MORE SPECIAL CASES IN SPECIFYING THE DEVIATION OF 3D REFERENCE AXES

Size: px

Start display at page:

Download "MORE SPECIAL CASES IN SPECIFYING THE DEVIATION OF 3D REFERENCE AXES"

Marianna Caldwell
5 years ago
Views:

1 5 th International & th All India Manufacturing Technology, Design and Research Conference (AIMTDR 014) December 1 th 14 th, 014, IIT Guwahati, Assam, India MORE SPECIAL CASES IN SPECIFYING THE DEVIATION OF 3D REFERENCE AXES T S R Murthy 1*, T Shravan Kumar 1* Kothiwal Institute of Technology & Professional Studies, Moradabad44001, MurthyTSR@yahoo.com Daimler Commercial Vehicles Pvt ltd. Chennai0009, Shravan.T@daimler.com Abstract In many precision machines and equipments, there are two or three reference axes intersecting at a theoretically as per the drawing. One example is gyro spin axes construction. The other examples are N/C machine tool spindle axis and table axis as reference axes. Yet another example is the three bevel gear reference axes of space equipment like helicopter, intersecting at a common. In all the above examples, theoretically the axes have to meet at a common. But at manufacturing stage one has to specify the acceptable deviation. Presently each axis is measured separately for its straightness or perpendicularity with some surface. Though the axes are measured separately one cannot say to what extent they are meeting or how closely they are approaching to the theoretical intersection. Further there are no standards to specify this error. The authors have earlier established and published different ways of fitting axis. Some six methods of specifying the deviation/error were also defined. Out of these six specifications, Tangential Spherical deviation (TSD) has more significance. This results in special cases when the viewing direction for specifying the error coincides with one of the three axes. A simplified method for evaluating such cases is discussed in this paper. Keywords GD & T, Evaluation of reference axes, Error evaluation 1 Introduction Measurement of straightness, flatness etc related to one dimension or two dimensions is well established. These are dependent on the availability of a suitable measuring instrument and an evaluation technique based on available D geometric tolerancing standard and evaluation technique. In any precision equipment or machine, there are two or three mutually perpendicular axes, corresponding to the slides and/or spindle. The present practice is to measure and evaluate each axis for its linearity/straightness, because of the limitations of a measuring instrument and because of the nonavailability of a suitable evaluation standard. Presently the trend is complete elimination of D drawings and migration to 3D with dimensioning and tolerancing in 3D drawings. In this regard a new ASME Y , STANDARD FOR CAD in to the digital domain is already available (Digital product definition data practices and X dimensioning and tolerancing Y14.41 & Y14.5M). In this context there is a need for suitable evaluation techniques for application in D and 3D tolerancing. A method for evaluation of two mutually perpendicular axes has been reported by Murthy ([1],1994), which evaluates an artefact with number of holes along two perpendicular rows for its hole positioning accuracy and alignment of the holes. A method for evaluation of three mutually perpendicular/nonperpendicular & intersecting axes has been reported by Murthy ([,3], 005, 00), which evaluates the three likely intersecting axes of precision equipments. After developing different methods for evaluation of the axes supposed to passing through a six different ways of specifying the deviation are indicated. Among these six different specifications the method of specifying the error one of the ways of specifying the error namely Tangential Spherical Deviation (TSD) has more significance. This specification leads to special cases when the viewing direction of the axis for specifying the error coincides with one of the axis. In this paper an algorithm for dealing with such cases is presented. 1

2 MORE SPECIAL CASES IN SPECIFYING THE DEVIATION OF 3D REFERENCE AXES Matlab has been used for developing the algorithm and for simulation and verification of the proposed method. Problem definition The problem definition and the work done so far are explained with the help of a figure. Fig.1 shows a Cartesian coordinate system with three likely intersecting axes and the measurements corresponding to the three axes of a machine like helicopter gear box. The problem is: how to findof these axes and out the likely of intersection how closely they are approaching and how to evaluate and specify the deviations/error? This has been solved by specifying a viewing direction and a projection plane perpendicular to the viewing direction, to project the axis on to the plane. This results in three lines on the projected plane forming a triangle. The diameter of the incalled TSD. This circle of the triangle so formed is results in special cases resulting in no triangle when the viewing direction coincides with one of the axes. This paper presents an algorithm for such cases. 3 Analysis To analyse the problem different methods were proposed by Murthy ([3,4],Nov 00, Dec 00). In one of the methods best fit axis for each reference axis is fitted and shortest distance is treated as the error. There are three shortest distances taking two lines at a time. If we consider the end s of these shortest distance lines, we get six s (say P 1 to P ). In another method three best fit planes are fitted taking two axes at a time and the intersection of the three planes gives a of intersection. This can be considered as seventh (P ) to approximate the actual theoretical intersection. The location of this from the theoretical one is the deviation. The above methods have been applied for different data and the results are satisfactory ([5], Nov 00). Theoretically all the s mentioned above should converge to the theoretical with zero deviation. However, the following deviations are defined for error specification. 1. Enclosed Spherical Deviation (ESD): It is defined as the diameter of the minimum sphere enclosing all the coordinates of the end s of the three shortest distance lines. Fig. 1. Reference coordinate system along with the three axes to be evaluated.

