Kinematics & Dynamics

Critical Speed & Natural Frequency Calculator — Resonance Separation Margin

Calculate the natural frequency and critical speed of an equivalent stiffness–mass system, then screen 1×, 2×, gear-mesh or external cyclic excitation for resonance overlap, amplification and separation margin.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Operating speed range

Equivalent SDOF system

Speed-dependent excitation

02
Output

Results

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Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Natural frequency and critical speed

The calculator models the system as an equivalent single-degree-of-freedom stiffness–mass system. The undamped angular natural frequency is:

ωn = √(k / m)

where:

  • ωn — angular natural frequency [rad/s]
  • k — equivalent system stiffness [N/m]
  • m — equivalent mass [kg]

The natural frequency in hertz is:

fn = ωn / (2π)

The corresponding 1× speed-equivalent critical speed is:

ncrit = 60 fn

where ncrit is expressed in revolutions per minute.

Excitation frequency

The excitation frequency depends on the selected source:

  • 1× rotational: fexc = n / 60
  • 2× rotational: fexc = 2n / 60
  • Gear mesh: fexc = zn / 60
  • External cyclic: fexc = fcycle

where n is rotational speed [rpm], z is the number of gear teeth, and fcycle is the entered external excitation frequency [Hz].

The calculator evaluates excitation at both operating speed and maximum speed for the rotational and gear-mesh cases.

Frequency ratio

The excitation-to-natural-frequency ratios are:

r = fexc,operating / fn

rmax = fexc,max / fn

A ratio approaching 1 means the excitation frequency is approaching the calculated natural frequency.

Damping and resonance band

Using the entered damping ratio ζ, the calculator defines the resonance bandwidth as:

Δf = 2ζfn

The resonance-band limits are:

flower = max(0, fn − Δf / 2)

fupper = fn + Δf / 2

The corresponding frequency-equivalent resonance-zone limits in rpm are:

navoid,min = max(0, ncrit − 30Δf)

navoid,max = ncrit + 30Δf

Dynamic amplification factor

For each evaluated frequency ratio, the calculator uses:

A(r) = 1 / √[(1 − r2)2 + (2ζr)2]

The reported amplification factor is the larger value calculated at the operating and maximum excitation frequencies.

Separation margin

The excitation frequency is converted to an equivalent rpm value:

nexc = 60fexc

The separation at each evaluated endpoint is:

SM = |ncrit − nexc| / ncrit × 100%

The calculator reports the smaller separation obtained from the operating and maximum excitation conditions:

SMreported = min(SMoperating, SMmax)

For 2× and gear-mesh excitation, nexc is an rpm-equivalent representation of excitation frequency. It is not the physical shaft speed itself.

Resonance overlap

For speed-dependent excitation, resonance overlap is detected when the excitation-frequency range from the operating condition to the maximum-speed condition intersects the calculated resonance band:

fexc,operating ≤ fupper and fexc,max ≥ flower.

02
Application

When to use this calculator

When to use this calculator

Use this calculator for an initial mechanical resonance screening when a system can reasonably be represented by an equivalent stiffness k, equivalent mass m and damping ratio ζ.

It is useful for checking whether a known operating-speed range or periodic excitation approaches the calculated natural frequency of the equivalent system.

The calculator can evaluate:

  • 1× rotational excitation associated with shaft or rotor speed,
  • 2× rotational excitation,
  • gear-mesh excitation using rotational speed and number of teeth,
  • an external cyclic excitation specified directly in hertz,
  • natural frequency and its 1× rpm-equivalent critical speed,
  • frequency ratio at operating and maximum conditions,
  • damping-based resonance bandwidth,
  • dynamic amplification factor,
  • separation margin from the natural frequency,
  • whether the evaluated excitation range overlaps the calculated resonance band.

Typical engineering applications

The calculation can be used as a preliminary resonance check for machinery, drives, mounted equipment and other mechanical systems where an equivalent lumped stiffness and mass are known or have been estimated.

