ME 321 Kinematics and Dynamics of Machines

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1 .0 INTRODUCTION ME Kinematics and Dynamics of Machines All Text References in these notes are for: Mechanism Design: Analysis and Synthesis, Volume, Fourth Edition, Erdman, Sandor and Kota, Prentice-Hall, 00. Students should review the introductory chapter of the text.. Definitions Kinematics is the study of motion, without regard to forces. This is usually the first step in the analysis or design of a mechanism. Kinetics is the study of forces on systems in motion. Dynamics is the combination of kinematics and kinetics. A Mechanism is a combination of rigid or resilient bodies joined together to provide a specific absolute motion. A machine is a mechanism capable of performing useful work or capable of transmitting significant forces. An engine is a machine which converts energy from one form to another. Links are rigid or flexible members have at least two nodes (points of attachment). Example links are: Binary link: Ternary link: Quaternary link: ( nodes) ( nodes) ( nodes) Figure.: Example link configurations.

2 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 The degrees of freedom (dof) of a system are the number of independent coordinates which are required to uniquely define its position. For example, if each of the above links are restricted to move in a plane (as part of a planar mechanism), then they would each have dof (translation in x and y, and rotation ). A joint is a connection between two or more links (at their nodes), which allows some motion between the links, i.e., permits a particular dof. Conversely, a joint may be considered to restrict motion of links, i.e., reduce the number of degrees of freedom of a system of links. Example joints for planar mechanisms are illustrated in Figure.. Pin (or revolute) joint: - dof This joint permits relative rotation ( ) between two links. Sliding (or prismatic) joint: x dof This joint permits a relative translation (x) between two links. Rolling (with or without sliding) joint: With sliding, this is refered to as a dof joint since it permits two independent motions ( and ). Without sliding, it is referred to as a dof joint, since only one rotation is independent ( or ). Figure.: Example joints for planar mechanisms. Links are combined using joints to form kinematic chains, or just linkages. A linkage with at least one link fixed is a mechanism. We will deal almost exclusively with planar motion. Such motion can be further classified into the following cases: Rectilinear translation: points in the body move in parallel straight lines (ex., piston in Figure.).

3 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 Curvilinear translation: points in the body move along identical curves, and so the link does not rotate with respect to the ground (ex., link connecting two disks in Figure.). Rotation: points in the body rotate about a single point, which is usually fixed to the ground (ex., disks in Figure.). General planar motion: a general combination of rotation and translation (ex., connecting rod joining piston and disk in Figure.). Figure.: Example mechanism showing different types of planar motion. We classify linkages in terms of the number of links. A dyad, Figure., is two links with one joint, usually a pin joint. By itself, it does not usually constitute a useful linkage, but the term is sometimes used to indicate parts of a more complex linkage. In addition, a dyad may be added to an existing linkage to transmit motion from a motor (rotation) to the linkage. Figure.: Example dyad. Similarly, three links (a triad), Figure., can not usually be used alone to form a useful mechanism. It either represents a part of a more complex mechanism, or a structure. Figure.: Example traids. The most useful mechanism has four links the four-bar mechanism. There are two basic configurations. In one case, the four links are joined by pin-joints. In the other

4 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 case, a slider joint replaces one of the pin joints. Schematic and simplified (skeleton) diagrams of each are shown in Figure.. An mechanism inversion is said to occur when the fixed link is allowed to move, and an alternative link is fixed. The relative motion between the links remains unchanged, but the absolute motion, and the function of the mechanism is changed. This is most dramatically seen in the various inversions of the slider-crank mechanism, Figure.7. Figure.: Example four-bar linkage and slider crank.

5 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 Figure.7: Various inversions of the slider-crack mechanism, from top to bottom: conventional engine, rotary engine, quick-return mechanism, and pump.. Degrees of Freedom Text Reference: Degrees of freedom and Gruebler s equation are covered in section.7 of the text, pages -0.

