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  • Editura

    NAUTICA

  • II

  • III

    Ioan CLIMNESCU Lucian GRIGORESCU

    Computer aided design of mechanisms

    /Proiectarea asistat a mecanismelor/

    Editura

    NAUTICA 2013

  • IV

    Referent tiinific: Prof. univ. dr. ing. Nicolae ZIDARU

    Note: This Volume was developed by the Authors in the following proportions: /Not: Acest Volum a fost conceput de ctre Autori n urmtorul procentaj/:

    Ioan Climnescu-80% Lucian Grigorescu-20%

    Editura NAUTICA, 2013

    Editur recunoscut CNCSIS

    Str. Mircea cel Btrn nr.104

    900663 Constana, Romnia

    tel.: +40-241-66.47.40

    fax: +40-241-61.72.60

    e-mail: [email protected]

    Descrierea CIP a Bibliotecii Naionale a Romniei:

    IOAN CLIMANESCU

    Computer Aided Design of Mechanisms

    /Proiectarea asistat a mecanismelor/-Ioan

    Climanescu , Lucian Grigorescu-Constanta;

    Nautica, 2013

    Bibliogr. ISBN CLIMNESCU, IOAN; GRIGORESCU, LUCIAN

  • V

    INDEX/CUPRINS/ 1.0 Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ ..1-21.1 Field of Study/Obiectul de studiu/ ...1-2 1.2 Basic concepts/Concepte de baz/ ...1-6 1.2.1 Pairs (Joints)/Cuple cinematice/ ..1-6 1.2.2 Degrees of Freedom/Grade de libertate/ ..1-8 1.2.3 Link (Kinematic elements)-Kinematic Chain/Elemente cinematice-Lan cinematic/.1-11 1.3 Degree of Freedom of Mechanisms/Gradele de libertate ale mecanismelor/ ....1-14 1.4 Constrained Unconstrained Mechanism/Mecanism Constrns-Neconstrns/ ....1-19 1.5 Kinematic Inversion/Inversiunea Cinematic/ ...1-20 1.6 Grbler's Equation/Ecuaia lui Grbler/ 1-22 1.7 Enumeration of Kinematic Chains in Mechanisms/Enumerarea Lanurilor Cinematice n Mecanisme/ 1-24 1.8 Spherical and Two Dimensional Space/Spaiul n Coordinate Sferice i Spaiul Bi-Dimensional/ ....1-27 1.9 Classification of Mechanisms/Clasificarea Mecanismelor/ ......1-28 2.0 Positional Analisys of Mechanisms /Analiza Poziional a Mecanismelor/ ..2-22.1 Position of a Particle/Poziia Punctului Material/ .2-2 2.2 Kinematics of a Rigid Body in Plane/Cinematica corpului rigid n plan/ ..2-4 2.3 Coincident Points/Puncte coincidente/ ..2-6 2.4 Vector Loops of a Mechanism /Conturul vectorial al unui mechanism/ ..2-8 2.5 Graphical Solution of Loop Closure Equations/Soluiile grafice ale ecuaiilor de contur/2-18 2.6 Step-Wise Solution of the Loop Closure Equation/Soluiile iterative ale ecuaiilor contururilor nchise/ .2-25 2.7 Position Analysis of Mechanisms By Means of Complex Numbers/Analiza poziional a mecanismelor cu numere complexe/ .2-29 2.8 Numerical Solution of the Loop Closure Equations/Soluii numerice ale ecuaiilor de contur/...2-32 3.0 Kinematic Analisys of Mechanisms /Analiza Cinematic a Mecanismelor/...3-2 3.1 Velocity and Acceleration Analysis of a Rigid/Analiza Vitezelor i Acceleratiilor ale unui rigid/ ...3-2 3.1.1 Translation of a rigid body/Translaia rigidului/ ....3-2 3.1.2 Rotation of a rigid body/Rotaia rigidului/ .3-3 3.1.3 General Plane Motion/Micarea Plan Paralel/ ....3-6 3.2 Velocity and Acceleration Vectorial Analysis of Mechanisms/Analiza Vectorial a Vitezelor i Acceleraiilor n Mecanisme/ ..3-16 3.3 Position, Velocity and Acceleration Analytical Calculation of Mechanisms/Calculul Analitic al Poziiei, Vitezelor i Acceleraiilor n Mecanisme/ ...3-36 3.3.1 Kinematic analysis of a crank/ Analiza Cinematic a Elementului Conductor/.......3-39 3.3.2 Kinematic analysis of a link in general motion/Analiza cinematic a unui element n miscare plan-paralel/ ....3-43 3.3.3 The Kinematic Analysis of a Dyad with 3 Rotation Joints Using Distance Method/Analiza Cinematic a Grupei Structurale RRR Prin Metoda Distantelor/ ..3-46 3.3.4 The Kinematic Analysis of RRT or TRR Dyads using Contour method / Analiza Cinematica a Diadei RRT-TRR prin Metoda Contururilor/ ....3-55 3.3.5 The Kinematic Analysis of RTR Dyad using Contour method / Analiza Cinematica a Diadei RRT- TRR prin Metoda Contururilor/ .3-57 3.3.6 The Kinematic Analysis of TRT Dyad using Contour method / Analiza Cinematica a

  • VI

    Mecanismului ce Contine Diada TRT prin Metoda Contururilor/ ..3-58 3.3.7 The Kinematic Analysis of RTT-TTR Dyad using Contour method /Analiza Cinematica a Mecanismului ce Contine Diada RTT-TTR prin Metoda Contururilor/ ..3-58 4.0 Four-Bar Mechnisms Analisys/Analiza Mecanismelor Patrulatere/ ..4-24.1 Introduction/Introducere /.....4-2 4.2 Dead-Centre Positions of Crank-Rocker Mechanisms/Poziia moart a mecanimelor manivel-balansier/ ..4-6 4.3 Transmission Angle/Unghiul de transmisie/ ..4-7 4.4 Slider Crank Mechanisms/Mecanismele Biel-Manivel/ .4-10 5.0 Force Analisys in Mechanisms/Analiza Dinamic a Mecanismelor/ ..5-25.1 Introduction/Introducere / 5-2 5.1.1 Principles of Dynamics/Pricipiile dinamicii/ .5-2 5.1.2 Forces and Couples/Fore i Momente/ ..5-3 5.2 Forces in Mechanisms/Forele din Mecanisme/ .5-7 5.2.1 Static Equilibrium/Analiza static /.5-10 5.3 Static Force Analysis of Machinery/Anliza static a mecanismelor/ .5-12 5.3.1 Systems without Resisting Force/Sisteme fr fore rezistente/ .5-12 5.3.2 Principle of Superposition/Metoda Superpoziiei/ ...5-16 5.3.3 Systems with Resisting Force/Mecanisme cu fore rezistente/ .5-22 5.4 Dynamc Force Analyss/Analiza dinamic a mecanismelor/ ..5-40 5.4.1 Center of Mass and Moment of Inertia of a Rigid Body/Centrul de mas si momentul de inerie al rigidului/ ..5-40 5.4.2 Newton's Second Law of Motion for a Rigid Body/Legea a doua a dinamicii aplicat micrii rigidului/ ..5-42 5.4.3 D'Alambert's Principle/Principiul lui DAlambert/ ...5-46 5.5 Dynamc Force Analyss of Machinery/Analiza dinamic a mecanismelor/ 5-48 5.6 Dynamc Force Analyss of a Four-Bar Mechansm/Analiza dinamic a unui mecanism patrulater/ ....5-53 6.0 Gear Trains/Trenuri de Roi Dinate/ .....6-2 6.1 General Issues, Clasification/Generalitai, Clasificri/ . 6-2 6.2 The Fudamental Law of Gearing/Legea Fundamental a Angrenrii/. ...6-4 6.3 Curves to be Used for Teeth Profiles Generation /Curbele Folosite Pentru Construcia Profilului Danturii Roilor Dinate/ ...6-7 6.4 The Gear Basic Rack/Cremaliera de Referin/ .....6-9 6.5 Geometrical characteristics of the simple gears/ Elementele Geometrice ale Roilor Dinate Cilindrice cu Dini Drepi/ ..6-10 6.6 Simple Gear Trains/ Trenuri de Roi dinate cilindrice/ ......6-12 6.7 Planetary Gear Trains/Trenuri de roi dinate planetare/ ..6-16 6.8 Gear Trains with Bevel Gears/Trenuri de roi dinate conice/ ..6-22 7.0 Cams/Came/ .....7-2 7.1 Cam Types and Classification of Cams/Tipuri de came i clasificarea lor/ ..7-3 7.2 Cam Design/Proiectarea camelor/ ...7-4 7.3 Basic Cam Motion Curves/Curbe folosite la profilele camelor/ ...7-8 7.3.1 Linear motion/Curba micrii liniare/ ..7-8 7.3.2 Simple Harmonic Motion (SHM)/Curba armonic simpl/ ..7-11 7.3.3 Parabolic or Constant Acceleration Motion Curve/Curba parabolic sau de acceleraie constant/ ....7-11 7.3.4 Cycloidal Motion Curve/Curba cicloidal/ 7-15 7.3.5 Combined Straight Line-Circular arc motion curve/Curba combinat Linie dreapt-Arc circular/ ....7-18 7.3.8 Cubic or Constant Pulse Motion Curve/Curba cubic sau de impuls constant/ ..7-20

  • VII

    7.3.9 Double Harmonic motion curve/Curba dublu armonic/ ..7-22 7.3.10 Polynomial Motion Curves/Curbele polinomiale/ ...7-24 7.4 Cam Size Determination/Determinarea mrimii camei/ ..7-27 7.4.1 Pressure angle/Unghiul de presiune/ ....7-28 7.4.2 Cam Curvature/Curbura camei/ .....7-33 7.5 Construction of the Cam Profile/Construcia profilului camei/ ..7-37

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-1

    INDEX/CUPRINS/

    1.0 STRUCTURAL ANALISYS OF MECHNISMS /ANALIZA STRUCTURAL A MECANISMELOR/ ____________________________________________________________________2

