Test Piece for 5-axis Machining Centers
|
|
- Elmer Spencer
- 5 years ago
- Views:
Transcription
1 Test Piece for 5-axis Machining Centers S. Bossoni, W. Knapp, K. Wegener Institute of Machine Tools and Manufacturing, ETH Zurich, Switzerland, and inspire AG für mechatronische Produktionssysteme und Fertigungstechnik, Zurich, Switzerland Abstract The only standardized test piece for simultaneous 5-axis machining is defined in NAS 979 which does not specify detailed setup conditions. In this paper, the influence of test piece parameters and location (position and orientation in the machine coordinate system) on the movements of the machine axes is analyzed for simultaneous 5-axis milling applications. To conclude recommendations of parameter sets are presented and the influence of geometric machine errors is shown for one specific machining center. Keywords: Machine; Test piece; 5-axis machining 1 INTRODUCTION More and more 5-axis machine tools are used in industry, but today a standardized test piece for 5-axis machine tools, directly applicable to different machine tool designs, is still missing. The Aerospace Industries Association of America, Inc. (AIA) published a National Aerospace Standard (NAS) describing uniform cutting tests about 40 years ago [1]. It includes a profile cone frustum cutting test that can be performed to verify accuracies of five combined axes of motion (three linear and two rotary) for five axis machines under finishing conditions. The description in [1] is well suited for profiler machines common in the aerospace sector addressed by the standard. For such configurations, the description requires a simultaneous 5-axis movement without any further definition of the setup. On the contrary, for other types of machine designs, the setting conditions have to be defined more precisely in order to require a simultaneous 5-axis movement and to enable comparability. Moreover, no research has been found about the effects of individual geometric machine errors on such test piece geometry. The prediction and compensation of geometric machine errors of 5-axis machining centers on the geometry of a cone frustum test piece is described in [2]. 2 TOOL PATH REQUIRED FOR A CONE GEOMETRY When analyzing a required movement of a 5-axis machine a series of tool positions and directions have to be analyzed. For the case of a tool path for a cone geometry, the set of tool directions describe also a cone. The type and position of this cone depends on the cone geometry, the position of the workpiece and on the milling strategy. For end milling the cone formed by the tool directions is parallel to the cone of the workpiece at a distance of the tool radius from the workpiece (same cone angle α). The analysis presented in this work is focused on simultaneous end milling test pieces on a machining center with a tilting rotary table setup. More specific a machine with a t-(c)-z-y-b-x-b-c-w kinematic chain (structure code according to [3]) is used as an example. In this structure code, the upper case letters stand for the machine axes according to ISO 841 [4]. An upper case letter in brackets indicates an axis without positioning control. The lower case letters represent: tool (t), bed (b), and workpiece (w). Figure 1: Example of a right circular cone (α = 30 ) inclined by an angle β = 20 with the two frames Σ 0 and Σ The Proceedings of MTTRF 2009 Annual Meeting 235
2 2.1 Parameters of an inclined right circular cone The form of a right circular cone is defined only by the cone angle α (also referred to as half apex angle). The position and orientation in a specific frame can be defined either by definition of the position of the apex A and the direction of the cone axis or by definition of a circular section of the cone with the center M, the radius r and the direction of the cone axis towards the apex. When analyzing the cone describing the tool path it makes sense to use the circular base represented by the nominal tool positions instead of the apex. For the definition of a point P on the perimeter of the cone (representing a point of the tool path), a radial angle of the cone φ is needed. The orientation of a cone depends on the inclination angle β (wedge angle). The cone is inclined relative to a frame Σ 1 with the angle β around Y 1 (Y-direction of frame Σ 1 that coincides with Y 0, according to figure 1). The coordinates of a point 1 P in frame Σ 1 result according to figure 1 in: 1 r cos β cos φ P = rsin φ r sin β cos φ 2.2 Movement of rotational axes The movements of the rotational axes of a 5-axis machine, required to manufacture a cone geometry, depend only on the cone form (angle α) and the inclination of the cone axis relative to the nominal direction of the rotational axis of the machine nearest to the workpiece (seen in the kinematic chain). The movement of the rotary axes of all 5-axis machining centers with a serial kinematic setup with two orthogonal rotary axes is the same for a given combination of cone angle α and inclination angle β. In case of the analyzed tilting rotary table machine setup the inclination angle β is used to describe the angle between the C-axis (rotary table axis) and the cone axis. Therefore, the Z-axis of frame Σ 1 is defined to be parallel to the nominal direction of the axis C of the rotary table. The orientation of frame Σ 1 relative to the angle C of the rotary table only has an influence on the starting angle of the rotary table. This is of reduced importance since the rotary table has an unlimited angular range and therefore an arbitrary point of origin. The zero of the C-axis is defined to be equal to the zero of the radial cone angle φ. The translational position of the cone has only an influence on the movement of the linear axes Required angles of the tilting axis B The origin of the angle B of the tilting axis (B = 0 ) is defined to be the position of the axis B where the axis C is parallel to the direction Z 1. The angle of the tilting axis B corresponds to the angle between the tool direction and the direction of the rotary table axis C. This must be the angle between a slant s and the Z-axis of frame Σ 1 (see figure 1) which can be expressed as: B = arccos cos αcos β + sin αsin βcos φ (2) ( ) (1) Two important facts can be concluded from equation (2): The sign of the angle B is not defined. The extreme values of the angle of the tilting axis B for a given cone and setup (α and β are constant) are at φ = 0 and φ = 180. This means that the extreme values of the tilting axis B are in the XZ-plane of frame Σ Required angles of the rotary table C The rotary table needs to turn the slant into the plane perpendicular to the tilting axis. The angle C of the rotary table can be derived when analyzing the projection of the cone in the plane of the rotary table (plane perpendicular to the axis of the rotary table, which corresponds to the XY-plane of frame Σ 1 ). The angle C equals the angle between the projection of the height of the cone (in this case in positive X-direction of frame Σ 1 ) and the projection of the slant (see figure 1). Equation (3) results for the angle C. sin φ C = arctan (3) cos β cos φ sin β cot α For the interpretation of the resulting movements of the rotational axes of the machine, three situations are distinguished: α > β, α = β, α < β. 2.3 Cone angle α larger than inclination angle β For α > β the projection of the apex A into the XY-plane of frame Σ 1 is inside the projection of the base as shown in figure 1. Therefore, the range of the angle C of the rotary table is from 0 to 360 since it depends on the direction of the projection of the slants in the XY-plane of frame Σ 1. The minimal displacement of the tilting axis B min results to be the difference between the cone angle α and the inclination angle β whereas the maximal displacement of the tilting axis B max is equal to the sum of α and β. Therefore the range of the tilting axis is equal to twice the inclination angle β and independent of the cone angle α (except for special setups with α = β where the range of the tilting axis B can be up to four times the inclination angle β). Generally for α > β the tilting axis B has a reversal but the rotary axis C has no reversal. 