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?
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
Definitions
All of the above contain linkages which consist of:
•
•
Links
Joints
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.
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
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
Union College
Mechanical Engineering
MER 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 nodes
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
JOINTS
Joints
allow
DOF
between
links
Union College
Mechanical Engineering
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.
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
JOINTS
Joints
allow
DOF
between
links
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
2.5: Determining DOF or Mobility
Joints Reduce
System DOF
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
2.5: Determining DOF or Mobility
Joints Reduce
System DOF
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
Kinematics Diagrams
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
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.
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
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
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
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)
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
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)
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
Kinematics Diagrams
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
Applying Mobility Equations - 1
M = 3 (L-1) –2(J1) – J2 = 3(8-1) –2(10) – 0 = 1
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT
Applying Mobility Equations - 2
M = 3 (L-1) –2(J1) – J2 = 3(6-1) –2(7) – 1 = 0
Union College
Mechanical Engineering
MER 312: Dynamics and Kinematics
(of Mechanisms) / AT