DOCSLIB.ORG

204 Screw Outputs Influence of Screw and Barrel Wear on Output

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
204 Screw Outputs Influence of Screw and Barrel Wear on Output 204 3 Plasticizing Materials of Construction formula assumes pumping against low pres- sure, giving no consideration to melt quality Different materials of construction are and leakage flow of worn screws. used to meet the requirements of the dif- With all these and other limitations, the for- ferent plastics being processed. As an exam- mula can still provide guidance as follows: ple, bimetallic barrels offer extensive durabil- ity when processing abrasive materials such 1. It can serve as a general guide to the as glass- and mineral-filled plastics, certain output of the screw. engineering plastics, and granulatedhecycled 2. If the actual output of the screw is signifi- plastics. In contrast, when processing unfilled cantly greater than calculated, it is caused by nylon continuously, the probability is that you high compression ratios that overpump the will have to replace a worn out screw about metering section. Sometimes this is desirable, every six months. These questions are dis- but it can lead to surging and rapid screw wear cussed below under the subheading Screw if it is excessive. Wear Protection. 3. If the output is a lot less, it usually indi- cates a feed problem or a worn screw or bar- rel. The latter can be determined by measure- Screw Outputs ment. A feed problem can, on occasion, be corrected by changes in barrel temperature The rate of output (throughput), or the settings. More often, the problem is caused by speed at which plastic is moved through other items, such as screw design, shape and the plasticator, has been pushed continu- bulk density of the feedstock, surface condi- ally higher as a result of design advances in tion of the screw root and barrel ID in the screws, IMM equipment, and plastic materi- feed area, feed-throat design, or screw tem- als. Output rates generally range from a few perature. kilograms to tons per hour on single-screw machines. (With twin-screw extruders using large diameters, output rates range from a Influence of Screw and Barrel Wear few kilograms to at least 30 tons per hour.) on Output A rough estimate for output rate (OR) in lb/h can be calculated by using the barrel’s There are two types of wear. One is me- ID in inches and using the following equation: chanical, such as adhesive and abrasive wear. OR = 16 ID2; for kg/h multiply by 0.4536. The other is corrosion, which produces pitted The output of a screw is fairly predictable, surfaces. Adhesive wear is caused by contact provided that the melt is under control and between the flight and the barrel. The screw reasonably repeatable. With a square-pitch and barrel are engineered to minimize such screw (a conventional screw where the dis- contact, but some is unavoidable. The plastic tance from flight to flight is equal to the di- material being processed can significantly in- ameter), a simplified formula for output is fluence the abrasive and/or corrosive actions. R = 2.3D2hgN, where R is the rate or out- Wear does not occur suddenly but builds put in lb/h (kg/h), D is the screw diameter in up over months of machine operation. It fi- in. (mm), h is the depth in the metering sec- nally shows itself in one of several ways. The tion in in. (mm) (for a two-stage screw use examples to be discussed concern recipro- the depth of the first metering section), g is cating screw machines, since they have the the specific gravity of the melt, and N is the major wear problems; but wear also occurs screw rotation speed (rpm). in two-stage IMMs, including the breaking This formula does not take into account of nonreturn valve rings. There can be loss back flow and leakage flow over the flights. of shot control or consistency, requiring in- These flows are not usually a significant factor creased feed to make up for melt slippage unless the plastic has a very low viscosity dur- back over the valve and screw. Screw recov- ing processing or the screw is worn out. The ery time can increase. There can be a decrease 3 Plasticizing 205 in product quality. The cycle time increases that of the screw’s root is usually about 2 : 1. due to higher-temperature melt. The barrel ID enlarges at twice the rate of the Mechanical wear is usually in the part of decrease of the root of the screw. This differ- the screw where the feed section ends and the ence occurs because the heat source comes transition starts. It is usually caused by the use from the outside of the barrel. The screw root of high back pressure, an improper heat pro- is not as hot as the barrel’s interior surface. file, or a worn nonreturn valve that restricts When a nonreturn valve is new, it will fit flow. the barrel closely, preventing leakage of melt Often the plastic is filled with talc, glass, during injection. When the barrel ID changes or other materials that do not melt but form due to wear, leakage begins to occur during slugs and can cause scrubbing of the screw injection. If the barrel wear reaches 0.010 to and barrel, particularly the roots of screw 0.012 in. (0.025 to 0.030 cm), the ring on cer- flights. This action continues until the flight tain nonreturn valves can break. Then the is worn away. The barrel is likewise worn in IMM