For a geared system, the gear-mesh mode is useful when the tooth-passing frequency rather than the basic shaft rotational frequency is the excitation of interest.

When not to use this calculator

This calculator does not derive shaft critical speed from shaft diameter, span, bearing positions, rotor geometry or static deflection. It therefore should not be treated as a shaft-whirling or full rotordynamic analysis tool.

It also does not calculate multiple mode shapes, speed-dependent eigenfrequencies, bearing or seal dynamic coefficients, gyroscopic effects, rotor unbalance response, torsional mode shapes or structural modal behavior. Systems in which those effects are significant require an appropriate modal, rotor-dynamic, torsional-vibration or finite-element analysis.

The entered required separation margin is a user-defined design criterion. The calculator does not automatically determine a required margin from a particular machinery standard.

03
Decision support

How to interpret the result

Critical speed

ncrit is the natural frequency expressed as an equivalent 1× rotational speed in rpm. Increasing stiffness raises this value according to the square root of stiffness, while increasing equivalent mass lowers it according to the inverse square root of mass.

For 2× and gear-mesh excitation, do not interpret ncrit directly as the shaft speed at which resonance must occur. The calculator performs the resonance comparison in the frequency domain using the selected excitation order.

Separation margin

SM is the smallest percentage separation between the calculated natural frequency and the evaluated excitation at the operating or maximum condition.

A larger separation margin means the tested excitation endpoint is farther from the modeled natural frequency. The calculated margin is compared with the user-entered required margin SMreq.

If the calculated separation margin falls below the entered requirement, the calculator returns a warning unless a more severe unsafe condition has already been detected.

Frequency ratio

r and rmax compare excitation frequency with natural frequency.

  • r < 1 — excitation is below the natural frequency.
  • r ≈ 1 — excitation approaches the natural frequency and resonance becomes important.
  • r > 1 — excitation is above the natural frequency.

Resonance bandwidth

Δf defines the damping-based resonance band used by the calculator. Increasing the entered damping ratio widens this implemented band around fn.

The reported navoid,min and navoid,max values are frequency-equivalent rpm limits for this resonance band. They correspond directly to physical shaft-speed limits only for 1× rotational excitation. For higher excitation orders or gear mesh, the shaft-speed relationship must include the excitation multiplier.

Amplification factor

A is the largest dynamic amplification factor calculated at the operating and maximum excitation frequencies using the implemented damped single-degree-of-freedom response equation.

As the excitation ratio approaches resonance, amplification can increase strongly when damping is low. In the calculator's decision logic, an amplification factor greater than 20 produces an unsafe result.

Resonance overlap

C = 1 indicates that the evaluated excitation-frequency range overlaps the calculated resonance band. This condition is classified as unsafe by the calculator.

RB = 1 indicates that an evaluated endpoint lies within the resonance band.

Result status

  • Safe: no resonance overlap is detected, amplification does not exceed the implemented unsafe threshold, and the calculated separation margin satisfies the entered requirement.
  • Warning: the evaluated condition does not trigger an unsafe criterion, but the calculated separation margin is below the required value or another implemented resonance-band warning applies.
  • Unsafe: the excitation range overlaps the resonance band or the calculated amplification factor exceeds 20.
  • Invalid: one or more required inputs violate the calculator's validation rules.

A safe result confirms only the criteria evaluated by this calculator. It does not demonstrate that every vibration mode or failure mechanism of the actual machine is acceptable.

