Formula sheet

Mechanical Vibration and Rotating Machinery Reliability Formula Sheet

Vibration formulas for forcing frequencies, natural frequency, damping, transmissibility, unbalance force, speed separation, sampling, fatigue, and validation.

Branch: Mechanical Engineering
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass relationships used to calculate vibration risk and rotating-machinery reliability. Use it to convert running speed into forcing frequencies, estimate natural frequency, screen resonance separation, calculate isolation transmissibility, estimate unbalance force, set sampling requirements, connect vibration exposure to fatigue, and document validation evidence.

The formulas are intentionally practical. They do not replace machine-specific standards, API or ISO acceptance criteria, bearing catalog rules, rotor-dynamic models, finite element modal analysis, torsional vibration studies, balancing procedures, or site safety requirements. They are useful when an engineer needs a traceable calculation before commissioning, troubleshooting, design review, or maintenance release.

Unit Conventions and Notation

Use consistent SI units unless a practical unit form is stated.

Symbol	Meaning	Typical unit
$n$	rotational speed	$\text{rpm}$
$f$	cyclic frequency	$\text{Hz}$
$\omega$	angular speed	$\text{rad/s}$
$m$	supported or rotating mass	$\text{kg}$
$k$	stiffness	$\text{N/m}$
$c$	viscous damping coefficient	$\text{N s/m}$
$\zeta$	damping ratio	dimensionless
$r$	frequency ratio, $f/f_n$	dimensionless
$T_R$	force transmissibility	dimensionless
$U$	unbalance, $me$	$\text{kg m}$
$F_u$	rotating unbalance force amplitude	$\text{N}$
$v$	vibration velocity	$\text{m/s}$ or $\text{mm/s}$
$X$	displacement amplitude	$\text{m}$ or $\text{mm}$
$f_s$	sampling frequency	$\text{Hz}$
$N$	number of samples or fatigue cycles	dimensionless

State whether vibration values are RMS, peak, peak-to-peak, filtered, unfiltered, casing, shaft-relative, horizontal, vertical, axial, synchronous, order-tracked, or broadband. Many vibration disputes are unit and convention disputes.

Speed, Orders, and Forcing Frequencies

Rotational frequency from speed:

\displaystyle f_{1x}=\frac{n}{60}

Angular speed:

\displaystyle \omega=2\pi f=\frac{2\pi n}{60}

The $m$ th order forcing frequency:

f_{m}=m f_{1x}

Gear mesh frequency:

f_{mesh}=N_t f_{shaft}

where $N_t$ is the number of teeth on the gear rotating at $f_{shaft}$ .

Blade-passing or vane-passing frequency:

f_{BPF}=N_b f_{shaft}

where $N_b$ is the number of blades, vanes, lobes, or repeated rotating features.

Worked Example: Mapping a Fan Spectrum

A fan runs at $1785\ \text{rpm}$ . The impeller has 12 blades, and a gear stage has 32 teeth on the shaft being measured.

\displaystyle f_{1x}=\frac{1785}{60}=29.75\ \text{Hz}

\omega=2\pi(29.75)=186.9\ \text{rad/s}

The common synchronous orders are:

f_{2x}=2(29.75)=59.5\ \text{Hz}

f_{BPF}=12(29.75)=357\ \text{Hz}

f_{mesh}=32(29.75)=952\ \text{Hz}

The result says that a strong peak near $29.75\ \text{Hz}$ is synchronous 1x vibration, while peaks near $357\ \text{Hz}$ or $952\ \text{Hz}$ should be compared with blade-passing and gear-mesh mechanisms. This does not diagnose the fault by itself. Phase, direction, load dependence, bearing data, process conditions, and speed dependence must agree with the mechanism.

Single-Degree-of-Freedom Natural Frequency

For a simple mass and equivalent stiffness:

\displaystyle \omega_n=\sqrt{\frac{k}{m}}

\displaystyle f_n=\frac{1}{2\pi}\sqrt{\frac{k}{m}}

Static deflection under weight:

\displaystyle \delta_s=\frac{mg}{k}

so an approximate vertical natural frequency can also be estimated from:

\displaystyle f_n=\frac{1}{2\pi}\sqrt{\frac{g}{\delta_s}}

These equations assume a single dominant mode, linear stiffness, small displacement, rigid supported equipment, and no strong coupling with pipes, ducts, frames, soil, foundations, or nearby machines.