does not operate efficiently. the corresponding area. A change in the heat profile in the rear and center zones of the bar- rel will usually eliminate this problem. The Screw Wear plastic must start to melt as it reaches the end of the feed zone in order to move easily into The wear in screw plasticators generally the transition zone. If the screw returns in an causes an increase in the clearance between erratic manner, the plastic does not have the screw flight and barrel (Fig. 3-37). It often oc- required temperature, and screw and barrel curs toward the end of the compression sec- wear result. tion. This type of wear is more likely to oc- Another complication can be that the shot cur when the screw has a high compression size and the cycle time do not allow sufficient ratio. Regardless of where it occurs, the plas- residence time for the plastic to melt properly ticator’s melting capacity is reduced. If the as it passes through the barrel. A usual guide wear is serious enough, it will cause the prod- is that if the residence time is less than 1 min, ucts to exit at a slower rate or (more likely) there may be a cold condition. to have lower quality. In addition to adhesive wear (caused by metal to metal contact under high stress), abrasive wear (galling), and cor- Influence of the Material on Wear rosion wear (chemical reaction/mechanical attack on the sliding surfaces), screws are sub- Corrosive wear due to plastic materials ject to laminar wear (affecting thin outer lay- usually occurs in the front of the screw and ers of metal at interfaces) and surface-fatigue barrel. The major wear is in the metering sec- wear (micro- or macroscopic separation from tion of the screw, at times extending a little the surfaces). into the transition section. Most of the wear problem is with the barrel core and the screw root. It leads to darkening of the screw and Production Variations pitting of its surface. Note that certain plastics and fillers (nylon, phenolic, etc.) when heated As screw flights and the insides if bar- degrade, giving off corrosive gases and/or rels wear, the pumping ability of the screw liquids. is diminished. Some materials and some ad- This wear usually occurs during startup and ditives will cause higher wear than others; shutdown when the melt is not moving on for example, linear low-density polyethylene cycle and sits in the barrel under heat. The (LLDPE) will cause more wear than conven- long soaking time of the barrel on startup and tional LDPE or polypropylene. Many fillers, shutdown can cause degradation of the plas- such as titanium dioxide (used for white tic when it is in contact with the steel screw coloring) and reinforcing fibers, also create and barrel. The ratio of wear of the barrel to high-wear situations. Under some conditions, 206 3 Plasticizing ,- - -.- -. .. .- -_ T' F Fig. 3-37 Example of severe screw wear. screw/barrel wear can lead to instability of the extent of deterioration. The wear pattern machine output, but typically the main effect then can be plotted to show the screw and is output reduction. At some point, wear will barrel life for the given production case. It create an unacceptable situation that necessi- has been estimated that a 10% reduction in tates rebuilding or replacement of machinery screw surface thickness can cause as much as parts, such as the screws, barrel, and feed sec- a 25% reduction in the output of an IMM. tions. Changes in the ability to increase the In plastics processing, there are two major screw speed and still produce an acceptable types of wear with which to contend: abra- melt contribute to the decision about when sive and adhesive. Abrasive wear is a func- wear has passed acceptable limits. tion of the type of material being processed Variations involved in production oper- and the volume of material pumped through ations, including the materials run, screw the machine. There are two types of abrasive speeds used, melt pressures, barrel set tem- wear: two-body and three-body.
Load more

Recommended publications
  • Screw and Lie Group Theory in Multibody Dynamics Recursive Algorithms and Equations of Motion of Tree-Topology Systems
    Multibody Syst Dyn DOI 10.1007/s11044-017-9583-6 Screw and Lie group theory in multibody dynamics Recursive algorithms and equations of motion of tree-topology systems Andreas Müller1 Received: 12 November 2016 / Accepted: 16 June 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Screw and Lie group theory allows for user-friendly modeling of multibody sys- tems (MBS), and at the same they give rise to computationally efficient recursive algo- rithms. The inherent frame invariance of such formulations allows to use arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit–Hartenberg convention) and to avoid introduction of joint frames. The com- putational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formula- tions. In this paper, recursive O(n) Newton–Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These for- mulations are related to the corresponding algorithms that were presented in the literature. Two forms of MBS motion equations are derived in closed form using the Lie group formu- lation: the so-called Euler–Jourdain or “projection” equations, of which Kane’s equations are a special case, and the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives.
