Degree Of Freedom
Degree of FreedomNo. of independent coordinates needed to define position of bodyHow many DOF does a body in three-space (3-D) have?Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
DefinitionsAll of the above contain linkages which consist of: LinksJointsUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
LinksA link is a nominally rigid body thatpossess at least 2 nodes.A node is an attachment point to otherlinks via joints.Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Links- The order of a link indicates the number ofjoints to which the link is connected (or thenumber 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 nodesUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Links- Are assumed to be rigid bodies- Have nodes for attachment- Can be any shape (not just those shown)-Link order number of nodesUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
JOINTSJointsallowDOFbetweenlinksUnion CollegeMechanical EngineeringA joint is a connection betweentwo or more links at their nodes,which allowsmotion to occur between thelinks.A pivot is a joint that allowsrotary motion,A slider is a joint that allowslinear motion.MER 312: Dynamics and Kinematics(of Mechanisms) / AT
JOINTSJointsallowDOFbetweenlinksUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
2.5: Determining DOF or MobilityJoints ReduceSystem DOFUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
2.5: Determining DOF or MobilityJoints ReduceSystem DOFUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Kinematics DiagramsUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Determining Mobility or DOFGruebler & Kutzbach EquationsGruebler’sEquation is themost commonlyused equationfor evaluatingsimplelinkages.Kutzbach’s equationis modifiedGruebler’s Equationthat takes intoconsideration full (1DOF) and half (2DOF) joints.Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Determining Mobility or DOFGrubler & Kutzbach EquationsLower pairs (first order joints) or full-joints (counts as J 1 inGruebler’s Equation) have one degree of freedom (only one motion canoccur):–- Revolute (R): Also called a pin joint or a pivot, take care to ensure that theaxle member is firmly anchored in one link, and bearing clearance is presentin the other link , washers make great thrust bearings, snap rings keep it alltogether A rolling contact joint also counts as a one-degree-of-freedom revolutejoint- Prismatic (P): Also called a slider or sliding joint, beware Saint-Venant!- Helical (H): Also called a screw, beware of thread strength, friction andefficiencyUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Determining Mobility or DOFGruebler & Kutzbach EquationsJoints: Multiple Degree-of-Freedom Lower Pair joints with multiple degrees of freedom:– Cylindrical (C) 2 DOF A multiple-joint (J 2)– Spherical (S) 3 DOFA multiple-joint not used in planar mechanisms (J 3)– Planar (F) 3 DOF A multiple-joint (J 3)Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Determining Mobility or DOFGruebler & Kutzbach EquationsJoints: 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 createoscillating 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)Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Kinematics DiagramsUnion CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Applying Mobility Equations - 1M 3 (L-1) –2(J1) – J2 3(8-1) –2(10) – 0 1Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Applying Mobility Equations - 2M 3 (L-1) –2(J1) – J2 3(6-1) –2(7) – 1 0Union CollegeMechanical EngineeringMER 312: Dynamics and Kinematics(of Mechanisms) / AT
Union College Mechanical Engineering MER 312: Dynamics and Kinematics (of Mechanisms) / AT 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.
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