2.4 Cone angle α equal to inclination angle β If the cone angle α is equal to the inclination angle β the slant at φ = 0 is parallel to the Z-direction Z 1 of frame Σ 1 and the projection of the apex A into the XY-plane of frame Σ 1 is on the projection of the base. Since the angle B of the tilting axis corresponds to the angle between a slant of the cone and the direction Z 1, B must be zero for φ = 0 for α = β (tool axis parallel to C). The angle C of the rotary table is not defined if the projection of the slant in the XY-plane of frame Σ 1 has no length. Therefore the angle C for φ = 0 depends on the angles C for φ close to 0. The limit of the angle C for the radial cone angle approaching 0 differs by 180, depending on the direction of approach The Proceedings of MTTRF 2009 Annual Meeting
3 ' Z 1 Y 1 = Y 0 Z 1 Z 0 Z 0 A A C X 1 X 1 A Y 1 X 1 X 0 Figure 2: Example of a configuration with α < β for the geometrical analysis of the range of the angles of the rotary table C 2.5 Cone angle α smaller than inclination angle β For α < β the projection of the apex A into the XY-plane of frame Σ 1 is outside the projection of the base as shown in figure 2. The range ΔC of the angles of the rotary table are equal to the angle between the two tangents to the ellipse of the projection of the tool path into the XY-plane of frame Σ 1 trough the projection of the apex A into the XY-plane of frame Σ 1. Therefore ΔC must be smaller than 180 for α < β. The minimum range ΔC of the angle of the rotary table is twice the cone angle α. This is the case if the inclination angle β is 90. A general expression of the range ΔC of the angle of the rotary table is: 1 ( an ) Δ C = 2arctan cos β tan arccos(t α cot β ) 2.6 Recommended angular parameters The range of the rotary axes should be the same on all types of machines in order to have a fair comparison. Therefore, the cone angle α and the inclination angle β are recommended to be the same, independent of the machine design. In order to have a full rotation of the rotary axis C, the cone angle α has to be larger than the inclination angle β. The test piece should be applicable on machines with a range of the tilting axis of 90 or more. The range needed during the simultaneous movement for the cone geometry is chosen to be 2/3 of the minimal range of 90, which is 60. This is obtained with an inclination angle β of 30. The range of the tilting axis is selected to be centered between 0 and 90. For an inclination angle β = 30 a cone angle of α = 45 results. 3 MOVEMENT OF LINEAR AXES The movement of the linear axes of 5-axis machines can be divided into a component needed for the relative translational displacement of the tool and a second component, which is needed in order to change the tool direction. This second compensational component consists of the linear displacement of the rotary axes in order to change the tool direction without changing the location of the tool center point (TCP). It is proportional to the distance between the rotational axes and the TCP. (4) 3.1 Tilting rotary table For the analysis of the movement of the linear machine axes of 5-axis machines with both rotational axes on the work piece side (tilting rotary table), the position of the workpiece relative to the crossing of the two rotary axes is relevant. A third frame Σ 2 is introduced according to figure 3. It is attached to the rotary table with its origin O 2 at the nominal crossing of the rotary table axis C with the tilting axis B. A Point 2 P in frame Σ 2 results through translation according to: 2 1 x cos cos + 0 r β φ x0 P = P+ y = rsin φ + y 0 0 z sin cos + 0 r β φ z0 In order to obtain the machine coordinates M P the rotations of the two workpiece sided rotary axes have to be taken into account to yield: P = R R M 2 tilt table P cos B 0 sin B cos C sin C 0 = sin cos 0 sin B 0 cos B O 2 Z 2 X 2 Y 2 x 0 y0 2 C C P O 1 β Z 1 z 0 A Z 0 (5) (6) Figure 3: Example of a right circular cone (α = 30 ) inclined by an angle β = 20 in frame Σ 2 with the frames Σ 0 and Σ 1. X 0 Y 1 X The Proceedings of MTTRF 2009 Annual Meeting 237
4 The expression of equation (6) as a function of the cone parameters with the radial angle φ (instead of the angles of the rotary axes) becomes very complex. Instead, some special configurations and setups are mentioned: If the apex A of the cone formed by the tool directions coincides with the crossing of the two rotary axes of the machine, no movement of the linear axes is required. (2- axis movement) If the cone axis crosses the point O 2 (figure 3, crossing of the axes B and C), the Z-axis of the machine has no movement (4-axis movement) The range of the Y-axis (axis parallel to the tilting axis) corresponds to twice the distance of the apex A from the C-axis. If the cone axes crosses the C-axis (y 0 = 0) the movement of the machine axes have a symmetry to the XZ-plane of frame Σ 2 (see figure 3). 4 PROPOSED SIZE AND SETUP FOR A SPECIFIC MACHINE DESIGN For a given machine design the size and position of the test piece is limited by the ranges of the machine axes and by collisions. To avoid collisions the test piece is recommended to be placed within the surface of the work piece pallet on the rotary table (r Table = 100 mm diameter for the analyzed vertical machining center). Since the test goal of the test piece is to show the relevant geometric errors of the machine no unnecessary machine dynamics is wanted for the movement. Therefore, the lateral displacement y 0 is recommended to be zero to avoid dynamics of the linear axes due to an asymmetric movement. The diameter of the cone limits the range of the linear axes because with an increase of the diameter, the cone has to be positioned closer to the crossing of the two rotary axes (point O 2 in figure 3). In order to be still well manageable (fixation, measurement) the diameter of the tool path is proposed to be 80 mm. The movement of the linear axis parallel to the tilting axis (Y-axis) is not influenced by the motion of the tilting axis B. Thus, it only needs a range over the dimensions of the surface of the work piece pallet. The other two axes (X and Z) need a larger range of movement for the compensational movements. Therefore, the range of the linear axes should be larger for the X- and Z-axis than for the Y-axis. This can be reached by placing the cone at a negative distance x 0 = - r Table + r TCP cosβ = mm and at a maximal height above the tilting axis B. For the given ranges a height z 0 = 150 mm is chosen in order to stay within the limits of the ranges of the X- and Z-axes. The resulting movement of the machine axes is shown in figure 4 as a function of the radial angle φ. To give a better idea of the path of the TCP, figure 5 shows the path relative to the machine frame. 5 INFLUENCE OF GEOMETRIC MACHINE ERRORS For the simulation of the geometric errors and identifying the resulting test piece errors, three basic steps are carried out [5, 6]: 1. Transform points of the work piece geometry into the nominal kinematic model of the machine. 2. Transform points back into the work piece coordinate system with a general kinematic model containing all potential geometric errors. 3. Identify the resulting cone form: 238 a. Least squares plane and circle through the actual coordinates of the points. b. Cone axis through center of least squares circle normal to least squares plane. Least squares circle evaluation of the distances of points from cone axis in direction normal to the cone surface results in a measure for the cone form deviation. X M, Y M, Z M [mm] X M B C Y M φ [ ] Figure 4: Positions of the machine axes as a function of the radial cone angle φ (α = 45, β = 30, r = 40mm, x 0 = -65.4mm, y 0 = 0, z 0 = 150mm) Z M z 0 x 0 Z 1 Z 2 X 0-50 X 2 0 X M Y 150 M -50 Figure 5: Illustration of the cone position in the machine coordinate system when the TCP is at the bottom position of the cone (φ = 0). The red curve corresponds to the path of the TCP in the machine coordinate system and represents the movement of the machine axes The Proceedings of MTTRF 2009 Annual Meeting Z 0 X 1 Z M Y B, C [ ]
