KEVOS® MAKEOS · Power Transmission

Countershaft Bearing Check (Gear Forces, Bearing Loads in Two Planes, L10 Life, Shaft Stress)

For a shaft carrying a driven gear and a driving gear between two bearings, what load does each bearing carry, how long will it last, and is the shaft stressed within limits?

Check of a countershaft with two spur gears: the tangential and radial tooth forces on each gear from the torque and pitch diameter (as the gear tooth forces tool), resolved into two planes from the angular position of each mesh, the bearing reactions in both planes and their resultants, the L10 rating life of each bearing (as the bearing L10 life tool) and its static safety factor (as the bearing static ratio tool), and the largest bending moment with the torque giving the combined stress in the shaft, against your life, safety factor and stress limits.

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How this tool works

Inputs, units and where to find them

InputUnitTypical source
Torque carried by the shaftN·mTorque the shaft transmits from the driven gear to the driving gear, including any service factor you use.
Shaft speedrpmRotational speed of the countershaft.
Bearing spacing (A to B)mmDistance between the centres of the two bearings.
Pressure angle°Pressure angle of the gear teeth (20° is the usual standard).
Driven gear (power in)
Pitch diametermmPitch diameter of the gear that receives power from its mating gear.
Position from bearing AmmDistance of that gear's centre from bearing A along the shaft (negative or beyond the spacing for an overhung gear).
Mesh position angle°Angle around the shaft, from a reference direction and measured in the direction of rotation, at which this gear meshes with its mating gear.
Driving gear (power out)
Pitch diametermmPitch diameter of the gear that delivers power to the next gear.
Position from bearing AmmDistance of that gear's centre from bearing A along the shaft.
Mesh position angle°Angle around the shaft at which this gear meshes with its mating gear, measured the same way.
Bearings and shaft
Bearing typechoiceBall bearings use a life exponent of 3, roller bearings 10/3.
Dynamic load rating C of each bearingkNBasic dynamic load rating of each bearing, from the bearing maker's catalogue.
Static load rating C0 of each bearingkNBasic static load rating of each bearing, from the catalogue.
Required L10 lifehRating life the bearings must reach, from your design brief.
Required static safety factorLeast ratio of static rating to static load you require (from the bearing maker's guidance for the duty).
Shaft diameter at the gearsmmShaft diameter at the gear seats where the bending moment is largest.
Allowable combined stress in the shaftMPaAllowable combined (von Mises) stress for the shaft material and duty.

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: Torque carried by the shaft
Illustration: Torque carried by the shaftA shaft with a curved arrow around it: torque Torque carried by the shaft is the twisting effect, force times lever arm.Ttwisting effect= force × lever armExample: 150 N·mTorque carried by the shaft
What it is
The torque the countershaft carries from the gear that drives it to the gear that drives the next shaft.
Units and format
Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m to 1000000 N·m. Required: the tool does not calculate until every input is valid.
Example
The form opens with 150 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
All gear forces, bearing loads and shaft stresses rise in proportion.
Get it right
Use the largest steady torque, with any service factor your design practice applies for shock.
What this means: Shaft speed
Illustration: Shaft speedA turning disc with a marker; Shaft speed is how many full turns it makes in each minute.turns of the shaftper minutenExample: 500 rpmShaft speed
What it is
The rotational speed of the countershaft, used to turn the bearing rating life in revolutions into hours.
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 500 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Life in hours falls in proportion to speed; the loads are unchanged.
Get it right
Use the speed at which the shaft spends most of its running time.
What this means: Bearing spacing (A to B)
Illustration: Bearing spacing (A to B)A bar with a dimension line and arrows from one end to the other, marking the length Bearing spacing (A to B).LExample: 200 mmBearing spacing (A to B)
What it is
The distance between the centres of the two bearings that support the shaft.
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 200 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 span shares the load differently and raises the bending moment.
Get it right
Measure between bearing centres, not between housing faces.
What this means: Pressure angle
Illustration: Pressure angleTwo lines meeting at a point with an arc between them, marking the angle Pressure angle.φExample: 20°Pressure angle
What it is
The pressure angle of the gear teeth, which sets how much of each tooth force pushes the gears apart.
Units and format
Enter a number in °: degrees of angle. The form accepts 10° to 35°. Required: the tool does not calculate until every input is valid.
Example
The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A larger angle raises the radial forces on the bearings.
Get it right
20° is the usual standard; older gears may be 14.5°.