    [Show full text]
  • Nanocrystalline Nanowires: I. Structure
    Nanocrystalline Nanowires: I. Structure Philip B. Allen Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800∗ and Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973-5000 (Dated: September 7, 2006) Geometric constructions of possible atomic arrangements are suggested for inorganic nanowires. These are fragments of bulk crystals, and can be called \nanocrystalline" nanowires (NCNW). To minimize surface polarity, nearly one-dimensional formula units, oriented along the growth axis, generate NCNW's by translation and rotation. PACS numbers: 61.46.-w, 68.65.La, 68.70.+w Introduction. Periodic one-dimensional (1D) motifs twinned16; \core-shell" (COHN)4,17; and longitudinally are common in nature. The study of single-walled car- heterogeneous (LOHN) NCNW's. bon nanotubes (SWNT)1,2 is maturing rapidly, but other Design principle. The remainder of this note is about 1D systems are at an earlier stage. Inorganic nanowires NCNW's. It offers a possible design principle, which as- are the topic of the present two papers. There is much sists visualization of candidate atomic structures, and optimism in this field3{6. Imaging gives insight into provides a template for arrangements that can be tested self-assembled nanostructures, and incentives to improve theoretically, for example by density functional theory growth protocols. Electron microscopy, diffraction, and (DFT). The basic idea is (1) choose a maximally linear, optical spectroscopy on individual wires7 provide detailed charge-neutral, and (if possible) dipole-free atomic clus- information. Device applications are expected. These ter containing a single formula unit, and (2) using if possi- systems offer great opportunities for atomistic modelling.
    [Show full text]
  • Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators
    PHYSICAL REVIEW X 8, 031070 (2018) Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators Eslam Khalaf,1,2 Hoi Chun Po,2 Ashvin Vishwanath,2 and Haruki Watanabe3 1Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan (Received 15 February 2018; published 14 September 2018) The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs), featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with “hinge” surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.
    [Show full text]
  • Chapter 1 Rigid Body Kinematics
    Chapter 1 Rigid Body Kinematics 1.1 Introduction This chapter builds up the basic language and tools to describe the motion of a rigid body – this is called rigid body kinematics. This material will be the foundation for describing general mech- anisms consisting of interconnected rigid bodies in various topologies that are the focus of this book. Only the geometric property of the body and its evolution in time will be studied; the re- sponse of the body to externally applied forces is covered in Chapter ?? under the topic of rigid body dynamics. Rigid body kinematics is not only useful in the analysis of robotic mechanisms, but also has many other applications, such as in the analysis of planetary motion, ground, air, and water vehicles, limbs and bodies of humans and animals, and computer graphics and virtual reality. Indeed, in 1765, motivated by the precession of the equinoxes, Leonhard Euler decomposed a rigid body motion to a translation and rotation, parameterized rotation by three principal-axis rotation (Euler angles), and proved the famous Euler’s rotation theorem [?]. A body is rigid if the distance between any two points fixed with respect to the body remains constant. If the body is free to move and rotate in space, it has 6 degree-of-freedom (DOF), 3 trans- lational and 3 rotational, if we consider both translation and rotation. If the body is constrained in a plane, then it is 3-DOF, 2 translational and 1 rotational. We will adopt a coordinate-free ap- proach as in [?, ?] and use geometric, rather than purely algebraic, arguments as much as possible.