5 The effect of an individual error depends on the error magnitude and on the direction of the error. The component of the direction of the error in the sensitive direction (i.e. normal to the cone surface) affects the form of the cone. However, only changing effects can be seen as a form deviation since constant effects just change the cone angle or the cone position. When analyzing the components of the normal directions (n x, n y, n z ) of the cone surface in the machine frame as a function of the radial cone angle φ, the following points can be seen: n x : the component of the unit length normal direction of the cone surface in X-direction is always positive and has a length between 1 and 0.7. n y : the component of the unit length normal direction of the cone surface in Y-direction has a length between -0.7 and 0.7 n z : the normal direction of the cone surface has no component in Z-direction since it corresponds to the tool direction (length = 0). An example of a typical cone form deviation due to a constant error acting in X-direction, as well as in Y-direction is shown in figure Geometric errors with negligible influence on the form of an end milled cone surface on a machine with the tilting rotary table: Nomenclature of errors follows [7]. Vertical straightness error motions EZX, EZY, Z positioning error EZZ, Z zero position Z0Z, radial error motion EZB: geometric machine errors acting only in tooldirection (Z) have no influence on an end milled cone surface (nor on any end milled surface on a machine tool with tilting rotary table) Hysteresis of C positioning ECC hyst : the axis of the rotary table has no reversal point for setups with cone angles α larger than the inclination angle β. Therefore, the hysteresis of the positioning error of the rotary table cannot have any influence. If an analysis of the influence of ECC hyst on the cone form is desired, a setup with α < β needs to be chosen. Z tilt error motions EAZ, EBZ are rotations of the tool axis around the TCP, which have a negligible influence on the cone form for a small cone height of 5 mm. Y yaw error motion ECY and Z roll error motion ECZ are rotations around the tool axis and have no additional influence on the workpiece geometry (additional to straightness error motions). Squareness of B to X axis C0B: influence negligible because B max and B min are reached at Y M = 0, thus the sensitive direction is in X and not in Y. C tilt error motions EAC, EBC and B tilt error motion ECB: no additional influence to radial error motion of the axis C, respectively B for small cone heights. 5.2 Direct measurement of the 5-axis movement for a cone geometry In different publications [2, 8] the measurement of the deviations of a 5-axis movement for a cone geometry with a double ball bar (DBB) measuring device is presented. It should be noted that the results of a measurement setup with the DBB square to the cone axis does not represent the errors of the cone form because errors parallel to the cone surface are also measured as form errors. For a more realistic representation of the form error, the DBB should be set up perpendicular to the cone surface. This way also collision problems are avoided [8]. On a machine with tilting rotary table, an alternative would be to use an R-Test [9] and to position the precision sphere at the apex of the cone. In any case, the measurement cannot replace the cutting of a test piece but it is very well suited for a study of parameter influence, such as changing contouring feed. 6 CONCLUSION A test piece for 5-axis machining centers has been presented and the influence of the test piece and the setup parameters are analyzed. The following points can be concluded: The required movements of the rotary machine axes are the same for all machine tools with two orthogonal rotary axes (independent of the machine design and the translatory position of the test piece) A combination of a cone angle α = 45 and an inclination angle β = 30 represents a recommendable test piece setup for machines with a tilting axis with a range of at least 90 and a second rotary axis with an unlimited range. For machines with different ranges of the linear axes the Figure 6: Typical form error due to a constant geometric error acting in X-direction (a), e.g. X-position of B-axis, and in Y- direction (b), e.g. Y-position of the axis C. (blue: result with positive error value, red: result with negative error value) The Proceedings of MTTRF 2009 Annual Meeting 239
6 procedure how to decide on the angular parameters to be chosen was shown, a procedure that is also applicable for conical ball bar tests. The influence of the translatory position of the test piece on the movement of the linear axes has been shown for the most complex machine design (tilting rotary table machine). A recommended setup has been shown for a specific machine. The presented test piece is a good way to test the performance of a 5-axis machine for the specific milling strategy (end milling). If another milling strategy (e.g. face or ball nose end milling) is of interest, the same analysis has to be carried out for a test piece manufactured with the other milling strategy since then the influence of the geometric machine errors of the test piece errors changes. REFERENCES [1] NAS 979, 1969, Uniform Cutting Tests - NAS Series Metal Cutting Specification. Aerospace Industries Association of America, Inc. [2] Uddin, M. S.; Ibaraki, S.; Matsubara, A., Matsushita, T., Prediction and compensation of machining geometric errors of five-axis machining centers with kinematic errors, Precision Engineering, 2009, 33, [3] ISO/WD , 2008, Test conditions for machining centres Part 6: Accuracy of feeds, speeds, and interpolations. working document ISO/TC 39/SC 2/WG3 N123. [4] ISO 841, 2001, Industrial automation systems and integration - Numerical control of machines - Coordinate system and motion nomenclature. [5] Bringmann B., 2007, Improving Geometric Calibration Methods for Multi-axes Machining Centers by Examining Error Interdependencies Effects, 2/664. Fortschritt-Berichte VDI, Düsseldorf. [6] Bringmann B., Besuchet J.P., and Rohr L., 2008, Systematic evaluation of calibration methods. Annals of the CIRP 57/1: [7] ISO 230-1, 1996, Test Code for Machine Tools Part 1: Geometric Accuracy of Machines Operating Under No-Load or Finishing Conditions. [8] Matano K., Ihara Y., 2007, Ball bar measurement of five-axis conical movement. 8th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM & Robotic Performance, Lamdamap. [9] Weikert S, Knapp W., 2004, R-Test: A New Device for Accuracy Measurements on Five Axis Machine Tools. Annals of the CIRP 53/1: The Proceedings of MTTRF 2009 Annual Meeting