Driven gear (power in)

What this means: Pitch diameter
Illustration: Pitch diameterA circle with a line straight through its centre from edge to edge, marking the diameter Pitch diameter.d₁measured acrossthe centreExample: 200 mmPitch diameter
What it is
The pitch diameter of the gear on the countershaft that receives power from its mating gear.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 5000 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A smaller gear carries larger tooth forces for the same torque.
Get it right
Pitch diameter is the module times the number of teeth, not the outside diameter.
What this means: Position from bearing A
Illustration: Position from bearing AA bar with a dimension line and arrows from one end to the other, marking the length Position from bearing A.x₁Example: 50 mmPosition from bearing A
What it is
Where the receiving gear sits along the shaft, measured from bearing A.
Units and format
Enter a number in mm: millimetres. The form accepts -10000 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moving a gear towards a bearing shifts its load onto that bearing.
Get it right
Enter a negative value, or one beyond the bearing spacing, for a gear overhung outside the bearings.
What this means: Mesh position angle
Illustration: Mesh position angleTwo lines meeting at a point with an arc between them, marking the angle Mesh position angle.θ₁Example: 0°Mesh position angle
What it is
The angle around the shaft, measured from a chosen reference direction in the direction of rotation, at which the receiving gear meshes with its mating gear.
Units and format
Enter a number in °: degrees of angle. The form accepts 0° 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
Changes how the two gears' forces add or cancel at the bearings.
Get it right
Measure both mesh angles from the same reference and in the direction the shaft turns.

Driving gear (power out)

What this means: Pitch diameter
Illustration: Pitch diameterA circle with a line straight through its centre from edge to edge, marking the diameter Pitch diameter.d₂measured acrossthe centreExample: 80 mmPitch diameter
What it is
The pitch diameter of the gear on the countershaft that delivers power to the next gear.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 5000 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
A smaller gear carries larger tooth forces for the same torque.
Get it right
Pitch diameter is the module times the number of teeth.
What this means: Position from bearing A
Illustration: Position from bearing AA bar with a dimension line and arrows from one end to the other, marking the length Position from bearing A.x₂Example: 150 mmPosition from bearing A
What it is
Where the delivering gear sits along the shaft, measured from bearing A.
Units and format
Enter a number in mm: millimetres. The form accepts -10000 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 150 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moving a gear towards a bearing shifts its load onto that bearing.
Get it right
Use the gear's face centre-line.
What this means: Mesh position angle
Illustration: Mesh position angleTwo lines meeting at a point with an arc between them, marking the angle Mesh position angle.θ₂Example: 180°Mesh position angle
What it is
The angle around the shaft at which the delivering gear meshes with the next gear, measured from the same reference and in the direction of rotation.
Units and format
Enter a number in °: degrees of angle. The form accepts 0° to 360°. Required: the tool does not calculate until every input is valid.
Example
The form opens with 180° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Changes how the two gears' forces add or cancel at the bearings.
Get it right
Meshes on opposite sides of the shaft are 180° apart.

