BIZOMAT MAKEOS · Mechanics
Inverted Slider-Crank Position, Velocity and Acceleration (Inversion 3)
For an inverted slider-crank, where the slider block pivots on the output link and the coupler slides through it, what are the output link angle, the slider position along the coupler, their rates and accelerations, and the Coriolis acceleration at the slider?
Analytical kinematics of Norton’s inversion 3 of the fourbar slider-crank: the output angle θ4 from P sin θ4 + Q cos θ4 + R = 0 with P = a sin θ2 sin γ + (a cos θ2 − d) cos γ, Q = −a sin θ2 cos γ + (a cos θ2 − d) sin γ, R = −c sin γ, solved through tan(θ4/2), the coupler angle θ3 = θ4 ± γ, the varying length b from the loop equations, ω3 = ω4 = aω2 cos(θ2 − θ3) ÷ [b + c cos(θ4 − θ3)], the slip velocity ḃ, and α3 = α4 and the slip acceleration b̈ from the acceleration loop equations including the Coriolis term 2ḃω3.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Crank a (link 2) | mm | Length of the crank from O2 to the crank pin A. |
| Output link c (link 4, pivot to slider) | mm | Length of the output link from its pivot O4 to the slider block pivot B. |
| Ground link d (link 1) | mm | Distance between the fixed pivots O2 and O4. |
| Slider angle γ between links 3 and 4 | ° | Fixed angle between the slide line (link 3) and link 4 at the block, 90° when the slide is square to link 4. |
| Crank angle θ2 | ° | Crank angle measured anticlockwise from O2O4. |
| Crank angular velocity ω2 (+ anticlockwise) | rad/s | Crank angular velocity; anticlockwise is positive. |
| Crank angular acceleration α2 | rad/s² | Crank angular acceleration; anticlockwise is positive. |
| Circuit | choice | Which of the two assemblies, given by the sign of γ in θ3 = θ4 ± γ. |
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: Crank a (link 2)
- What it is
- Length of the crank from its fixed pivot O2 to the crank pin 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: Output link c (link 4, pivot to slider)
- What it is
- Length of the output link from its fixed pivot O4 to the pivot of the slider block.
- 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 30 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 swing of the output link.
- Get it right
- Measure from the fixed pivot to the block pin centre.
What this means: Ground link d (link 1)
- What it is
- Distance between the two fixed pivots O2 and O4.
- 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 output swing and the slip.
- Get it right
- Measure centre to centre of the frame pivots.
What this means: Slider angle γ between links 3 and 4
- What it is
- Fixed angle between the sliding line of the coupler and the output link at the block.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.1° to 179.9°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° 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 solution and the coupler angle.
- Get it right
- Use 90° when the slide is square to link 4.
What this means: Crank angle θ2
- What it is
- Angle of the crank measured anticlockwise from the 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 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 the crank, 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 the crank, 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: Circuit
- What it is
- Which of the two ways the linkage is assembled, by the sign of γ.
- Units and format
- Choose one of 2 options from the list: Open (θ3 = θ4 + γ); Crossed (θ3 = θ4 − γ).
- Example
- The form opens with “Open (θ3 = θ4 + γ)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the position solution.
- Get it right
- Open adds γ to the output angle; crossed subtracts it.
Method and formulas
- P, Q, R as above; S = R − Q, T = 2P, U = Q + R; θ4 = 2 arctan[(−T ± √(T² − 4SU)) ÷ 2S] (eqs. 4.21, 4.22), taking the root that gives a positive b.
- θ3 = θ4 + γ (open) or θ4 − γ (crossed) (eq. 4.18); b = (a sin θ2 − c sin θ4) ÷ sin θ3 (eq. 4.20a).
- ω4 = ω3 = aω2 cos(θ2 − θ3) ÷ [b + c cos(θ4 − θ3)]; ḃ from eq. 6.29 (eqs. 6.25, 6.30).
- α4 = α3 and b̈ by solving the two scalar acceleration equations together, which include the Coriolis term 2ḃω3 (eqs. 7.22, 7.25).
- Example: θ4 = 99.02°, b = 61.415 mm and ω4 = -6.0813 rad/s.
Limitations
- If T² − 4SU is negative the links cannot assemble at this crank angle.
- Angles are measured anticlockwise from the X axis along O2O4; the analysis is for one position.
- Velocities and accelerations are in mm/s and mm/s² when lengths are in millimetres.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Analyse linkages driven through a sliding block, such as an oscillating cylinder
- Find the slip velocity and acceleration at a sliding joint
- Find the Coriolis acceleration in a rotating slider
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Output and coupler angles θ4 and θ3
- Slider position b along the coupler
- Angular velocity ω4 and slip velocity ḃ
- Angular acceleration α4, slip acceleration b̈ and the Coriolis acceleration
Interpreting the result
- The Coriolis acceleration 2ḃω3 is present whenever the slider slips along a rotating link.
- The slip velocity is lost motion along the slide; only the transmission component does useful work.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 161–163, 276–278, 315–318. Topic reference; no source tables, figures or examples are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
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Open the Inverted Slider-Crank Position, Velocity and Acceleration (Inversion 3) calculator