7 MTTRF 2008 Test piece for 5-axis machining centers S. Bossoni, K. Wegener IWF ETH Zurich, Switzerland, and inspire AG, Zurich, Switzerland Outline Existing test pieces for 5-axis milling and their deficiencies Requirements for test piece Definition of test piece Movements of rotational axes and of linear axes Suggested test parameters Influence of geometric machine tool errors Summary 1 Existing Test Pieces for milling The Proceedings of MTTRF 2009 Annual Meeting 241
8 Deficiencies of the state of the art Standardized test piece for a 5-axis machine tool, directly applicable to different machine tool designs, is still missing. No detailed analysis of the adequacy of a cone geometry and possible combinations with other elements on a test piece has been published. Effects of geometric machine errors on such test pieces have not been shown. Publications with different variations of test pieces can be found in literature but no reasons are given for the selected parameters. Influence of different parameters (test piece + setup) is not understood yet. 3 5-axis milling machines The Proceedings of MTTRF 2009 Annual Meeting
9 Goals and requirements of the test piece The ideal test piece for acceptance tests of 5-axis machining should have the following features: easy and fast to manufacture easy and fast to measure easy to evaluate measuring results Set up in a way that it must be manufactured using 5 simultaneous axes on different types of 5-axis machining centers. Give comparable and quantitative results. Show the influence of the machine and not of the tool and the tool setup. Show the influence of errors relevant for the analyzed manufacturing strategy (e.g. end milling) with sufficient sensitivity. 5 Geometry of test piece For simultaneous 5-axis motion: normal and tangent vectors not all parallel to each other Possible geometries: sphere cone toroid Cone best because: only geometry that can be generated with straight lines can be manufactured with one closed movement with end mill easy and fast to measure and evaluate measuring results The Proceedings of MTTRF 2009 Annual Meeting 243
10 General transformation for 5-axis machines 7 Movement of linear machine axes Compensational movement of linear machine axes: Movements of the linear axes needed in order to change the relative orientation between tool and workpiece without changing the relative position of the tool centre point between tool and workpiece. Depends on configuration of rotary axes Offset of the rotary axes from the TCP: workpiece sided rotary axes: depends on position of TCP first tool sided rotary axis: constant second tool sided rotary axis: function of angle of first rotary axis Most complex for tilting rotary table because both rotary axes are workpiece sided The Proceedings of MTTRF 2009 Annual Meeting
11 Inclined right circular cone Set of tool directions for manufacturing of cone also describe a cone 9 Ranges of the angle C of the rotary table The Proceedings of MTTRF 2009 Annual Meeting 245
12 Angle C of rotary table depending on relation of α and β 11 Range of angle C for α< β The Proceedings of MTTRF 2009 Annual Meeting
13 Movements of rotary machine axes for α=β 13 Recommended angular parameters for all machine setups The Proceedings of MTTRF 2009 Annual Meeting 247
14 Proposed test piece parameters for machine example (slide 9) 15 Positions of machine axes The Proceedings of MTTRF 2009 Annual Meeting
15 Positions of machine axes (animation) 17 Radial representation of positions of machine axes The Proceedings of MTTRF 2009 Annual Meeting 249
16 Geometric errors of machines component errors: 6 for each machine axis location errors 3 per linear axis 5 per rotary axis 6 location errors are set zero for machine frame total of 43 geometric errors for 5-axis machine without the working spindle 19 2 types of resulting error forms Type 1: top-bottom B0Z B0C X0X X0C EXX lin EBX lin EYY lin EAY lin ECC lin Type 2: asymmetric C0Y EBY lin A0Z A0B A0C Y0Y ECX lin EAX lin The Proceedings of MTTRF 2009 Annual Meeting
17 Effects of location errors 21 Effects of 1000 random combinations of errors The Proceedings of MTTRF 2009 Annual Meeting 251
18 Summary of geometric evaluation Test piece for a 5-axis machine tool, directly applicable to different machine tool designs. Influence of parameters analyzed for any machining center with three orthogonal linear axes and two orthogonal rotary axes. Proposed test piece parameters are deduced from the analysis. Effects of geometric machine errors on test piece have been shown. Novel visualization forms presented. Influence of different parameters is understood. 23 Application for Mori Seiki NMV 5000 (project) Geometric errors of machine. Simulation of test piece for measured geometric errors. Machining of test piece. Variations (e.g. feed, position, orientation) with conical circular test, R-test The Proceedings of MTTRF 2009 Annual Meeting
Squareness C0Y A0Z B0Z
ETALON offers system solutions for highly accurate and complete geometric analysis, monitoring and accuracy improvement of machine tools, measuring machines, robots and structures. Innovative technologies
More informationAvailable online at ScienceDirect. 6th CIRP International Conference on High Performance Cutting, HPC2014
Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 14 ( 214 323 328 6th CIRP International Conference on High Performance Cutting, HPC214 A machining test to evaluate geometric errors
More informationDOI: /MMSJ.2017_12_
APPLICATION ON-THE-FLY MEASUREMENT OF CNC MACHINE TOOLS MICHAL HOLUB, JAN VETISKA FRANTISEK BRADAC, MICHAL VALA Dept. of Production Machines, Brno, Czech Republic DOI: 10.17973/MMSJ.2017_12_201791 e-mail:holub@fme.vutbr.cz
More informationAvailable online at ScienceDirect. 6th CIRP International Conference on High Performance Cutting, HPC2014
Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 14 ( 2014 ) 317 322 6th CIRP International Conference on High Performance Cutting, HPC2014 Investigation of a Multi-Cone Variant of
More informationDr. Heinrich Schwenke, CEO Etalon AG
The latest trends and future possibilities of volumetric error compensation for machine tools Errors of machine tools Kinematic chain Conventional measurements Etalons Multilateration approach Compensation
More informationDetermining the 5-axes machine tool contouring performance with dynamic R-test measurements
Determining the 5-axes machine tool contouring performance with dynamic R-test measurements H.A.M. Spaan 1, G.H.J. Florussen 1 1 IBS Precision Engineering, the Netherlands spaan@ibspe.com Abstract Current
More informationComputer aided error analysis for a threedimensional precision surface mapping system
Computer aided error analysis for a threedimensional precision surface mapping system M. Hill, J.W. McBride, D. Zhang & J. Loh Mechanical Engineering, U~riversity of Southampton, UK Abstract This paper
More informationTEST METHODS FOR COORDINATED MOTION OF FIVE-AXIS MACHINING CENTERS
TEST METHODS FOR COORDINATED MOTION OF FIVE-AXIS MACHINING CENTERS Shawn P. Moylan, Ronnie R. Fesperman, and M. Alkan Donmez Manufacturing Engineering Laboratories National Institute of Standards and Technology
More informationTitle calibration of five-axis machine to. Citation Precision Engineering (2013), 37(1)
Title Non-contact R-test with laser displ calibration of five-axis machine to Author(s) Hong, Cefu; Ibaraki, Soichi Citation Precision Engineering (2013), 37(1) Issue Date 2013-01 URL http://hdl.handle.net/2433/162006
More informationIf the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2)
Sphere Definition: A sphere is the locus of a point which remains at a constant distance from a fixed point. The fixed point is called the centre and the constant distance is the radius of the sphere.