    [Show full text]
  • Script Unit 4.4 (Screw Axes in Crystal Structures).Docx
    Script Unit 4.4 Welcome back! Slide 2 In the last unit, we introduced helical or screw symmetry in general; now, we want to investigate, what different kinds of screw axes, that is, what kind of different helical symmetries are present in crystal structures. Slide 3 Let’s look at a first example - here you see a unit cell of a crystal in two different views, a crystal, which should belong to the hexagonal crystal system; the right view is a projection along the c- direction. We see, that this crystal is composed of atoms, that build these helices running along all corners of the unit cell; if we look from above onto this plane, then these screws look like flat hexagons. Because this is a crystal we have translational symmetry, these helices are repeated again and again as a whole, here along the a- and b-direction. But there is - likewise as in glide planes - another symmetry element present, which has a translational component being smaller than a whole unit cell - and this is a screw axis. Slide 4 Along this axis the atoms are symmetry related to each other by applying a screw rotation: if we rotate first this atom by 60 degrees and then translate it parallel to the screw axis by one-sixth of the unit cell, then it will be mapped onto this atom. This can be done with all other atoms as well with this one, that one and so forth… This is an example of a six-fold screw axis, meaning the rotational part is 60 degrees - analogous to pure rotations.
    [Show full text]
  • A Computational Analysis of Line-Oriented Screw Transformations in Robotics
    University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science October 1988 A Computational Analysis of Line-Oriented Screw Transformations in Robotics Janez Funda University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Janez Funda, "A Computational Analysis of Line-Oriented Screw Transformations in Robotics", . October 1988. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-88-83. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/665 For more information, please contact [email protected]. A Computational Analysis of Line-Oriented Screw Transformations in Robotics Abstract This paper contains a computational analysis and comparison of various representations of a general rigid body spatial screw displacement. Point transformations and line transformations are treated separately. In the context of point transformations, only a brief summary of the known techniques (i.e., homogeneous transforms and quaternion/vector pairs) and their computational behavior is given. Among line transformations, which comprise the primary focus of this paper, four mathematical formalisms for effecting a general spatial screw displacement are presented and analyzed in terms of computational efficiency in performing (a) general screw displacements of lines, and (b) compositions of screw displacement operators. Both sequential and parallel algorithms are given for each operation. The four formalisms considered are: (1) dual orthogonal 3 x 3 matrix, (2) dual unit quaternion, (3) dual special unitary 2 x 2 matrix, and (4) dual Pauli spin matrices. The conclusion reached is that quaternion/vector pairs are the most economical of the point transformation operators, whereas dual unit quaternions represent the most compact and most efficient line ansformationtr formalism.
    [Show full text]
  • 3. Rigid Body Motion and the Euclidean Group
    V. Kumar 3. Rigid Body Motion and the Euclidean Group 3.1 Introduction In the last chapter we discussed points and lines in three-dimensional space, their representations, and how they transform under rigid body. In this chapter, we will develop the fundamental concepts necessary to understand rigid body motion and to analyze instantaneous and finite screw motions. A rigid body motion is simply a rigid body displacement that is a function of time. The first derivative of the motion will give us an expression for the rigid body velocity and this will lead us to the concept of an instantaneous screw. Similarly, higher order derivatives will yield expressions for the acceleration and jerk. 3.2 The Euclidean group In Chapter 2, we saw that the displacement of a rigid body B can be described in reference frame {A}, by establishing a reference frame {B} on B and describing the position and orientation of {B} in {A} via a homogeneous transformation matrix: é A A O' ù A R B r A B = ê ú ëê 01´ 3 1 ûú ( 1 ) A O’ A where r is the position vector of the origin O’ of {B} in the reference frame {A}, and R B is a rotation matrix that transforms the components of vectors in {B} into components in {A}. Recall from Chapter 2, the composition of two displacements, from {A} to {B}, and from {B} A B to {C}, is achieved by the matrix multiplication of AB and AC : A A O¢¢ A é R C r ù AC = ê ú ëê 0 1 ûú é A R A rO¢ ù é B R B r O¢¢ ù = ê B ú ´ ê C ú ëê 0 1 ûú ëê 0 1 ûú é A R ´ B R A R ´ B rO¢¢ + A r O¢ ù = ê B C B ú ëê 0 1 ûú ( 2 ) where the “´” refers to the standard multiplication operation between matrices (and vectors).
