How To Understand The Relationship Between A Body And A Joint
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1 Degree of Freedom No. of independent coordinates needed to define position of body How many DOF does a body in three-space (3-D) have?
2 Definitions All of the above contain linkages which consist of: Links Joints
3 Links A link is a nominally rigid body that possess at least 2 nodes. A node is an attachment point to other links via joints.
4 Links -The order of a link indicates the number of joints to which the link is connected (or the number of nodes per link). There are binary (2 nodes), ternary (3 nodes), and quaternary (4 nodes) links. - Can be any shape (not just those shown) -Link order = number of nodes
5 Links - Are assumed to be rigid bodies - Have nodes for attachment - Can be any shape (not just those shown) -Link order = number of nodes
6 JOINTS Joints allow DOF between links A joint is a connection between two or more links at their nodes, which allows motion to occur between the links. A pivot is a joint that allows rotary motion, A slider is a joint that allows linear motion.
7 JOINTS Joints allow DOF between links
8 2.5: Determining DOF or Mobility Joints Reduce System DOF
9 2.5: Determining DOF or Mobility Joints Reduce System DOF
10 Kinematics Diagrams
11 Determining Mobility or DOF Gruebler & Kutzbach Equations Gruebler s Equation is the most commonly used equation for evaluating simple linkages. Kutzbach s equation is modified Gruebler s Equation that takes into consideration full (1 DOF) and half (2 DOF) joints.
12 Determining Mobility or DOF Grubler & Kutzbach Equations Lower pairs (first order joints) or full-joints (counts as J = 1 in Gruebler s Equation) have one degree of freedom (only one motion can occur): - Revolute (R): Also called a pin joint or a pivot, take care to ensure that the axle member is firmly anchored in one link, and bearing clearance is present in the other link, washers make great thrust bearings, snap rings keep it all together A rolling contact joint also counts as a one-degree-of-freedom revolute joint - Prismatic (P): Also called a slider or sliding joint, beware Saint-Venant! - Helical (H): Also called a screw, beware of thread strength, friction and efficiency
13 Determining Mobility or DOF Gruebler & Kutzbach Equations Joints: Multiple Degree-of-Freedom Lower Pair joints with multiple degrees of freedom: Cylindrical (C) 2 DOF A multiple-joint (J = 2) Spherical (S) 3 DOF A multiple-joint not used in planar mechanisms (J = 3) Planar (F) 3 DOF A multiple-joint (J = 3)
14 Determining Mobility or DOF Gruebler & Kutzbach Equations Joints: Higher Pair Multiple Degree-of-Freedom Higher Pair joints with multiple degrees of freedom: Link against a plane and a force is required to keep the joint closed (force closed) A half-joint (J 2 = 1 in Kutzbach s equation) The link may also be pressed against a rotating cam to create oscillating motion Pin-in-slot Geometry keeps the joint closed (form closed) (Slide and pin) A multiple-joint (J = 2 in Gruebler s equation) Second order pin joint, 3 links joined, 2-DOF A multiple-joint (J = 2 in Gruebler s equation)
15 Kinematics Diagrams
16 Applying Mobility Equations - 1 M = 3 (L-1) 2(J1) J2 = 3(8-1) 2(10) 0 = 1
17 Applying Mobility Equations - 2 M = 3 (L-1) 2(J1) J2 = 3(6-1) 2(7) 1 = 0
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