    1.1 FIELD OF STUDY/OBIECTUL DE STUDIU/ _________________________________________________2 1.2 BASIC CONCEPTS/CONCEPTE DE BAZ/ __________________________________________________6

    1.2.1 Pairs (Joints)/Cuple cinematice/ __________________________________________________6 1.2.2 Degrees of Freedom/Grade de libertate/ ____________________________________________8 1.2.3 Link (Kinematic elements)-Kinematic Chain/Elemente cinematice-Lan cinematic/ __________11

    1.3 DEGREE OF FREEDOM OF MECHANISMS/GRADELE DE LIBERTATE ALE MECANISMELOR/ ___________14 1.4 CONSTRAINED UNCONSTRAINED MECHANISM/MECANISM CONSTRNS-NECONSTRNS/ _________19 1.5 KINEMATIC INVERSION/INVERSIUNEA CINEMATIC/ ______________________________________20 1.6 GRBLER'S EQUATION/ECUAIA LUI GRBLER/ __________________________________________22 1.7 ENUMERATION OF KINEMATIC CHAINS IN MECHANISMS/ENUMERAREA LANURILOR CINEMATICE N MECANISME/________________________________________________________________________24 1.8 SPHERICAL AND TWO DIMENSIONAL SPACE/SPAIUL N COORDINATE SFERICE I SPAIUL BI-DIMENSIONAL/ ______________________________________________________________________27 1.9 CLASSIFICATION OF MECHANISMS/CLASIFICAREA MECANISMELOR/ __________________________28

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-2

    1.0 Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/

    1.1 Field of Study/Obiectul de studiu/

    The aim of mechanisms study is to determine the general motion principles which are common to all machinery and to describe the general synthesis and analysis techniques that can be applied for the design of machinery.

    A machine may be defined as a combination of resistant bodies so arranged that by their means the mechanical forces of nature can be compelled to do work accompanied by certain determinate motion. Notice that in the above definition we are only concerned with the mechanical machines. The definition does not include electrical or heat machines. The main characteristics of a mechanical machine are that there is force (or torque) accompanied with motion.

    A mechanism may be defined as a group of rigid bodies connected to each other by rigid kinematic pairs (joints) to transmit force and motion. A machine structure is constructed to perform a particular task, such as a sewing machine, a lath, a packaging machine.

    A mechanism is considered to be more general. It is an isolated group of rigid bodies through the study of which we can understand the basic structure of any machine and can design machines that are not in existence.

    A machine may also involve a number of mechanisms and certain elements that are not rigid (although resistant). For example, in a machine we may have hydraulic drives, springs, dashpots, flexible elements, etc. which are not considered as bodies that can be included in a mechanism (although we shall see how we can include these elements into the mechanism). You can see such a machine in many construction sites performing the digging or moving the dirt Fig.1.1.

    Scopul disciplinei Mecanisme este de a determina legile generale de micare comune tuturor mainilor i tehnicile de sintez i analiz care pot fi aplicate proiectrii lor. O main poate fi definit ca fiind o combinaie de corpuri rigide aranjate astfel nct forele de origine mecanic s produc un lucru mecanic util pe anumite traiectroii determinate. Definiia de mai sus acoper doar mainile mecanice excluzndu-le pe cele electrice, hidraulice sau termice. n consecin o main mecanic are drept caracteristici prezena unei fore/moment transmis de-a lungul unei traiectorii determinate. Un mecanism se definete ca fiind un grup de corpuri rigide (elemente cinematice) legate ntre ele prin cuple cinematice care transmite o for/moment pe o traiectorie determinat de micare. O main este construit s execute o anumit operaie aa cum fac mainile precum strungurile, mainile de mpachetat, mainile de cusut etc.

    Un mecanism este considerat a fi un concept ceva mai general. El este un grup de corpuri rigide care dac este studiat poate face neleas structura unei maini care exist sau poate face posibil proiectarea unor maini noi.

    O main poate cuprinde unul sau mai multe mecanisme alturi de alte elemente cinematice rigide/rezistente. De pild ntr-o main pot exista mpreun arcuri, elemente hidraulice, elemente flexibile, etc. care s nu fac parte dintr-un mecanism anume dar care concur la funcionarea mainii. Mainile pot fi vzute de pild pe antierele de construcii precum excavatorul din Fig.1.1.

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-3

    Fig.1.1

    The internal combustion engine is the prime mover which gave the human kind an unthinkable mobility. Within the last century, the classical engine (Fig.1.2-a) with a centric slider-crank arrangement has reached perfection. Although it has not found a wide acceptance, the Wankel engine (Fig.1.2-b) is used in applications where the volume and the weight of the engine is quite important

    Motoarele cu combustie intern este invenia care a dat omenirii o mobilitate fr precedent n istorie. n ultimul secol motoarele cu combustie intern cu piston (Fig.1.2-a) au atins aproape perfeciunea. Dei mai puin rspndite, motoarele rotative tip Wankel (Fig.1.2-b) sunt folosite n aplicaii unde volumul i greutatea motorului sunt importante.

    a. b.

    Fig.1.2 Rock crusher (Fig.1.3)-In this type of machinery, the mechanical advantage must be very high to create forces to crush the rocks. Mechanisms of very high mechanical advantage are called toggle mechanisms and this action is known as the toggle action. Input force is amplified several times for crushing rocks between the jaws

    Concasorul (Fig.1.3) este caracterizat prin amplificarea forei de acionare care trebuie s devin suficient de mare s poat sfrma piatra. Are n compunere un mecanism cu genunchi care amplific fora generat de motorul de acionare de cteva ori astfel nct ntre flcile concasorului se dezvolt o for capabil s sfrme piatra.

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-4

    Fig.1.3

    In a number of machinery such as machine tools, cars, etc., we would like to obtain several speeds from a constant speed input. The gears are engaged or disengaged by shifting or there are clutches or slide-keys to engage or disengage gears to a shaft (Fig.1.4).

    n alte tipuri de maini ca de pild strungurile sau autovehiculele este necesar s existe la universal sau la roile autovehiculului viteze diferite pentru aceeai vitez constant intrat de la motorul electric sau de la respectiv motorul cu ardere intern. Pentru aceasta sunt folosite cutiile de viteze care prin intemediul unui cuplaj transmit viteze deiferite la elementul motor (Fig.1.4).

    Fig.1.4

    The crane shown in (Fig.1.5) is also known as Demag jib-crane. It is very often used at ports. After the load is lifted up we would like the tip of the crane to move horizontally. A four-bar arrangement approximately satisfies this requirement while its load carrying capacity is acceptable

    Macaraua din Fig.1.5 numit i macara pivotant tip Demag este des folosit n porturi. Dup ce se leag sarcina vrful macaralei trebuie micat n plan orizontal. Un mecanism patrulater poate asigura acest tip de micare n condiiile unei sarcini de ridicat relativ mare.

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-5

    Fig.1.5

    To shape metals we need large forces. Hydraulic, mechanical (Fig.1.6) or pneumatic presses are used. The mechanism shown has a high mechanical advantage.

    Pentru prelucarea prin deformare a tablelor este nevoie de fore mari de aceea se folosesc prese hidraulice, mecanice (Fig.1.6) sau pneumatice.

    Fig.1.6

    Using Trucks you carry very heavy loads reaching 40-50 tons (such as rocks). Dumping the load is also a hard task. The mechanism shown is one of the solutions found.

    Pentru ncrcarea sarcinilor grele se folosesc ncrctoare ca mai jos unde un mecanism simplu poate asigura executarea acestei operaiuni.

    Fig.1.7

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-6

    1.2 Basic concepts/Concepte de baz/

    1.2.1 Pairs (Joints)/Cuple cinematice/ Kinematic pair (joint)

    The main characteristics of a mechanism are not the rigid bodies (links) but the kinematic pairs that join these rigid bodies.

    Kinematic element, is that part of a rigid body which is used to connect it to another rigid body such that the relative motion between the two rigid bodies can occur.

    Kinematik pair (or simply joint) is the joining of two or more kinematic elements. The types of kinematic pairs and their distribution within the mechanism determine the main characteristics of a mechanism.

    Kinematic pairs may be classified in several different forms:

    Closed Kinematic pairs are those in which the contact between the kinematic elements is maintained within all possible positions of a mechanism. Figure below shows a closed kinematic pair:

    Cuple cinematice

    Mecanismele sunt caracterizate nu att de elementele cinematice ct de cuplele cinematice care leag ntre ele elementele cinematice. Elementul cinematic este un corp rigid care se leag de alte elemente cinematice rigide astfel nct s existe o micare relativ dup o anumit lege de micare.

    Cuplele cinematice au rolul de a lega ntre ele dou sau mai multe elemente cinematice. Tipul de cuple cinematice i modul lor de aezare n mecanism determin caracteristicile principale ale mecanismului.

    Cuplele cinematice pot fi clasificate dup mai multe criterii:

    Cuple cinematice nchise sunt acele cuple la care contactul dinte elementele cinematice este meninut pentru fiecare i oricare poziie a mecanismului, ca n figura de mai jos:

    Fig.1.8

    Open kinematic pair is one whose pairing and unpairing of its kinematic elements that form the joint are controlled. In the Geneva mechanism shown the contact between the pin and the slot is not continuous:

    Cuplele cinematice deschise sunt acele cuple intermitente care cupleaz/decupleaz controlat n mecanism. O asemenea cupl apare n mecanismul Cruce de Malta (sau mecanismul Geneva) la care contactul dintre tift i canalele mecanismului este intermitent:

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-7

    Fig.1.9

    In closed kinematic pairs the contact between the two kinematic elements is due to a normal forces acting at the contacting surfaces (Fig.1.12). Such pairs are force closed kinematic pairs. If one of the kinematic elements envelopes the other and the contact is due to the geometric shape than such joints are form closed kinematic pairs (Fig.1.10,11)

    n cuplele cinematice nchise contactul dintre dou elemente cinematice se face prin fore normale care acioneaz pe suprafeele de contact. Aceste cuple sunt cuple nchise prin fore (Fig.1.12). Dac unul dintre elementele cinematice cuprinde elementul cinematic cu care se afl n contact atunci cupla este nchis prin form (Fig.1.10-11).