    [Show full text]
  • Screw-Based Relative Jacobian for Manipulators Cooperating in a Task
    id6943924 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com ABCM Symposium Series in Mechatronics - Vol. 3 - pp.276-285 Copyright °c 2008 by ABCM SCREW-BASED RELATIVE JACOBIAN FOR MANIPULATORS COOPERATING IN A TASK Luiz Ribeiro, [email protected] Raul Guenther, [email protected] Daniel Martins, [email protected] Universidade Federal de Santa Catarina ânica Departamento de Engenharia Mec ório de Robótica Laborat Caixa Postal 476 ário – Campus Universit Trindade – ópolis, SC – 88040 900 Florian Brazil Abstract. A current trend in Robotics is the analysis and the design of cooperative manipulation system, i.e. systems composed of multiple manipulator units interacting one another in a coordinated way. A cooperative system is characterized by higher manipulability and load capacity with respect to single-manipulator systems. The concept of the relative Jacobian is used to generate trajectories in the joint space, for two robots in cooperation executing a specified task. This paper proposes an alternative method to derive the relative Jacobian, here designated screw-based relative Jacobian, that relates the velocities of a tool (attached to one manipulator), relative to a blank (attached to another manipulator) as a function of the manipulators joint velocities, using the screw representation of differential kinematics, and the method of successive screw displacements. It is presented a systematic method to calculate the relative Jacobian in a compact, direct and simple form. The presented method is specially suitable when the geometry of manipulators becomes more general and for systems with spatial manipulators. A Cartesian-space tool path, defined relative to the blank, is resolved in the joint space.
    [Show full text]
  • Lecture 5 Spatial Kinematics. Constraint
    Lecture 5 Spatial Kinematics. Constraint. Spherical kinematics Euler’s theorem Spatial kinematics Lecture 5 Classifying displacements Chasles’s theorem Screws Spatial Kinematics. Constraint. Cones Kinematic constraint Counting variables and equations Matthew T. Mason General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula Mechanics of Manipulation Lecture 5 Outline Spatial Kinematics. Constraint. Spherical kinematics Spherical Euler’s theorem kinematics Euler’s theorem Spatial kinematics Spatial kinematics Classifying displacements Chasles’s theorem Classifying displacements Screws Chasles’s theorem Cones Kinematic Screws constraint Counting variables and Cones equations General form Classifying constraints Mobility, Grübler Kinematic constraint Mobility and connectivity Counting variables and equations Grübler’s formula General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula Lecture 5 About spherical kinematics Spatial Kinematics. Constraint. Spherical kinematics Euler’s theorem Spatial kinematics Classifying displacements Chasles’s theorem Screws I Why study motions of the sphere? Because it Cones 3 Kinematic corresponds to rotations about a given point of E . constraint Counting variables and I There is a close connection to planar kinematics. Let equations General form the radius of the sphere approach infinity ::: Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula Lecture 5 Two not-antipodal points enough Spatial Kinematics. Constraint. Spherical kinematics Euler’s theorem Spatial kinematics Classifying displacements Chasles’s theorem Theorem (2.5) Screws Cones A displacement of the sphere is completely determined Kinematic constraint by the motion of any two points that are not antipodal. Counting variables and equations General form Classifying constraints Proof. Mobility, Grübler Mobility and connectivity Construct a coordinate frame ::: Grübler’s formula Lecture 5 Euler’s theorem Spatial Kinematics.
