BIZOMAT MAKEOS · Mechanics
Engine Lumped Masses, Shaking Force and Inertia Torque
From the masses of the crank, connecting rod and piston of a single-cylinder engine, what are the lumped masses at the crankpin and wrist pin, and what shaking forces and inertia torque does the engine produce at a crank angle and speed?
Dynamic model of the single-cylinder engine by Norton’s method: the conrod split into m3a = m3 lb ÷ (la + lb) at the crankpin and m3b = m3 la ÷ (la + lb) at the wrist pin with la and lb the distances from its centre of gravity, the centre of percussion lp = IG3 ÷ (m3 lb) for comparison, the crank as m2a = m2 rG2 ÷ r at the crankpin, mA = m2a + m3a and mB = m3b + m4; the shaking force Fx ≈ mA rω² cos ωt + mB rω² (cos ωt + (r/l) cos 2ωt) and Fy = mA rω² sin ωt, and the inertia torque Ti ≈ ½ mB r²ω² ((r/2l) sin ωt − sin 2ωt − (3r/2l) sin 3ωt).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Crank radius r (half the stroke) | mm | Crank throw, half the piston stroke. |
| Connecting rod length l (centre to centre) | mm | Connecting rod length between the crankpin and wrist pin centres. |
| Crankshaft speed | rpm | Crankshaft speed, taken as constant. |
| Crank angle ωt from top dead centre | ° | Crank angle measured from top dead centre in the direction of rotation. |
| Crank mass m2 (unbalanced part) | kg | Mass of the crank throw that is not balanced out, with its CG at rG2; 0 for a fully balanced crank. |
| Crank CG distance from the main bearing rG2 | mm | Distance from the main bearing axis to the CG of the crank mass m2. |
| Connecting rod mass m3 | kg | Total mass of the connecting rod. |
| Conrod CG to crankpin la | mm | Distance from the conrod CG to the crankpin centre; lb is the rod length less this. |
| Conrod moment of inertia about its CG (0 = skip) | kg·m² | Mass moment of inertia of the conrod about its CG, if known, to place the centre of percussion. |
| Piston assembly mass m4 | kg | Mass of the piston, rings, wrist pin and clips. |
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 radius r (half the stroke)
- What it is
- Radius of the crank throw, which is half the piston stroke.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 45 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Piston velocity rises in proportion and the r/l ratio changes.
- Get it right
- Measure from the main bearing centre to the crankpin centre, or halve the stroke.
What this means: Connecting rod length l (centre to centre)
- What it is
- Length of the connecting rod between crankpin and wrist pin centres.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 157.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer rod lowers r/l and the second harmonic.
- Get it right
- Measure centre to centre of the two bores in the rod.
What this means: Crankshaft speed
- What it is
- Rotational speed of the crankshaft, taken as steady.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3400 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Accelerations and inertia forces rise with its square.
- Get it right
- Use the speed you want to check, often the highest running speed.
What this means: Crank angle ωt from top dead centre
- What it is
- Angle the crank has turned from top dead centre.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -720° to 720°. 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 where in the cycle the values are taken.
- Get it right
- Measure in the direction of rotation; 0° is the piston at the top of its stroke.
What this means: Crank mass m2 (unbalanced part)
- What it is
- Mass of the crank throw that is not balanced by a counterweight.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds to the rotating mass at the crankpin.
- Get it right
- Enter 0 if the crank is fully counterweighted.
What this means: Crank CG distance from the main bearing rG2
- What it is
- Distance from the main bearing axis to the centre of gravity of the crank mass.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 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 crank’s share of the crankpin mass.
- Get it right
- Take it from CAD or by balancing the part on a knife edge.
What this means: Connecting rod mass m3
- What it is
- Total mass of the connecting rod with its cap and bolts.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.6 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Split between the crankpin and wrist pin masses.
- Get it right
- Weigh the rod assembly.
What this means: Conrod CG to crankpin la
- What it is
- Distance from the conrod centre of gravity to the crankpin centre.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 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 how much rod mass goes to each end.
- Get it right
- Find it by weighing each end with the rod level; usually about a third of the length.
What this means: Conrod moment of inertia about its CG (0 = skip)
- What it is
- Mass moment of inertia of the conrod about its centre of gravity.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0 kg·m² to 100000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0015 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Places the centre of percussion for checking the model.
- Get it right
- Take it from CAD or a pendulum test; enter 0 to skip the check.
What this means: Piston assembly mass m4
- What it is
- Mass of the piston with its rings, wrist pin and clips.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds to the reciprocating mass at the wrist pin.
- Get it right
- Weigh the piston assembly.
Method and formulas
- lb = l − la; m3a = m3 lb ÷ (la + lb), m3b = m3 la ÷ (la + lb) (eq. 13.10); centre of percussion lp = IG3 ÷ (m3 lb) (eq. 13.9e).
- m2a = m2 rG2 ÷ r (eq. 13.11); mA = m2a + m3a, mB = m3b + m4 (eq. 13.12).
- Fx ≈ mA rω² cos ωt + mB rω² (cos ωt + (r/l) cos 2ωt); Fy = mA rω² sin ωt (eq. 13.14, shaking force on the frame, sign as the inertia force).
- Ti ≈ ½ mB r²ω² ((r/2l) sin ωt − sin 2ωt − (3r/2l) sin 3ωt) (eq. 13.15e).
- Example: mA = 0.66984 kg, mB = 0.65238 kg; the inertia torque is -102.42 N·m at this angle.
Limitations
- The lumped model puts the second conrod mass at the crankpin instead of the centre of percussion, an approximation Norton accepts for early design.
- The equations keep harmonics up to the third (torque) and second (force).
- Constant crank speed is assumed.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Estimate the shaking forces of a single-cylinder engine or compressor
- Split a connecting rod into crankpin and wrist pin masses
- See the inertia torque that adds to the gas torque
Who it is for
- Engine and compressor designers
- Machine designers
- Mechanical engineering students
What you get
- Conrod masses at the crankpin and wrist pin
- Rotating mass mA and reciprocating mass mB
- Shaking force components and their primary and secondary amplitudes
- Inertia torque
Interpreting the result
- The rotating mass mA can be balanced by a counterweight; the reciprocating mass mB cannot be fully balanced in a single cylinder.
- Inertia torque averages zero over a revolution but adds to the torque fluctuation.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 614–622. 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
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