    Fig.1.10

    Closed kinematic pairs are classified according to the type of contact between the elements:

    Lower kinematic pairs are those in which the contact between the two elements is along a surface (Fig.1.11).

    Cuplele cinematice nchise pot fi clasificate i dup tipul de contact dintre elementele cinematice: Cuple cinematice inferioare sunt acelea la care contactul dintre elemente se face pe o suprafa (Fig.1.11).

    Fig.1.11

    Higher kinematic pairs are those in which the contact between the kinematic elements is along a line or at a point (Fig.1.12).

    Cuple cinematice superioare sunt acelea la care contactul dintre elemente se face pe o linie sau punct (Fig.1.12).

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-8

    The contact stresses created in higher kinematic pairs are usually unfavorable. Therefore, especially for mechanisms that must transmit forces of high magnitude (which are called power mechanisms), lower kinematic pairs must be preferred. However, in certain applications higher kinematic pairs may be used to reduce the number of parts in a mechanism.

    Tensiunea de contact care apare la cuplele superioare este mare i de regul au un impact nefavorabil asupra funcionrii. La mecanismele care transmit sau dezvolt fore mari (mecanisme de putere) cuplele cinematice inferioare sunt preferate. n anumite aplicaii cuplele superioare pot reduce numrul de elemente ale mecanismului ceea ce poate fi avantajos.

    Fig.1.12

    1.2.2 Degrees of Freedom/Grade de libertate/

    Some of the classifications used in the previous page are important in terms of force transmission (i.e. lower or higher kinematic pairs) or in terms of physical construction (i.e. form or force closed kinematic pairs). However the most important characteristic of the kinematic pairs is the type of motion that may exist between the kinematic elements. Depending on the type of kinematic pair used, there are different motion characteristics between the mating parts. Since there are two rigid bodies connected by a kinematic pair, they will have different relative motions with respect to each other according to the type of the kinematic pair. In order to classify this relative motion we have to understand the degree of freedom concept.

    The degree-freedom of space is the number of independent parameters to define the position of a rigid body in that space.

    Clasificrile de mai sus sunt relevante pentru modul de transmitere a forelor sau de construcia fizic a cuplelor. Ceea ce este cel mai important ns estensa tipul de micare care se poate transmite ntre elementele cinematice. Funcie de tipul de cupl cinematic folosit vor exista diferite tipuri de micri ntre elementele constitutive ale mecanismelor. Cum de regula o cupl cinematic leag ntre ele dou elemente cinematice, elementele vor avea o micare relativ aa cum o va permite/impune cupla cinematic. Pentru a putea clasifica din acest punct de vedere cuplele se introduce conceptul de grad de libertate. Spaiul gradelor de libertate este dat de numrul de parametri independeni care pot defini poziia unui corp rigid n spaiul fizic.

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-9

    Let us define the position of a rigid body in a three dimensions space. The first thing we must do is define a reference axis such as X,Y,Z in the figure. One form of defining the position of the rigid body is that we may arbitrarily select any three non-collinear points, (P1, P2, P3) and determine their location with respect to the reference plane (see figure on below). Once the location of these points are known with respect to the reference axis, the location of any other point can be determined since the distance of the particular point we are considering from P1, P2 and P3 is constant (rigid body), ai =const., i=1;2;3. For each one of the three points we have to define three parameters P1(x1, y1, z1), P2 (x2, y2, z2), P3 (x3, y3, z3). However, due to the rigid body concept we also have the following three equations relating these parameters:

    Fie un spaiu tridimensional n care se definete un sistem de axe Oxyz ca n figura de mai jos. Un mod de a defini poziia unui corp rigid n acest spaiu este de a lua trei puncte arbitrare necoliniare din corp (P1, P2, P3) crora li se va determina poziia fat de sistemul de axe. Dac poziia acestor puncte este cunoscut atunci poziia oricrui punct al rigidului poate fi determinat de vreme ce distanele n rigid sunt constante. Pentru fiecare dintre punctele P1, P2, P3 trebuiesc determinai cte trei parametri de poziie (coordonate) astfel: P1(x1, y1, z1), P2 (x2, y2, z2), P3 (x3, y3, z3). Dat fiind ipoteza distanelor ai =const., i=1;2;3, constante ntr-un rigid, vom avea trei ecuaii ca mai jos:

    Fig.1.13

    ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) 232232232213

    23

    213

    213

    213

    21

    212

    212

    212

    azzyyxx

    azzyyxx

    azzyyxx

    =++=++=++

    We have nine parameters (xi, yi, zi : i =1,2,3) and three relations among them. Therefore, if we define any six of these nine parameters, the location of the rigid body is known. Therefore In the general space the degree of freedom is six-three rotations around the axes X, Y, Z and 3 translations along the axes X, Y, Z. We can also use angles to locate the rigid body: this case, we may take a point A and a line passing through this point A. From the rigid body concept, if we know the position of A and the orientation of the line, the position of the rigid body is determined. In this case we have

    n consecin pentru punctele P1, P2, P3 vor exista 9-3=6 parametri independeni care definesc poziia unui rigid n spaiu. n spaiul fizic numrul maxim de grade de libertate este 9-3=6, trei rotaii i trei translaii n jurul i de-a lungul axelor X, Y, Z. Poziia unui rigid n spaiu poate fi definit i printr-un punct A i o linie care trece prin acest punct. Dac se cunoate poziia punctului A i direcia liniei atunci poziia rigidului este determinat. n acest caz vom avea 7 parametri

    (1.1)

  • Structural Analisys of Mechnisms /Analiza Structural a Mecanismelor/ _________________________________________________________________________________________________

    1-10

    seven parameters (xa, ya, za, 1 , 2 , 3 , ). We also have a relation between i in the form below (eq.1.2). Hence, by defining six parameters (xa, ya, za, ) and any two of the parameters 1 , 2 , 3 , we can locate the rigid body completely.

    (xa, ya, za, 1 , 2 , 3 , ) iar relaia existent ntre i este dat n ecuaia (1.2) drept care numrul de grade de libertate va fi 7-1=6. Dac se cunosc (xa, ya, za, ) i doi dintre parametrii

    1 , 2 , 3 , se poate determina complet poziia rigidului n spaiu.

    Fig.1.14

    1coscoscos 32

    22

    12 =++

    If the space that we consider is a plane, then we only need 3 parameters. Different forms of selecting these three parameters are shown below (polar, rectangular).

    Dac spaiul tridimensional devine un plan bidimensional numrul maxim de grade de libertate se reduce la 3 n plan sistemele de referin pot fi carteziene sau polare.

    Fig.1.15

    The degree-of-freedom of a kinematic pair is defined as the number of independent parameters that is required to determine the relative position of one rigid body with respect to the other connected by the kinematic pair. It is this characteristic that is used to classify the kinematic pairs. If the degree-of-freedom of a kinematic pair is 6, there is no joint involved. If the degree-of-freedom is 5, the kinematic pair must constrain one of the freedoms of space. There is no joint that can constrain the rotational degree-of-freedom while permitting the translational freedom in all directions. We

    Gradele de libertate ale unei cuple cinematice sunt definite ca fiind numrul de parametri independeni necesari pentru a se putea defini poziia relativ a unui element cinematic fat de cellalt element cinematic, ele fiind legate prin cupla cinematic respectiv. Prin acest nou concept se pot clasifica cuplele cinematice astfel: dac numrul gradelor de libertate ale unei cuple este 6 atunci nu exist nici o cupl de vreme ce elementele cinematice pot ocupa relativ orice poziie unul fa de cellalt. Dac numrul gradelor de libertate este 5 atunci cupla anuleaz/constrnge o micare

    (1.2)

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    1-11

    can constrain one of the translational freedoms. The joint thus obtained is sphere between parallel planes. Methodically, all the possible kinematic pairs are shown in Table 1-1 and Table 1-2. Note that the shapes of the kinematic pairs shown are of no importance. It is the degree-of-freedom and the type of freedom that is important.

    posibil ntre elementele cinematice. Dar fizic nu e posibil o cupl care s permit o rotaie n jurul unei axe concomitent cu translaii posibile de-a lungul tuturor axelor. Se pot anula ns toate translaiile concomitent cu existena tuturor rotaiilor (articulaia sferic). n Tabelul 1-1 i 1-2 sunt date tipurile de cuple posibile funcie de gradele de libertate.

    Tab 1.1 and 1.2

    1.2.3 Link (Kinematic elements)-Kinematic Chain/Elemente cinematice-Lan cinematic/ If a rigid body contains at least two kinematic elements we shall call it a link. A link may have more than two kinematic elements (but not less than two). One can classify links according to the number of kinematic elements it contains. These are binary, ternary or quarternary, etc.

    Dac un anumit rigid leag ntre ele cel puin dou (sau mai multe) elemente cinematice atunci rigidul se numete cupl. Cuplele se pot clasifica dup numrul de elemente cinematice pe care le leag. Ele pot alctui mpreun cu elementele adiacente elemente simple, Diade, Triade, Tetrade etc ca n figura de mai jos.

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    Fig.1.16

    The kinematic dimensions of a link in a mechanism are those dimensions which define the relative positions of the kinematic elements on that link and when these dimensions are specified, the link dimensions are known for motion analysis. These dimensions can be distances or angles. For the manufacture of the link or for the dynamics, etc other physical dimensions such as the width, height, thickness, etc. may be important. For the kinematic analysis we shall be interested only with the kinematic dimensions. The distance between revolute joints A and B (a), the angle slot makes with respect to the line AB ( ) and perpendicular distance from point A to the slot axis (b) are the kinematic dimensions of this link. Once these four parameters are known, the link is kinematically defined.