    [Show full text]
  • Introduction to Crystals Symmetry
    Introduction to crystals symmetry Symmetry in 3D 5/1/2013 L. Viciu| AC II | Symmetry in 3D 1 Outlook • Symmetry elements in 3D: rotoinversion, screw axes • Combining symmetry elements with the lattice • Space group symbols/representation • Wyckoff positions • Crystallographic conventions • Symmetry in crystal systems 5/1/2013 L. Viciu| AC II | Symmetry in 3D 2 Definitions - a remainder • Unit cell: The smallest volume that can generate the entire crystal structure only by means of translation in three dimensions. • Lattice: A rule of translation. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 3 Bravais Basis/ Crystal + lattice Motif structure 1. single atom: Au, Al, Cu, Pt 1. FCC 2. molecule: solidCH4 2. FCC + - + 3. ion pairs: Na /Cl 3. Rock salt 4. atom pairs: Carbon, Si, Ge 4. diamond •A crystal system is described only in terms of the unit cell geometry, i.e. cubic, tetragonal, etc •A crystal structure is described by both the geometry of, and atomic arrangements within, the unit cell, i.e. face centered cubic, body centered5/1/2013 cubic, etc. L. Viciu| AC II | Symmetry in 3D 4 The Bravais lattices 5/1/2013 L. Viciu| AC II | Symmetry in 3D 5 From 2D to 3D • Bravais lattices may be seen as build up layers of the five plane lattices: cubic and tetragonal stacking of square lattice layers orthorhombic P and I stacking of rectangular layers orthorhombic C and F stacking of rectangular centered layers rhombohedral stacking of hexagonal layers hexagonal stacking of hexagonal layers monoclinic stacking of oblique layers triclinic stacking of oblique layers 5/1/2013 L.
    [Show full text]
  • Review of the Geometry of Screw Axes
    Review of the geometry of screw Axes Nasser M. Abbasi Dept. of Mechanical engineering, UCI. 2006 Compiled on May 19, 2020 at 4:18am Contents 1 Denitions, Important formulas, and terminology 2 1.1 Planar displacement . 2 1.2 Spatial displacement . 2 1.3 Homogeneous Transformation . 2 1.4 The rotation axis . 2 1.5 A Pole or a Fixed point . 2 1.6 Rotational displacement . 3 1.7 Rodrigues vector 푏 .................................. 4 1.8 Screw.......................................... 4 1.9 Plücker coordinates of a line . 4 1.10 screw defined in terms of the plücker coordinates of its axis . 5 1.11 Important relations between Cayley Matrix 퐵 .................. 5 2 Introduction to the rotation matrix in 3D 6 3 Screw displacement 10 3.1 Derivation for expression for finding reference point for screw axis . 13 4 The Screw Matrix 16 5 The spatial displacement of screws 17 6 The screw axis of a displacement 24 7 References 25 Report writing at UCI during work on my MSc in Mechanical Engineering. 2006 1 2 1 Denitions, Important formulas, and terminology 1.1 Planar displacement General motion of a body in 2D that includes rotation and translation. We use [퐴] for the rotation matrix, the vector 풅 for the translation and the matrix [푇] for displacement. Hence we write 푿 = [퐴]풙 + 풅 1.2 Spatial displacement General motion of a body in 2D that includes rotation and translation. Another name for this is rigid displacement. Distance between points in a body remain unchanged before and after spatial displacement. 1.3 Homogeneous Transformation Recall that the transformation that defines spatial displacement, which is given by 푿 = [퐴]풙 + 풅 is non-linear due to the presence of the translation term 풅.
    [Show full text]
  • Fourier Analysis on Motion Using Screw Parameters
    Fourier Analysis on Motion Using Screw Parameters Gregory S. Chirikjian Department of Mechanical Engineering Johns Hopkins University Baltimore, Maryland 21218, USA email: [email protected] Abstract In this paper, the author presents the bi-invariant (Haar) integral for the group of rigid-body motions (Euclidean group) in three-dimensional space in terms of finite screw parameters, and proceeds to develop the matrix elements of irreducible unitary representations in this parameterization. This allows one to integrate and expand functions of motion described in terms of screw parameters. In robot kinematics, the screw-parameter description of a finite rigid-body motion is well known. This description of motion in three-dimensional space (which follows from Ball’s work on finite screws) provides an elegant way to view rigid-body kinematics. In contrast, the theoretical physics community usually uses Euler angles and spherical coordinates to parameterize rigid-body motions. It therefore comes as no surprise that in the field of Fourier analysis on groups, which has been developed in large part by theoretical physicists, that the Euler-angle/spherical coordinate description of rigid-body motions is most common. The contribution of this paper is to formulate Fourier analysis on the group of rigid-body motions in terms of screw parameters. The geometrically mean- ingful nature of the screw parameters combined with the group Fourier trans- form provides a tool for insight into problems that can be posed as convolutions ontheEuclideangroup.Suchproblemsincludeworkspacegenerationofserial
    [Show full text]