Chapter 1 Introduction

Transcription

1 Chapter 1 Introduction Generally all considerations in the force analysis of mechanisms, whether static or dynamic, the links are assumed to be rigid. The complexity of the mathematical analysis of mechanisms with elastic links has been a deterrent against giving up the rigidity assumption. Vibrations in mechanism links are often disregarded by the designers. Machine elements are often overdesigned and quasistatic, rigid-body analysis is preferred for its deformations under dynamic conditions may contribute to a machine s failure to perform adequately at higher speeds. The area of research pertaining to the motion of mechanisms, with consideration of link elasticity and mass distribution has been called the kineto-elastodynamic (KED) of mechanisms. Kineto-elasto dynamic is the study of the motion of mechanisms consisting of elements which may deflect due to external loads or internal body forces. The requirement for machines to run at higher speeds brought to the surface many problems, such as balancing and vibrations, which were not serious factors at lower speeds. Increasing demand of productivity, subsequently increase in machine speeds and stepping in of computers in manufacturing have speed up the study of elastodynamic effect in mechanisms is basically the problem of vibration, therefore the formulation is the differential equation. The effect of mass distribution and elasticity in mechanisms become significant at high speeds. High speed may imply different speeds for different mechanisms. One interpretation of this terminology may be the speed at which inertial forces become so large that they cannot be ignored. Further, at certain speeds these inertial forces may excite one or more modes of vibration of the mechanism. Researchers had observed that for speed more than 300 rpm the inertia effect of links cannot be neglected, therefore this speed is considered as high speed for four bar mechanism. The resulting defections at some critical locations in the mechanism may render the performance of the machine unacceptable. High stress levels together with the large number of stress reversals may cause early failure from fatigue. Other 1.1

2 problems associated with high-speed operation are difficulties in balancing and assurance of stability. 1.1 Basic concepts of mechanisms This section will present concepts fundamental and definitions of a number of terms used in synthesis and analysis of mechanisms Classification of link A mechanism is made of a number of resistant bodies out of which some may have motion relative to the others. A resistant body or a group of resistant bodies with rigid connections preventing their relative movement is known as a link. A link may also be defined as a member or a combination of members of mechanism, connecting other members and having motion relative to them. Thus a link may consist of one or more resistant bodies. To transmit motion, the input link (driver) and output link (driven) are connected by various types of links. Rigid link is one which does not undergo any deformation while transmitting motion. In the strictest sense rigid links do not exit. However, as the deformation of connecting rod, crank etc. of an internal combustion engine is not appreciable; they can be considered as rigid. (a) Binary link (b) Ternary link (c) Quaternary link Fig. 1.1 Different types/shapes of links in mechanism 1.2

3 A link, as shown in Fig. 1.1, is rigid body which possesses at least two nodes which are points for connections to other links. Flexible link which while transmitting motion is partly deformed in a manner not to affect transmission of motion. Thus it transmits motion in one direction only, so as to pull or push, e.g. belts and springs. Such a link is also known as a band. Fluid link is formed by having fluid in a receptacle and the motion is transmitted through the fluid by pressure or by compression only as in a hydraulic press Kinematic pair A pair is a joint of two links that permits relative motion. The relative motion between the elements or links that form a pair is required to be completely constrained or successfully constrained. Kinematic pairs are classified on the basis of any of the following considerations: (a) Nature of relative motion between the elements or links. (b) Nature of contact between the elements or links. (c) Nature of the mechanical arrangement for complete or successful constraint between the elements or links. (a) Nature of relative motion between the links The relative motion of a point on one element relative to the other on mating element can be that of turning, sliding, screw, planar, cylindrical or spherical. The controlling factor that determines the relative motions allowed by a given joint is the shapes of the mating surfaces or element. Each type of joint has its own characteristic shapes for the elements, and each permits a particular type of motion, which is determined by the possible ways in which these elemental surfaces can move with respect to each other. 1.3

4 i. Turning pair (Revolute pair): When connections of the two links are such that only a constrained motion of rotation of one link with respect to the other is possible, the pair constitutes a turning pair. It is also called hinged pair. It allows only relative motion of rotation which can be expressed by a single coordinate θ. Thus a turning pair has single degree of freedom. This is most common type of kinematic pair and is designated by letter R. ii. The sliding pair (Prismatic pair): This type of pair permits relative motion of sliding only in one direction (along a line) and, as such has only one degree of freedom. Pairs between piston and cylinder. This is also a common type of pair and is designated as P. iii. Cylindrical pair: A cylindrical pair permits a relative motion which is a combination of rotation and translation parallel to the axis of rotation, between the contacting links. The pair has thus two degree of freedom and is designated by a letter C. iv. Rolling pair: When the links of a pair have a rolling motion relative to each other, they form a rolling pair, e.g. a rolling wheel on a flat surface, ball and roller bearings, etc. v. Screw pair (Helical pair): The pair permits relative motion between coincident points, on mating links, along a helix curve. Both, axial sliding and rotational are involved. But as the sliding and rotational motions are related through helix angle, the pair has only one degree of freedom. The pair is commonly designated by letter S. Example of such pair is screw jack. vi. Globular or Spherical pair: The pair permits relative motion such that coincident points on working surfaces of links move along spherical surface. In other words, for a position of spherical pair, the joint permits relative rotation about three mutually perpendicular axes. It has tree degree of freedom and designated by a letter G. 1.4

