BIZOMAT MAKEOS · Mechanics
Offset Slider-Crank Position, Velocity and Acceleration (Analytical)
For an offset (or in-line) slider-crank at a given crank angle, speed and acceleration, where is the slider, how fast is it moving and accelerating, and what are the connecting rod angle, angular velocity and angular acceleration?
Analytical kinematics of the fourbar offset slider-crank (inversion 1) by Norton’s vector loop: θ3 = arcsin[(a sin θ2 − c) ÷ b] for the first circuit or arcsin[−(a sin θ2 − c) ÷ b] + π for the second, slider position d = a cos θ2 − b cos θ3, ω3 = (a cos θ2 ÷ b cos θ3) ω2, slider velocity ḋ = −aω2 sin θ2 + bω3 sin θ3, α3 = (aα2 cos θ2 − aω2² sin θ2 + bω3² sin θ3) ÷ (b cos θ3) and slider acceleration d̈ = −aα2 sin θ2 − aω2² cos θ2 + bα3 sin θ3 + bω3² cos θ3, with the offset c measured square to the slide axis.
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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. |
| Connecting rod b (link 3) | mm | Length of the connecting rod from A to the slider pin B. |
| Offset c of the slide axis | mm | Offset of the slide axis from the crank pivot, measured square to the slide; 0 for an in-line slider-crank. |
| Crank angle θ2 | ° | Crank angle measured anticlockwise from the slide direction. |
| Crank angular velocity ω2 (+ anticlockwise) | rad/s | Angular velocity of the crank; anticlockwise is positive. |
| Crank angular acceleration α2 | rad/s² | Angular acceleration of the crank; anticlockwise is positive. |
| Circuit | choice | Which side of the crank pivot the slider runs on; the two sides are Norton’s two circuits. |
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 pivot to the crank pin.
- 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
- Half the stroke of an in-line slider-crank.
What this means: Connecting rod b (link 3)
- What it is
- Length of the connecting rod between the crank pin and the slider pin.
- 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.
What this means: Offset c of the slide axis
- What it is
- Offset of the slide axis from the crank pivot, measured square to the slide.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 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
- Makes the forward and return strokes unequal.
- Get it right
- Enter 0 for an in-line slider-crank such as most engines.
What this means: Crank angle θ2
- What it is
- Crank angle measured anticlockwise from the direction of the slide axis.
- 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 45° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the slider and rod positions.
- Get it right
- Zero when the crank points along the slide towards the +X side.
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 100 rad/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Slider velocity rises in proportion and acceleration with its square.
- 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 side of the crank pivot the slider runs on.
- Units and format
- Choose one of 2 options from the list: Slider on the +X side of the crank pivot (as Norton’s Figure 4-9); Slider on the −X side of the crank pivot.
- Example
- The form opens with “Slider on the +X side of the crank pivot (as Norton’s Figure 4-9)” 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
- Norton’s Figure 4-9 has the slider on the +X side.
Method and formulas
- θ3 = arcsin[−(a sin θ2 − c) ÷ b] + π with the slider on the +X side, or arcsin[(a sin θ2 − c) ÷ b] with it on the −X side (eqs. 4.16a, 4.17); d = a cos θ2 − b cos θ3 (eq. 4.16b).
- ω3 = (a cos θ2 ÷ b cos θ3) ω2; ḋ = −aω2 sin θ2 + bω3 sin θ3 (eq. 6.22).
- α3 = (aα2 cos θ2 − aω2² sin θ2 + bω3² sin θ3) ÷ (b cos θ3); d̈ = −aα2 sin θ2 − aω2² cos θ2 + bα3 sin θ3 + bω3² cos θ3 (eqs. 7.16d, 7.16e).
- θ3 is measured at the slider pin B, as Norton defines it for this inversion.
- Example: the slider is at 144.9 mm moving at -3.5144 m/s.
Limitations
- The rod cannot reach the slide axis when |a sin θ2 − c| > b.
- Angles are measured anticlockwise from the slide axis through the crank pivot; the analysis is for one position.
- Velocities and accelerations are in mm/s and mm/s² when the lengths are in millimetres.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the piston or slider position, velocity and acceleration at a crank angle
- Check the effect of an offset on a slider-crank
- Get the connecting rod motion for dynamic forces
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Connecting rod angle θ3
- Slider position d
- Slider velocity and acceleration
- Connecting rod angular velocity and acceleration
Interpreting the result
- With no offset the slider positions repeat symmetrically about the crank pivot axis; an offset makes the strokes unequal in time.
- The slider acceleration is greatest near the top dead centre of an in-line slider-crank.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 159–161, 274–276, 311–313. 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.
Useful alongside · BIZOMAT original book
The crank press and its force curve
In Power Presses by the Numbers
What it covers: Why a crank press loses force with height above the bottom of the stroke, the rating point, Press 4's curve from 400 kN at 3 mm to 184 kN at 25 mm, and a pad it could not carry.
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