    The links connected to each other by kinematic

    Dimensiunea cinematic a unui Element simplu, Diade, Triade etc. dintr-un mecanism este dat de poziiile relative ale cuplelor din alctuire. Poziiile dintre elementele cinematice date de dimensiuni liniare sau unghiulare. Pentru fabricantul elementului este important nu numai poziia dintre cuple ci i grosimea, limea, lungimea etc. dar pentru proiectarea mecanismelor sunt de interes doar dimensiunile cinematice. Din figura de mai sus se poate vedea c dimensiunile cinematice sunt a, b, i i dac sunt cunoscute atunci elementul cinematic este complet definit. Elementele cinematice conectate ntre ele prin

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    pairs will form a kinematic chain. If all the kinematic pairs are closed, than we have a closed kinematic chain. If one of the kinematic pair is of open type, the kinematic chain is an open kinematic chain.

    Kinematic chain is a representation of the mechanism structure. We are not concerned with the dimensions of the links. Each link is represented as a line or as a polygon and at each vertex we have a kinematic element which joins with another element on another link. The dimensions of the edges are not important.

    There are certain joints where more than two links are connected. For such cases we define the degree-of a joint as the number of links connected at the joint minus one. One must assume that there joints at this point equal to the degree of the joint, (please do not confuse the degree of the joint with the degree-of-freedom of a joint).

    If all the links forming a kinematic chain are in the same plane or in parallel planes, the kinematic chain formed is said to be Planar kinematic chain . If all the points on all the links move on concentric spheres than the kinematic chain formed is Spherical kinematic chain . If some of the links have a general spatial motion than we have Spatial kinematic chain.

    If one of the links in a kinematic chain is fixed, then the system thus obtained is called a mechanism.

    cuple cinematice formeaz lanuri cinematice. Dac elementele cinematice nu au nici o cupl legat cu alt element atunci lanul cinematic este nchis, iar dac una sau mai multe cuple sunt nelegate atunci lanul este deschis. Lanul cinematic este o reprezentare simbolic a mecanismului fr a fi interesai de dimensiunile elementelor cinematice. Fiecare element cinematic este reprezentat simbolic printr-o linie, triunghi, poligon avnd la capete cuple prin care elementul s se lege cu alte elemente cinematice. n anumite cazuri o cupl leag ntre ele mai mult de dou elemente cinematice n care caz gradul cuplei ca fiind numrul de elemente legate prin cupl minus 1 (a nu se confunda gradul cuplei cu gradele de libertate ale cuplei). Dac toate elementele care formeaz un lan cinematic sunt aezate n acelai plan sau n plane paralele atunci lanul cinematic este plan. Dac elementele cinematice se pot mica n sfere concentrice atunci lanul cinematic este sferic. Dac elementele se pot mica n spaiu atunci lanul cinematic este spaial. Dac unul din elementele cinematice ale unui lan cinematic este fix atunci lanul cinematic devine un mecanism.

    Fig.1.17

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    1.3 Degree of Freedom of Mechanisms/Gradele de libertate ale mecanismelor/

    The degree of freedom of a mechanism is the number of independent parameters required to define the position of every link in that mechanism.

    As an example consider a very simple mechanism with four links connected to each other by four revolute joints.

    Assuming that the link lengths are known, if the value of the angle is given, then the position of every link can be determined by determining the coordinates of two points on each link: (A0B0 (Link 1), A0A (Link 2), AB (Link 3) ve BB0 (Link 4)). This is due to the fact that when is given the triangle A0B0A is known (Side-Angle-Side) and the distance AB0 can be calculated. Next, the triangle ABB0 is known completely (Side-Side-Side) we only need one parameter to locate the position of every link. For a Four-Bar mechanism, the degree of freedom of the mechanism is 1.

    Gradele de libertate ale unui mecanism sunt date de numrul de parametri independeni necesari pentru definirea poziiei fiecrui element cinematic al mecanismului. De pild fie un mecanism simplu cu patru elemente conectate ntre ele cu patru cuple de rotaie. Dac se cunosc lungimile elementelor i dac se cunoate unghiul , atunci poziia fiecrui element cinematic poate fi determinat dac se determin coordonatele a dou puncte aparinnd fiecrui element astfel: A0B0 (Element 1), A0A (Element 2), AB (Element 3), BB0 (Element 4). Dac n triunghiul A0B0A se cunoate (Latur-Unghi-Latur) atunci se poate calcula distana AB0 . Dac mai apoi se cunosc lungimile laturilor triunghiului ABB0 atunci cunoscnd se vor cunoate poziiile tuturor elementelor cinematice ale mecanismului. n consecin gradul de libertate a unui mecanism patrulater este 1.

    Fig.1.18

    Consider a mechanism with five links connected to each other by five revolute joints as a second example. If the angle is defined we can solve for the triangle A0AC0. However, the remaining links is a quadrilateral (ABCC0) which will require an additional parameter (Such as angle ) to locate the links. In such a case, since the number of parameters required to determine the position of the links is 2, the degree of freedom of this five link mechanism is 2.

    Fie un mecanism cu 5 elemente de lungimi cunoscute, conectate ntre ele cu 5 cuple de rotaie. Dac se definete unghiul se poate rezolva triunghiul A0AC0 . Elementele care rmn formeaz patrulaterul ABCC0 care pentru a fi determinat necesit cunoaterea unui parametru suplimentar: unghiul pentru a se putea determina poziia fiecrui element al patrulaterului. n consecin este nevoie de cunoaterea a 2 parametri suplimentari pe lng lungimile elementelor pentru a se putea preciza poziia tuturor elementelor mecanismului cu 5 laturi. n acest caz numrul de grade de

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    In the above examples:

    1. Instead of the angles and , other angles can be used as a free parameter. But in every case, for a particular mechanism the number of parameters required is unique. For example in the first example the angle BB0 makes with the horizontal can be selected as a free parameter and the position of each link will be uniquely determined.

    2. The number of parameters required is not a function of the link lengths. For example if the length a2 is 5 units instead of 4, the degree of freedom of the four-bar mechanism is still 1.

    libertate al mecanismului este 2. n ambele exemple de mai sus: 1. n locul unghiurilor i pot fi folosite oricare alte unghiuri ale laturilor ca i parametri independeni pentru determinarea poziiei elementelor. De pild n locul se poate lua unghiul format de elemental 4 (BB0) cu orizontala. 2. Numrul de parametri independeni nu este funcie de lungimea elementelor cinematice. Dac a2 are lungimea de 5 uniti de msur n loc de 4, gradul de libertate al mecanismului cu 4 bare rmne 1.

    Fig.1.19

    We must be able to determine an equation that relates the degree of freedom of a mechanism with the number of links, number of joints and the degree of freedom of the joints. To express these quantities in mathematical terms let us define:

    =Degree of freedom of space

    (=3 planar space; = 6 general (spatial) space)

    l= The number of links in a mechanism (including the fixed link)

    j = The number of joints in a mechanism

    fi =The degree of freedom of the ith joint in the mechanism

    F = The degree of freedom of the mechanism

    First consider l links floating freely (no joints!!)

    Trebuie s se determine o ecuaie care s lege gradele de libertate ale unui mecanism de numrul de cuple i elemente cinematice ale acestuia. Se definesc urmtoarele cantiti: =Numrul de grade de libertate din spaiu (n plan 3, n spaiu 6). l= Numrul de elemente cinematice ale mecanismului incluznd elementele fixe tip batiu, j = Numrul de cuple fi = Gradele de libertate ale elementului i din mecanism. F= Numrul de grade de libertate ale mecanismului. Fie mai nti l numrul de elemente cinematice

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    in a space with freedom. In such a case, apart from the fixed link (the fixed reference frame is attached to one of the links, therefore no parameter is required to determine the position of this fixed link), we will need parameters for each link. Since there are l-1 floating links left, and when there are no joints in the mechanism, the number of parameters required to determine the position of every link is: (l-1).

    nelegate care plutesc liber n spaiul cu grade de libertate. n acest caz i afar de elementele fixe tip batiu, vom avea nevoie de parametri pentru fiecare element n parte pentru a i se putea determina poziia. De vreme ce sunt l-1 elemente cinematice (unul dintre ele este fix) care plutesc n spaiu fr cuple de legtur ntre ele, atunci numrul de parametri independeni necesari pentru determinarea poziiei elementelor plutitoare este : (l-1).

    Fig.1.20

    Now, consider the joints by utilizing a simple example. In the above figure there are four floating links in planar space. If there are no joints, then the number of parameters requires determining the position of these links will be 3 x 4=12. If link 2 is connected to links 4 and 5 by revolute joints, and if there is a cylinder in slot joint between links 2 and 3, the number of parameters required determining the position of these four links will be less. We will still need 3 parameters to determine the position of link 2 (for instance the x, y, z coordinates of its centre of gravity). Once the position of link 2 is known, links 4 and 5 can only rotate relative to link 2, and link 3 can rotate and translate along the slot axis relative to link 2. To locate the position of link 4 we need angle (one known parameter), to locate the position of link 5 we need (1 knew parameter) and to locate the position of link 3 we need 2 more parameters (b and ). Thus the total number of parameters required is 3+1+1+2=7 instead of 12, when there were no joints. In planar space we need 2 less parameters for the revolute joint and one less parameter for the cylinder in slot joint. This means, we dont need to define a

    Fie acum un exemplu simplu. n figura de mai sus sunt 4 elemente cinematice care plutesc n spaiul bidimensional. Dac nu exist cuple active atunci numrul de parametri necesari pentru definirea poziiei lor este 3 x 4=12. Dac elementul 2 este conectat de elementele 4 i 5 prin cuple de rotaie i dac elementul 2 se leag cu un tift poziionat n canalul elementului 2, atunci numrul de parametri necesari definirii poziiei elementelor scade. Pentru aceasta vor fi necesari 3 parametri independeni (de pild coordonatele carteziene x, y, z ale centrului de greutate n spaiu) pentru definirea poziiei elementului 2. Dac se tie poziia elementului 2 atunci elementele 4 i 5 se pot roti faa de elementul 2 iar elementul 3 se poate roti i simultan poate translata n canal faa de elementul 2. Pentru a localiza elementul 4 avem nevoie s tim unghiul , pentru a localiza elementul 5 avem nevoie s tim unghiul i pentru a localiza elementul 3 fa de 2 trebuiesc cunoscuti parametri b i . n consecin pentru a determina complet poziia elementelor mecanismului sunt necesari 3+1+1+2=7 parametri n loc de 12 necesari cnd elementele sunt flotante. Cu alte cuvinte

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    parameter in the direction for which the motion is constrained. If the degree of freedom of space is , a kinematic joint with fi degrees of freedoms constrains, ( - fi) degrees of freedom and we need not define this many number of parameters. Since there are different joints with different degrees of freedom, in a mechanism with j joints, the total number of freedoms constrained by all the joints will be:

    nu aven nevoie de parametri pentru acele direcii de translaie/rotaie unde micarea este stopat de ctre cupl. Dac numrul de grade de libertate al spaiului este atunci o cupl cinematic cu ( - fi) grade de libertate va anula tot atia parametri. Cum exist mai multe cuple diferite fiecare anulnd un numr diferit de grade de libertate, atunci ntr-un mecanism cu j cuple numrul total de grade de libertate constrnse va fi:

    ( ) ==

    =j

    1ii

    j

    1ii ...fjf

    The degree of freedom of the mechanism will then be the degree of freedom of all the links without joints minus the degrees of freedom constrained by the joints:

    F = Degrees of freedom without any joint Constraints imposed by the joints

    The equation (1.4) is known as the General Degree-of-freedom Equation.

