Joints Reduce System DOF

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Joints (kinematic pairs) – Connection between 2 or more links that allow motion or potential motion between the connected links.

Classification:

1 – contact between the elements, line, point or surface

2 – # of dof allowable

3 – force or form closed

4 – order of the joint, i.e. the # of the links joined

1 - Types of Contact

Lower pair: a joint, e.g. pin in a hole (Revolute, prismatic, cylindrical, screw, planar, spherical (ball)

Higher pair: a joint with line, curve or surface contact, e.g. cam or gear.

2 - DOF

One-freedom joint: rotation (revolute) or translation (prismatic slider) also referred to as full joints (full = 1 DOF) – lower pairs.

Two-freedom joint or (half joints): ½ (2 being the DOF) or roll-slide joint – higher pair.

Note: full joint (lower pair) and half joint (higher pair)

Joints can be locked, e.g. applying brakes on icy conditions. This can change the # of dof.

3 - Closure

Force-Closed: a joint that requires some external force to keep it together or closed.

Form-Closed : a joint that is closed by its geometry.

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4 – Joint Order

Joint Order: # of links joined minus one – remember for one joint you need at least 2 links. You may add more links per joint

Determining Mobility or DOF : Gruebler & Kutzbach Equations

DOF is provided or calculated in order to create or expect predictable output. It is also the # of independent coordinates required to define a position.

M = 3L - 2J - 3G

Where: M = degree of freedom or mobility

L = number of links

J = number of joints

G = number of grounded links

Thus Gruebler's equation becomes: M = 3(L-1) - 2J

Kutzbach's modification of Gruebler's equation in this form: M = 3(L-1) - 2J₁ -J₂

_{Where: M = degree of freedom or mobility}

_{L = number of links}

J_{1 = number of 1 DOF (full) joints}

J_{2 = number of 2 DOF (half) joints}

Copyrights

Created for "Design of Machinery" by R.L. Norton and " The Multimedia Handbook of Mechanical Devices" by S.Wang Copyright © The McGraw-Hill Education. Permission required for reproduction or display.