More informationSoftware correction of geometric errors for multiaxis
Journal of Physics: Conference Series PAPER OPEN ACCESS Software correction of geometric errors for multiaxis systems o cite this article: V eleshevskii and V Sokolov 2017 J. Phys.: Conf. Ser. 858 012037
More information1. The Pythagorean Theorem
. The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be
More informationJournal of Advanced Mechanical Design, Systems, and Manufacturing
A Study on Error Compensation on High Precision Machine Tool System Using a 2D Laser Holographic Scale System* (First Report: Scale Development and Two Dimensional Motion Error Compensation Method) Toru
More informationRESEARCHES REGARDING ROTARY-TILTING TABLE PRECISION EVALUATION METHOD USING THE BALL-BAR SYSTEM
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 16, Number 4/2015, pp. 513 521 RESEARCHES REGARDING ROTARY-TILTING TABLE PRECISION EVALUATION METHOD USING
More informationBuds Public School, Dubai
Buds Public School, Dubai Subject: Maths Grade: 11 AB Topic: Statistics, Probability, Trigonometry, 3D, Conic Section, Straight lines and Limits and Derivatives Statistics and Probability: 1. Find the
More informationInvestigation of the kinematic system of a 450 kv CT scanner and its influence on dimensional CT metrology applications
Investigation of the kinematic system of a 450 kv CT scanner and its influence on dimensional CT metrology applications Frank Welkenhuyzen 1, Bart Boeckmans 1, Ye Tan 1,2, Kim Kiekens 1,2, Wim Dewulf 1,2,
More informationACTA TECHNICA CORVINIENSIS Bulletin of Engineering Tome X [2017] Fascicule 2 [April June] ISSN:
Tome X [2017] Fascicule 2 [April June] ISSN: 2067 3809 1. Yung-Cheng WANG, 2. Bean-Yin LEE, 1. Chih-Hao HUANG, 3. Chi-Hsiang CHEN DEVELOPMENT OF THE AXIAL PRECISION INSPECTION SYSTEM FOR SPINDLES IN TOOL
More informationSINUMERIK live: Programming dynamic 5-axis machining directly in SINUMERIK Operate
SINUMERIK live: Programming dynamic 5-axis machining directly in SINUMERIK Operate Basics, possibilities, and limits siemens.com/cnc4you Programming dynamic 5-axis machining directly in SINUMERIK Operate
More informationA Novel Kinematic Model of Spatial Four-bar Linkage RSPS for Testing Accuracy of Actual R-Pairs with Ball-bar
A Novel Kinematic Model of Spatial Four-bar Linkage RSPS for Testing Accuracy of Actual R-Pairs with Ball-bar Zhi Wang 1, Delun Wang 1 *, Xiaopeng Li 1, Huimin Dong 1 and Shudong Yu 1, 2 1 School of Mechanical
More informationRESEARCHES REGARDING ROTARY TILTING TABLE PRECISION EVALUATION METHOD USING THE BALLBAR SYSTEM
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 16, Number 1/215, pp RESEARCHES REGARDING ROTARY TILTING TABLE PRECISION EVALUATION METHOD USING THE BALLBAR
More informationMulti-Axis Surface Machining
CATIA V5 Training Foils Multi-Axis Surface Machining Version 5 Release 19 January 2009 EDU_CAT_EN_MMG_FI_V5R19 1 About this course Objectives of the course Upon completion of this course you will be able
More informationAvailable online at ScienceDirect. onal Scientific. the
Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 14 ( 214 ) 269 274 6th CIRP International Conference on High Performance Cutting, HPC214 Influencee of Motion Error of Translational
More informationMultivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
More informationI. SETS AND FUNCTIONS. 3. Which of the following relations are function from A = { 1,4,9,16 } to B = { -1,2,-3,-4,5,6 }?.
TWO MARK QUESTIONS I. SETS AND FUNCTIONS 1. If A ={4,6.7.8.9}, B = {2,4,6} and C= {1,2,3,4,5,6 } then find i. A U (B C) ii. A \ (C \ B) 2. Draw venn diagram for the following. i. (A B)' ii. A (BUC) iii.
More informationDesign and Communication Graphics
An approach to teaching and learning Design and Communication Graphics Solids in Contact Syllabus Learning Outcomes: Construct views of up to three solids having curved surfaces and/or plane surfaces in
More informationRotating Table with Parallel Kinematic Featuring a Planar Joint
Rotating Table with Parallel Kinematic Featuring a Planar Joint Stefan Bracher *, Luc Baron and Xiaoyu Wang Ecole Polytechnique de Montréal, C.P. 679, succ. C.V. H3C 3A7 Montréal, QC, Canada Abstract In
More informationResearch on Measuring and Optimization Method of Dynamic Accuracy of CNC Machine Tools
Sensors & Transducers 2014 by IFSA Publishing, S. L. http://www.sensorsportal.com Research on Measuring and Optimization Method of Dynamic Accuracy of CNC Machine Tools 1, 2 Zhiming FENG, 2 Guofu YIN,
More informationGEOMETRIC ERRORS COMPENSATION OF CNC MACHINE TOOL MICHAL HOLUB, PETR BLECHA, FRANTISEK BRADAC TOMAS MAREK, ZDENEK ZAK
GEOMETRIC ERRORS COMPENSATION OF CNC MACHINE TOOL MICHAL HOLUB, PETR BLECHA, FRANTISEK BRADAC TOMAS MAREK, ZDENEK ZAK Brno University of Technology Faculty of Mechanical Engineering Brno, Czech Republic
More informationCalypso Construction Features. Construction Features 1
Calypso 1 The Construction dropdown menu contains several useful construction features that can be used to compare two other features or perform special calculations. Construction features will show up
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com January 2007 2. Figure 1 A a θ α A uniform solid right circular cone has base radius a and semi-vertical angle α, where 1 tanα = 3. The cone is freely suspended by a string attached
More informationResearch Collection. Indirect Measurement of Volumetric Accuracy for Three-Axis and Five-Axis Machine Tools A Review. Journal Article.
