Mechanisms, Kinematic Pairs and Degrees of Freedom

A mechanism is a set of rigid links joined so that they move in a controlled way. Before finding velocities or forces, GATE often asks how many inputs a mechanism needs: its degrees of freedom.

Links, pairs and chains

• A link is a rigid body that connects to others. The fixed link is the frame.
• A kinematic pair is a joint between two links.
• Lower pairs have surface contact: revolute (turning), prismatic (sliding), screw, cylindrical, spherical, planar.
• Higher pairs have point or line contact: cam and follower, gear teeth, a wheel rolling on a rail.
• A kinematic chain is a set of links joined so that relative motion is possible. Fixing one link makes it a mechanism.

Degrees of freedom (Grübler / Kutzbach)

For a planar mechanism with n links (including the frame), j₁ lower pairs (each removes 2 freedoms) and j₂ higher pairs (each removes 1):

F = 3(n − 1) − 2j₁ − j₂

• F = 1: a constrained mechanism (one input drives it).
• F = 0: a structure. F < 0: a statically indeterminate structure.
• Watch for redundant freedoms, like a roller follower spinning about its own pin, which add 1 without affecting the output.

Grashof's law (four-bar chain)

With link lengths s (shortest), l (longest), p and q:

• If s + l ≤ p + q, at least one link can rotate fully (a Grashof chain).
• Shortest link fixed → double crank (drag link).
• Link next to the shortest fixed → crank–rocker.
• Link opposite the shortest fixed → double rocker.
• If s + l > p + q, no link can rotate fully: all inversions are triple rockers.

Inversions

Fixing different links of a chain gives its inversions:

• Four-bar chain: beam engine (crank–rocker), coupled locomotive wheels (double crank), Watt's indicator (double rocker).
• Single slider-crank chain: reciprocating engine, Whitworth quick return, crank-and-slotted-lever quick return, rotary engine, hand pump.
• Double slider-crank chain: elliptical trammel, Scotch yoke, Oldham coupling.

Quick-return ratio = time of cutting stroke / time of return stroke, which is greater than 1 for shapers.

Worked example

A planar mechanism has 6 links and 7 revolute joints. F = 3(6 − 1) − 2 × 7 = 15 − 14 = 1: it needs one input. (Stephenson and Watt six-bar chains are of this type.)

Practice