3 5 th International & th All India Manufacturing Technology, Design and Research Conference (AIMTDR 014) December 1 th 14 th, 014, IIT Guwahati, Assam, India. Maximum Distance Deviation (MDD): It is defined as the maximum distance among the 15 distances ( C = 15, distance between six different s P 1 to P taken two at time) 3. Maximum Shortestdistance Deviation (MSD): It is defined as the maximum of the 3 shortest distances 4. Tangential Spherical Deviation (TSD): It is defined as the diameter of the smallest sphere tangential to all the three lines 5. Maximum Throat Deviation (MTD): It is defined as the maximum throat of the hyperboloids constructed with each of the lines about a theoretical mean axis.. Simple Spherical Deviation (SSD): It is defined as twice the maximum distance computed from the theoretical axis intersection to the end s of shortest distance lines. The approach for computations of ESD is given by the author []. The computation of MDD, MSD and MTD and methods for minimum zone evaluation are and explained in [3, 4, 5, & 10]. The TSD has some importance. The method of computing TSD is interesting and is explained by Murthy et al ([], 011) and given below in brief, as it is partly required in the solution of special cases. A. Method to compute TSD In this method the following logic is used. Let us assume a theoretical mean axis or the specified mean axis for measurement and evaluation purpose to be specified in GD & T. When all the three axes are projected on to a plane perpendicular to the mean axis or specified axis, we get three lines in the 3D plane. The three lines give three s of intersection forming a triangle as shown in Fig.. The incircle tangent to all the three sides of the triangle represents the sphere tangent to all the three axes in space. Its diameter is the TSD. The method to compute TSD is already explained []. B. Special cases in TSD evaluation. When the viewing or specified direction (for indicating the error) coincides with one of the axis then the projection of the three axes on to the projected plane will be represented as two lines and a. Thus no triangle is formed as shown in Fig.3, for computation of TSD easily. The algorithm for such cases is given below. 1) Mathematical analysis: Fig. 4 shows an axis (line) AB in dotted lines passing through two s A(a 1,b 1,c 1 ) and B(a,b,c ). The figure also shows a plane which is normal to a mean or specified line passing through origin (specified line shown as dot and dash type) Projection of axis (slope m) Projection of axis 3 as a Projection of axis 1(slope m1) Fig.3. Projection of three axes as lines & a on the projection plane normal to the viewing direction. A (x1, y1, z1) line Incircle B (a, b, c) lines A (a1, b1, c1) p lx + my + nz = p B (x3, y3, z3) Fig. Axes projected in the plane perpendicular to the specified mean axes. Fig. 4 Axes AB projected as A B in the plane perpendicular to specified axis 3

4 MORE SPECIAL CASES IN SPECIFYING THE DEVIATION OF 3D REFERENCE AXES The algorithm to find the projection of s A and B namely A and B etc are a given by Murthy([], 011). When all the three projected lines intersect to form a triangle the computation of TSD is easy. Now in the special cases there are two lines and a. Now to find TSD, we need to construct a circle tangent to two lines and passing through a. We generally feel that that there is only one circle which is tangent to two lines and passing through a. But from Fig.3 we can find two circles satisfying the above condition (namely tangent, tangent, ). Obviously we need the diameter of the smaller circle to specify TSD. An extensive step by step procedure to arrive at the radius of the circle has been presented Murthy([], 01). Now an alternative simple method is presented as explained below. Let θ be the angle between the projected intersecting lines OA and OB as shown in Fig.5. Here A is also marked as the tangent. Let α be the angle between OA and OP, where P is the projection of one line as a P. 300 coordinate system. Also y c is the radius of the circle in the local coordinate system As (1) Also as the angle between OA and OB is θ, the angle between OA and OC is θ/. Also tan = k (a constant & slope) Or () Substituting equation () in equation (1) and solving the quadratic equation for we get 1 As α is the angle between lines OA and OP (3) cos (4) sin (5) B The angles θ and α are computed from the dot product relation of the D or 3D vectors of those intersecting lines namely OA, OB, and OP. Thus equation (3) gives two solutions for the circle centre y C A x O Fig.5. axes with local coordinate system. P If C is the centre of the required circle in the projected plane and P is the line projected as a, then, then distance CA and CP are equal being the radius of the circle. Considering a local coordinate system with x axis along OA, the coordinates of C and P are C(x c,y c ) and P(x p,y p ) in the local 4 Results Table 1 show coordinates of the axis or line end coordinates (a, b, c) along with projected end coordinates along different viewing directions. When the viewing direction is same as the axis direction the line will be projected as a where start and end coordinates are same. They are shown as Bold Italic numbers. The last row shows the coordinates of the of intersection of the intersecting lines in the plane of projection. Table shows the results of the analysis for a specific simulated data. The first six rows show the input data for all the three lines. The remaining data shows the computed data as per the proposed method. Rows to 10 show the diameters of the two circles when the viewing directions are considered along each axis (line). Row 11 shows a random direction where, in the projected plane, all the three axis intersect and form a triangle giving a single inscribed circle as TSD. This proposed method was found convenient for solving the special cases. This also 4