Worked Example: Mount Natural Frequency

A fan skid has supported mass $m=1600\ \text{kg}$ and four installed isolators with total vertical stiffness $k=12.8\ \text{MN/m}$ .

\displaystyle \omega_n=\sqrt{\frac{12.8\times10^6}{1600}}=89.4\ \text{rad/s}

\displaystyle f_n=\frac{89.4}{2\pi}=14.2\ \text{Hz}

The static deflection check is:

\displaystyle \delta_s=\frac{1600(9.81)}{12.8\times10^6}=0.00123\ \text{m}=1.23\ \text{mm}

If the fan runs at $900\ \text{rpm}$ , then $f_{1x}=15.0\ \text{Hz}$ . The installed mount frequency is close to the forcing frequency, so the support may amplify vibration instead of isolating it. The calculation is simplified, but it correctly flags the need for a speed sweep, operating deflection shape, mount data check, and floor response measurement.

Frequency Ratio, Separation, and Critical Speed Screening

Frequency ratio:

\displaystyle r=\frac{f}{f_n}=\frac{\omega}{\omega_n}

Separation margin from a mode or critical speed:

\displaystyle M_{sep}=\frac{|f_n-f_{exc}|}{f_{exc}}

Critical speed ratio:

\displaystyle R_c=\frac{n_{run}}{n_{crit}}

A first-pass design review normally tries to keep steady operating speeds away from lightly damped structural modes, rotor critical speeds, and torsional natural frequencies. The required separation is not universal. It depends on damping, consequence of failure, uncertainty in stiffness and mass, transient speed crossings, and the applicable machinery standard.

Worked Example: Is the Separation Acceptable?

A measured structural mode is at $34\ \text{Hz}$ and the operating shaft speed is $1785\ \text{rpm}$ .

f_{exc}=1785/60=29.75\ \text{Hz}

\displaystyle M_{sep}=\frac{|34-29.75|}{29.75}=0.143=14.3\%

A 14.3 percent separation may be acceptable for a well-damped, low-consequence structure, but it is marginal for a lightly damped machine with large uncertainty or high consequence. The engineering response should be to confirm damping and mode shape, not to treat the percentage as an automatic pass.

Damping Ratio and Logarithmic Decrement

Critical damping coefficient:

c_c=2\sqrt{km}=2m\omega_n

Damping ratio:

\displaystyle \zeta=\frac{c}{c_c}

Logarithmic decrement from two successive free-decay peaks:

\displaystyle \delta=\ln\left(\frac{x_1}{x_2}\right)

For lightly damped systems:

\displaystyle \zeta\approx\frac{\delta}{2\pi}

More exactly:

\displaystyle \zeta=\frac{\delta}{\sqrt{4\pi^2+\delta^2}}

Worked Example: Estimating Damping from Decay

A coast-down bump test gives two successive displacement peaks of $8.0\ \text{mm}$ and $5.0\ \text{mm}$ .

\delta=\ln(8.0/5.0)=0.470

\displaystyle \zeta=\frac{0.470}{\sqrt{4\pi^2+0.470^2}}=0.0746

The estimated damping ratio is about 7.5 percent. That is enough to reduce the resonance peak compared with an almost undamped support, but it does not make operation near resonance harmless. The estimate is only reliable if the decay is dominated by one mode and the sensor is measuring the same response coordinate in both peaks.

Forced Vibration Magnification

For harmonic force applied to a damped single-degree-of-freedom system, the displacement magnification factor is:

\displaystyle H(r)=\frac{1}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}

where $r=f/f_n$ . Near $r=1$ , small damping and small modeling errors can cause large changes in response.

At exact resonance, the approximate displacement magnification is:

\displaystyle H(1)\approx\frac{1}{2\zeta}

for low damping.

Engineering Comment

This magnification applies to a very simplified model. Real machines can have multiple modes, anisotropic bearing stiffness, base flexibility, nonlinear mounts, pipe loads, casing modes, rotor modes, aerodynamic forcing, looseness, and speed-dependent excitation.

Vibration Isolation Transmissibility

For a force-excited single-degree-of-freedom isolator, force transmissibility is commonly written as:

\displaystyle T_R=\frac{\sqrt{1+(2\zeta r)^2}}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}

Isolation begins only when the transmissibility drops below 1. For low damping, that requires approximately:

r>\sqrt{2}

Adding damping reduces the resonance peak but can increase high-frequency transmitted force. Isolation design is therefore a tradeoff, not simply “more damping is better.”

