Links, pairs, kinematic chains and degrees of freedom

Links, kinematic pairs and chains, and how to count planar degrees of freedom with the Kutzbach/Gruebler equation, including triple joints and redundant roller spin.

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Why it matters

Before you analyse the velocity of a piston, the force on a wiper-linkage pin or the motion of a suspension arm, you have to know whether the assembly can move at all and how many inputs it needs. Counting links, pairs and degrees of freedom answers that in a minute, and it is the first check a designer makes on any new linkage, from a steering mechanism to a robot arm.

Key ideas

Link (kinematic link or element). A resistant body (or an assembly of rigidly joined parts) that has relative motion with respect to other parts of the machine and transmits motion or force. A connecting rod with its cap, bolts and bushes is one link because the parts move as one. Links need not be rigid in the strict sense: a belt or chain is resistant in tension, and a fluid column in compression, so they also count as links. By the number of pairing elements (joints) a link carries it is called binary (2), ternary (3) or quaternary (4).

Kinematic pair. Two links in contact such that their relative motion is constrained in a definite way.

  • By type of contact: a lower pair has surface contact (pin in a hole, slider in a guide); a higher pair has point or line contact (cam and follower, gear teeth, ball on a race, a wheel rolling with slip).
  • By relative motion: turning (revolute, R), sliding (prismatic, P), screw, cylindrical, spherical, planar. Only R and P act in a plane mechanism.
  • By constraint: closed pair (held together mechanically, like a pin joint) or unclosed/force-closed pair (held by gravity or a spring, like a cam follower).
  • By the motion allowed: completely constrained (one definite motion, like a square bar in a square hole), incompletely constrained (more than one possible motion, like a round shaft in a round hole with no collars) and successfully constrained (made definite by external means, like a valve held on its cam by a spring).

Kinematic chain. Links joined by pairs so that the last link joins the first, forming a closed loop in which the relative motion of every link is definite. The simplest constrained chain has four links and four lower pairs.

Mechanism, machine and structure. Fix one link of a kinematic chain and you have a mechanism. A mechanism that transmits useful work (power) is a machine. An assembly whose links have no relative motion is a structure (a truss, a frame).

Inversion. Fixing different links of the same chain gives different mechanisms (inversions). The relative motion between any two links does not change with inversion; only the absolute motions do. This is the subject of the next topic.

Degrees of freedom (mobility). The number of independent input motions (coordinates) needed to define the position of every link. A free rigid body in a plane has 3 DOF (x, y and rotation); in space it has 6.

  • Fixing one link removes its 3 DOF, so n links start with 3(n − 1).
  • Every lower pair in a plane (R or P) removes 2 DOF and leaves 1.
  • Every higher pair removes 1 DOF and leaves 2 (rolling plus sliding at the contact). Pure rolling without slip removes 2 and is counted as a lower pair.
  • F = 1 means one input drives the whole mechanism (constrained motion). F = 0 means a structure; F < 0 means a statically indeterminate (redundant) structure; F = 2 or more needs that many independent inputs (a five-bar linkage, a differential).

Counting rules that decide most answers.

  • When k links meet at one pin, count (k − 1) joints there, not one.
  • The frame counts as one link even if it is drawn as several fixed points.
  • A roller on a follower adds a link and a revolute pair but its spin does not change the follower motion. This is a redundant (idle) DOF; subtract it to get the effective mobility.
  • Gruebler's equation ignores geometry. Special proportions (parallel equal links, a double-parallelogram) can let a chain with calculated F = 0 move; these are exceptions you must recognise, not violations of the rule.

Formulas

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

  • F: degrees of freedom (dimensionless); n: number of links including the frame; j: number of lower pairs (single-DOF joints: R, P, or rolling without slip); h: number of higher pairs (rolling with slip, cam contact). Applies to planar mechanisms in which every link moves in parallel planes.

3n − 2j − 4 = 0 (Gruebler's criterion, h = 0, F = 1)

  • The condition for a planar chain of n links and j single-DOF lower pairs to have constrained motion. For n = 4 it gives j = 4; for n = 6, j = 7.