Research Collection Journal Article Indirect Measurement of Volumetric Accuracy for Three-Axis and Five-Axis Machine Tools A Review Author(s): Ibaraki, Soichi; Knapp, Wolfgang Publication Date: 2012 Permanent
More informationJinying. factors to. proposed. Introduction. Error element. solutions. same time, Figure
International Conference on Applied Science and Engineering Innovation (ASEI 2015) Error element analysis and detection for BV755 CNC milling machine based on Ball-bar Jinying Chen 1, Xiaotie Niu 2, Lijuan
More informationAbstract. Introduction
The efficient calculation of the Cartesian geometry of non-cartesian structures J.M. Freeman and D.G. Ford Engineering Control and Metrology Research Group, The School of Engineering, University of Huddersfield.
More informationUNITEST - A new device for the static and dynamic testing of unconventional machine structures
UNITEST - A new device for the static and dynamic testing of unconventional machine structures K.E. Meitz Department of Production Engineering, Technical University Graz, Kopernikusgasse 24, A-8010 Graz,
More informationAccuracy for CMMs and Machine Tools
Systematic machine errors Rapid Calibration and Compensation of CMM and Machine ools: Recent Developments of the Lasertracer echnology Motivation Geometry errors he LaserRACER RAC-CAL for Error Mapping
More informationPolar coordinate interpolation function G12.1
Polar coordinate interpolation function G12.1 On a Turning Center that is equipped with a rotary axis (C-axis), interpolation between the linear axis X and the rotary axis C is possible by use of the G12.1-function.
More informationRobotics (Kinematics) Winter 1393 Bonab University
Robotics () Winter 1393 Bonab University : most basic study of how mechanical systems behave Introduction Need to understand the mechanical behavior for: Design Control Both: Manipulators, Mobile Robots
More informationFall 2016 Semester METR 3113 Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics
Fall 2016 Semester METR 3113 Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics Lecture 5 August 31 2016 Topics: Polar coordinate system Conversion of polar coordinates to 2-D
More informationLaser diagonal testing
Laser diagonal testing H-5650-2056-01-B 20/08/2018 1 Introduction This presentation explains how Renishaw laser calibration systems can be used to check machine positioning performance along machine diagonals,
More information6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.
Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct
More informationSTRAND E: Measurement. UNIT 13 Areas Student Text Contents. Section Squares, Rectangles and Triangles Area and Circumference of Circles
UNIT 13 Areas Student Text Contents STRAND E: Measurement Unit 13 Areas Student Text Contents Section 13.1 Squares, Rectangles and Triangles 13. Area and Circumference of Circles 13.3 Sector Areas and
More informationADJUSTABLE GEOMETRIC CONSTRAINTS 2001 MIT PSDAM AND PERG LABS
ADJUSTABLE GEOMETRIC CONSTRAINTS Why adjust kinematic couplings? KC Repeatability is orders of magnitude better than accuracy Accuracy = f ( manufacture and assemble ) Kinematic Coupling Accuracy Adjusted
More informationSurface Roughness Control Based on Digital Copy Milling Concept to Achieve Autonomous Milling Operation
Available online at www.sciencedirect.com Procedia CIRP 4 (2012 ) 35 40 3rd CIRP Conference on Process Machine Interactions (3rd PMI) Surface Roughness Control Based on Digital Copy Milling Concept to
More informationFast checking of CMM geometry with a patented tool
17 International Congress of Metrology, 13012 (2015) DOI: 10.1051/ metrolo gy/201513012 C Owned by the authors, published by EDP Sciences, 2015 Fast checking of CMM geometry with a patented tool Jean-François
More informationResearch on error detection technology of numerical control machine tool. Cao YongJie
Joint International Mechanical, Electronic and Information Technology Conference (JIMET 2015) Research on error detection technology of numerical control machine tool Cao YongJie Shanghai University of
More informationBirkdale High School - Higher Scheme of Work
Birkdale High School - Higher Scheme of Work Module 1 - Integers and Decimals Understand and order integers (assumed) Use brackets and hierarchy of operations (BODMAS) Add, subtract, multiply and divide
More information3D MODELLING OF CYLINDRICAL CUTTING TOOL GEOMETRY WITH HELICAL TEETH
Annals of the University of Petroşani, Mechanical Engineering, 15 (013), 45-55 45 3D MODELLING OF CYLINDRICAL CUTTING TOOL GEOMETRY WITH HELICAL TEETH IOSIF DUMITRESCU 1, SORIN MIHĂILESCU, VILHEM ITU 3
More informationUNIT 2 2D TRANSFORMATIONS
UNIT 2 2D TRANSFORMATIONS Introduction With the procedures for displaying output primitives and their attributes, we can create variety of pictures and graphs. In many applications, there is also a need
More informationEmpirical Modeling of Cutting Forces in Ball End Milling using Experimental Design
5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India Empirical Modeling of Cutting Forces in
More information4.10 INVOLUTE INTERPOLATION (G02.2, G03.2)
B 63014EN/02 POGAMMNG 4. NTEPOLATON FUNCTONS 4.10 NVOLUTE NTEPOLATON (G02.2, G03.2) nvolute curve machining can be performed by using involute interpolation. nvolute interpolation ensures continuous pulse
More informationThe Jitterbug Motion
The By 80-4 Poplar St. Rochester, NY 460 Copyright, September 00 09-9-00 Introduction We develop a set of equations which describes the motion of a triangle and a vertex of the Jitterbug. The Jitterbug
More informationImproving Vision-Based Distance Measurements using Reference Objects
Improving Vision-Based Distance Measurements using Reference Objects Matthias Jüngel, Heinrich Mellmann, and Michael Spranger Humboldt-Universität zu Berlin, Künstliche Intelligenz Unter den Linden 6,
More informationDynamic and geometric error assessment of a XYC axis subset on fiveaxis. high speed machine tools using programmed endpoint constraint
Dynamic and geometric error assessment of a XYC axis subset on fiveaxis high speed machine tools using programmed endpoint constraint measurements Slamani M, Mayer* J.R.R, Balazinski M, Zargarbashi S.H.H,
More informationAP CALCULUS BC 2014 SCORING GUIDELINES
SCORING GUIDELINES Question The graphs of the polar curves r = and r = sin ( θ ) are shown in the figure above for θ. (a) Let R be the shaded region that is inside the graph of r = and inside the graph
More informationError Budget as a Design Tool For Ultra-Precision Diamond Turning Machines Form Errors
Error Budget as a Design Tool For Ultra-Precision Diamond Turning Machines Form Errors Mark Walter, Bruce Norlund, Robert Koning, Jeff Roblee, Precitech, Inc. Keene, NH 3431 USA Abstract This paper describes
More information11 cm. A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a depth of 8 cm.
