Linkages and Mechanisms
Mobility, the Grashof condition, four-bar and slider-crank kinematics, toggle mechanisms and transmission angle.
Links, joints and degrees of freedom
A mechanism is a set of rigid links joined by joints so that a given input motion produces a useful output motion: a crank turns into a rocker swing, a rotation into a straight-line stroke. Its degrees of freedom (DOF, mobility) say how many inputs are needed to fix it. For a planar mechanism, Gruebler’s equation is M = 3 (n − 1) − 2 j₁ − j₂, with n the number of links (including the fixed frame), j₁ the number of joints with one DOF (pin, slider) and j₂ the number with two (cam, rolling contact).
| Mechanism | Links n | Pin or slider joints j₁ | Mobility M |
|---|---|---|---|
| Four-bar linkage | 4 | 4 | 1 |
| Slider-crank | 4 | 4 (3 pins and 1 slider) | 1 |
| Five-bar linkage (two inputs) | 5 | 5 | 2 |
| Cam and follower | 3 | 2 pins + 1 slider, 1 cam contact (j₂ = 1) | -1 |
| Rigid triangle (a truss) | 3 | 3 | 0 (a structure) |
M = 1 means one input moves the whole machine; M = 0 is a structure, and M < 0 is over-constrained (it jams unless the dimensions are exact). Over-constraint is sometimes useful (it adds stiffness) but needs tight tolerances and the clearances of Fits and Tolerances.
The four-bar linkage and the Grashof condition
The four-bar has a fixed ground link, an input crank, a floating coupler and an output link. Grashof’s condition says that at least one link can rotate fully if the sum of the shortest and longest links is no more than the sum of the other two: s + l ≤ p + q. Then, with the shortest link as the input crank next to the ground, the linkage is a crank-rocker (continuous rotation in, oscillation out); with the shortest link as the ground it is a double crank (drag link); opposite the ground it is a double rocker. If s + l > p + q no link can turn fully (all double rockers, non-Grashof). The transmission angle μ between the coupler and the output link measures how well the force is passed on: keep it between 40° and 140°; outside that range the force goes into the joint, not the output.
Worked example: a crank-rocker for the guard door
Worked example: crank 40 mm, coupler 90 mm, rocker 80 mm, ground 100 mm
- Grashof: shortest s = 40, longest l = 100, others 80 and 90: s + l = 140 ≤ p + q = 170: Grashof. The shortest link (the 40 mm crank) is adjacent to the ground, so the crank turns fully and the 80 mm link rocks: a crank-rocker.
- Mobility: M = 3 (4 − 1) − 2 × 4 = 1.
- Rocker swing: when the crank lies along the ground link the diagonal is 60 mm (crank pointing at the rocker pivot, θ = 0°) and 140 mm (pointing away, θ = 180°). The rocker angle (measured from the ground line at its pivot) is 78.6° and 37.0° at those two positions, and over the whole turn it ranges from 37.0° to 437.9°: a swing of 401.0°.
- Transmission angle: μ = arccos[(b² + c² − a² − d² + 2 a d cos θ) ÷ (2 b c)]: 40.8° at θ = 0° and 110.7° at 180°. Over the full turn μ ranges from 40.8° to 110.7°: inside the 40° to 140° limits, but only just at the low end; a longer coupler would improve it.
- The motion is not uniform: the rocker goes out slowly and back faster (a quick-return effect) or the reverse, depending on the geometry. For a door a slow, strong closing stroke is wanted.
The slider-crank
The slider-crank turns the rotation of a crank of radius r into the straight-line motion of a slider through a connecting rod of length l (or the reverse): engines, compressors, pumps and punch presses. The slider position is x = r cos θ + √(l² − r² sin² θ), and the stroke is exactly 2 r. The velocity is nearly sinusoidal; the longer the rod (large l ÷ r) the closer to a sine wave.
Worked example: a 50 mm crank and 150 mm rod at 1 000 rpm
- Stroke = 2 r = 100 mm; mean piston speed = 2 × stroke × n ÷ 60 = 3.33 m/s. The peak velocity is 5.52 m/s at θ = 73° (a little before 90°, not exactly at 90°).
- Inertia forces: a reciprocating mass of 0.5 kg gives a primary force m r ω² = 274 N (once per revolution) plus a secondary force of m r ω² (r ÷ l) = 91 N at twice the crank speed. These forces shake the frame: balance them with counterweights or a second cylinder (Flywheels, Balancing and Rotors).
- Engine rule: keep the mean piston speed under about 20 m/s to limit inertia and wear.