Bearings and shaft

What this means: Bearing type
Illustration: Bearing typeThe options for Bearing type, with the one currently selected highlighted.Ball (life exponent 3)Roller (life exponent 10/3)Example: Ball (life exponent 3)Bearing type
What it is
Whether the shaft runs in ball bearings or roller bearings, which sets how steeply life falls as load rises.
Units and format
Choose one of 2 options from the list: Ball (life exponent 3); Roller (life exponent 10/3).
Example
The form opens with “Ball (life exponent 3)” selected as a worked example. Choose the option that matches your case.
Effect on the result
Roller bearings use a life exponent of 10/3 instead of 3.
Get it right
Use the type of the bearing that will be fitted at the more heavily loaded end.
What this means: Dynamic load rating C of each bearing
Illustration: Dynamic load rating C of each bearingA block with an arrow pushing on it; the arrow is the force Dynamic load rating C of each bearing.CExample: 30 kNDynamic load rating C of each bearing
What it is
The basic dynamic load rating of each bearing: the load at which it reaches a rating life of one million revolutions.
Units and format
Enter a number in kN: kilonewtons (1 kN = 1000 N). The form accepts 0.000001 kN to 100000 kN. Required: the tool does not calculate until every input is valid.
Example
The form opens with 30 kN as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Life rises with the cube (or 10/3 power) of the rating.
Get it right
Take it from the bearing maker's catalogue for the exact bearing number.
What this means: Static load rating C0 of each bearing
Illustration: Static load rating C0 of each bearingA block with an arrow pushing on it; the arrow is the force Static load rating C0 of each bearing.C₀Example: 16 kNStatic load rating C0 of each bearing
What it is
The basic static load rating of each bearing, used to check it against permanent dents under heavy or shock loads.
Units and format
Enter a number in kN: kilonewtons (1 kN = 1000 N). The form accepts 0.000001 kN to 100000 kN. Required: the tool does not calculate until every input is valid.
Example
The form opens with 16 kN as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
The static safety factor rises in proportion.
Get it right
Take it from the catalogue for the exact bearing number.
What this means: Required L10 life
Illustration: Required L10 lifeA timeline from a start mark to an end mark; Required L10 life is the time between them.startendL10Example: 20000 hRequired L10 life
What it is
The L10 rating life in hours the bearings must reach for the machine's design life.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h to 10000000 h. Required: the tool does not calculate until every input is valid.
Example
The form opens with 20000 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Only moves the life pass line.
Get it right
Use your design brief or the machine maker's guidance for this kind of machine.
What this means: Required static safety factor
Illustration: Required static safety factorTwo bars: a base value and the same value multiplied by Required static safety factor, the factor.base valuewith factors₀Example: 2Required static safety factor
What it is
The least ratio of each bearing's static rating to its load that you accept.
Units and format
Enter a plain number with no unit. The form accepts 0.1 to 20. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Only moves the static pass line.
Get it right
Bearing makers suggest higher values for shock loads and quiet running; use their guidance.
What this means: Shaft diameter at the gears
Illustration: Shaft diameter at the gearsA circle with a line straight through its centre from edge to edge, marking the diameter Shaft diameter at the gears.dmeasured acrossthe centreExample: 35 mmShaft diameter at the gears
What it is
The diameter of the shaft at the gear seats, where the bending moment is largest.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 35 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Shaft stresses fall with the cube of the diameter.
Get it right
Use the smallest diameter at a gear seat, not a larger shoulder.
What this means: Allowable combined stress in the shaft
Illustration: Allowable combined stress in the shaftA test bar pulled from both ends: stress Allowable combined stress in the shaft is the force carried by each square millimetre of its cross-section.σastress = force ÷ areaExample: 100 MPaAllowable combined stress in the shaft
What it is
The highest combined bending and torsional stress you allow in the shaft material for this duty.
Units and format
Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 3000 MPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 100 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Only moves the shaft pass line.
Get it right
Take it from your design standard for the shaft steel; fatigue needs a separate check.

Method and formulas

  1. Tooth forces (as the gear tooth forces tool): F_t = 2T ÷ d, F_r = F_t tan φ.
  2. At mesh angle θ the radial force points to the shaft centre and the tangential force acts along the direction of rotation on the driven gear and against it on the driving gear; each force is resolved into two planes.
  3. Bearing reactions from moments about each bearing in each plane: R_B = −Σ F x ÷ L, R_A = −Σ F − R_B; resultant load √(R_y² + R_z²).
  4. L10 life (as the bearing L10 life tool) = (C ÷ P)^p × 10⁶ ÷ (60 n) h with p = 3 or 10/3; static factor C₀ ÷ P (as the bearing static ratio tool).
  5. Bending moment at each gear from the reactions and forces to its left, resultant of the two planes; combined stress √(σ² + 3τ²) with σ = 32M ÷ (π d³), τ = 16T ÷ (π d³).
  6. Example: the bearings carry 2,063.6 N and 3,308.7 N, giving L10 lives of 102,410 h and 24,848 h; the shaft sees 49.971 MPa combined.

Limitations

  • Spur gears only, with the shaft simply supported on two bearings; helical axial forces, bearing preload and gear weight are not included.
  • Steady torque; fatigue of the shaft and the effect of keyways or shoulders are not checked.
  • Example values are illustrations. Replace them with values for your actual task.

Typical uses

  • Check the bearings of an intermediate gearbox shaft
  • See how the mesh positions change the bearing loads
  • Check the shaft diameter against bending and torsion

Who it is for

  • Machine designers
  • Mechanical engineers
  • Gearbox and maintenance engineers

What you get

  • Tooth forces on each gear
  • Bearing loads in each plane and resultant
  • L10 life and static safety factor of each bearing
  • Largest bending moment and combined shaft stress

Interpreting the result

  • Where the two meshes sit around the shaft matters as much as the forces: meshes on opposite sides can add the tangential forces at one bearing, while meshes on the same side partly cancel them.
  • Bearing life falls with the cube of the load, so the more heavily loaded bearing nearly always governs.

References

  • Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2160, 2404–2412, 257–277. Topic reference; no source tables or recommended settings are reproduced.

Learn the Concepts Behind This Tool

These KEVOS® Library readings explain the method this tool uses, with worked examples.

KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.

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