mechanism manufactured in the lot. It is equally difficult to find exact reason for deterioration of performance. If designer can analyze effect of possible errors, he can modify the mechanism such that in event of any normal inaccuracy, performance of mechanism can be within expected range. Main objective of the analysis is to determine the sensitivity of the mechanism to dimensional inaccuracies. Performance of mechanism is normally expressed as ratio of effort and load. This ratio of effort and load is used to estimate possible deviation in performance of a mechanism in this paper. Unlike referred in [1], where the set of tolerances are proposed on various links in a mechanism for acceptable percentage variation of performance. In this paper an attempt is made to identify the most sensitive link/s, named as Control Link/s in a mechanism. Narrow tolerances on this link can allow the designer to provide broad tolerances on remaining links. This will reduce the cost of manufacturing. CONCEPT: The performance of a mechanism in a torque or force manipulator is the ratio of effort and load. The ideal ratio can be identified or specified. If the same ratio is obtained in the fabricated mechanism, performance of the mechanism is ideal. Any variation in the ratio can be expressed as percentage variation in the expected ratio., Performance deterioration = (Variation in the ratio / Ideal ratio) * 100 Permissible deterioration can be specified at the time of defining requirement of the mechanism. If the performance deterioration of the fabricated mechanism is within the permissible limits, the mechanism should be acceptable. Tolerances on the mechanism should be specified such that when the link dimensions are within the tolerances, performance deterioration should be positively within the limits. Tolerances are normally specified in relation to the manufacturing process of the link. Based on the permissible tolerance on the link lengths maximum possible deterioration of the performance can be estimated. If it is within the limits, the manufacturing process is acceptable. However if the deterioration is not within the limit, further analysis is required to find out ideal combination of tolerances. The procedure for identifying such combination is explained in next part. The ratio of load to effort is Mechanical Advantage of the mechanism. This ratio is inversely proportional to the ratio of angular velocities of the two links. Ratio of angular velocities can be easily calculated using the method of instantaneous centers.

3 Load/Efforts = ωi/ωo= Ilo Iio / Ili Iio Where w1 = Angular velocity of input link I1j =Instantaneous centers between frame & input link. I1o =Instantaneous centers between frame & output link Ii o =Instantaneous centers between input & output link. Refer figure no. 1 in the last for 4R & 6R mechanism and table no. 4 of I1i, I1o & Iio for different output links of same mechanism i.e. 1) four bar chain link 4 as output link 2) Six bar chain link 6 as output link 3 ) Six bar chain link 5 as output link. When performance of the mechanism is based on the ratio at the input & output forces, the mechanical advantage is a direct parameter of the performance evaluation. When the mechanism acts as a function generator variation in mechanical advantage changes the slope of the function and thus is a significant parameter. This parameter may not be of much significant in path generation or guidance PROCEDURE: Ideal mechanical advantage can be calculated using the specified link lengths. This Mechanical advantage will change if dimensions change for one or more links. It is not possible to perfect which combination will produce maximum change. Maximum variation of length of one link will cause maximum change in mechanical advantage because the tolerances are very small as compared to length of the link. It is possible that changes in mechanical advantage due to variation in length of one link can be compensated by variation in the length of the other link. Hence the maximum change in mechanical advantage may not occur when all the links are at their maximum or minimum length. To find maximum change in the Mechanical Advantage it is required to find the change in all combinations of the link lengths. For simplicity, only three possibilities are considered for each link. The link can be of standard length or it can be of lengths with extreme tolerances on both sides (L± t will give three length L, L+t. L-t). This will give 3 n combinations for which changes in Mechanical Advantage are to be calculated and its maximum value can be thus found. This maximum value will now tell whether for given combination of tolerances the mechanism performance is within limits or not. Now if the performance is within limit it can be said that the tolerance scheme is acceptable. If the performance is not within the permissible limit, it is possible to find number of combinations, which are not acceptable. The ratio of not acceptable combinations to the total number of combinations can roughly predict probability of poor performance of the mechanism. The performance can be improved by providing closer tolerances, which will reduce the number of rejected combinations. The choice remains with the designer to select closer tolerances