04
Worked case

Calculation example

Example — 1× rotational resonance screening

Consider an equivalent mechanical system with the following data:

  • Operating speed: n = 600 rpm
  • Maximum speed: nmax = 800 rpm
  • Required separation margin: SMreq = 15%
  • Equivalent stiffness: k = 1,000,000 N/m
  • Equivalent mass: m = 100 kg
  • Damping ratio: ζ = 0.05
  • Excitation type: 1× rotational speed

1. Calculate the natural frequency

ωn = √(k / m)

ωn = √(1,000,000 / 100) = 100.000 rad/s

fn = ωn / (2π)

fn = 100 / (2π) = 15.915 Hz

2. Convert the natural frequency to critical speed

ncrit = 60fn

ncrit = 60 × 15.915 = 954.93 rpm

3. Calculate the excitation frequencies

For 1× rotational excitation:

fexc,operating = 600 / 60 = 10.000 Hz

fexc,max = 800 / 60 = 13.333 Hz

4. Calculate the frequency ratios

r = 10.000 / 15.915 = 0.628

rmax = 13.333 / 15.915 = 0.838

5. Determine the resonance bandwidth

Δf = 2ζfn

Δf = 2 × 0.05 × 15.915 = 1.592 Hz

The resonance band is therefore approximately:

flower = 15.915 − 1.592 / 2 = 15.120 Hz

fupper = 15.915 + 1.592 / 2 = 16.711 Hz

The maximum 1× excitation frequency of 13.333 Hz remains below this band, so the evaluated operating range does not cross the calculated resonance band.

6. Calculate the amplification factor

At the operating condition:

A = 1.643

At maximum speed:

Amax = 3.229

The calculator therefore reports an amplification factor of approximately 3.229.

7. Calculate the separation margin

The smaller separation occurs at the maximum-speed condition:

SM = |954.93 − 800| / 954.93 × 100%

SM = 16.22%

This is greater than the entered requirement of 15%.

8. Resonance-zone limits

The equivalent resonance-zone limits are approximately:

907.18 rpm to 1002.68 rpm

Interpretation

For this 1× excitation case, the 600–800 rpm operating range remains below the calculated resonance zone, no resonance overlap is detected, and the calculated separation margin exceeds the entered 15% requirement. Under the calculator's implemented criteria, the evaluated condition is classified as safe.

This result is a single-degree-of-freedom resonance screening result. It does not establish the absence of other structural, torsional or rotor-dynamic critical speeds.

05
Model boundaries

Assumptions and limitations

Model assumptions

  • The mechanical system is represented by one equivalent mass and one equivalent linear stiffness.
  • The calculated natural frequency is the undamped single-degree-of-freedom value ωn = √(k/m).
  • The entered equivalent stiffness and mass are assumed to represent the dynamic mode of interest adequately.
  • Stiffness, mass and damping ratio are treated as constant over the evaluated operating-speed range.
  • The damping ratio is used to define the implemented resonance bandwidth and dynamic amplification equation but does not shift the calculated natural frequency.
  • 1×, 2× and gear-mesh excitation frequencies are assumed to scale directly with rotational speed.
  • External cyclic excitation is treated as a fixed frequency independent of shaft speed.
  • The required separation margin is supplied by the user and is not automatically derived from a standard.

Important limitations

  • The calculator does not derive shaft stiffness, equivalent mass or damping from component geometry.
  • It does not calculate shaft whirling speed from static deflection or beam geometry.
  • It does not solve a multi-degree-of-freedom eigenvalue problem or identify multiple natural modes.
  • It does not account for speed-dependent natural frequencies, gyroscopic effects, anisotropic bearings, bearing or seal dynamic coefficients, distributed shaft flexibility or rotor-support coupling.
  • The implemented amplification equation is a simplified damped single-degree-of-freedom response model and is not a complete rotor unbalance-response analysis.
  • The resonance-zone rpm values are frequency-equivalent limits. For 2× and gear-mesh excitation they must not be interpreted directly as physical shaft-speed limits without applying the corresponding excitation order.
  • Gear-mesh screening uses tooth count and shaft speed only; detailed gear-mesh stiffness variation, transmission error, sidebands and multiple gear stages are not modeled.
  • Torsional, lateral and structural resonances outside the represented equivalent mode are not evaluated.
  • The model does not establish compliance with API, ISO or another machinery design standard.