Worked Example: Why a Soft Mount Still Amplifies

Use the fan skid from the natural-frequency example:

$f_n=14.2\ \text{Hz}$
operating forcing frequency $f=15.0\ \text{Hz}$
damping ratio $\zeta=0.08$

\displaystyle r=\frac{15.0}{14.2}=1.06

\displaystyle T_R=\frac{\sqrt{1+(2(0.08)(1.06))^2}}{\sqrt{(1-1.06^2)^2+(2(0.08)(1.06))^2}}\approx5.0

The mount transmits about five times the static-force amplitude predicted by this simplified model. That explains why a machine on isolators can make a floor vibrate more after a mount replacement. The corrective action is not just “rebalance the fan”; it may require lower stiffness, different damping, flexible services, frame checks, and measured validation at operating speed.

Rotating Unbalance Force

Unbalance is:

U=me

where $m$ is the unbalanced rotating mass and $e$ is eccentricity.

The harmonic force amplitude from unbalance is:

F_u=U\omega^2=me\omega^2

Because unbalance force scales with speed squared, a modest increase in speed can strongly increase bearing, foundation, and casing response.

Worked Example: Unbalance Force at Running Speed

A residual unbalance is estimated as $U=0.012\ \text{kg m}$ at $1800\ \text{rpm}$ .

\displaystyle f=\frac{1800}{60}=30.0\ \text{Hz}

\omega=2\pi(30.0)=188.5\ \text{rad/s}

F_u=0.012(188.5)^2=426\ \text{N}

The force is not enormous compared with the static weight of many machines, but it is rotating, cyclic, and applied at bearing locations through a flexible structure. If the speed passes close to a mode, the response can be much larger than the force alone suggests.

Velocity, Displacement, and Acceleration for Harmonic Motion

For sinusoidal displacement:

x(t)=X\sin(\omega t)

Velocity amplitude:

V=\omega X

Acceleration amplitude:

A=\omega^2X

RMS value for a pure sine:

\displaystyle x_{RMS}=\frac{X}{\sqrt{2}}

and similarly for velocity and acceleration.

Worked Example: Converting Velocity RMS to Displacement

A bearing housing has $8.0\ \text{mm/s RMS}$ at $29.75\ \text{Hz}$ , dominated by one sinusoidal 1x component.

v_{peak}=\sqrt{2}(8.0)=11.3\ \text{mm/s}

\omega=2\pi(29.75)=186.9\ \text{rad/s}

\displaystyle X=\frac{v_{peak}}{\omega}=\frac{11.3}{186.9}=0.0605\ \text{mm}

The displacement amplitude is about $0.061\ \text{mm}$ for the synchronous component. This conversion is useful when checking runout, clearance, casing motion, or shaft-relative probes, but it is only valid for a narrowband sinusoidal component. Broadband vibration cannot be converted this way without spectral information.

Sampling, Nyquist Limit, and FFT Resolution

Nyquist requirement:

f_s>2f_{max}

In practice, allow margin for anti-alias filtering and instrument roll-off:

f_s\geq 2.5f_{max} \quad \text{to} \quad 4f_{max}

Frequency resolution for an FFT record:

\displaystyle \Delta f=\frac{f_s}{N}=\frac{1}{T_{record}}

where $T_{record}$ is the time duration of the sampled record.

Number of revolutions captured:

N_{rev}=f_{1x}T_{record}

Worked Example: Measurement Setup for a 1 kHz Spectrum

A commissioning test must capture vibration components up to $1000\ \text{Hz}$ and separate peaks about $1\ \text{Hz}$ apart.

Choose:

f_s=2560\ \text{Hz}

This gives a Nyquist frequency of:

\displaystyle f_N=\frac{f_s}{2}=1280\ \text{Hz}

For $1\ \text{Hz}$ resolution:

\displaystyle T_{record}\geq\frac{1}{1}=1.0\ \text{s}

So:

N=f_sT_{record}=2560(1.0)=2560\ \text{samples}

The setup is enough for a basic spectrum, but it may not be enough for order tracking during speed changes. If shaft speed drifts during the record, a tachometer reference and order analysis may be needed to prevent spectral smearing.

Fatigue Exposure from Vibration

Number of cycles accumulated by a periodic vibration component:

N=f t

If a vibration mode creates a stress amplitude $\sigma_a$ , a simple Miner damage estimate is:

\displaystyle D=\sum_i \frac{n_i}{N_i}

where $n_i$ is the number of applied cycles in bin $i$ , and $N_i$ is the fatigue life at that stress amplitude from the relevant S-N curve.