Diagram NOT accurately drawn 10 cm 11 cm 12 cm 3.5 cm A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a depth of 8 cm. A metal sphere of radius 3.5
More informationA new slant on diagonal laser testing
A new slant on diagonal laser testing T.J. Morris Cincinnati Machine U.K. Ltd., Birmingham, England Abstract In the early 1990s the Americans introduced diagonal laser tests as a quick method for evaluating
More informationParallel Lines Investigation
Year 9 - The Maths Knowledge Autumn 1 (x, y) Along the corridor, up the stairs (3,1) x = 3 Gradient (-5,-2) (0,0) y-intercept Vertical lines are always x = y = 6 Horizontal lines are always y = Parallel
More informationQ-1 The first three terms of an AP respectively are 3y 1, 3y +5 and 5y +1. Then y equals
CBSE CLASS X Math Paper-2014 Q-1 The first three terms of an AP respectively are 3y 1, 3y +5 and 5y +1. Then y equals (A) -3 (B) 4 (C) 5 (D) 2 Q-2 In Fig. 1, QR is a common tangent to the given circles,
More information120 cm respectively. Its base is a rectangle with
[ In this exercise, give your answers correct to significant figures if necessary. ] 1. The base of a pyramid is an isosceles right-angled triangle where the lengths of the two equal sides are. The height
More informationVisualization and Analysis of Inverse Kinematics Algorithms Using Performance Metric Maps
Visualization and Analysis of Inverse Kinematics Algorithms Using Performance Metric Maps Oliver Cardwell, Ramakrishnan Mukundan Department of Computer Science and Software Engineering University of Canterbury
More informationTime: 3 hour Total Marks: 90
Time: 3 hour Total Marks: 90 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 34 questions divided into four sections A, B, C, and D. 3. Section A contains of 8
More informationEdge and local feature detection - 2. Importance of edge detection in computer vision
Edge and local feature detection Gradient based edge detection Edge detection by function fitting Second derivative edge detectors Edge linking and the construction of the chain graph Edge and local feature
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c
More informationISO INTERNATIONAL STANDARD. Test code for machine tools Part 4: Circular tests for numerically controlled machine tools
INTERNATIONAL STANDARD ISO 230-4 Second edition 2005-04-01 Test code for machine tools Part 4: Circular tests for numerically controlled machine tools Code d'essai des machines-outils Partie 4: Essais
More information2. INTRODUCTION TO CNC
Q. Define NC Machines. 2. INTRODUCTION TO CNC A method of automation, in which various functions and processing of machine tools are controlled by letters and symbols. The general objective of NC technology
More informationUsing Capacitance Probes to Measure the Limit of Machine Contouring Performance
Using Capacitance Probes to Measure the Limit of Machine Contouring Performance Don Martin, Lion Precision, 563 Shoreview Park Road, St. Paul, NIN 55126 Most machine tools used for discrete part manufacturing
More informationPrecision cylindrical face grinding
Precision Engineering 23 (1999) 177 184 Precision cylindrical face grinding Albert J. Shih a, *, Nien L. Lee b a Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh,
More informationWORKSPACE AGILITY FOR ROBOTIC ARM Karna Patel
ISSN 30-9135 1 International Journal of Advance Research, IJOAR.org Volume 4, Issue 1, January 016, Online: ISSN 30-9135 WORKSPACE AGILITY FOR ROBOTIC ARM Karna Patel Karna Patel is currently pursuing
More informationContouring accuracy assessment of CNC machine tools - a user's view M. Gull
Contouring accuracy assessment of CNC machine tools - a user's view M. Gull Abstract The purpose of this paper is to discuss the methods used at AWE Aldermaston to assess the contouring accuracy of computer
More informationUnit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
More informationK.S.E.E.B., Malleshwaram, Bangalore SSLC Model Question Paper-2 (2015) MATHEMATICS
Max Marks: 80 No. of Questions: 40 K.S.E.E.B., Malleshwaram, Bangalore SSLC Model Question Paper- (015) MATHEMATICS 81E Time: Hours 45 minutes Code No. : 81E Four alternatives are given for the each question.
More informationSupporting planning for shape, space and measures in Key Stage 4: objectives and key indicators
1 of 7 Supporting planning for shape, space and measures in Key Stage 4: objectives and key indicators This document provides objectives to support planning for shape, space and measures in Key Stage 4.
More informationMach4 CNC Controller Mill Programming Guide Version 1.0
Mach4 CNC Controller Mill Programming Guide Version 1.0 1 Copyright 2014 Newfangled Solutions, Artsoft USA, All Rights Reserved The following are registered trademarks of Microsoft Corporation: Microsoft,
More informationWEEKS 1-2 MECHANISMS
References WEEKS 1-2 MECHANISMS (METU, Department of Mechanical Engineering) Text Book: Mechanisms Web Page: http://www.me.metu.edu.tr/people/eres/me301/in dex.ht Analitik Çözümlü Örneklerle Mekanizma
More informationProperties of a Circle Diagram Source:
Properties of a Circle Diagram Source: http://www.ricksmath.com/circles.html Definitions: Circumference (c): The perimeter of a circle is called its circumference Diameter (d): Any straight line drawn
More informationTrig Practice 09 & Nov The diagram below shows a curve with equation y = 1 + k sin x, defined for 0 x 3π.
IB Math High Level Year : Trig: Practice 09 & 0N Trig Practice 09 & Nov 0. The diagram below shows a curve with equation y = + k sin x, defined for 0 x. The point A, lies on the curve and B(a, b) is the
More information3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane?
