Does the worksheet calculate the first critical speed of a real rotor?

No. It calculates the undamped natural-frequency marker of one documented effective point mass and one linear lateral stiffness. A real rotor-bearing-seal train can have multiple speed-dependent modes and requires a rotor-dynamic model and response evidence.

Is the result an ISO or API acceptance verdict?

No. ISO 21940-11 and ISO 21940-12 concern rotor balancing behaviour and tolerances; ISO 21940-12 states that structural resonances are outside its scope. Exact API equipment criteria depend on the applicable licensed edition, equipment scope and clauses.

When may the solid-shaft helper be used?

Only for the displayed ideal Euler-Bernoulli case: a uniform solid circular massless beam, constant documented Young modulus, one point mass at the stated location, and ideal support boundary conditions. It does not include shaft mass, real bearings, damping, gyroscopic effects, seals or housings.

Why is there no safe or danger zone?

A universal percentage band would ignore the applicable machine standard, mode, damping, excitation order, operating range, transient passage and response analysis. The worksheet reports arithmetic comparison only and does not issue an engineering verdict.

Documented one-DOF linear model · no equipment verdict

Documented Point-Mass Lateral-Frequency Worksheet

Calculate the equivalent undamped natural-frequency marker of one documented effective point mass and one linear lateral stiffness at the same degree of freedom. The result is not automatically the first critical speed of a real rotor and is not an ISO or API acceptance calculation.

ω_n = √(k_eff/m_eff)Direct stiffness or ideal beam helperNo universal separation bandNo pass/fail

Documented model inputs

Stiffness input methodChanging method clears method-specific values and prior results

Effective point mass m_eff

At the selected lateral degree of freedom; not automatically total rotor mass

Optional documented synchronous forcing speedNon-negative magnitude; output is arithmetic only

Equivalent linear lateral stiffness k_eff

At the same degree of freedom and direction as m_eff

Ideal boundary and point-mass locationSupport choice also fixes the assumed mass/load location

Ideal beam span/length L

Between ideal supports, or clamp to free end

Uniform solid diameter dUses the same length unit as L

Documented Young modulus E

Use alloy-, temperature- and state-specific controlled data

Machine, lateral degree of freedom and direction

Effective-mass source and reduction record

Stiffness or beam-property source and idealization record

Optional forcing-speed source and applicable operating/excitation record

I confirm that m_eff and k_eff describe the same linear lateral degree of freedom and direction; both are positive and applicable to the documented state. If the beam helper is used, I accept its uniform solid massless Euler–Bernoulli beam, constant E, ideal support and single point-mass assumptions. I will treat N_echivalent only as an undamped one-DOF speed-equivalent marker—not as a certified critical speed, safe operating speed, ISO/API verdict or substitute for rotor-dynamic response analysis.

Equivalent undamped point-mass result

—

Speed-equivalent marker N_echivalent

—

Natural frequency f_n

—

Angular natural frequency ω_n

—

Normalized effective mass

—

Normalized effective stiffness

—

Compliance 1/k_eff

—

Ideal beam I and L/d

Not used

Optional forcing-speed comparison

Not entered

Limita modelului: this is one linear, undamped, stationary-coordinate point-mass/stiffness model. It omits distributed shaft mass, multiple discs and modes, bearing/support/housing flexibility and damping, cross-coupling and anisotropy, seals and fluids, gyroscopic and rotary-inertia effects, shear deformation, coupling/train interaction, temperature/state changes, nonlinearities and speed-dependent coefficients. Do not use its marker as a machine clearance, run/avoid, or acceptance verdict.

Direct documented-stiffness model

m_effẍ + k_effx = 0
ω_n = √(k_eff/m_eff) rad/s
f_n = ω_n/(2π) Hz
N_echivalent = 60f_n rpm

The ratio k/m has units (N/m)/kg = s⁻²; its square root is rad/s, with the radian dimensionless. N_echivalent is only the rotational speed whose 1× frequency equals f_n. It is not a claim that an actual rotor will have its first response peak at that speed.

Optional ideal solid circular beam helper

I = πd⁴/64
k_beam = CEI/L³
C = 48: pin–pin, central point mass/load
C = 192: clamped–clamped, central point mass/load
C = 3: clamped–free, point mass/load at free end

The helper derives static point-load stiffness from Euler–Bernoulli deflection formulas and then uses the same one-DOF frequency equation. The shaft itself is treated as massless; E, I and L are constant; and the support and point-mass location are ideal. The arithmetic does not become a distributed-beam eigenfrequency calculation.

Entered quantity	Internal SI conversion	Limită
lbm	1 lbm = 0.45359237 kg exactly	Mass, not pound-force
N/mm	1 N/mm = 1000 N/m	Linear stiffness
lbf/in	1 lbf/in = 4.4482216152605/0.0254 = 175.126835246… N/m	Uses the standard pound-force and international inch
în	1 in = 0.0254 m exactly	L and d must use the same selected unit
GPa	1 GPa = 10⁹ Pa	E is not selected from a universal material preset
ksi	1 ksi = 1000 lbf/in² = 6,894,757.293168… Pa	Pressure/stress unit for E

The exact pound and inch basis follows NIST SP 811 Appendix B.8. Changing a unit clears the affected numeric field and prior result rather than silently reinterpreting a number.

NASA one-DOF rotor boundary

NASA/CR—2004-213069, Disk Crack Detection for Seeded Fault Engine Test, presents a simplified one-degree-of-freedom Jeffcott rotor with disk mass M and shaft stiffness k_s. It gives ω_cr = √(k_s/M) and limits that representation to single-disk assemblies under relatively rigid bearings at relatively low speeds near or below the first bending critical. The report also shows that response depends on speed ratio and damping; it does not support the former page’s statement that vibration grows “exponentially.”

