Calculate the critical (resonance) speed of a rotating shaft using the single-DOF rotor model. ω_n = √(k/m), n_cr = (60/2π)·√(k/m) rpm. Also returns the natural frequency f_n in Hz.
Execution
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Key facts
Category
Math & Numbers
Input types
number
Output type
json
Sample coverage
4
API ready
Yes
Overview
The Shaft Critical Speed Calculator helps mechanical engineers and rotor dynamics specialists determine the critical (resonance) speed of a rotating shaft using a single-degree-of-freedom (single-DOF) rotor model. By entering the shaft stiffness (k) and rotor mass (m), you can instantly calculate the angular frequency (ω_n), natural frequency (f_n), and critical whirling speed (n_cr) in RPM to prevent destructive mechanical resonance.
When to use
When designing rotating machinery, such as turbines, pumps, or drive shafts, to ensure operating speeds do not match the shaft's natural frequency.
During rotor dynamics analysis to identify the whirling threshold and avoid destructive resonance.
When troubleshooting vibration issues in existing industrial rotating equipment by verifying the system's stiffness-to-mass ratio.
How it works
1Enter the equivalent stiffness (k) of the shaft in Newtons per meter (N/m).
2Input the concentrated mass (m) of the rotor or disc in kilograms (kg).
3Adjust the decimal places for precision and click calculate to compute the angular frequency, natural frequency, and critical speed in RPM.
Use cases
Designing industrial electric motor shafts to operate safely below or above the first critical whirling speed.
Verifying the structural safety of centrifugal pump shafts under varying rotor mass configurations.
Teaching mechanical engineering students the fundamentals of rotor dynamics and single-DOF vibration models.
Examples
1. Verifying a Small Fan Rotor Shaft
Mechanical Design Engineer
Background
An engineer is designing a cooling fan assembly where a 5 kg impeller is mounted on a steel shaft with an estimated lateral stiffness of 200,000 N/m.
Problem
The fan needs to run at 1,800 RPM. The engineer must ensure this operating speed does not coincide with the shaft's critical speed.
How to use
Input 200000 for Stiffness (k) and 5 for Mass (m), then set decimal places to 4.
Outcome
The calculator outputs a critical speed of 1909.8593 RPM. Since the operating speed of 1,800 RPM is close to this resonance point, the engineer decides to stiffen the shaft to raise the critical speed safely above 2,200 RPM.
2. Analyzing a Heavy Industrial Rotor
Vibration Analyst
Background
A maintenance team is troubleshooting a large industrial blower. The rotor mass is 120 kg, and the shaft stiffness is calculated to be 4,800,000 N/m.
Problem
Determine the natural frequency in Hz and critical speed in RPM to set safe variable frequency drive (VFD) lockout zones.
How to use
Enter 4800000 for Stiffness (k) and 120 for Mass (m).
FAQ
What is critical speed in a rotating shaft?
Critical speed is the rotational speed at which a shaft becomes unstable and begins to vibrate violently (whirl) due to resonance with its natural frequency.
What formula does this calculator use?
It uses the single-DOF rotor model where angular frequency is ω_n = √(k/m) rad/s, natural frequency is f_n = ω_n / 2π Hz, and critical speed is n_cr = (60/2π)·ω_n RPM.
Can I calculate multiple critical speeds for multi-mass systems?
No, this calculator uses a single-degree-of-freedom model, which calculates the primary (first) critical speed for a single concentrated mass.
What units should I use for stiffness and mass?
Use Newtons per meter (N/m) for stiffness (k) and kilograms (kg) for mass (m).
How do I prevent a shaft from operating at critical speed?
You can change the operating speed, increase shaft stiffness (e.g., by increasing diameter or shortening span), or reduce the rotor mass.
The calculator determines the natural frequency is 31.8310 Hz and the critical speed is 1909.8593 RPM, allowing the analyst to program the VFD to skip the 1850–1970 RPM range.