    Sau altfel spus numrul de grade de libertate al mecanismului va fi egal cu numrul de grade de libertate al elementelor necuplate minus numrul de grade de libertate anulate de ctre cuple. F=Gradele de libertate fr cuplare ale elementelor-Gradele de libertate anulate de cuple Ecuaia de mai jos (1.4) este cunoscut ca fiind Ecuaia general a gradelor de libertate.

    +=

    =

    ==

    j

    1ii

    j

    1ii f)1jl(Ffj)1l(F

    Mechanism

    (1.3)

    (1.4)

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    Planetary Gear with a slotted lever

    Quick Return Mechanism

    Spatial four-bar

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    Adjustable Drive

    Grabber

    Application of the Degree-of-Freedom Equation

    1.4 Constrained Unconstrained Mechanism/Mecanism Constrns-Neconstrns/

    Constrained mechanism may mean two different things in mechanisms literature:

    1. It may refer to mechanisms in which F = 1.

    2. It may refer to mechanisms whose degree-of-freedom may be greater or equal to one, but the number of inputs (the number of independent parameters) defined is equal to the degree-of freedom.

    With Unconstrained mechanisms, we mean those mechanisms with more than one degree-of-freedom and the number of inputs defined is less than the degree-of-freedom of the mechanism, but the motion is constrained by the forces and dynamic characteristics of the system. A good example is the differential of a car where the rotation of the wheels is governed by the moment acting on them. Due to these characteristics, when taking a turn, the inner

    Noiunea de Mecanism Constrns poate avea dou nelesuri: 1. Poate defini un mecanism la care F=1; 2. Poate defini un mecanism al crui numr de

    grade de libertate poate fi egal sau mai mare de 1 dar la care numrul de parametri independeni prin care mecanismul primete micare este egal cu numrul de grade de libertate.

    Prin Mecanism neconstrns se nelege acel mecanism care are mai mult de un grad de libertate i la care numrul de ci prin care mecanismul primete micare este mai mic dect numrul de grade de libertate dar care are micarea constrns de caracteristicile dinamice ale sistemului. Un exemplu ar fi mecanismul diferenial al unui autovehicul la care rotirea celor dou roi este guvernat i corelat de momentul care acioneaz asupra fiecreia. Astfel un asemenea mecanism la un

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    wheel rotates less than the outer wheel.

    Below, two other examples for unconstrained mechanisms are given. One of the freedoms of the mechanism is usually controlled by a continuous motion. The other freedom is usually controlled by a spring and a key (not shown in the above examples). The motion is governed by the continuous input PLUS the force or moment acting on the links under the spring force.

    viraj, roata care este la interior se va roti mai puin dect cea de la exterior. Mai jos sunt date alte dou exemple de mecanisme neconstrnse la care unul dintre gradele de libertate este controlat de micarea primit (elementul motor) iar micarea ultimului element cinematic este determinat de elementul motor PLUS fora de inerie amortizat/controlat de arcuri.

    Fig.1.21

    1.5 Kinematic Inversion/Inversiunea Cinematic/

    Kinematic inversion is the process of fixing different links in a kinematic chain (or assuming any one of the links, other than the fixed link as fixed). It is a good method of generating some new mechanisms and it is very often used for the synthesis and analysis of the mechanisms to determine the relative motion between the links. In the figures below the kinematic inversions of a four-link chain with three revolute and one prismatic pairs are shown.

    Although the joints and the link length dimensions are the same, four different mechanisms results.

    Oldham Coupling: This is the practical application of the mechanism shown in Fig.1.23. Notice that links 1 and 3 are in the form of a cylinder. Link 4 is also a cylinder which forms two prismatic joints with the extensions on each side.

    Inversiunea cinematic este procesul de fixare a unor elemente din lanul cinematic (altul afar de elementul considerat fix prin ipotez) i folosete la definirea altor tipuri de mecanisme dect cel iniial. Este folosit mai ales la sinteza/analiza mecanismelor i pentru a determina micarea relativ dintre diferitele elemente cinematice. Mai jos sunt date exemple de inversiune cinematic a unui mecanism cu 4 elemente i cu 3 cuple de rotaie i o cupl de translaie. Dei dimensiunile cuplelor i elementelor sunt identice, prin inversiune cinematic se obtin 4 tipuri de mecanisme diferite Cuplajul Oldham este un alt exemplu de mecanism inversat dat n Fig.1.23. Elementele 1 i 3 sunt cilindrice iar 4 este deasemenea cilindric formnd dou cuple prismatice cu extensii pe fiecare latur.

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    Fig.1.22

    Fig.1.23

    Fig.1.24

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    1.6 Grbler's Equation/Ecuaia lui Grbler/

    General degree-of freedom equation can be simplified for certain special cases and certain special conclusions can be derived. For the case that we are going to analyze, we have one degree of freedom (F=1) planar (=3) mechanisms that contain only prismatic and revolute joints (fi=1, fi=j). If we use these values in the general degree-of freedom equation, we have:

    Ecuaia general a gradelor de libertate poate fi simplificat pentru anumite cazuri la care se pot trage i anumite concluzii. Pentru cazul n care avem F=1, =3 (n plan) i sunt implicate doar cuple de translaie/rotaie cu fi=1, fi=j, ecuaia (1.4) poate fi scris:

    3l-2j-4=0

    This equation is known as Grbler's equation.

    We can conclude the following for the mechanisms that satisfy Grubler's equation:

    1. The number of links in the mechanism must be even. Proof: l and j are integers, whatever the value of j, 2j is an even number. Also (2j+4) is an even number. Since 3l = 2j + 4, in order this equation to be true, 3l must be even. Since 3 is an odd number, multiplication with an even number can only give an even number. Therefore l must be even.

    2. The number of binary links in the mechanism must be greater or equal to four. Proof: If a number of links contain k kinematic elements, let us denote this number of links by lk. We cannot have l1 (since there can be no link with one kinematic element). The total number of links, l, in the mechanism will then be equal to:

    Aceast ecuaie este cunoscut sub numele de ecuaia lui Grubler. Se poate concluziona c: 1. Numrul de elemente cinematice dintr-un

    mecanism trebuie s fie par. Demonstraie: cum l i j sunt numere ntregi oricare ar fi valoarea lui j deducem c 2j este un numr par. Deasemenea 2j+4 va fi un numr par. Cum 3l=2j+4, pentru ca l s fie ntreg este necesar la 3l s fie deasemenea un numr par. Cum 3 este un numr impar, atunci ca 3l s fie par trebuie pe cale de consecin ca l s fie un numr par.

    2. Numrul de elemente binare (elemente simple cu dou cuple la capete) dintr-un mecanism trebuie s fie egal sau mai mare de 4. Demonstraie: Dac un element cinematic complex (diat, triad etc.) conine k elemente cinematice simple, se poate nota cu lk tipul de element cinematic. l1 nu poate exista de vreme ce un element cinematic simplu trebuie s aibe cuple la ambele capete pentru a exista ntr-un mecanism. Numrul total de elemente cinematice simple dintr-un mecanism alctuit din elemente simple, diade, triade, tetrade, pentade etc. este:

    l=l2+l3+l4+l5+.....ln or/sau/

    3l=3l2+3l3+3l4+3l5+.....3ln

    (In this equation l2 refers to the number of links with 2 kinematic elements- i.e. number of binary links-, l3 is the number of links with three kinematic elements - i.e. ternary links-

    Mai sus s-a notat cu l2 elementul cinematic simplu, l3 diada, l4 triada etc. Numrul de elemente cinematice simple dintr-un astfel de mecanism va fi:

    (1.5)

    (1.6)

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    ,etc.). The number of kinematic elements in the mechanism will be:

    2l2+3l3+4l4+.....nln=Number of Kinematic Elements/Numr elemente cinematice/

    Since two kinematic elements are joined to form a kinematic pair:

    Cum dou elemente cinematice simple sunt conectate cu cuple ntre ele:

    2j = 2l2+3l3+4l4+.....nln

    If we now substitute eqs (1.6) and (1.8) into Grubler's Equation (1.5):

    Dac se nlocuiete (1.6) i (1.8) n ecuaia lui Grubler's (1.5):

    l2-(l4+2l5+3l6+.....+(n-3)ln=4

    l2=4-P P = l4+2l5+3l6+.....+(n-3)ln

    P is always a positive quantity. It can at most be zero, if all the links in the mechanism are binary or ternary links. Hence, the number of binary links (l2) can at least be 4 if P = 0, otherwise it is greater than 4.