Toggle mechanisms and mechanical advantage
A toggle is a linkage that multiplies force enormously as it nears a straight-line (dead-center) position: two links of equal length hinged at the center, pushed from the side. The output force along the line is Fout = Fin ÷ (2 tan α), where α is the angle each link makes with the line; as α → 0, the force tends to infinity (limited by link deflection and the pin friction). Toggle clamps, rock crushers, stamping presses and vices use it. The same idea in a lever gives Fout = Fin × b ÷ a: a lever with arms 100 mm and 400 mm has a mechanical advantage of 4, paid for by the distance moved.
| Angle α from the straight line | Mechanical advantage 1 ÷ (2 tan α) | Output for a 400 N input |
|---|---|---|
| 10° | 2.8 | 1.1 kN |
| 5° | 5.7 | 2.3 kN |
| 2° | 14.3 | 5.7 kN |
| 1° | 28.6 | 11.5 kN |
A toggle clamp at α = 5° gives 5.7 times the hand force. The clamp locks over center (past the straight line) so it stays closed without any force. Real mechanisms stop short of exact dead center, and flexibility, pin friction and tolerances set the final force: check the links in compression and buckling (Columns and Buckling) and the pins in bearing and shear.
Designing a linkage
- Function: what path or angle is wanted (a rocking of 30° for a door, a straight line, a closing force)?
- Type selection: crank-rocker for oscillation, slider-crank for a stroke, double rocker for a limited motion, a Watt or Stephenson six-bar for more complex timing.
- Dimensioning: choose link lengths by the Grashof condition, check the transmission angle over the full range, find positions graphically or with a script, and then check for interference with the frame and between the links.
- Forces: static first (joint reactions from the load), then dynamic (inertia forces, as above); size the pins, links and bearings for the peak and the fatigue cycle.
- Tolerance and clearance: every pin joint has backlash; the sum of clearances in a four-bar can add up to a visible error. Use bushings or needle bearings, preload where needed and specify the fits.
- Simulate and prototype: a motion study in CAD or a cardboard model with paper-clip pivots finds problems quickly (see Prototyping Techniques).
Other mechanisms
| Mechanism | Output | Examples |
|---|---|---|
| Geneva wheel | intermittent rotation (indexing) | film projector, assembly indexing tables |
| Ratchet and pawl | one-way stepping | jacks, wrenches, winches |
| Scotch yoke | pure sinusoidal reciprocation | valve actuators, test machines |
| Pantograph, scissor lift | amplified or copied motion; lifting | copy mills, lifts, trolleys |
| Straight-line linkages (Watt, Chebyshev, Peaucellier) | approximate or exact straight-line motion | windscreen wipers, parallel motion guides |
| Differential and planetary gears | summing motions; high ratios | vehicle axles, gearboxes (see Gears and Power Transmission) |
| Flexure (compliant) mechanisms | motion by bending a thin section: no pins, no backlash | precision stages, instruments, 3D-printed grippers |
Related
- Previous: Cams and Followers. Next: Vibration, Isolation and Noise.
- Industrial Robotics for serial and parallel mechanisms.
- Gears and Power Transmission.
- Machine Design: the whole guide.
FAQ
What is the Grashof condition for?
It tells you whether a four-bar can turn fully, and so what type of motion it gives (crank-rocker, double crank, double rocker). If s + l > p + q nothing turns fully.
What is a dead-center position?
A position in which the output link and the coupler (or the crank and the connecting rod) are in line: the mechanism cannot be driven from the input and has to be passed by inertia, a second cylinder or a flywheel.
Why does my linkage bind?
It is over-constrained, a clearance is too small, or the transmission angle has gone below 40°. Check the mobility, the tolerances and the geometry at the worst position.
Should I use a linkage or a servo axis?
A linkage is cheap, robust and fast but fixed; a servo is flexible but needs a drive and a control. Many machines use both: servo-driven linkages.
References
- Norton, R. L., Design of Machinery, McGraw-Hill; Myszka, D. H., Machines and Mechanisms: Applied Kinematic Analysis, Pearson; Erdman, A. G., Sandor, G. N., Mechanism Design, Prentice Hall.
- Shigley, J. E., Uicker, J. J., Theory of Machines and Mechanisms, Oxford University Press; Hartenberg, R. S., Denavit, J., Kinematic Synthesis of Linkages.
- Grashof, F., Theoretische Maschinenlehre (1883); ISO 3952 (kinematic diagrams: graphical symbols).
Last reviewed: October 2026. Link lengths and masses are illustrative; the kinematic equations are exact.