〰 Mechanical Vibration

Natural Frequency Calculator

Four calculators for vibration analysis. Natural frequency, damping, forced response, and beam modes. Free, no login required.

Stiffness (k) *

N/m

Spring or structural stiffness. Beam: k = 3EI/L³

Mass (m) *

Lumped mass at the vibrating point

System type (optional, affects result display)

System Model Undamped gives pure natural frequency

Damping Coefficient (c)

N·s/m

Leave blank for undamped calculation

Enter valid positive values for stiffness and mass.

Natural Frequency (fn)

—

〰️

Angular Frequency

—

rad/s

Period

—

seconds

Damped Freq (fd)

—

Critical Damping (cc)

—

N·s/m

Damping Ratio (zeta)

—

ratio

📐

Formulas Used

fn = (1/2π) × sqrt(k/m)

📊

Typical Stiffness Values for Common Systems Click row to fill k

System	Stiffness Formula	Typical k (N/m)	Notes

Mass (m) *

Vibrating body mass

Stiffness (k) *

N/m

Spring or structural stiffness

Damping Coefficient (c) *

N·s/m

Viscous damping constant of the system

Input Mode Switch between known and unknown

Enter valid values for all required fields.

Damping Ratio (zeta)

—

ratio

📉

Critical Damping (cc)

—

N·s/m

Natural Frequency

—

Damped Frequency

—

Log Decrement (delta)

—

per cycle

Decay per 10 cycles

—

% amplitude left

📐

Formulas Used

cc = 2 × sqrt(k × m) = 2m × wn
zeta = c / cc
wd = wn × sqrt(1 - zeta²)
delta = 2 × pi × zeta / sqrt(1 - zeta²)

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Damping Classification

Underdamped (zeta less than 1) System oscillates with decreasing amplitude. Most mechanical systems fall here. Typical structural zeta: 0.01 to 0.05.

Critically Damped (zeta = 1) Returns to equilibrium fastest without oscillating. Used in door closers, some instrument suspensions.

Overdamped (zeta greater than 1) Returns to rest slowly, no oscillation. Heavy oil-filled dashpots, hydraulic shock absorbers at high load.

Typical zeta values Steel structures: 0.01 to 0.02. Concrete: 0.03 to 0.05. Rubber mounts: 0.05 to 0.15. Vehicle suspensions: 0.2 to 0.4.

Stiffness (k) *

N/m

System stiffness

Mass (m) *

Vibrating mass

Damping Coefficient (c) *

N·s/m

Enter 0 for undamped forced response

Forcing Amplitude (F0) *

Peak amplitude of the applied harmonic force

Forcing Frequency (f) *

Frequency of the applied force

Static Deflection (xst)

xst = F0/k, auto-filled on calculate

Enter valid values for all required fields.

Dynamic Amplitude (X)

—

📡

Magnification Factor

—

X / xst

Freq Ratio (r)

—

omega / wn

Phase Angle

—

degrees

Natural Frequency (fn)

—

Static Deflection

—

📐

Formulas Used

xst = F0 / k
r = omega / wn
MF = 1 / sqrt((1 - r²)² + (2·zeta·r)²)
X = xst × MF
phi = atan(2·zeta·r / (1 - r²))

📖

Resonance and Magnification

At resonance (r = 1) MF = 1/(2·zeta). For zeta = 0.05, MF = 10. The amplitude is 10 times the static deflection at resonance.

Isolation region (r greater than 1.41) MF drops below 1 once r exceeds sqrt(2). This is the vibration isolation region. Machine mounts are designed to operate here.

Phase angle meaning Below resonance: response nearly in phase with force. At resonance: 90 degrees lag. Above resonance: nearly 180 degrees lag (opposing).

Practical tip Run machines above 1.41 times natural frequency for isolation, not below. A stiffer mount raises fn, which may push the operating point closer to resonance.

Young's Modulus (E) *

GPa

Steel: 200 GPa, Aluminium: 70 GPa, Concrete: 30 GPa

Second Moment of Area (I) *

mm⁴

Rectangle: bh³/12 · Circle: pi·d⁴/64

Beam Length (L) *

Effective span between supports

Linear Mass Density (rho·A) *

kg/m

Mass per unit length = density × cross-section area

Boundary Condition * Support conditions determine mode shape coefficients

Mode Number (n) Higher modes have larger beta·L values

Enter valid values for all fields.

Beam Natural Frequency

—

🏗️

Angular Freq (wn)

—

rad/s

Beta·L coefficient

—

dimensionless

Flexural Rigidity (EI)

—

N·m²

📐

Formula Used

fn = (betaL)² / (2pi·L²) × sqrt(EI / rhoA)

📏

Beta·L Values for Common Boundary Conditions Euler-Bernoulli beam theory

Boundary Condition	Mode 1	Mode 2	Mode 3
Simply Supported (SS)	3.1416	6.2832	9.4248
Fixed-Fixed (CC)	4.7300	7.8532	10.996
Cantilever (Fixed-Free)	1.8751	4.6941	7.8548
Fixed-Pinned (CP)	3.9266	7.0686	10.210

📖

Second Moment of Area Reference

Solid Rectangle (b × h) I = b·h³ / 12 (bending about horizontal axis). For a 50×100 mm section: I = 50×100³/12 = 4,166,667 mm⁴

Solid Circle (diameter d) I = pi·d⁴ / 64. For a 50 mm bar: I = pi×50⁴/64 = 306,796 mm⁴

Hollow Rectangle I = (B·H³ - b·h³) / 12 where B×H is outer and b×h is inner dimension

Hollow Circle (D outer, d inner) I = pi·(D⁴ - d⁴) / 64. Used for pipes, shafts, structural hollow sections.

👨‍🔧

Vaibhav Dhokpande

Builder, TaskJunction

I built this because vibration analysis should not require a Rs 50,000 software license or a university library login.

Natural frequency, damping ratio, forced response, beam modes. These are textbook formulas that every mechanical engineer learns in their third year. But when you actually need them on the job, everything is locked behind Matlab, ANSYS, or some European standards body.

The formula for natural frequency is over 150 years old. It belongs to everyone.

So I built this. No account. No ads in your face. No upgrade popup. You open it, type your numbers, and get the answer. That is the whole idea behind TaskJunction.

This is for the design engineer checking if a bracket will resonate with a pump, the student finishing a vibration assignment at midnight, and the maintenance guy who needs to know if that machine is running too close to resonance before it shakes itself apart.

Vaibhav

taskjunction.org