BIZOMAT MAKEOS · Mechanics
Fourbar Linkage Complete Force Balance (Counterweights on Crank and Rocker)
What counterweight mass-radius products, and at what angles, must I add to the crank and rocker of my fourbar so that its overall centre of mass stays still and the shaking force on the frame vanishes?
Complete force balance of a fourbar linkage by the Berkof-Lowen method Norton presents: the global mass centre is made stationary when m2b2 e^{jφ2} = m3 (b3 (l2/l3) e^{jφ3} − l2) and m4b4 e^{jφ4} = −m3 b3 (l4/l3) e^{jφ3}, giving (m2b2)x = m3 (b3 (l2/l3) cos φ3 − l2), (m2b2)y = m3 b3 (l2/l3) sin φ3, (m4b4)x = −m3 b3 (l4/l3) cos φ3, (m4b4)y = −m3 b3 (l4/l3) sin φ3, with the coupler mass and centre of gravity you enter and the existing unbalance of the crank and rocker subtracted to give the counterweights to add.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Crank length l2 (O2 to A) | mm | Length of the crank between pin centres O2 and A. |
| Coupler length l3 (A to B) | mm | Length of the coupler between pin centres A and B. |
| Rocker length l4 (O4 to B) | mm | Length of the rocker between pin centres O4 and B. |
| Coupler mass m3 | kg | Mass of the coupler including its pins. |
| Coupler CG distance from A, b3 | mm | Distance from joint A to the coupler’s centre of gravity. |
| Coupler CG angle from line AB, φ3 | ° | Angle from the line AB to the line from A to the coupler CG, measured anticlockwise. |
| Existing crank unbalance m2b2 (0 if none) | kg·mm | Mass times CG distance from O2 of the crank as it is, before counterweighting. |
| Angle of the existing crank unbalance from O2A, φ2 | ° | Angle of the crank CG from the line O2A. |
| Existing rocker unbalance m4b4 (0 if none) | kg·mm | Mass times CG distance from O4 of the rocker as it is. |
| Angle of the existing rocker unbalance from O4B, φ4 | ° | Angle of the rocker CG from the line O4B. |
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 length l2 (O2 to A)
- What it is
- Length of the crank between its pivot O2 and 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 127 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales the crank counterweight.
- Get it right
- Measure pin centre to pin centre.
What this means: Coupler length l3 (A to B)
- What it is
- Length of the coupler between joints A and 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 381 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Divides the coupler mass between the crank and rocker.
- Get it right
- Measure pin centre to pin centre.
What this means: Rocker length l4 (O4 to B)
- What it is
- Length of the rocker between its pivot O4 and 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 254 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales the rocker counterweight.
- Get it right
- Measure pin centre to pin centre.
What this means: Coupler mass m3
- What it is
- Mass of the coupler link including its pins and anything it carries.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 1000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Both counterweights rise in proportion.
- Get it right
- Weigh the coupler assembly or take it from CAD.
What this means: Coupler CG distance from A, b3
- What it is
- Distance from joint A to the centre of gravity of 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 190 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes both counterweights and their angles.
- Get it right
- Take it from CAD or by balancing the link.
What this means: Coupler CG angle from line AB, φ3
- What it is
- Angle from the line AB to the line from A to the coupler’s centre of gravity.
- 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 angles of both counterweights.
- Get it right
- Measure anticlockwise at A; 0° if the CG lies on AB.
What this means: Existing crank unbalance m2b2 (0 if none)
- What it is
- Mass of the crank as it is now, multiplied by the distance of its centre of gravity from its pivot.
- Units and format
- Enter a number in kg·mm: kg·mm. The form accepts 0 kg·mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 kg·mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Subtracted from the required total to give the counterweight to add.
- Get it right
- Find it from CAD or by weighing; enter 0 if the crank's CG is on the pivot.
What this means: Angle of the existing crank unbalance from O2A,…
- What it is
- Angle of the crank's centre of gravity measured from the line O2A.
- 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
- Sets the direction of the existing unbalance.
- Get it right
- Measure anticlockwise from the line through the link’s pin centres.
What this means: Existing rocker unbalance m4b4 (0 if none)
- What it is
- Mass of the rocker as it is now, multiplied by the distance of its centre of gravity from its pivot.
- Units and format
- Enter a number in kg·mm: kg·mm. The form accepts 0 kg·mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 kg·mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Subtracted from the required total to give the counterweight to add.
- Get it right
- Find it from CAD or by weighing; enter 0 if the rocker's CG is on the pivot.
What this means: Angle of the existing rocker unbalance from O4B…
- What it is
- Angle of the rocker's centre of gravity measured from the line O4B.
- 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
- Sets the direction of the existing unbalance.
- Get it right
- Measure anticlockwise from the line through the link’s pin centres.
Method and formulas
- Required crank m2b2: x = m3 (b3 (l2/l3) cos φ3 − l2), y = m3 b3 (l2/l3) sin φ3 (eq. 12.8c).
- Required rocker m4b4: x = −m3 b3 (l4/l3) cos φ3, y = −m3 b3 (l4/l3) sin φ3 (eq. 12.8d).
- Angles are measured from each link’s line of centres (O2A, O4B); counterweight to add = required − existing, converted to magnitude and angle.
- With these the global mass centre is stationary and the shaking force is zero; the shaking moment and input torque are not balanced.
- Example: the crank needs 157.59 kg·mm at 156.3° and the rocker 253.33 kg·mm at 210°.
Limitations
- The method balances shaking force only; Norton notes that force balance usually increases the shaking moment and the input torque.
- Pin forces at the crank and rocker pivots remain; they become equal and opposite.
- Masses and CG positions of the links are entered by you from the design.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size counterweights that cancel the shaking force of a fourbar
- Find where on the crank and rocker the counterweights go
- Check the existing unbalance of the crank and rocker
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Total mass-radius product needed on the crank and its angle
- Total mass-radius product needed on the rocker and its angle
- Counterweights to add after the existing unbalance
- x and y components
Interpreting the result
- A counterweight mass-radius product can be any combination of mass and radius that gives the product.
- The required angle on each link is measured from its own line of centres, so the counterweight moves with the link.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 580–583. Topic reference; no source tables, figures or examples are reproduced.
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Open the Fourbar Linkage Complete Force Balance (Counterweights on Crank and Rocker) calculator