BIZOMAT MAKEOS · Mechanics
Fourbar Linkage Position, Velocity and Acceleration (Analytical)
For a pin-jointed fourbar linkage at a given input angle, speed and acceleration, what are the coupler and output angles, their angular velocities and accelerations, the transmission angle, and the position and velocity of a coupler point?
Analytical kinematics of the pin-jointed fourbar by Norton’s vector loop solution: the output angle from Freudenstein’s constants K1 = d/a, K2 = d/c, K3 = (a² − b² + c² + d²) ÷ 2ac and θ4 = 2 arctan[(−B ∓ √(B² − 4AC)) ÷ 2A] for the open (−) or crossed (+) circuit, the coupler angle from the loop equations, ω3 = (aω2 ÷ b) sin(θ4 − θ2) ÷ sin(θ3 − θ4) and ω4 = (aω2 ÷ c) sin(θ2 − θ3) ÷ sin(θ4 − θ3), α3 = (CD − AF) ÷ (AE − BD) and α4 = (CE − BF) ÷ (AE − BD) with Norton’s acceleration constants, the transmission angle |θ3 − θ4| taken acute, and the position and velocity of a coupler point at distance p and angle δ3 from A.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Input link a (link 2) | mm | Length of the input link from O2 to A. |
| Coupler b (link 3) | mm | Length of the coupler from A to B. |
| Output link c (link 4) | mm | Length of the output link from O4 to B. |
| Ground link d (link 1) | mm | Distance between the fixed pivots O2 and O4, along the X axis. |
| Input angle θ2 (from the ground line) | ° | Angle of the input link measured anticlockwise from the line O2O4. |
| Input angular velocity ω2 (+ anticlockwise) | rad/s | Angular velocity of the input link; anticlockwise is positive. |
| Input angular acceleration α2 | rad/s² | Angular acceleration of the input link; anticlockwise is positive. |
| Circuit (assembly) | choice | Which of the two ways the linkage is assembled at this input angle. |
| Coupler point distance p from A (0 = none) | mm | Distance from joint A to a point of interest on the coupler. |
| Coupler point angle δ3 from line AB | ° | Angle of that point from the line AB, measured anticlockwise at A. |
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: Input link a (link 2)
- What it is
- Length of the input link 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
- Changes the motion and the angles of the whole linkage.
- Get it right
- Measure pin centre to pin centre.
What this means: Coupler b (link 3)
- What it is
- Length of 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 motion and the angles of the whole linkage.
- Get it right
- Measure pin centre to pin centre, not the overall link length.
What this means: Output link c (link 4)
- What it is
- Length of the output link from its fixed pivot O4 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 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 motion and the angles of the whole linkage.
- Get it right
- Measure pin centre to pin centre.
What this means: Ground link d (link 1)
- What it is
- Distance between the two fixed pivots O2 and O4 of the fourbar.
- 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 motion and the angles of the whole linkage.
- Get it right
- Measure centre to centre of the frame pivots.
What this means: Input angle θ2 (from the ground line)
- What it is
- Angle of the input link measured anticlockwise from the ground line O2O4.
- 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 40° 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, not from horizontal, if the frame is tilted.
What this means: Input angular velocity ω2 (+ anticlockwise)
- What it is
- Angular velocity of the input link, 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: Input angular acceleration α2
- What it is
- Angular acceleration of the input link, 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 turning at constant speed.
What this means: Circuit (assembly)
- What it is
- Which of the two ways the links are assembled at this input angle.
- Units and format
- Choose one of 2 options from the list: Open; Crossed.
- Example
- The form opens with “Open” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the solution of the position equation.
- Get it right
- Open: the links next to the shortest do not cross; crossed: they do.
What this means: Coupler point distance p 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
- Gives the path and speed of that point.
- Get it right
- Enter 0 if you do not need a coupler point.
What this means: Coupler point angle δ3 from line AB
- What it is
- Angle of the coupler point from the line AB, measured anticlockwise at A.
- 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
- Places the coupler point on the coupler.
- Get it right
- Measure on the coupler drawing from line AB to line AP.
Method and formulas
- K1 = d/a, K2 = d/c, K3 = (a² − b² + c² + d²) ÷ 2ac; A = cos θ2 − K1 − K2 cos θ2 + K3, B = −2 sin θ2, C = K1 − (K2 + 1) cos θ2 + K3; θ4 = 2 arctan[(−B ∓ √(B² − 4AC)) ÷ 2A], minus for the open circuit (eqs. 4.8, 4.10).
- θ3 from b cos θ3 = −a cos θ2 + c cos θ4 + d and b sin θ3 = −a sin θ2 + c sin θ4 (eqs. 4.6c, 4.6d).
- ω3 = (aω2 ÷ b) sin(θ4 − θ2) ÷ sin(θ3 − θ4); ω4 = (aω2 ÷ c) sin(θ2 − θ3) ÷ sin(θ4 − θ3) (eq. 6.18).
- α3 = (CD − AF) ÷ (AE − BD), α4 = (CE − BF) ÷ (AE − BD) with A = c sin θ4, B = b sin θ3, C = aα2 sin θ2 + aω2² cos θ2 + bω3² cos θ3 − cω4² cos θ4, D = c cos θ4, E = b cos θ3, F = aα2 cos θ2 − aω2² sin θ2 − bω3² sin θ3 + cω4² sin θ4 (eq. 7.12).
- Transmission angle = |θ3 − θ4|, taken as π minus it when above 90° (eq. 4.28); coupler point R_P = R_A + p e^{j(θ3 + δ3)} (eq. 4.27), V_P = V_A + jpω3 e^{j(θ3 + δ3)}.
- Example: θ3 = 20.298°, θ4 = 57.325°, ω4 = 2.7992 rad/s.
Limitations
- A negative discriminant means the links cannot be connected at this input angle (beyond a toggle position or impossible lengths).
- Angles are measured anticlockwise from the X axis along the ground link O2O4; the analysis is for one position — run it at each angle of interest.
- Lengths may be in any one unit; velocities and accelerations of points are reported in mm/s and mm/s² of the length unit entered.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the output angle and speed of a fourbar at one crank position
- Check the angular accelerations for dynamic force analysis
- Find where a coupler point is and how fast it moves
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Coupler and output angles θ3 and θ4
- Angular velocities ω3 and ω4
- Angular accelerations α3 and α4
- Transmission angle
- Velocities of A, B and the coupler point
Interpreting the result
- The open and crossed circuits are the two ways the same links can be assembled; a linkage cannot pass from one to the other without being taken apart.
- Transmission angles below about 40° give poor force transmission.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 156–159, 169–170, 271–274, 309–311. Topic reference; no source tables, figures or examples are reproduced.
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Open the Fourbar Linkage Position, Velocity and Acceleration (Analytical) calculator