BIZOMAT MAKEOS · Mechanics
Drive Shaft Torsional Check (Natural Frequency, Resonance Margins, Stress, Twist)
Does the torsional natural frequency of my motor, shaft and load fall clear of the running-speed excitations, and are the shaft stress and twist acceptable?
Check of a two-inertia drive (motor and load joined by a shaft): torsional stiffness of the shaft, effective inertia and natural frequency (as the torsional vibration tool), the excitation bands for the first few orders of running speed over the speed range, the separation of the natural frequency from each band against your margin, the critical speeds, and the shaft shear stress and angle of twist at the drive torque.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shear modulus | GPa | Shear modulus of the shaft material (steel about 79 GPa). |
| Shaft outside diameter | mm | Outside diameter of the shaft. |
| Shaft bore (0 = solid) | mm | Bore of a hollow shaft; 0 for solid. |
| Shaft length | mm | Length of shaft between the motor and the load. |
| Motor inertia | kg·m² | Moment of inertia of the motor rotor and coupling half. |
| Load inertia | kg·m² | Moment of inertia of the driven load. |
| Operation | ||
| Lowest running speed | rpm | Lowest speed the drive runs at continuously. |
| Highest running speed | rpm | Highest speed the drive runs at. |
| Excitation orders to check (1 to …) | Number of running-speed orders that excite the system (1× from unbalance, 2× from universal joints and so on). | |
| Separation margin | % | How far the natural frequency must stay from each excitation band. |
| Drive torque | N·m | Torque the shaft transmits. |
| Allowable shear stress | MPa | Allowable torsional shear stress of the shaft. |
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: Shear modulus
- What it is
- The shear modulus of the shaft material, from the material data. It sets how much the shaft twists under torque.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 79.3 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Natural frequency rises with the square root of the shear modulus.
- Get it right
- Type gigapascals; do not use the elastic (Young’s) modulus.
What this means: Shaft outside diameter
- What it is
- The outside diameter of the shaft that joins the motor and the load.
- 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 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stiffness rises with the fourth power of diameter; shear stress falls roughly with its cube.
- Get it right
- If the diameter changes along the shaft, this tool does not cover it; use the uniform section.
What this means: Shaft bore (0 = solid)
- What it is
- The inside diameter of a hollow shaft. Type 0 for a solid shaft.
- 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 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Shaft length
- What it is
- The free length of shaft between the motor and the load inertias, the part that twists.
- 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 800 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stiffness falls in proportion to length, so natural frequency falls with its square root.
What this means: Motor inertia
- What it is
- The moment of inertia of the driving end: the motor rotor plus the coupling half and anything else fixed to it, from the motor maker’s data.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 1000000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use kg·m²; GD² values must be divided by four.
What this means: Load inertia
- What it is
- The moment of inertia of the driven end: the load and everything fixed to it, referred to the shaft speed.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 1000000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Parts on other shafts must be referred through the square of the speed ratio.
Operation
What this means: Lowest running speed
- What it is
- The lowest speed at which the drive runs continuously. It sets the bottom of each excitation band.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Highest running speed
- What it is
- The highest speed at which the drive runs. It sets the top of each excitation band.
- 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 1500 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must not be below the lowest running speed.
What this means: Excitation orders to check (1 to …)
- What it is
- How many multiples of running speed to check as sources of excitation, starting at once per revolution (unbalance) and going up (universal joints, gear meshes and so on).
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 12. Use a whole number. 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.
- Get it right
- Include every order your drive actually produces, such as twice per revolution for a universal joint.
What this means: Separation margin
- What it is
- How far the natural frequency must stay below or above each excitation band, as a percentage of the band edge.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 100%. 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.
- Get it right
- Take it from your design standard or the equipment maker.
What this means: Drive torque
- What it is
- The torque the shaft transmits in normal running, used for the shear stress and the angle of twist.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m to 100000000 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
- Shear stress and twist rise in proportion to torque.
What this means: Allowable shear stress
- What it is
- The torsional shear stress you allow in the shaft, from the material and your design standard or safety factor.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- K = G π (d⁴ − d_i⁴) ÷ 32L; J_eff = J₁ J₂ ÷ (J₁ + J₂); f_n = √(K ÷ J_eff) ÷ 2π (as the torsional vibration tool).
- Order k excites from k n_min ÷ 60 to k n_max ÷ 60 Hz; f_n must lie below (1 − margin) × the band start or above (1 + margin) × the band end; critical speed = 60 f_n ÷ k.
- τ = 16 T d ÷ π (d⁴ − d_i⁴); twist = T ÷ K.
- Example: f_n = 66.277 Hz; the critical speed nearest the running range is 1,988.3 rpm; the shaft carries 28.294 MPa and twists 1.0903°.
Limitations
- Two-inertia model with a massless shaft; couplings and gear meshes add flexibility.
- Damping is ignored, so margins should be generous.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a long drive shaft or line shaft
- Check a variable-speed drive for resonance
- Size a shaft for stiffness as well as strength
Who it is for
- Design engineers
- Machine designers
- Vibration analysts
What you get
- Stiffness, effective inertia and natural frequency
- Excitation bands, critical speeds and margins
- Shaft stress and twist
Interpreting the result
- Stiffening the shaft raises f_n with √K; a larger load inertia barely lowers it once J₂ ≫ J₁.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 3412–3413, 296–302. Topic reference; no source tables or recommended settings 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
Starting, stopping and torsion
In Whirling Shafts by the Numbers
What it covers: Getting a heavy rotor up to speed: inertia, run-up time and the energy a direct-on-line start puts into a motor, resonances passed through on the way, and torsional checks of the drive.
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