BIZOMAT MAKEOS · Mechanics
Geared Fivebar Linkage Position, Velocity and Acceleration
For a geared fivebar linkage with my link lengths, gear ratio and phase angle, where are links 3, 4 and 5 at a given crank angle, what are their angular velocities and accelerations, and where is a coupler point and how fast is it moving?
Analytical kinematics of Norton’s geared fivebar: links 2 and 5 are pivoted on the ground and geared together so that θ5 = λθ2 + φ, which reduces the fivebar to two unknowns. θ4 comes from the half-angle solution θ4 = 2 arctan[(−E ± √(E² − 4DF)) ÷ 2D] with Norton’s A, B, C, D, E and F, θ3 from the loop equations, ω3 and ω4 from the two velocity equations with ω5 = λω2, and α3 and α4 from Norton’s acceleration equations with α5 = λα2. It also gives the transmission angle between links 3 and 4 and the position and speed of a point on the coupler.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Link 2 (geared crank) a | mm | Length of link 2 from its fixed pivot O2 to the joint A with the coupler. |
| Link 3 (coupler) b | mm | Length of link 3 from joint A to joint B. |
| Link 4 c | mm | Length of link 4 from joint C on link 5 to joint B. |
| Link 5 (geared link) d | mm | Length of link 5 from its fixed pivot O5 to joint C. |
| Ground link O2O5 f | mm | Distance between the fixed pivots O2 and O5, along the X axis. |
| Gear ratio λ (θ5 = λθ2 + φ; negative for external gears) | Ratio between the rotations of the two geared links; external gears turn opposite ways, so λ is then negative. | |
| Phase angle φ | ° | Angle of link 5 when link 2 is at 0°, set by how the gears are meshed. |
| Crank angle θ2 | ° | Angle of link 2 measured anticlockwise from the line O2O5. |
| Crank angular velocity ω2 (+ anticlockwise) | rad/s | Angular velocity of link 2; anticlockwise is positive. |
| Crank angular acceleration α2 | rad/s² | Angular acceleration of link 2; anticlockwise is positive. |
| Assembly (sign of the root in eq. 4.24h) | choice | Which of the two assemblies of links 3 and 4, by the sign taken in the θ4 solution. |
| Coupler point distance from A (0 = none) | mm | Distance from joint A to a point of interest on the coupler. |
| Coupler point angle from line AB | ° | Angle from line AB to the coupler point, anticlockwise. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Link 2 (geared crank) a
- What it is
- Length of link 2, the geared crank, from its fixed pivot O2 to joint A.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the size of the input motion.
- Get it right
- Measure pin centre to pin centre.
What this means: Link 3 (coupler) b
- What it is
- Length of link 3, the coupler, from joint A to joint B.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 120 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the position of B and the coupler curve.
- Get it right
- Measure pin centre to pin centre.
What this means: Link 4 c
- What it is
- Length of link 4 from joint C on link 5 to joint B.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the position of B and the coupler curve.
- Get it right
- Measure pin centre to pin centre.
What this means: Link 5 (geared link) d
- What it is
- Length of link 5, the second geared link, from its fixed pivot O5 to joint C.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the size of the second input motion.
- Get it right
- Measure pin centre to pin centre.
What this means: Ground link O2O5 f
- What it is
- Distance between the two fixed pivots O2 and O5 of the geared links.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the reach the coupler and link 4 must span.
- Get it right
- For gears in mesh this is the sum of their pitch radii.
What this means: Gear ratio λ (θ5 = λθ2 + φ; negative for extern…
- What it is
- Gear ratio between the two geared links, with sign for direction of turning.
- Units and format
- Enter a plain number with no unit. The form accepts -10 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes how fast link 5 turns against link 2.
- Get it right
- External gears turn opposite ways: λ = −(gear 2 diameter ÷ gear 5 diameter).