5 (b) Nature of contact between the links Classification of kinematic pairs on the basis of nature of contact, was suggested by the great kinematician Reuleaux, and is the best known. He classified kinematic pairs in two categories: i. Lower pair: A pair of links having surfaced or area contact between the members is known as a lower pair. The contact surfaces of the two links are similar. All revolute pairs, sliding pairs, screw pairs, globular pairs and cylindrical pairs fall in this category. ii. Higher pair: When a pair has a point or line contact between the links, it is known as a higher pair. The contact surfaces of the two links are dissimilar. Meshing gear teeth, cam follower, wheel on surface, ball and roller bearing are a few examples of higher pairs. (c) Nature of the mechanical arrangement for complete or successful constraint between the links Another important way of classifying is to group them as i. Closed pair: When the elements of a pair are held together mechanically, it is known as a closed pair. The two elements are geometrically identical; one is solid and full and the other is hollow or open. The latter not only envelops the former but also encloses it. The contact between the two can be broken only by destruction of at least one of the members. ii. Unclosed pair: When two links of a pair are in contact either due to force of gravity or some spring action, they constitute an unclosed pair. In this, the links are not held together mechanically Degree of freedom (DOF) An unconstrained rigid body moving in space can describe the following independent motions: 1. Translational motions along any three mutually perpendicular axes, and 1.5

6 2. Rotational motions about these axes The number of degree of freedom (DOF) is equal to the number of independent coordinates required to specify its configuration, i.e., the relative positions of all the links Kinematic chain, mechanism and machine A kinematic chain can be defined as an assemblage of links which are interconnected through pairs, permitting relative motion between links. A chain is called closed chained when links are so connected in sequence that first link is connected to last, ensuring that all pairs are complete because of mated elements forming surfaces at joints. As against this, when links are connected in a sequence, with first link not connected to the last, the chain is called an open chain. Examples of planar open chain are not many but they have many applications in the area of robotics. If one of the links of a constrained kinematic chain is fixed, the result is a mechanism. If a number of bodies (links) are assembled in such a way that the motion of one causes constrained and predictable motions to the other, it is known as a mechanism. Thus, mechanism transmits and modifies a motion. A machine is a mechanism or a combination of mechanism which, apart from imparting definite motions to the parts, also transmits and modifies the available mechanical energy into some kind of desired work. It is neither a source of energy nor a producer of work but helps in proper utilization of the same. The motive power has to be derived from external sources. Many kinematicians of repute prefer to reserve the term linkage to describe mechanisms consisting of lower pairs only. But on number of occasion this term has been used rather loosely as a term synonymous to the term mechanisms Determining DOF of mechanism The concept of DOF is fundamental to both the synthesis and analysis of mechanism. 1.6

7 Kinematic chains or mechanisms may be either open or closed as shown in Fig A closed mechanism will have no open attachment points or nodes and may have one or more degree of freedom. An open mechanism of more than one link will always have more than one degree of freedom, thus requiring as many actuators (motors) as it has DOF. A common example of an open mechanism is an industrial robot. An open kinematic chain of two binary links and one joint is called dyad. (a) Open mechanism chain (b) Closed mechanism chain Fig. 1.2 Mechanism chains Reuleaux limited his definitions to closed kinematic chains and to mechanisms having only one DOF, which he called constrained. The somewhat broader definitions above are perhaps better suited to current day applications. A multi-dof mechanism, such as a robot, will be constrained in its motions as long as the necessary numbers of inputs are supplied to control all its DOF. 1.7