Published NPTEL arithmetic reproduced

NPTEL/IIT Guwahati, Single-DOF Damped Rotor Model, examples 2.1–2.2, uses m = 10 kg and k = 100 kN/m to obtain ω_n = 100 rad/s. Direct mode reproduces 100 rad/s, 15.915494309… Hz and 954.929658551… rpm. This is a public university worked example of the one-DOF relation, not a universal machine acceptance criterion.

Beam-helper provenance

MIT OpenCourseWare 2.080, Structural Mechanics Lecture 5, gives central point-load deflections PL³/(48EI) for pin–pin support and, by its clamped–clamped point-load expression, PL³/(192EI) at midspan. MIT 1.050 Problem Set 11 reference sheet gives PL³/(3EI) for an end-loaded cantilever. Inverting deflection/load yields the three helper stiffnesses.

Referinţă	Verified public status/scope on 13 July 2026	Treatment here
ISO 21940-11:2016, Edition 1; Amendment 1:2022	Published; under systematic review at stage 90.20. It establishes balancing procedures and unbalance tolerances for rotors with rigid behaviour. ISO 1940-1:2003 is shown as withdrawn and replaced.	No structural critical-speed formula or “below 70%” rigid-rotor rule is attributed to it.
ISO 21940-12:2016, Edition 1	Published and confirmed in 2025 at stage 90.93. It concerns balancing rotors with flexible behaviour and explicitly places structural resonances and their modification outside its scope.	No claim that this worksheet classifies rigid/flexible behaviour or demonstrates balancing conformity.
API Std 610, 612 and 617	The official API Standards Plan lists Std 610 Edition 13 (29 June 2026), Std 612 Edition 8 (1 November 2020), and Std 617 Edition 9 (1 April 2022). API TR 684-1 Edition 1 was published in 2019 and Edition 2 is under development.	Former universal 115%/120% and ±20% claims are removed. Exact applicability and separation/response clauses remain NEEDS_LICENSED_SOURCE for the selected equipment, edition and contract.

No substitute standard: an owner, purchaser, equipment standard or project specification may require a particular lateral analysis, separation margin, damping assumption, unbalance response, train model or acceptance evidence. Identify and apply the licensed requirement; do not infer it from this worksheet.

A real rotor-bearing system can have several lateral modes. Natural frequencies and response may change with rotational speed because of gyroscopic effects, bearing and seal coefficients, temperature and operating state. Critical speed is tied to an excitation intersection and response, not merely a static solid-shaft stiffness divided by a disk mass.

Use a controlled mass/stiffness or finite-element rotor model with actual stations, distributed shaft mass, discs, couplings and overhangs.
Represent bearing, support, housing, seal and fluid coefficients with their applicable speed/load/state dependence.
Include rotary inertia, gyroscopic moments, shear deformation and anisotropy when material to the model.
Evaluate the applicable excitation orders on a Campbell diagram and calculate damped unbalance/forced response as required.
Validate with run-up/coast-down, modal or other controlled evidence where appropriate, and apply the licensed equipment/project acceptance criteria.

The optional forcing-speed output reports only r = N_forțare/N_echivalent and ΔN = N_forțare − N_echivalent. It deliberately does not convert either number into “OK,” “caution,” “danger,” “safe” or “unsafe.”

Former content or behaviour	Problem and correction
“Rayleigh method” and distributed-mass capability	The code used only √(k/m) with a point mass and three static stiffness formulas; no distributed shaft mass or Rayleigh quotient was implemented. The replacement names the actual point-mass model.
Result labelled first critical speed	The model cannot establish the first critical of a real rotor-bearing-seal system. It now reports an equivalent undamped speed-frequency marker.
Universal ±20%/±40% danger, caution and OK zones	No controlled source or machine scope supported those verdicts. They are removed; optional comparison is arithmetic only.
Universal API 610/612/617 percentage claims	Edition, equipment scope, clause and operating definition were not controlled. Exact criteria are not guessed and are marked NEEDS_LICENSED_SOURCE.
“Rigid rotor typically below 70% of critical” attributed around ISO balancing	ISO 21940-11/-12 address balancing behaviour; the public ISO 21940-12 scope excludes structural resonances. The unsupported shortcut and withdrawn ISO 1940 label are removed.
Steel/stainless/aluminium modulus presets	Young modulus depends on the controlled material, alloy, condition and temperature. The replacement requires a documented value and has no authoritative-looking preset.
Text example 6029 N/mm, 448.2 rad/s, 4280 rpm	The former code’s displayed inputs give 6040.049… N/mm, 448.703… rad/s and 4284.804… rpm, so prose and code disagreed. The new published example is generated by the same audited model and independently tested.
“Vibration amplifies exponentially”	The standard damped one-DOF response is a rational function of frequency ratio and damping, not exponential growth. The misleading wording is removed.
Defaults, presets, auto-calculation, partial parsing and persisted state	The page could issue an apparently authoritative answer without a controlled model record. It now starts blank, validates complete finite inputs, requires provenance and confirmation, and calculates only on explicit submit.

Only if the controlled one-DOF reduction shows that total mass is the correct effective mass at the selected lateral coordinate. For a real mode, effective/modal mass depends on the coordinate and mode shape.

No. It means an ideal beam with zero translation and zero slope at both ends. Real bearing, housing and pedestal flexibility must be represented in the actual rotor model or in a documented equivalent stiffness.

Only the ideal undamped linear steady-state model has an unbounded mathematical response exactly at resonance. Real response depends on damping, forcing, nonlinear limits and the complete system; this worksheet calculates no amplitude.

No. Identify the applicable licensed standard/specification and excitation orders, build the required rotor-dynamic model, evaluate damped response and transients, and obtain the responsible engineering review.