For critical rotating machinery, systems with several closely spaced modes, flexible supports or significant rotor-dynamic effects, use an appropriate modal or rotor-dynamic model and verify the final design against the governing engineering requirements.

06
Questions

Frequently asked questions

What is critical speed in this calculator?
Critical speed is reported as the 1× rotational-speed equivalent of the calculated natural frequency: ncrit = 60fn. It represents the rpm at which a 1× rotational excitation would coincide with the natural frequency of the equivalent stiffness–mass model. For 2× or gear-mesh excitation, resonance can occur at a different physical shaft speed because the excitation frequency is a multiple of shaft speed.
What is the natural frequency formula for a spring-mass system?
For the equivalent single-degree-of-freedom model used by this calculator, the undamped angular natural frequency is ωn = √(k/m), where k is system stiffness and m is equivalent mass. The natural frequency in hertz is fn = √(k/m)/(2π).
What is the difference between natural frequency and resonance?
Natural frequency is a property of the modeled stiffness–mass system. Resonance is the operating condition in which an excitation frequency approaches that natural frequency and the vibration response can be amplified. This calculator compares the selected excitation with the calculated natural frequency and a damping-based resonance band.
How is the resonance separation margin calculated?
The calculator converts each evaluated excitation frequency to an rpm-equivalent value and calculates SM = |ncrit − nexc| / ncrit × 100%. It evaluates both the operating and maximum conditions and reports the smaller separation margin.
Does this calculator calculate shaft critical or whirling speed from shaft geometry?
No. The calculator does not use shaft diameter, shaft length, bearing spacing, rotor mass locations, static shaft deflection or bending mode shapes. It calculates the natural frequency of an equivalent stiffness–mass model. A shaft critical-speed or whirling calculation based on actual rotor geometry requires a different mechanical model.
What excitation types can be checked?
The implemented excitation modes are 1× rotational speed, 2× rotational speed, gear-mesh frequency and an independently specified external cyclic frequency. Gear-mesh frequency is calculated from rotational speed multiplied by the entered number of gear teeth.
How does damping ratio affect the resonance calculation?
The entered damping ratio ζ affects both the calculated resonance bandwidth and the dynamic amplification factor. The implemented bandwidth is Δf = 2ζfn, while amplification is calculated from A = 1/√[(1 − r²)² + (2ζr)²]. The natural frequency and reported critical speed themselves are calculated from stiffness and mass and are not shifted by ζ in this model.
How does the calculator check gear-mesh resonance?
For gear-mesh excitation, the excitation frequency is fexc = zn/60, where z is the number of teeth and n is shaft speed in rpm. The calculator evaluates this frequency at the operating and maximum shaft speeds and checks whether that excitation range intersects the calculated resonance band.
What does resonance overlap mean?
Resonance overlap means that the evaluated excitation-frequency range intersects the damping-based resonance band around the calculated natural frequency. In the implemented decision logic, detected overlap produces an unsafe result.
What does a low frequency ratio mean?
The frequency ratio is r = fexc/fn. Values well below 1 indicate excitation below the modeled natural frequency, values near 1 indicate proximity to resonance, and values above 1 indicate excitation above the natural frequency. The complete result should also be interpreted using resonance overlap, amplification and separation margin.
Is the required separation margin determined automatically?
No. The required separation margin is an input selected by the user. The calculator compares the calculated margin against that entered requirement but does not automatically derive a mandatory value from a specific machinery code or standard.
Does a safe result prove that a rotating machine has no resonance problems?
No. A safe result means that the criteria implemented in this equivalent single-degree-of-freedom screening model are satisfied for the entered excitation. Other natural modes, shaft bending, torsional resonance, bearing and support dynamics, gyroscopic effects and other excitation sources may still require separate verification.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.