    3. The number of kinematic elements in one link cannot be greater than half of the number of links in the mechanism.

    Cum P este mereu o cantitate pozitiv (i la limit zero), dac toate elementele mecanismului sunt simple sau diade, i dac P=0 atunci l2 (elementele simple) trebuie s fie cel puin n numr de 4 (sau mai mare). 3. Numrul de elemente cinematice simple dintr-un element complex (diade, triade etc.) nu poate fi mai mare dect jumtate din numrul de elemente cinematice simple din mecanism.

    Fig.1.25

    Proof: For instance consider a link (a) with i kinematic elements and let this number of kinematic elements be the maximum that a link can have. A kinematic chain using this hyper-link can be formed if we attach links of type (b) (see figure), and if we join these links with links of type (c), in this case the number of kinematic elements on link (a) will be a maximum. There will be 1 link of type (a), i links of type (b) and (i-1) links of type (c). Then the number of links in this mechanism will be:

    Demonstraie: Fie un element a care cuprinde i elemente cinematice simple i fie i numrul maxim de elemente simple din comunerea unui element complex. Un lan cinematic care folosete un asemenea hiper-element a poate fi format prin ataarea elementelor tip b (triade) care dac se unesc ntre ele cu elemente tip c, atunci lanul cinematic avnd un element tip a va avea maximum de elemente simple posibile. Aadar avem 1 element tip a, i elemente tip b i i-1 elemente tip c. Numrul de elemente simple din lan va fi:

    (1.7)

    (1.8)

    (1.9)

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    l=1+i+(i-1)

    or/sau/ i=l/2

    I=l/2 is the maximum number of kinematic elements on any one link when the mechanism contains l links.

    Deci dac un mecanism conine l elemente simple atunci i=l/2 este numrul maxim de elemente simple care pot compune un element complex.

    1.7 Enumeration of Kinematic Chains in Mechanisms/Enumerarea Lanurilor Cinematice n Mecanisme/

    The meaning of Enumeration" is "to list one by one; to count. Kinematic enumeration is the process of determining all possible kinematic chains or mechanisms satisfying certain predetermined criteria. There are elegant mathematical methods for the enumeration of kinematic chains (such as graph theory).

    Enumerarea lanurilor cinematice nseamn procesul de determinare a tuturor lanurilor cinematice posibile (sau mecanisme posibile) care satisfac un anumit criteriu impus. Se pot folosi metode matematice precum teoria grafurilor.

    Fig.1.25

    Consider kinematic chains that satisfy Grbler's equation. If the number of links is restricted to four, than the number of joints is four and all the links are binary. Without considering the type of joint (revolute or prismatic) we have one type of kinematic chain (Figure 1.25-a). If we also consider the type of joint in our enumeration, we can have four different kinematic chains (Figures b, c, d, e). Note that we cannot possibly have three or four prismatic joints. Once the kinematic chain is obtained, using kinematic inversion, one can then enumerate all possible mechanisms.

    The next highest link number that satisfies Grbler's equation is l=6, and the number of joints must be 7. Using the conclusions that were derived, we can at most have links with 3 kinematic elements (ternary links) and we must also have 4 binary links. The only possible combination is 4 binary and 2 ternary links. (since 2j = 2l2 + 3l3 and l2= 4, l3 = 2 for j = 7 is

    Fie un lan cinematic care satisface ecuaia lui Grubler. Dac numrul de elemente este 4 atunci numrul de cuple este 4 i toate elementele sunt simple (binare). Dac nu se ine seama de tipul de cupl, schema structural apare ca n Fig.1.25-a. Dac se ine seama i de tipul cuplei (translaie T sau rotaie R) rezult 4 tipuri de lanuri cinematice diferite (Fig.1.25-b,c,d,e). S se observe c nu se poate avea 3 sau 4 cuple de translaie ntr-un astfel de mecanism. Dac se folosete apoi inversiunea cinematic se poate deduce ntreaga list de mecanisme posibile. Urmtorul numr de elemente cinematice simple care satisfac ecuaia lui Grubler (dup 4) este 6 iar numrul de cuple necesar este 7. Dup concluzia nr. 3 de mai sus putem avea maximum n componen elemente tip triad sau avem nevoie de 4 elemente binare simple. Combinaiile posibile sunt 4 elemente simple binare i 2 diade (cum 2j = 2l2 + 3l3 i l2= 4, l3

    (1.10)

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    the only possible integer solution to the equation). These links may be combined in two different forms, thus yielding two different types of kinematic chains without considering the type of joint (Figure below). In the "Watt's Chain" the two ternary links are connected to each other, whereas in "Steffenson's chain" the two ternary links are not directly joined.

    = 2 pentru j = 7 sunt singurele soluii ntregi). Aceste elemente pot fi combinate n dou soluii de lanuri cinematice (fr a se lua n calcul tipul de cuple (Fig.1.26). n lanul lui Watt avem 2 elemente ternare conectate direct unul de altul iar in lanul Steffenson sunt 2 elemente ternare cuplate ntre ele cu un element simplu.

    Fig.1.26

    If the distance between the two kinematic elements on a ternary link goes to zero, we have limitting cases of Watt and Steffenson chains. The joint degree will increase and the two limitting chains will be as shown below. These are the special cases of 6 link chains.

    Dac unul dintre elementele ternare se reduce dimensional la zero aprnd doar o cupla tripl n loc avem cazurile limit ale lanurilor lui Watt i Steffenson ca mai jos:

    Fig.1.27

    One can then obtain different kinematic chains (different in terms of the type of joint) by using sliding joints instead of revolute joints. As a simple example, consider a six-link mechanism driven by a piston cylinder. Let the cylinder be connected to the fixed link by a revolute joint. All other joints, except the piston cylinder are revolute joints.

    Dac n loc de cuple de rotaie se consider i cuple de translaie se obine alte tipuri de lanuri cinematice. Ca un exemplu simplu fie un mecanism cu 6 elemente acionat de un piston hidraulic. Fie pistonul conectat la elementyul fix tip batiu cu o cupl de rotaie. Toate cuplele cu excepia cilindrului sunt tot cuple de rotaie.

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    Fig.1.28

    A piston-cylinder can be considered as two binary links connected by a prismatic joint. Hence in Watt's chain or in Streffenson's chain a piston-cylinder must replace the two binary links connected to each other. In Steffenson's chain there is a unique location where the piston-cylinder can be placed. In case of Watt's chain although there two different locations, due to symmetry of the chain, there is no difference. Since the cylinder must be connected to the fixed link, there are two different six link mechanisms that satisfy these conditions. The result is as shown above.

    Un sistem piston-cilindru poate fi socotit ca legnd dou elemente simple binare cu o cupla de translaie. Deci n lanurile lui Watt i Steffenson trebuie nlocuit o cupl de rotaie cu cupla de translaie. n lanul lui Steffenson nu exist dect o singur locaie unde se poate poziiona pistonul iar n cazul lanului Watt dei exist dou poziii posibile lanul cinematic care rezult este acelai datorit simetriei. Cum pistonul trebuie legat de elementul fix date n Fig.1.28.

    Fig.1.29

    The special 6 link chains can also be used. There are two special chains where one or two link dimension of a ternary link is zero. In practice, the truck dump mechanism, cement pump boom, loaders and back-hoe systems all use the above chains. Two practical examples, log grabber and back-hoe, are shown below:

    Mecanismul cu 6 elemente poate avea i dou sub-cazuri speciale unde elementele ternare se reduc la zero dimensional i devin cuple triple. n practic excavatoarele sau alte maini de ncrcare folosesc aceast soluie cum se vede mai jos:

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    Fig.1.30

    1.8 Spherical and Two Dimensional Space/Spaiul n Coordinate Sferice i Spaiul Bi-Dimensional/

    Although we have not restricted the general degree-of-freedom equation for any particular space, the given examples were concerned with planar or spatial mechanisms where the degree-of freedom of space was 3 or 6 respectively. Another three dimensional space is the spherical space in which the links are restricted to move in concentric spheres (the degree of freedom of space is three). Such mechanisms are usually identified if all the revolute joint axes intersect at one point. A good example is the Hooke's joint (sometimes it called Cardan joint). Three different Cardan joint constructions are shown below:

    Cardan Joint-1 (such a construction is used to transmit heavy loads- in cars and trucks, for example).

    Pn acum nu s-a fcut nici o restricie explicit asupra tipului de spaiu n care evolueaz mecanismul, pentru spaiul plan gradele de libertate posibile fiind 3 i n spaiul tridimensional este 6. Un alt spatiu posibil este cel n coordonate sferice la care cuplele restrictioneaz elementele cinematice s se mite n coordonate sferice (astfel gradele de libertate ale acestui spaiu sferic sunt 3). Un asemenea mecanism poate fi alctuit din 2 cuple de rotaie decalate ntre ele la 900. Un exemplu este cupla tip Hooke numit i cupla Cardanic date mai jos: Cupla cardanic tip 1 poate transmite sarcini mari la autovehicule de pild,

    Fig.1.31

    Cardan Joint-2 (Such a construction is used in machine tools for easy assembly and disassembly).

    Cupla cardanic tip 2 este folosit mai ales la mainile-unelte,

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    Fig.1.32

    Cardan Joint-3 (This is the usual shape that you will see in books. this construction is used extensively in a variety of applications).

    Cupla cardanic tip 3 este cea mai comun i are o varietate mare de aplicaii,

    Fig.1.33

    In two dimensional space (=2). In planar motion two-dimensional space exists for screw mechanisms or for 3-link chains with sliding joints only (the degree of freedom of space is two (=2). In case of 3-link mechanisms in the case of screw mechanisms there is rotation and translation along one axis -screw axis- only but only one movement is independent since translation and rotation are correlated.

    n spaiul bidimensional cu =2 i n cazul mecanismelor cu urub-piuli avnd 3 elemente cu cuple de translaie micarea de rotaie nu este independent de cea de translaie, ele fiind corelate prin pasul filetului.

    Fig.1.34

    1.9 Classification of Mechanisms/Clasificarea Mecanismelor/

    According to Reuleaux, (Fig.1.35) mechanisms are classified into six basic types:

    Dup Reauleaux (Fig.1.35) mecanismele pot fi clasificate n 6 tipuri de baz:

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    1. Screw Mechanisms 2. Wheel mechanisms (gear mechanisms

    or roller mechanisms) 3. Cam mechanisms 4. Crank mechanisms (sometimes also

    called link mechanisms). 5. Belt mechanisms 6. Ratchet and lock mechanisms

    (including Geneva drives).