6 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 Each of the above four-bar mechanisms (or slider cranks) has degree of freedom (dof), or a mobility of. That is, input is required to define the position at any point in time. The number of dof s for a system can usually be determined from inspection. To do this, try to visualize how the mechanism might move. If it cannot move, then it has 0 dof. If it can move, restrain one dof and determine whether any part of the mechanism can still move. If not, the mechanism had dof. Keep restraining individual dof s until no motion is possible. The number of dof s which you had to restrain is the number of dof s of the system. A more systematic approach to determine the number of degrees of freedom is provided by Gruebler s equation. For a planar mechanism, the mobility, M, (or degrees of freedom) is given by: M n f f where, M is the mobility, n is the number of links (including the fixed link), f is the number of -dof joints (ex., pins and sliders), and f is the number of -dof joints (ex., rollers with sliding). Note that each link in a planar mechanism has dof: translation in x and y, and rotation about z. One is subtracted from the number of links to account for the fixed link. Each -dof joint restrains dof. Each -dof joint restrains dof. Example : Calculation of mobility. M = (-) () = dof M = (-) () = dof M = (-) () = dof

7 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 Example : Calculation of mobility (continued). roll-sliding contact M = (-) () = dof Note that if links and do not contact, f = 0 and M =. M = (-) () = 0 Note that this linkage has 0 dof. However, if links,, and are exactly the same length (and are at the same angle), then this mechanism can move with dof it is a special case because of the geometry. M = (-) () = - A Note that we had to count the pin at A twice once for joining links and, and once for joining links and (don t count it a third time!). A mobility less than zero indicates a statically indeterminate structure. Further examples from the text: Problem. has 0 sub-problems (Figures P.8 to P.7). Solutions to recommended problems will be provided in files corresponding to each (sub-)section.. Simple Mechanisms Text Reference: Four-bar linkages are introduced in section.. Six-bar mechanisms are discussed in section. 7

8 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 By working with Gruebler s equation, we can examine the possible combinations of links and joints which we can use to form a mechanism with a specified number of degrees of freedom. Generally, we are after dof. Consider a mechanism with links. Since link is assumed fixed, we start out with dof, and reduce this to by adding a single pin or slider joint, Figure.8. This is not a particularly useful mechanism, since we get out pretty much what we put in: a rotation in the first case and a translation in the second case. Note that we could also use two rollslider joints to constrain the second link, as shown at right in Figure.8. This is not particularly useful either. Figure.8: Example -bar mechanisms Now consider a mechanism with links. In this case, we start out with dof, and must add joints to constrain dof to be left with dof. This is possible with two pins and a roll-slider joint, since the latter only constrains dof. This is an example of a camfollower mechanism. Replacing one of the pins with a slider results in the more conventional cam-follower mechanism shown at right in Figure.9. Figure.9: Example -bar mechanisms. With four bars, and four pins, we get the standard -bar mechanism, Figure.0. Replacing one pin with a slider gives the slider-crank mechanism, Figure.0, and its various inversions, Figure.7. 8

9 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 Figure.0: Four-bar and slider-crank mechanisms. Note that, if we assume link in the four-bar linkage is the input, we may either take the output from link, or the coupler (link ). Often, the coupler is larger than is required simply to connect links and, and the output is the path traced by a point on this enlarged coupler, Figure.. Coupler curve (artist s impression) Figure.: Four-bar linkage used to generate path (coupler curve). It is not possible to generate a -dof mechanism with links and just pin or slider joints. Five links gives dof, which is an even number. Several six-bar linkages, however, are possible. Six links gives dof, which can be reduced to dof through the addition of 7 pin or slider joints. This cannot be accomplished, however, with only binary links, since connecting binary links together would require only pin joints. It turns out that we must use ternary links and binary links. If we place the ternary links adjacent to one another, we get a Watt linkage, Figure.. If we separate the two ternary links, we get a Stephenson linkage, Figure.. 9

10 ME Kinematics and Dynamics of Machines S. Lambert Winter 00 Watt I Watt II Figure.: Watt six-bar mechanisms (note that the ground is a ternary member for the Watt II mechanism. Stephenson I Stephenson II Stephenson III Figure.: Stephenson six-bar mechanisms. 0

Analytical and Applied Kinematics

Analytical and Applied Kinematics Vito Moreno moreno@engr.uconn.edu 860-614-2365 (cell) http://www.engr.uconn.edu/~moreno Office EB1, hours Thursdays 10:00 to 5:00 1 This course introduces a unified and