F = 6(n − 1) − 5j₁ − 4j₂ − 3j₃ − 2j₄ − j₅ (spatial Kutzbach)

  • jᵢ: number of joints that allow i degrees of freedom (R, P, screw: i = 1; cylindrical: i = 2; spherical: i = 3). Applies to spatial linkages; over-constrained spatial chains with special geometry can still move.

Worked examples

Example 1 (standard). A six-link Watt chain has two ternary and four binary links joined by seven revolute pairs. Find the mobility.

  1. Data: n = 6, j = 7, h = 0.
  2. Formula: F = 3(n − 1) − 2j − h.
  3. F = 3(6 − 1) − 2(7) − 0 = 15 − 14 = 1.
  4. Check with Gruebler: 3n − 2j − 4 = 18 − 14 − 4 = 0, so motion is constrained.

Answer: F = 1, one input drives the mechanism.

Example 2 (standard). A slider-crank mechanism: frame, crank, connecting rod and piston.

  1. n = 4; joints: frame–crank (R), crank–rod (R), rod–piston (R), piston–frame (P), so j = 4; h = 0.
  2. F = 3(4 − 1) − 2(4) − 0 = 9 − 8 = 1.

Example 3 (GATE level). A radial cam drives a roller follower. Links: frame, cam, roller, follower stem. The cam is pinned to the frame, the roller is pinned to the stem, the stem slides in a frame guide, and the roller touches the cam with rolling and slipping possible. Find the calculated and effective DOF.

  1. n = 4.
  2. Lower pairs: cam–frame (R), roller–stem (R), stem–frame (P), so j = 3.
  3. Higher pair: roller–cam, so h = 1.
  4. F = 3(4 − 1) − 2(3) − 1 = 9 − 6 − 1 = 2.
  5. One of these is the free spin of the roller about its own pin, which does not affect the follower position: a redundant DOF.
  6. Effective mobility = 2 − 1 = 1.

If the roller is assumed to roll without slipping, the roller–cam contact counts as a lower pair: j = 4, h = 0, F = 9 − 8 = 1 directly.

Example 4 (GATE level, coincident joints). A planar linkage has 8 links. Eight pins each join two links, and at one more pin three links are joined together. There are no sliders or higher pairs. Find F.

  1. The triple pin counts as 3 − 1 = 2 joints, so j = 8 + 2 = 10.
  2. F = 3(8 − 1) − 2(10) = 21 − 20 = 1, a constrained mechanism.
  3. Had the triple pin been counted once, j = 9 and F = 21 − 18 = 3: a wrong answer that makes the linkage look like it needs three inputs.

Common mistakes

  • Counting a pin that joins three links as one joint. It is two; four links at one pin is three.
  • Forgetting that the fixed link (frame) is one of the n links.
  • Treating a higher pair as removing 2 DOF. It removes 1; rolling without slip is the exception that removes 2.
  • Reporting the roller spin as a true DOF: subtract redundant DOF before calling a cam mechanism two-input.
  • Using F = 6(n − 1) − 5j for planar mechanisms. That is the spatial formula and gives nonsense for plane linkages.
  • Assuming F = 0 always means no motion. Special geometry (parallel equal links) can allow motion; Gruebler's count gives the general case.

For GATE ME

Expect direct mobility counts on a sketched linkage, often with a triple joint, a slider in a slotted link, a gear pair or a cam with a roller, where the trap is miscounting joints or redundant DOF. Also expect conceptual items on pair classification (lower or higher, closed or force-closed, type of constraint) and on which statements about chains, mechanisms and structures are true. Practise counting links and joints on 10 to 15 different sketches until it is automatic.

Quick check

  1. Three links are pinned at one point. How many joints does that pin contribute?
  2. What is the mobility of a four-bar chain with four revolute pairs?
  3. How many DOF does a higher pair remove in a planar mechanism?
  4. A chain gives F = 0 by Gruebler's equation. What is it, in general?
  5. Why is the spin of a roller follower not counted in the effective DOF?