    This classification, although it has certain important merits, does not really separate all the mechanisms that we see in practice. We usually have mechanisms that include more than one of the above six basic types.

    Other classifications may be made concerning the following topological characteristics:

    1. Degree-of-freedom of space of the mechanism (e.g. spatial, spherical, planar, etc.)

    2. Degree-of-freedom of the mechanism 3. Number of links in the mechanism 4. Number of joints in the mechanism 5. Types of joints in the mechanism

    1. Mecanisme urub-piuli, 2. Mecanisme cu roi (cu roi dinate sau

    roi de friciune), 3. Mecanisme cu came, 4. Mecanisme cu manivel, 5. Mecanisme cu curele, 6. Mecanisme cu clichet incluznd cele cu

    cruce de Malta, Aceast clasificare dei intuitiv nu separ clar ntre ele mecanismele care se pot ntlni n practic. Deseori mecanismele din practic cuprind dou sau mai multe mecanisme din tipologia de mai sus nseriate sau n paralel. Alte clasificri in cont de alte caracteristici precum:

    1. Tipul spaiului n care evolueaz mecanismul (plan, tridimensional, sferic),

    2. Numrul de grade de libertate al mecanismului,

    3. Numrul de elemente componente, 4. Numrul de cuple, 5. Tipurile de cuple.

    Fig.1.35

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    INDEX/CUPRINS/

    2.0 POSITIONAL ANALISYS OF MECHANISMS /ANALIZA POZIIONAL A MECANISMELOR/ ____________________________________________________________________2

    2.1 POSITION OF A PARTICLE/POZIIA PUNCTULUI MATERIAL/___________________________________2 2.2 KINEMATICS OF A RIGID BODY IN PLANE/CINEMATICA CORPULUI RIGID N PLAN/ _________________4 2.3 COINCIDENT POINTS/PUNCTE COINCIDENTE/ _____________________________________________6 2.4 VECTOR LOOPS OF A MECHANISM /CONTURUL VECTORIAL AL UNUI MECHANISM/_________________8 2.5 GRAPHICAL SOLUTION OF LOOP CLOSURE EQUATIONS/SOLUIILE GRAFICE ALE ECUAIILOR DE CONTUR/ ___________________________________________________________________________18 2.6 STEP-WISE SOLUTION OF THE LOOP CLOSURE EQUATION/SOLUIILE ITERATIVE ALE ECUAIILOR CONTURURILOR NCHISE/ ______________________________________________________________25 2.7 POSITION ANALYSIS OF MECHANISMS BY MEANS OF COMPLEX NUMBERS/ANALIZA POZIIONAL A MECANISMELOR CU NUMERE COMPLEXE/ __________________________________________________29 2.8 NUMERICAL SOLUTION OF THE LOOP CLOSURE EQUATIONS/SOLUII NUMERICE ALE ECUAIILOR DE CONTUR/ ___________________________________________________________________________32

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    2.0 Positional Analisys of Mechanisms /Analiza Poziional a Mecanismelor/

    2.1 Position of a Particle/Poziia Punctului Material/

    In order to determine the position of a point (particle) in plane, we must first locate a reference frame. For example, if we attach a coordinate frame to a moving body, the position of a point on the rigid body will be defined by constant dimensions. If our reference frame is attached to a fixed body or another body that is moving, the coordinates of a point on this rigid body will be variable. Once the reference frame is established, different co-ordinate systems (Cartesian or polar) and different parameters can be used to determine the position of a particle. In describing the location of a point (see Figure below), we must state either the distance from the origin of the reference frame to point P and give the angular orientation of the line OP in the reference frame, or the Cartesian coordinates x, y.

    Pentru a se determina poziia unui punct ntr-un plan trebuie mai nti definit un sistem de referin. Dac se ataeaz un sistem de referin unui rigid n micare atunci coordonatele unui punct aparinnd rigidului fa de sistemul ataat acestuia rmn constante n timp. Dac sistemul de referin este ataat unui corp fix atunci un punct aparinnd unui rigid n micare vor fi variabile n timp. Pot fi folosite diferite sisteme de referin (carteziene sau polare) implicnd diferite tipuri de parametri care pot fi folosii n definirea poziiei unui punct. Din figura de mai jos se poate vedea c n definirea poziiei punctului P se pot folosi fie coordonatele polare precum distana dintre originea sistemului i P i unghiul dintre OP i axa Ox, fie coordonatele carteziene x, y.

    Fig.2.1

    These two specifications (the magnitude and direction) are the properties that define a vector. Therefore, the position of a particle (point) is given by a position vector rPO

    rr = . The vector can be represented in Cartesian form as:

    Dac se d mrimea distanei de la origine la P i sensul, atunci se definete practic un vector de poziie, ca urmare poziia unui punct e dat de vectorul de poziie rPO

    rr = . n coordinate carteziene vectorul de poziie se poate exprima astfel:

    yjxirrrr +=

    In this equation i and j are the unit vectors for Ox and Oy axes, and x and y are the distances in horizontal and vertical axes which are to be measured by a certain scale.

    If polar form the position vector is determined by r and , where r is the distance from the origin to the particle and is the inclination of

    n ecuaia de mai sus i i j sunt vectorii unitate (sau versorii) pentru axele Ox i Oy iar x i y sunt distane pe direcii orizontale i verticale ce trebuie msurate. Dac sistemul de referin este polar atunci vectorul de poziie este dat de r i , unde r este distana de la origine la punct iar este

    (2.1)

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    the line OP from a reference line (Ox), measured in counter-clockwise direction.

    To convert from x, y to r, or vice versa:

    unghiul format de linia OP fa de axa Ox msurat n sesn trigonometric. Pentru a se trece de la sistemul de referin cartezian la cel polar sau invers:

    yxarctg;yxr

    ;sinry;cosrx

    22 =+===

    For the determination of the position of a particle, we can also make use of complex algebra. Although complex numbers are not vectors, they can be used to represent position vectors in plane if the axes of the reference frame are used as the real and the imaginary axes of the complex plane. In such a case the position of a particle can be expressed in terms of a complex number z which is:

    Pentru determinarea poziiei unui punct se pot folosi i numerele complexe. Dei acestea nu reprezint vectori ele pot fi folosite pentru reprezentarea poziiei unui vector n plan dac axele sistemului de referin devin x=Re(z) sau axa Ox devine ax real, i y=Im(z) sau axa Oy devine ax imaginar, z fiind numrul complex. n acest caz poziia unui punct material se poate scrie astfel:

    22

    222

    yxz

    zyx)iyx)(iyx(zz

    conjugatcomplex Numarul iyxziyxz

    +==+=+=

    =+=

    Fig.2.2

    where x and y are the distances measured along the real and imaginary axes and i is an operator which is defined as the unit imaginary number ( 1i = ). The usefulness of complex numbers is due to the fact that the transformation from the Cartesian parameters (x, y) to the polar parameters (r, ) or vice versa, can be performed with no additional burden. Since the complex number, z, which shows the position vector of a particle can be written as:

    unde x i y sunt distanele msurate de-a lungul axei reale i imaginare, iar i este un operator numit i numr unitate imaginar cu proprietatea c 1i = . Utilitatea numerelor complexe se datoreaz faptului c trecerea de la sistemul cartezian (x, y) la cel polar (r, ) i invers se poate face foarte uor. Un numr complex z care descrie poziia unui vector al unui punct, se poate scrie:

    (2.2)

    (2.3)

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    ( ) ( ) ==

    ==+==

    =+=+=+=

    =+===

    sinircosrrezsau z

    sinicosez

    sinicosez

    sinicosrz;sinicosrzsinircosrzyixz

    yxarctg;yxr;sinry;cosrx

    i

    i

    i

    22

    In the exponential form of the position vector

    ire where r is the distance of point P to O (OP, which is the magnitude of the vector) and

    ie is a unit vector along the direction of OP ( sincos ie i = ). In other words, In Fig.2.2 diagram if a number is multiplied by

    ie it rotates by an angle in counterclockwise direction.

    In general the position of the particle will change in time (it is a function of time) This change can be expressed either using rectangular or polar coordinates.

    n forma sa exponenial poziia unui vector va fi dat de ire unde r este distana rOP r (identic cu modulul vectorului de poziie) iar

    ie este vectorul unitate poziionat de-a lungul liniei OP ( sincos ie i = ). Altfel spus n diagrama din figura 2.2, dac se nmulete un numr real cu ie , vectorul corespondent se va roti cu unghiul n sens trigonometric. n cazul general poziia n timp a punctului material se va schimba. Aceasta schimbare poate fi descris fie n coordinate carteziene fie polare.

    2.2 Kinematics of a Rigid Body in Plane/Cinematica corpului rigid n plan/

    Rigidity is an assumption. This assumption simplifies the mathematical model to a very big extent. Due to rigidity assumption we can make the following important conclusions:

    1. The plane motion of a rigid body is completely described by the motion of any two points within the rigid body.

    Rigiditatea unui corp este o ipotez teoretic menit s simplifice mult modelele matematice din mecanic (sau mecanisme). Rigiditatea are drept consecin: 1. Micarea plan a unui corp rigid este complet descris dac se cunoate micarea a dou puncte oarecare ale corpului rigid.

    Fig.2.3

    (2.4)

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    Let us assume that the motions of points A and B are given. At any time instant t, the position of the rigid body is then known. Take any point S (Fig.2.3). At every position of the rigid body the position of point S is completely known, since the distances AS, BS and AB are fixed. For example if the rigid body has moved from A, B and S to a new location given by A', B', using the constant dimensions of AS, BS and AB=AB, one can locate S'. Hence every point on the rigid body can be determined one the position of any two points such as A and B are known.

