BIZOMAT MAKEOS · Mechanics
Torsional Natural Frequency of a Drive
At what frequency will this motor-and-load drive resonate in torsion, and is that safely above the control bandwidth?
Torsional stiffness of a connecting shaft and the torsional natural frequency of a single inertia on a shaft or of a motor and load joined by a shaft, compared with the servo or control bandwidth.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| System | choice | Two inertias on a free shaft, or one inertia on a shaft fixed at the other end. |
| Stiffness | choice | Calculate the stiffness from a plain shaft or enter it (couplings and gearboxes in series reduce it). |
| Torsional stiffness (entered) | N·m/rad | Overall torsional stiffness between the inertias. |
| Shear modulus | GPa | Shear modulus of the shaft. |
| Shaft outside diameter | mm | Outside diameter. |
| Shaft bore (0 = solid) | mm | Bore diameter for a hollow shaft. |
| Shaft length | mm | Twisting length of the shaft. |
| Motor inertia | kg·m² | Motor rotor inertia. |
| Load inertia | kg·m² | Load inertia referred to the same shaft. |
| Control bandwidth | Hz | Closed-loop bandwidth of the drive or controller. |
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: System
- What it is
- The vibration model: a motor and a load joined by a shaft with both ends free to turn, or a single load on a shaft whose far end is held fixed.
- Units and format
- Choose one of 2 options from the list: Motor and load joined by the shaft; Load on a shaft fixed at the far end.
- Example
- The form opens with “Motor and load joined by the shaft” selected as a worked example. Choose the option that matches your case.
- Get it right
- In the single-load model the motor inertia is not used.
What this means: Stiffness
- What it is
- Whether the torsional stiffness is calculated from a plain solid or hollow shaft of given size and material, or typed in for the whole connection.
- Units and format
- Choose one of 2 options from the list: From shaft dimensions; Enter the torsional stiffness.
- Example
- The form opens with “From shaft dimensions” selected as a worked example. Choose the option that matches your case.
What this means: Torsional stiffness (entered)
- What it is
- The overall twisting stiffness of the connection between the inertias: the torque needed to twist it through one radian. Used only when the stiffness is entered.
- Units and format
- Enter a number in N·m/rad: N·m/rad. The form accepts 0.000001 N·m/rad or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2500 N·m/rad as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The natural frequency rises with the square root of the stiffness.
- Get it right
- Couplings, keys and gearboxes in series reduce it: add the reciprocals of the individual stiffnesses and take the reciprocal of the sum.
What this means: Shear modulus
- What it is
- The shear modulus (modulus of rigidity) of the shaft material, used when the stiffness is calculated from the shaft dimensions.
- 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.
- Get it right
- Use G, not Young’s modulus E, from the material data sheet, and enter it in GPa.
What this means: Shaft outside diameter
- What it is
- The outside diameter of the plain shaft section that twists between 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 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The stiffness of a solid shaft rises with the fourth power of its diameter, so the natural frequency rises with its square.
What this means: Shaft bore (0 = solid)
- What it is
- The inside diameter of a hollow shaft; zero for a solid shaft. It must be smaller than the outside diameter.
- 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 length of shaft that actually twists, between the points where the motor and the load (or the fixed end) are attached.
- 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 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The stiffness is inversely proportional to the length, so a longer shaft lowers the natural frequency.
What this means: Motor inertia
- What it is
- The mass moment of inertia of the motor rotor and anything fixed to the motor end of the shaft, used only in the motor-and-load model.
- 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 0.002 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from the motor data sheet; add hubs or couplings mounted at the motor end.
What this means: Load inertia
- What it is
- The mass moment of inertia of the driven load at the other end of the shaft, referred to the shaft speed; in the single-load model it is the only inertia used.
- 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
- Inertia behind a gearbox must be divided by the ratio squared to refer it to the motor shaft.
What this means: Control bandwidth
- What it is
- The closed-loop bandwidth of the servo drive or speed controller, the frequency up to which it follows its commands; the natural frequency is compared with it.
- Units and format
- Enter a number in Hz: hertz (cycles per second). The form accepts 0.000001 Hz to 100000 Hz. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 Hz as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use hertz, not rad/s: divide a value in rad/s by 2π.
Method and formulas
- Shaft stiffness K = G J_p ÷ L, with J_p = π (d⁴ − dᵢ⁴) ÷ 32.
- Single inertia: f_n = (1 ÷ 2π) √(K ÷ J_L), as in the handbook.
- Two inertias: f_n = (1 ÷ 2π) √[K (J₁ + J₂) ÷ (J₁ J₂)].
- Example: the drive resonates at 181.15 Hz against a 50 Hz bandwidth.
Limitations
- Lumped inertias on a massless shaft; first mode only.
- Couplings, keys and gear mesh add compliance and lower the frequency.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a servo axis for torsional resonance
- Choose a coupling or shaft stiff enough for the bandwidth
- Diagnose oscillation in a motion axis
Who it is for
- Automation engineers
- Servo and controls engineers
- Machine designers
What you get
- Torsional stiffness
- Natural frequency
- Ratio to the control bandwidth
Interpreting the result
- If the resonance falls within the control bandwidth, stiffen the drive or add a notch filter.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 3412–3413. 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.
Method explained here · BIZOMAT original book
Twist: torsional modes of the fan drive
In Vibrating Machine Elements by the Numbers
What it covers: Model the RF-3 drive as inertias joined by torsional springs, find its twisting modes with and without the belt, and decide whether to stiffen the shaft or skip a speed.
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