4 or to permit probability of poor performance. Depending on the use and type of mechanism, its applications and the manufacturing process of mechanism, variation in the dimensional limits can be anticipated. Designer would always choose closer tolerance rather than permeating probability of poor performance. It will be ideal if a designer can identify a link/s those are sensitive to dimensional changes and change in their dimension/s leads to variation of performance. Mathematical modeling and computer programmer is prepared to study the effect of variation of performance due to designed tolerances [1]. Further the programme will identify the most sensitive link/s or combination, which will be called as "Control link/s". The analysis even suggests close tolerance on control link/s and wide tolerance on other links rather than generalised close tolerances on all the links. RESULT: A computer programme is prepared to find instantaneous centers of a mechanism, using graphical analysis. Instantaneous centers are used to find mechanical advantage. Effect of dimensional change due to tolerance on percentage variation of mechanical advantage is tabulated. Total 243 combinations are possible for 6-Link mechanism considering extreme limits. Results obtained are summerised in table no. 1,2 and 3.Table no.1 shows results obtained when ±0.5 tolerances is provided on all links one by one, keeping tolerances on other links to between ± 1 mm. Sr. Links Tolerance Total Rejected % of No. Rejection 1 Link-2 ± Link-3 ± Link-4 ± Link-5 ± Link-6 ± Table no. 1 Table no. 2 shows the results obtained when the tolerance on the combination of two links are kept ±0. 5 mm and remaining link tolerances are ±1 mm. Sr. no. Links Toleranc Total Rejected % of e Rejection

5 1 Link 2 and 3 ± Link 2 and 4 ± Link 2 and 5 ± Link 2 and 6 ± Link 3 and 4 ± Link 3 and 5 ± Link 3 and 6 ± Link 4 and 5 ± Link 4 and 6 ± Link 5 and Table no. 2 Table no. 3 shows the results obtained when the tolerance on link 2, link 3,and combination is kept ± 0.5 mm and below, remaining link tolerances are ±1 mm. Sr. Links Tolerance Total Rejected % of No combination Rejection 1 Link 2 ± Link 2 ± Link 3 ± Link 3 ± Link 3 ± Link 2 and 3 ±0.5, ± Link 2,3,4,5,6 ± Table no. 3 CONCLUSION: The proposed method is useful in determining the tolerances required for individual links in a mechanism. The mechanism studied shows that (table no 1) link no. 2 and 3 are sensitive links. Closer tolerances on these links reduces percentage of rejection to 7.4% and 4.5 % Table no.3 aptly clears that the rejection percentage is same, keeping ±0.5mm tolerance on all the links or keeping 0.2mm tolerance on link no. 3 and ± 1 mm on remaining links. Closer tolerances ± 0.5 and ± 0.2 on link no. 2 and 3 respectively, further reduces the rejection percentage to 1. 6% thus link no. 3 is identified as a "Control link" and combination link no.3 & link no. 2 is also a controlling combination to reduce rejection.

6 REFERENCES: 1) C..C..Handa & Dr. H. T. Thorat,..A Generalised Approach In Anticipating The Effect Of Link Tolerance on performance of mechanism using instantaneous center ", Nacomm-99, Bombay. 2) C.C. Handa, M Tech Project instantaneous centre A Tool For Measuring Sensitivity Of Mechanism" 1997, VRCE, Nagpur. 3) Lee Ming Yin A.G. Erdman And S. Faik ""A Generlised Performance Sensitivity Synthesis Methodology For Four-Bar Mechanism", J Of Mechanical Design, DE- Vol. 46,1992,ppl -4. 4) B. Roth L A T Yang." Application Of Instantaneous Invariances To The Analysis And Synthesis Of Mechanism" J Of Engineering Industry, Trans. ASME, Feb77, pp ) Keith Nisbelt & Sanjay Gupta, ""A Dynamic (Implementation Of Keller's Sketching Rules For Burmester Curves: An Approach To Sensitivity Analysis"', J Of Machine Design DEVol-46, pp Sr. No Conditions I 1i I 10 I io 01 Four bar chain, Link 2 input and link 4 output link I 12 I 14 I Six bar chain, Link 2 input and link 6 output link I 12 I 16 I Six bar chain, link 2 input & link 5 output floating link I 12 I 15 I 25 Table no. 4 Figure no. 1

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