What this means: Phase angle φ
- What it is
- Angle of link 5 when link 2 is at 0°, set by how the gears are meshed.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the coupler curve without changing any link.
- Get it right
- Re-meshing the gears by one tooth changes it by one tooth angle.
What this means: Crank angle θ2
- What it is
- Angle of link 2 measured anticlockwise from the line O2O5.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the position of every link.
- Get it right
- Measure from the line joining the fixed pivots.
What this means: Crank angular velocity ω2 (+ anticlockwise)
- What it is
- Angular velocity of link 2, positive anticlockwise.
- Units and format
- Enter a number in rad/s: rad/s. The form accepts -100000 rad/s to 100000 rad/s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 rad/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All velocities rise in proportion.
- Get it right
- rad/s = rpm × 2π ÷ 60.
What this means: Crank angular acceleration α2
- What it is
- Angular acceleration of link 2, positive anticlockwise.
- Units and format
- Enter a number in rad/s²: rad/s². The form accepts -10000000 rad/s² to 10000000 rad/s². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 rad/s² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds tangential acceleration terms.
- Get it right
- Enter 0 for a crank at constant speed.
What this means: Assembly (sign of the root in eq. 4.24h)
- What it is
- Which of the two ways links 3 and 4 can be assembled at this crank angle.
- Units and format
- Choose one of 2 options from the list: Minus root (−); Plus root (+).
- Example
- The form opens with “Minus root (−)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the position solution.
- Get it right
- Try both and keep the one that matches your drawing.
What this means: Coupler point distance from A (0 = none)
- What it is
- Distance from joint A to a point of interest on the coupler.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds the coupler point position and speed.
- Get it right
- Enter 0 if no coupler point is needed.
What this means: Coupler point angle from line AB
- What it is
- Angle from the line AB to the line from A to the coupler point.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the coupler point around A.
- Get it right
- Measure anticlockwise at A; 0° puts the point on AB.
Method and formulas
- θ5 = λθ2 + φ, ω5 = λω2, α5 = λα2 (eqs. 4.23c, 7.28c).
- A = 2c[d cos θ5 − a cos θ2 + f], B = 2c[d sin θ5 − a sin θ2], C = a² − b² + c² + d² + f² − 2af cos θ2 − 2d(a cos θ2 − f) cos θ5 − 2ad sin θ2 sin θ5; D = C − A, E = 2B, F = A + C; θ4 = 2 arctan[(−E ± √(E² − 4DF)) ÷ 2D] (eq. 4.24h).
- θ3 from b cos θ3 = −a cos θ2 + c cos θ4 + d cos θ5 + f and b sin θ3 = −a sin θ2 + c sin θ4 + d sin θ5 (eqs. 4.24e, f).
- ω3 and ω4 by solving the two velocity equations together (eqs. 6.32, 6.33); α3 and α4 from eqs. 7.29a and 7.29b.
- Example: θ3 = 28.149°, θ4 = 75.045° and ω4 = -4.4768 rad/s.
Limitations
- If E² − 4DF is negative the links cannot assemble at this crank angle.
- The gears are taken as exact: backlash and tooth errors are ignored.
- When λ is not a whole number the linkage takes more than one crank turn to repeat; the assembly check covers crank angles 0–360° only.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Analyse a geared fivebar at any crank angle
- Try gear ratios and phase angles for a coupler path
- Check the transmission angle of a geared fivebar
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Angles θ3, θ4 and θ5
- Angular velocities ω3, ω4 and ω5
- Angular accelerations α3, α4 and α5
- Transmission angle and the coupler point position and speed
Interpreting the result
- Changing the phase angle by re-meshing the gears changes the coupler curve without changing any link.
- Keep the transmission angle between links 3 and 4 well away from 0° and 180°.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 164–166, 277–278, 319–320. Topic reference; no source tables, figures or examples are reproduced.
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Open the Geared Fivebar Linkage Position, Velocity and Acceleration calculator