8 To determine the overall DOF of any mechanism, we must account for the number of links and joints, and for the interactions among them. The DOF of any assembly of links can be predicted from an investigation of the Gruebler condition. Any link in a plane has three DOF. Therefore, a system of L unconnected links in the same plane will have 3L DOF. Two unconnected links in the same plane have a total six DOF. When these two links are connected by lower pair its remove two DOF, leaving four DOF. In addition, when any grounded or attached to the reference frame, all three of its DOF will be removed. This reasoning leads to Gruebler s equation: DOF = 3 L 2 J 3 G (1.1) where, L = number of links J = number of joints G = number of grounded links Note that in any real mechanism, even if more than one link is grounded, the net effect will be to create one larger, higher order ground link, as there is only one ground plane. Thus G is always one, and Gruebler s equation becomes: DOF = 3 ( L 1) 2 J (1.2) The value of J in above Eqs. (1.1) and (1.2) must reflect the value of all joints in the mechanism. That is, higher pair counts as ½ because they only remove one DOF. It is less confusing if we use Kutzbach s modification of Gruebler s equation in this form: DOF = 3 ( L 1) 2 J J (1.3) 1 2 where, L = number of links J 1 = number of lower pair joints J 2 = number of higher pair joints Quite often, one or more links of a mechanism may have a redundant degree of freedom. If a link can be moved without causing any movement in the rest of the mechanism, then the link is said to have a redundant degree of freedom. Obviously 1.8

9 then the redundant degree of freedom of link gets eliminated. The effective degree of freedom of a mechanism can be expressed as DOF - = 3 ( L -1 ) - 2 J - 1 J 2 F r (1.4) where, F r = number of redundant degree of freedom A system may possess one or more links which do not introduce any extra constraint. Such links are redundant. A redundant joint is one that is unnecessary because other joints provide the needed position and/or orientation. DOF - F = 3 ( L - Lr -1) - 2 ( J1 - J r ) - J 2 r (1.5) where, L r = number of redundant links J r = number of redundant joints A rigid body moving in three dimensions has six degree of freedom, three of which are translational and the other three rotational. One or more of these is/are curtailed at kinematic pair; the number of DOF curtailed depends on the type of kinematic pair. The degrees of freedom of a spatial mechanism can be easily obtained as DOF = 6 ( L -1) - 5 ( R - P - H) - 4 C - 3(S+P ) - F r (1.6) where, R, P, H, C, S and P`stand, respectively, for the number of revolute, prismatic, screw (think of helix), cylinder, spherical, and planar pairs present in the mechanism The Grashof condition [68] The four bar planar mechanism has been simplest possible pin-joints mechanism for single degree of freedom controlled motion. It also appears in various disguises such as the slider-crank and the cam-follower. It is in fact the most common and ubiquitous device used in machinery. It is also extremely versatile in term of the types of motion which it can generate. Simplicity is one mark of good design. The fewest parts that can do the job will usually give the least expensive and most reliable solution. Thus the four bar mechanism should be among the first solutions to motion control problems to be 1.9

10 investigated. The Grashof condition is a very simple relationship which predicts the behavior of a four bar planar mechanism s inversions based only on the link lengths. Let us: S = length of shortest link L = length of longest link P = length of one remaining link Q = length of other remaining link Then if: S + L P + Q (1.7) The mechanism is Grashof and at least one link will be capable of making a full revolution with respect to the ground plane. If that inequality is not true, then the mechanism is non-grashof and no link will be capable of a complete revolution relative to the ground plane. Note that the above statements apply regardless of the order of assembly of the link. That is, the determination of the Grashof condition can be made on a set of unassembled links. Whether they are later assembled in S, L, P, Q order or S, P, L, Q or any other, will not change the Grashof condition. The motions possible from a four bar mechanism will depend on both the Grashof condition and the inversion chosen. The inversions will be defined with respect to the shortest link. The motions are: For the case, S + L < P + Q (1.8) Ground either link adjacent to the shortest and get a crank-rocker, in which the shortest link will fully rotate and the other link pivoted to ground will oscillate. Ground the shortest link and get a double- crank, in which both links pivoted to ground make complete revolutions as does the coupler. Ground the link opposite the shortest link and get a Grashof double rocker, in which both links pivoted to ground oscillate and only the coupler makes a full revolution. 1.10

11 For the case, S + L > P + Q : (1.9) All inversion will be double-rockers in which no link can fully rotate. For the case, S + L = P + Q (1.10) Referred to as special case Grashof, all inversions will be either double-crank or crank-rockers but will have change points twice per revolution of input crank when the links all become collinear. At these change points the output behavior will become indeterminate. The mechanism behavior is then unpredictable as it may assume either of two configurations. 1.2 Definition of terms used in analysis of mechanism[1] Terms use in analysis of mechanism is defined as follows and also listed with indicating the particular aspects of mechanisms in motion in Table Static analysis: Determination of internal forces, stress, strains and deflections in members and joints due to external and/or gravitational loading. 2 Kinematic analysis: Examination of the displacements, velocity ratios, acceleration ratios, etc., of a mechanism with all of its members regarded as rigid. The reference variable is usually a position parameter. 3 Dynamic analysis: Determination of the displacements, velocities, accelerations, etc., of a mechanism, including derivation of inertia forces of a mechanism made up of rigid members. The reference variable is time. 4 Elastic analysis: Examination of the stresses and defections of an elastic system due to static loads in order to determine system flexibilities or stiffnesses. 5 Elastodynamic analysis: Examination of displacements, velocities, accelerations, stresses, strains, etc., of a moving elastic mechanism. Inertia forces are calculated by assuming all of the members rigid. 6 Kineto-elastodynamic analysis: Examination of the displacements, velocities, accelerations, stresses, strains, etc., of a moving elastic 1.11