    Fie cunoscute micrile a dou puncte A i B. Prin urmare la un moment dat t, poziia rigidului este cunoscut. Se ia un punct oarecare S (Fig.2.3) aparinnd rigidului. Pentru fiecare poziia n timp a rigidului, poziia punctului S este mereu cunoscut de vreme ce distanele AS, BS, AB, prin ipoteza rigiditii corpului, sunt constante. Dac rigidul se mic ntr-o alt poziie A, B, S, cum AS, BS sunt constante i cum AB=AB , poziia lui S poate fi determinat. Deci dac se cunoate poziia a dou puncte aparinnd rigidului, oricare punct al rigidului va putea fi definit poziional.

    Fig.2.4

    The position of a rigid body can also be defined by a vector from one point of the rigid body to another such as A and B, which will be called

    BA vector (Fig.2.4).

    The vector AB is a fixed vector on the rigid body. Since A and B can be selected anywhere, the length of the vector is arbitrary. Relative to a fixed reference frame, the position of the rigid body can as well be defined by giving the position of the origin of this vector (point A which is a point on the rigid body) and its orientation (angle )

    2. Rigidity ensures that the particles lying on a straight line have equal velocity components in the direction of this line.

    Since the distance between any two points along this line remains constant, there can be no velocity difference for points along this line (otherwise they must come closer to each other or separate, which is against the hypothesis that the rigid body cannot deform). Velocity

    Poziia rigidului poate fi definit i printr-un vector care leag un punct al rigidului de altul, ca de pild A de B (Fig.2.4) notat BA . Vectorul BA este un vector fixat pe rigid i cum A i B sunt arbitrare atunci modului vectorului este i el arbitrar. Fa de un sistem fix de referin poziia rigidului poate fi descris dac se cunosc poziia punctului A care este originea vectorului i unghiul al vectorului. 2. Rigiditatea asigur faptul c punctele situate pe o linie dreapt n cuprinsul rigidului au componente/proiecii ale vitezei egale (vitezele lor din plan se proiecteaz pe aceast linie). Cum distanele intr-un rigid sunt fixe, atunci orice vitez ar avea rigidul i n oricare direcie, dac proieciile vitezelor fiecrui punct de pe o linie dreapta din rigid nu ar fi egale, punctele respective se vor apropia/departa unul fa de altul, violnd ipoteza nedeformabilitii

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    difference may be on perpendicular direction to the line.

    rigidului. Dac exist diferene n vitezele celor dou puncte acestea se pot manifesta doar pe direcie perpendicular pe aceasta linie.

    )V(pr)V(prVV BABA

    Fig.2.5

    3. If we are concerned with the kinematics of the rigid bodies only, it is sufficient to consider just a line on the rigid body (vector AB, for example).

    Since the actual boundaries of the body do not influence the kinematics (but it does influence the dynamics), the rigid body in plane motion is to be regarded as a large plane which embraces any desired two point in the plane.

    3. Dac suntem interesai doar de cinematica unui corp rigid atunci pentru a o studia este suficient s se studieze cinematica unei linii din rigid precum vectorul BA . Cum frontierele fizice ale corpului nu influeneaz cinematica sa (dei influeneaz dinamica sa), micarea plan a rigidului poate fi privit ca micarea unui plan n care se situeaz oricare 2 puncte dorite.

    2.3 Coincident Points/Puncte coincidente/

    In mechanisms we are not involved with the motion of one rigid body but of several rigid bodies. From the corollaries obtained for rigid bodies in plane motion, we can assume that each link in a mechanism is a plane of infinite dimensions which may be represented by a straight line. The relative motions of these links with respect to each other are governed by the joints that connect them. These planes are assumed to be superimposed on top of each other as shown below:

    n disciplina Mecanisme nu sunt studiate corpuri rigide separat ci sisteme de corpuri rigide. Din corolarele deduse mai sus pentru micarea plan, se poate presupune c fiecare element cinematic este un plan infinit dar care poate fi reprezentat printr-o linie dreapt. Micarea relativ a acestor elemente cinematice este impus de cuplele de legtur dintre ele. Fie planele unor elemente cinematice puse unul peste altul ca n figura de mai jos:

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    Fig.2.6

    Consider the motion of link 2 with respect to link 1. It will be a rotation about the axis of the revolute joint connecting these links, A0. The size of the joint or the shape of the joint is completely unimportant and, furthermore, it does not matter which link is fixed since we are concerned with the relative motion only. The relative motion between links 2 and 3 or 4 and 3 will again be a rotation about A or B respectively.

    Consider a general point P. In order to locate this point, let us assume that we have pierced a hole through all of the four planes involved. There will be a corresponding point P1 on plane l, P2 on plane 2, P3 on Plane 3 and P4 on plane 4. At the position considered, all these four points are coincident but at any other position these points will be at different relative positions. Consider point P2. It will trace a circle with origin A0 on plane 1 and it will trace another circle with origin A on plane 3. The path of P on plane 4 will not be a circle but a higher order curve (it is a fourth-order algebraic curve in general).

    Let us now consider point B. There will be four corresponding points B1, B2, B3 and B4. Since B is on the axis of the revolute pair joining links 3 and 4, B3 and B4 will be coincident at all positions. Such points (B3 and B4) will be called permanently coincident. Where as the other points (B1 and B2) are instantly coincident with B3 or B4.

    Fie c elementul 2 se mic relativ fa de elementul 1. Aceast micare este o micare de rotaie n jurul cuplei care leag cele dou elemente, A0. Mrimea i forma cuplei sunt complet neimportante i nici nu este important dac unul dintre elementele cinematice este fix sau nu de vreme ce intereseaz doar micarea relativ. Micarea relativ dintre elementele 2 i 3 sau 3 i 4 vor fi deasemenea rotaii n jurul cuplelor A sau B. Fie un punct oarecare P. Pentru a-i defini poziia se presupune c se d o gaur prin toate planele elementelor. Va exista un punct P1 n planul 1, P2 n planul 2 etc. n poziia iniial cele 4 puncte sunt coincidente dar odat cu evoluia n timp a mecanismului ele nu vor mai fi coincidente. Punctul P2 de pild va descrie un cerc cu originea n A0 din planul 1 i alt cerc cu originea n A pe planul 3. Traiectoria lui P pe planul 4 nu va mai fi o curba simpl ci una de ordin mai nalt (de regul polinomial de ordin 4). Fie punctul B. Vor exista patru puncte corespondente B1, B2, B3 i B4 pe planele respective. Cum B se situeaz pe axa cuplei care conecteaz elementele 3 i 4, atunci B3 i B4 vor fi permanent coincidente. B1 i B2 se nu vor mai fi coincidente odat cu evoluia mecanismului drept care ele sunt puncte coincidente instantaneu.

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    2.4 Vector Loops of a Mechanism /Conturul vectorial al unui mechanism/

    The main difference between freely moving bodies and the moving links in a mechanism is that they have a constrained motion due to the joints in between the links. The links connected by joints form closed polygons that we shall call a loop. The motion analysis of mechanisms is based on expressing the evolution of these loops in time, mathematically.

    In kinematic analysis we shall assume that all the necessary dimensions of each link is given and link length dimensions (i.e. the distance between the joints or the angles) can be determined from the given dimensions using the geometry of the link.

    We have seen that it is sufficient to represent the position of each link (rigid body) by describing the position of any two points on that link. One way of selecting these two points on a link is to use the permanently coincident points. It is obvious that in such a procedure, the origin of a vector will be defined by the previous vector and thus the number of parameters to define the link positions will be decreased.

    Diferena dintre un corp rigid care se mic liber i elementele cinematice ale mecanismelor este c micarea celor din urm este constrns de existena cuplelor ditre ele. Elementele cinematice legate prin cuple cinematice formeaz poligoane nchise numite uneori i bucle. Esena analizei mecanismelor st n exprimarea matematic a evoluiei acestor poligoane nchise n timp. n analiza cinematic se presupune c se cunosc toate dimensiunile elementelor cinematice sau orice puncte ale unor elemente complexe pot fi determinate geometric. S-a artat deja ca este suficient s se reprezinte poziia a numai dou puncte pentru a se deduce poziia fiecrui element cinematic al mecanismului. O cale de a selecta aceste dou puncte este de a fi selectate acele puncte coincidente permanent. ntr-o astfel de procedur originea vectorului de poziie al unui element va fi determinat de vectorul elementului precedent astfel nct numrul de parametri independeni scade.

    a. b.

    Fig.2.7

    Let us consider a four-bar mechanism as shown above (Fig.2.7-a) as a simple example. In this mechanism A0, is a permanently coincident point between links 1 and 2, A is permanently coincident point between links 2 and 3, B between 3 and 4 and B0 between 1 and 4. Let us disconnect joint B. In such a case we will obtain two open kinematic chains A0AB (links 2,3) with two degrees of freedom and A0B0B (links 1,4) with one degree of freedom

    Fie un mecanism patrulater dat ca mai sus (Fig.2.7-a), n care A0 este cupla permanent de rotaie ntre elementele 1 i 2, A cupla de rotaie dintre 2 i 3, B ntre 3 i 4 i B0 ntre 4 i 1. Dac se deconecteaz cupla B (Fig.2.7-b) se vor obine dou lanuri cinematice deschise A0AB (elementele 2,3) cu dou grade de libertate i A0B0B (elementele 1,4) cu un grad de libertate.

  • Position Analisys of Mechanisms /Analiza Poziional a Mecanismelor/ _________________________________________________________________________________________________

    2-9

    (Fig.2.7b).

    To determine the positions of the links we must have a reference frame. One obvious choice is to select the fixed pivots A0, B0 as one of the co-ordinate axes and select A0 or B0 as the origin. Next, in order to define the position of link 2 (its length is known), we must define angle 12, which is related with the degree of freedom of the joint between links 1 and 2. To determine the position of link 3, since the location of the permanently coincident point A between 2 and 3 can be determined when 12 defined, we must now define 13, which is related to the freedom of the joint between links 2 and 3. Similarly 14 must be defined to determine the position of link 4. Hence we need 3 parameters (12, 13 and 14) which are all related to the joint freedoms for the open kinematic ch