5 5 th International & th All India Manufacturing Technology, Design and Research Conference (AIMTDR 014) December 1 th 14 th, 014, IIT Guwahati, Assam, India suggests a method of specifying the TSD when no viewing direction is mentioned on the drawing. In such cases the maximum TSD obtained from viewing along all the three directions can be considered. In this case its value is 0.3. Thus this method provides a solution when the view direction is not known or not specified. Table 1 Coordinates of line(axis) end s when projected on to a plane in different normal directions Descripti on coordinated when Viewed along Line 1 coordinated when Viewed along Line coordinated when Viewed 3 coordinated when Viewed along given direction a b c x1 y1 z1 x1 y1 z1 x1 y1 z1 x1 y1 z1 1 Line 1 Start Line 1 End Line Start Line End Line 3 Start Line 3 End Coordina tes of intersecti on

6 MORE SPECIAL CASES IN SPECIFYING THE DEVIATION OF 3D REFERENCE AXES Table.. Summary of results Sr Description a b c No 1 Line 1 Start Line 1 End Line Start Line End Line 3 Start Line 3 End Direction l m n cosines of Line Line Line Line (View Direction) TSD D1 D Min of (D1, D) 1 TSD ) 13 TSD ) 14 TSD ) 15 TSD ) 1 TSD if view direction not specified 0.3 (Max of the above) 5 Conclusions Different methods developed to find best fit space axes satisfying the constraints to some extent for the randomly oriented three axes of a precision machine or equipment like helicopter gear box are briefly explained. Also different ways of specifying the error are defined. Finally a method for computation of TSD when the viewing direction coincides with one of the axis is explained. The procedural steps were found convenient to arrive at the solution. The results of the algorithm are also given in the form of a table. ACKNOWLEDGMENT The authors are thankful to their managements for permission to publish this work. REFERENCES [1] Murthy T S R, An approach for simultaneous interrelated tolerance evaluation, Proc. 1 th AIMTDR Conf, P (Dec 1994) [] Murthy T S R, Measurement and evaluation of three mutually perpendicular axes of precision equipment, 11 Dec 005, COPEN 005, Kolkatta, p 4445, 005. [3] Murthy T S R, Measurement and evaluation of three intersecting axes of precision equipment, 9th international symposium on measurement and quality control (9 th ISMQC) November 1 4, 00, IIT Madras, 00 [4] Murthy T S R, Measurement and evaluation of three closely approaching axes of precision equipment, COPEN 00, Dec 00, CET, Trivendrum, p3334, 00 [5] Murthy T S R, Methods for evaluation of 3D reference axes, ICAMT00, Nov 00, CMERI, Durgapur, p4450 [] Murthy T S R, Rao B R, Abdin S Z, Evaluation of Spherical Surfaces, Wear, vol.5, p114, 199 [] Murthy T S R & Shravan Kumar T, A Method for specifying the deviation of 3D reference axes, COPEN Dec 1011, 011, College of Engg, Pune, 011 [] Murthy TSR and T Shravan Kumar, Method for specifying the deviation of 3D reference axes. AIMTDR 01, Jadavpur University, Kolkatta, Dec 01,p44 4 [9] Murthy TSR and T Shravan Kumar, Special cases in specifying the deviation of 3D reference axes. COPEN, NIT Calicut. Dec 013 [10] Murthy T S R and S Z Abdin, Minimum zone evaluation of surfaces, Int. JMTDR, vol.0, p1313,190

CHAPTER 40 CARTESIAN AND POLAR COORDINATES

CHAPTER 40 CARTESIAN AND POLAR COORDINATES EXERCISE 169 Page 462 1. Express (3, 5) as polar coordinates, correct to 2 decimal places, in both degrees and in From the diagram, r = 32 + 52 = 5.83 y and 5