Math 4 Practice Problems for Midterm. A unit vector that is perpendicular to both V =, 3, and W = 4,, is (a) V W (b) V W (c) 5 6 V W (d) 3 6 V W (e) 7 6 V W. In three dimensions, the graph of the equation
More informationUse of Number Maths Statement Code no: 1 Student: Class: At Junior Certificate level the student can: Apply the knowledge and skills necessary to perf
Use of Number Statement Code no: 1 Apply the knowledge and skills necessary to perform mathematical calculations 1 Recognise simple fractions, for example 1 /4, 1 /2, 3 /4 shown in picture or numerical
More information13.1. Functions of Several Variables. Introduction to Functions of Several Variables. Functions of Several Variables. Objectives. Example 1 Solution
13 Functions of Several Variables 13.1 Introduction to Functions of Several Variables Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives Understand
More informationElements of three dimensional geometry
Lecture No-3 Elements of three dimensional geometr Distance formula in three dimension Let P( x1, 1, z1) and Q( x2, 2, z 2) be two points such that PQ is not parallel to one of the 2 2 2 coordinate axis
More informationChapter 15: Functions of Several Variables
Chapter 15: Functions of Several Variables Section 15.1 Elementary Examples a. Notation: Two Variables b. Example c. Notation: Three Variables d. Functions of Several Variables e. Examples from the Sciences
More informationCNC 8055 MC EXAMPLES MANUAL REF Ref. 0601
EXAMPLES MANUAL Ref. 0601 All rights reserved. No part of this documentation may be copied, transcribed, stored in a data backup system or translated into any language without Fagor Automation's explicit
More informationExtended Mathematics for Cambridge IGCSE by David Rayner. Chapter 1. Identify and use rational and irrational numbers, real numbers.
Schemes of Work Overview Structure There are two separate schemes of work laid out in the following units, one for students following the Core Curriculum and one for students following the Extended Curriculum.
More informationAccurate Trajectory Control for Five-Axis Tool-Path Planning
Accurate Trajectory Control for Five-Axis Tool-Path Planning Rong-Shine Lin* and Cheng-Bing Ye Abstract Computer-Aided Manufacturing technology has been widely used for three-axis CNC machining in industry
More informationA practical system for 5-axis volumetric compensation
A practical system for 5-axis volumetric compensation S.R. Postlethwaite, D.G. Ford 7%e School of Engineering, The University ofhuddersfield, Oueensgate, Huddersfield, HD1 3DH, England Email: s.r.postlethwaite@hud.ac.uk
More information12 m. 30 m. The Volume of a sphere is 36 cubic units. Find the length of the radius.
NAME DATE PER. REVIEW #18: SPHERES, COMPOSITE FIGURES, & CHANGING DIMENSIONS PART 1: SURFACE AREA & VOLUME OF SPHERES Find the measure(s) indicated. Answers to even numbered problems should be rounded
More informationOptimal Path Planning for Helical Gear Profile Inspection with Point Laser Triangulation Probes
Kevin B. Smith Professor, Brigham Young University, Dept. of Electrical and Computer Engineering, Provo, UT 84602 Yuan F. Zheng Professor, The Ohio State University, Dept. of Electrical Engineering, Columbus,
More informationDETC APPROXIMATE MOTION SYNTHESIS OF SPHERICAL KINEMATIC CHAINS
Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007 September 4-7, 2007, Las Vegas, Nevada, USA DETC2007-34372
More informationvolume & surface area of a right circular cone cut by a plane parallel to symmetrical axis (Hyperbolic section)
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter December 25, 2016 volume & surface area of a right circular cone cut by a plane parallel to symmetrical axis (Hyperbolic section) Harish
More informationNear-field time-of-arrival measurements for four feed-arms with a bicone switch
EM Implosion Memos Memo 37 February 2010 Near-field time-of-arrival measurements for four feed-arms with a bicone switch Prashanth Kumar, Serhat Altunc, Carl E. Baum, Christos G. Christodoulou and Edl
More information12 - THREE DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) = 2. ( d ) - 3. ^i - 2. ^j c 3. ( d )
- THREE DIMENSIONAL GEOMETRY Page ( ) If the angle θ between the line x - y + x + y - z - and the plane λ x + 4 0 is such that sin θ, then the value of λ is - 4-4 [ AIEEE 00 ] ( ) If the plane ax - ay
More informationGame Engineering: 2D
Game Engineering: 2D CS420-2010F-07 Objects in 2D David Galles Department of Computer Science University of San Francisco 07-0: Representing Polygons We want to represent a simple polygon Triangle, rectangle,
More informationUNIT IV - Laser and advances in Metrology 2 MARKS
UNIT IV - Laser and advances in Metrology 2 MARKS 81. What is interferometer? Interferometer is optical instruments used for measuring flatness and determining the lengths of slip gauges by direct reference
More informationGrade 9 Math Terminology
Unit 1 Basic Skills Review BEDMAS a way of remembering order of operations: Brackets, Exponents, Division, Multiplication, Addition, Subtraction Collect like terms gather all like terms and simplify as
More information11.0 Measurement of Spindle Error Motion
11.0 Measurement of Spindle Error Motion 11.1 Introduction The major spindle error motion is caused by the alignment of the spindle rotational axis, the centerline of the tool holder and the centerline
More informationQuaternions and Dual Coupled Orthogonal Rotations in Four-Space
Quaternions and Dual Coupled Orthogonal Rotations in Four-Space Kurt Nalty January 8, 204 Abstract Quaternion multiplication causes tensor stretching) and versor turning) operations. Multiplying by unit
More informationCAD/CAM/CAE Computer Aided Design/Computer Aided Manufacturing/Computer Aided Manufacturing. Part-9 CNC Fundamentals
CAD/CAM/CAE Computer Aided Design/Computer Aided Manufacturing/Computer Aided Manufacturing Part-9 CNC Fundamentals CNC Fundamentals All CNC machine tools follow the same standard for motion nomenclature
More informationStrategy. Using Strategy 1
Strategy Using Strategy 1 Scan Path / Strategy It is important to visualize the scan path you want for a feature before you begin taking points on your part. You want to try to place your points in a way
More informationDETERMINATION OF ALLOWABLE DEVIATION OF AXES OF ASSEMBLED DETAILS WITH CYLINDRICAL SURFACES AT ASSEMBLY WITH VIBRATIONS
7 th Research/Expert Conference with International Participations QUALITY 2011, Neum, B&H, June 01 04, 2011 DETERMINATION OF ALLOWABLE DEVIATION OF AXES OF ASSEMBLED DETAILS WITH CYLINDRICAL SURFACES AT
More information