Kinematic pairs, chains and mechanisms

Why it matters

Every machine on a shop floor, from a shaper's quick-return drive to a robot gripper, is built from links joined by pairs. Before you calculate velocities, forces or stresses you must know whether the linkage moves at all, and how many inputs it needs to move in a definite way. Mobility counting and Grashof's rule are also among the quickest marks in the theory-of-machines part of the paper.

Key ideas

Link (element). A resistant body (rigid, or one that only carries tension/compression like a belt or a fluid column) that has relative motion with other links. A link with two joints is binary, with three ternary, with four quaternary. The fixed link is called the frame.

Kinematic pair. Two links in contact so that their relative motion is constrained. Pairs are classified in three ways:

• By contact. Lower pair: surface contact (revolute/turning pair, prismatic/sliding pair, screw pair, cylindrical, spherical, planar). Higher pair: point or line contact (cam and follower, gear teeth, ball on a race, wheel on rail).
• By relative motion. Turning, sliding, rolling, screw (helical), spherical.
• By closure. Self-closed (form-closed) pairs are held together by their geometry, like a pin in a hole. Force-closed pairs need an external force, like a spring keeping a follower on a cam.

Constraint. In a plane a free body has 3 degrees of freedom (two translations, one rotation). A revolute or prismatic pair removes 2 of them and leaves 1. A planar higher pair with slipping (cam and flat follower) removes 1 and leaves 2. A gear or rolling pair with no slip behaves like a lower pair and removes 2.

Kinematic chain. Links connected by pairs so that the last link joins the first and relative motion is possible and definite. If the assembly cannot move at all it is a structure (F = 0) or a redundant/superstructure (F < 0). An open chain (robot arm) does not close the loop and needs one actuator per joint.

Mechanism and machine. A mechanism is a kinematic chain with one link fixed. A machine is a mechanism (or a set of them) that transmits force and does useful work. A clock's gear train is a mechanism; a lathe is a machine.

Inversion. Fixing a different link of the same chain gives a different mechanism, but the relative motion between any two links does not change. The four inversions of the single slider-crank chain give the reciprocating engine, the Whitworth quick-return and rotary engine, the slotted-lever quick-return and oscillating-cylinder engine, and the hand pump. The double slider-crank chain gives the elliptical trammel, the Scotch yoke and the Oldham coupling.

Grashof's law (four-bar chain). Let s = shortest, l = longest, p and q the other two links. If s + l ≤ p + q at least one link can make a full revolution (Grashof chain):

• shortest link adjacent to the frame (s is the crank) → crank-rocker,
• shortest link fixed → double-crank (drag link),
• link opposite the shortest fixed → double-rocker. If s + l > p + q, no link can rotate fully, whichever link is fixed: every inversion is a triple-rocker.

Limits of the mobility formula. Grübler/Kutzbach counting ignores geometry. Special proportions (parallel equal links, as in a parallelogram linkage with an extra link) can move even when the count says F = 0, and idle freedoms such as the free spin of a roller follower are counted although they do not affect the output.

Formulas

F = 3(n − 1) − 2j − h (Kutzbach criterion, planar mechanisms)

• F = degrees of freedom (mobility), dimensionless
• n = number of links including the frame
• j = number of lower pairs (single-DOF pairs), counted as binary joints
• h = number of higher pairs that allow 2 DOF (rolling with slip)
• Applies to planar mechanisms. When k links meet at one pin, count it as (k − 1) joints.

3n − 2j − 4 = 0 (Grübler criterion for a constrained mechanism with only single-DOF lower pairs and F = 1)

• Consequence: n must be even for such a mechanism, so the simplest are the four-bar (n = 4, j = 4) and six-bar chains (n = 6, j = 7).

j = (3/2)·n − 2 (Grübler's equation for a kinematic chain made only of lower pairs, as in the classical Indian texts)

s + l ≤ p + q (Grashof condition; lengths in m or mm, any consistent unit)

F = 6(n − 1) − 5j₁ − 4j₂ − 3j₃ − 2j₄ − j₅ (spatial Kutzbach), where jᵢ = number of pairs with i degrees of freedom.

Worked examples

Example 1 (standard). A Watt six-bar linkage has 6 links (two ternary, four binary) and 7 revolute pairs. Find its mobility.

1. Formula: F = 3(n − 1) − 2j − h
2. n = 6, j = 7, h = 0.
3. F = 3(6 − 1) − 2(7) − 0 = 15 − 14 = 1.
4. The six-bar needs one input. F = 1

Example 2 (GATE level). (a) A disc cam drives a roller follower that slides in a vertical guide. The roller can roll and slip on the cam. Find F and interpret it. (b) A four-bar chain has links of 30 mm, 70 mm, 50 mm and 60 mm, and the 50 mm link is fixed with the 30 mm link adjacent to it. Classify the mechanism.

(a)

1. Links: frame, cam, follower, roller → n = 4.
2. Lower pairs: cam–frame (revolute), follower–frame (prismatic), roller–follower (revolute) → j = 3.
3. Higher pair: cam–roller → h = 1.
4. F = 3(4 − 1) − 2(3) − 1 = 9 − 6 − 1 = 2.
5. One freedom is the cam input; the other is the free spin of the roller about its pin, an idle (redundant) freedom that does not change the follower motion. The useful mobility is F = 2, of which 1 is effective.

(b)

1. s = 30 mm, l = 70 mm, p + q = 50 + 60 = 110 mm.
2. s + l = 30 + 70 = 100 mm ≤ 110 mm → Grashof chain.
3. The shortest link (30 mm) is adjacent to the fixed link, so it rotates fully while the opposite link oscillates.
4. Crank-rocker mechanism.

Common mistakes

• Forgetting to count the frame as a link in n.
• Counting a pin shared by three links as one joint instead of two.
• Counting a pure-rolling (no slip) contact as a higher pair with h = 1; it removes two freedoms and is counted in j.
• Applying Grashof's rule to choose the inversion without checking the condition first; if s + l > p + q, every inversion is a triple-rocker.
• Reading F < 0 as "jams sometimes"; by count it is a statically indeterminate structure (unless special geometry allows motion).
• Calling a belt or chain "not a link"; links need only resist the load they carry.

For GATE PI

Expect one-mark questions on mobility of a drawn linkage (count links and joints carefully, especially multiple joints and sliders), the type of pair (lower/higher, self/force-closed), inversions of the slider-crank and double slider-crank chains, and Grashof classification of a four-bar chain with given link lengths. Practise sketching each inversion and drawing linkages where three links meet at one pin.

Quick check

1. A planar linkage has 8 links and 10 single-DOF joints. What is F?
2. Is a cam with a spring-loaded follower self-closed or force-closed?
3. Links 20, 45, 35 and 40 mm form a four-bar chain. Can any link rotate fully?
4. Which inversion of the double slider-crank chain is used to connect two parallel, slightly offset shafts?

Answers: 1. F = 21 − 20 = 1; 2. force-closed; 3. yes, 20 + 45 = 65 ≤ 75, so it is a Grashof chain; 4. the Oldham coupling.