Answers: 1. Two. 2. 1. 3. One. 4. A structure (no relative motion), unless special geometry allows motion. 5. It is a redundant DOF: the roller can spin without changing the position of any other link.

Interview questions

All Theory of Machines and Vibrations interview questions

Try answering each one aloud before you open it.

  1. 1.What is a kinematic link in the context of mechanical systems?Concept

    A kinematic link is a resistant body, or a group of parts joined rigidly so that they move as one, that has relative motion with respect to other parts of a machine and transmits motion or force. It need not be perfectly rigid: a belt or chain is resistant in tension and so counts as a link. A connecting rod with its cap, bolts and bushes is a single link. Links are named binary, ternary or quaternary by the number of joints they carry.

  2. 2.Explain the difference between a kinematic pair and a kinematic chain.Concept

    A kinematic pair is two links in contact whose relative motion is constrained in a definite way, for example a pin in a hole (turning pair) or a slider in a guide (sliding pair). A kinematic chain is a set of links joined by pairs to form a closed loop in which the relative motion of every link is definite. The simplest constrained chain is the four-bar chain with four lower pairs. Fixing one link of a chain turns it into a mechanism.

  3. 3.What are the degrees of freedom in a mechanical system, and why are they important?Concept

    Degrees of freedom (mobility) is the number of independent input coordinates needed to fix the position of every link of a mechanism. For a planar mechanism it is found from the Kutzbach equation F = 3(n − 1) − 2j − h. F = 1 means one input, such as a motor on the crank, drives the whole mechanism; F = 0 means a structure; F = 2 needs two inputs, as in a five-bar linkage or a differential. It is the first check that a proposed linkage can actually be driven the way the designer intends.

  4. 4.Why is a four-bar linkage commonly used in mechanical systems?Application

    The four-bar chain is the simplest closed chain with constrained motion (n = 4, j = 4, F = 1), so one input gives a definite output. By choosing link lengths (Grashof's condition) it gives crank-rocker, double-crank or double-rocker action, and with a slider it becomes the slider-crank and its inversions. It is cheap to make with pin joints, carries load well and appears in windscreen wipers, steering linkages, suspension arms and door closers.

  5. 5.What happens if a kinematic chain has more degrees of freedom than necessary?Application

    If a linkage has more degrees of freedom than inputs, its motion is not determined: for the same input position the other links can take many positions, so the output is unpredictable. A five-bar linkage with one motor, for example, needs a second input or an extra link to constrain it. The fix is to add a link or a joint constraint until F equals the number of actuators, or to add an actuator.

  6. 6.Calculate the degrees of freedom for a planar mechanism with 5 links and 6 pairs.Numerical

    Using the Kutzbach equation with only lower pairs, F = 3(n − 1) − 2j = 3(5 − 1) − 2(6) = 12 − 12 = 0. Zero degrees of freedom means the assembly is a structure: no link can move relative to another, so it is not a mechanism. This assumes all six pairs are single-DOF lower pairs and that no pin joins more than two links.

  7. 7.What is the significance of a higher pair in a kinematic chain?Concept

    A higher pair has point or line contact instead of surface contact, as between a cam and follower or two gear teeth. In a planar mechanism it removes only one degree of freedom and leaves two (rolling and sliding at the contact), whereas a lower pair removes two, so it appears as the h term in F = 3(n − 1) − 2j − h. Higher pairs let a designer get complex output motions from simple input, but the small contact area gives high contact stress and wear.

  8. 8.Determine the degrees of freedom for a spatial mechanism with 8 links and 10 pairs.Numerical

    For a spatial mechanism the Kutzbach equation is F = 6(n − 1) − 5j₁ − 4j₂ − 3j₃ − 2j₄ − j₅, where jᵢ counts joints allowing i degrees of freedom. If all ten pairs are single-DOF joints (revolute or prismatic), F = 6(8 − 1) − 5(10) = 42 − 50 = −8. A negative value means the chain is over-constrained: in general it is a redundant structure, though special geometries (such as the Bennett linkage) can still move.

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