Mean-stress screening can use a Goodman-type relation:

\displaystyle \frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_{ut}}\leq\frac{1}{N_f}

This is a screening equation, not a substitute for a qualified fatigue assessment, weld detail classification, fracture mechanics, material scatter, surface finish, corrosion, temperature, residual stress, or inspection planning.

Worked Example: Cycle Count Behind a Vibration Alarm

A machine runs for $2000\ \text{h}$ with a dominant 1x vibration at $29.75\ \text{Hz}$ .

t=2000(3600)=7.20\times10^6\ \text{s}

N=29.75(7.20\times10^6)=2.14\times10^8\ \text{cycles}

The cycle count is already in the high-cycle fatigue range. Even if nominal stress looks modest, welded brackets, impeller weld repairs, cracked guards, sensor mounts, baseplate corners, and keyway regions deserve attention. A vibration alarm is therefore a reliability signal, not only a comfort or noise issue.

Trend Ratios, Alarm Evidence, and Reliability Screening

Trend ratio for a monitored indicator:

\displaystyle R_{trend}=\frac{x_{current}}{x_{baseline}}

Rate of change:

\displaystyle \dot{x}\approx\frac{x_2-x_1}{t_2-t_1}

Risk priority number:

RPN=SOD

where $S$ is severity, $O$ occurrence, and $D$ detection rating.

Availability from mean time between failures and mean time to repair:

\displaystyle A=\frac{MTBF}{MTBF+MTTR}

For an exponential reliability approximation:

R(t)=e^{-t/MTBF}

For Weibull reliability:

R(t)=e^{-(t/\eta)^\beta}

Use reliability formulas carefully. Vibration faults are often condition-dependent, censored by maintenance action, and influenced by operating regime. Do not fit a Weibull curve to a small, inconsistent failure sample and treat it as proof.

Worked Example: RPN After Corrective Action

Before field balancing, a fan has severity $S=8$ , occurrence $O=5$ , and detection $D=4$ .

RPN_{before}=8(5)(4)=160

After correction and verification, occurrence is reduced to $O=2$ , and detection improves to $D=3$ because the machine now has a repeatable tachometer-based vibration baseline.

RPN_{after}=8(2)(3)=48

The RPN reduction is useful for prioritization, but it is not a physical probability. The release decision still needs measured vibration after correction, phase stability, bearing temperature, motor current, process condition, and follow-up trend data.

Validation Checklist for Calculations

For vibration or rotating-machinery release, record:

Check	Why it matters
speed and tachometer reference	distinguishes 1x, 2x, gear mesh, blade passing, and non-synchronous content
sensor location and axis	prevents comparing casing, foundation, shaft, radial, and axial values as if identical
RMS, peak, and filtering convention	avoids false acceptance or false alarm due to unit mismatch
operating point	vibration can change with speed, load, flow, pressure, temperature, and control state
natural frequency or speed sweep evidence	confirms whether a high response is excitation or resonance amplification
phase stability	helps separate imbalance from looseness, rubs, intermittent faults, and measurement error
bearing and lubrication data	vibration may be symptom, cause, or consequence of bearing distress
repeat measurement after correction	proves the change worked and creates a new baseline

Common Mistakes

Treating a 1x peak as imbalance without checking resonance, runout, looseness, misalignment, process forcing, and sensor error.
Comparing RMS velocity with peak displacement or acceleration without converting conventions.
Designing isolators from static load capacity only, without checking natural frequency and transmissibility.
Sampling below the needed bandwidth and then trusting an aliased spectrum.
Ignoring transient speed crossings through critical speeds.
Using a global vibration limit without considering machine class, location, foundation, sensor mounting, bearing type, and consequence.
Assuming that a balance correction also fixes cracked supports, soft foot, duct constraint, thermal growth, or bearing defects.
Reporting a formula result without uncertainty, measurement setup, operating point, and validation evidence.

How to Use This Sheet in the Cluster

Use the topic page to understand the diagnostic workflow and physical mechanisms. Use the exercise set to practice full calculations. Use the imbalance and isolator case studies to see how these formulas support engineering decisions. Use the stress, fatigue, machine design, reliability, instrumentation, and digital-twin references when vibration must be converted into structural risk, maintenance action, measurement design, or model validation.

REF

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