12 mechanism. Effects of elastic deformation upon the inertia forces are included in the analysis. 7 Kinematic synthesis: Creation of a mechanism which satisfies various combinations of prescribed positions, velocity ratios, acceleration ratios, etc., assuming all members as rigid and massless. The reference variable is a position parameter. 8 Kineto-elastostatic synthesis: Creation of a mechanism which satisfies various combinations of prescribed positions, velocity ratios, acceleration ratios, force and torque transmissions, etc. The reference variable is a position parameter. Mechanism members are assumed to be elastic. 9 Dynamic synthesis: Creation of a mechanism which satisfies various combinations of prescribed positions, velocities, accelerations, etc., considering members as rigid and as having concentrated or distributed masses. The reference variable is time. 10 Dynamic balancing: Creation of a mechanism which satisfies various combination of prescribed positions, velocities, accelerations, etc., considering members as rigid and as having concentrated or distributed masses, including minimization of shaking forces and/or moments within the mechanism and those transmitted to its supports. The reference variable is time. 11 Kineto-elastodynamic synthesis: Creation of a mechanism which satisfies various combinations of positions, velocities, accelerations, force and torque transmissions, stresses, strains, etc., at a predetermined running speed. Mechanism members are assumed to be elastic and have concentrated or distributed mass. 12 Kineto-elastodynamics: The study of the motion of mechanisms consisting of elements which may deflect due to external loads or internal body forces. 1.12

13 Table 1.1 Nomenclature in kineto-elastodynamics [1] Types of performance Forces in members and joints Stresses and strains in members and joints Rigid body position, velocity, etc. Aspects of mechanism in motion Inertia forces due to rigid body motion Elastic deformation Static analysis Inertia forces due to elastic deformation Time as a reference variable Torque or force balancing Stability Kinematic analysis Dynamic analysis Elastic analysis Elasto-dynamic analysis Kineto-elastodynamic analysis Kinematic synthesis Kineto-elastostatic synthesis Dynamic synthesis Dynamic balancing Kineto-elastodynamic synthesis Kineto-elastodynamic balancing Kineto-elastodynamic stability of motion indicate presents 1.13

14 1.3 Outline of thesis This thesis is organized into seven chapters. The current chapter deals with an introduction to the subject and the basic definitions and laws used in theory of machine and mechanism. The remaining chapters are as follows: Chapter 2 critical reviews of the work done in pasts with reference to the topic have been selected for research. Since there are different aspects to the research, the review is not limited to papers on the elastodynamic analysis of four bar planner mechanism. After this the objectives of the present research work have been defined. Chapter 3 deals with the kineto-dynamic (rigid) analysis of four bar planar mechanism. In this chapter the links are to be considered as rigid. In first phase, position and kinematic analysis, such as velocities and accelerations are derived. It is discussed about the transmission angle and toggle position of mechanism. As a second phase method for dynamic analysis of four bar mechanism has been discussed. For this case, speed is assumed to be constant and only forces being applied to the mechanism are inertia forces. The outcomes of this analysis are, pin joint forces and torque required to drive the mechanism at different crank angle. Chapter 4 gives the detailed presentation of the application of the finite element method to the analysis of the four bar planar mechanism. In this analysis, links are assumed as elastic links. The links of mechanism are to be modeled as beam element in axial and bending deflection. Mass and stiffness matrix developed for beam element. After this same matrix has been defined for the mechanism by considering the one link as a one beam element with nine DOF and equation motion is solve in MATLAB. Chapter 5 presents the procedure for FEM analysis in ANSYS software. Various parameter of mechanism i.e. dimension of the links, material of links, crank speed and stiffness behavior of link are specified in this part. Various steps involved in analysis of four bar planar mechanism in ANSYS software are discussed in detail. Details of finite element modeling like number node, numbers element per links and 1.14

15 material properties of link, etc. have been presented. For the different mechanism and with different consideration meshed model figure are presented. Chapter 6 describes with the results and discussion of different types of analysis carried out in MATLAB and ANSYS. Results obtained by programming in MATLAB and simulation in ANSYS have been validated with experimental results mentioned in literatures. Afterwards, to study the effect of various parameters of mechanism on strain developed in flexible coupler are presented in a form of graph of bar chart for comparison. Chapter 7 draws the conclusions and provides some recommendations for the future scope of this research. References used for this research work are mentioned at the end. 1.15