MEC-31

Natural frequency ω=√(k/m)

ωn = √(k/m), fn = ωn/2π. One-mass, one-spring, no damper.

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VibrationUndamped SDOF

Governing equation

\omega_n=\sqrt{k/m},\quad f_n=\omega_n/2\pi

$m \in \left[0.1, 80\right]$

where

k: Stiffness (N/m)
m: Mass (kg)
\omega_n: Natural ω (rad/s)
f_n: Natural frequency (Hz)

The figure, part by part

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①

Governing law — red

\omega_n=\sqrt{k/m},\quad f_n=\omega_n/2\pi

$m \in \left[0.1, 80\right]$

$m = 0.1$ $\omega_n = 22.51$

Along the curve

②

Domain of the variable — blue

m_{\min} = 0.1,\quad m_{\max} = 80

$m \in \left[0.1, 80\right]$

$m = 0.1$ $m = 0.1$

Along the curve

③

Response curve — green

\omega_n=\sqrt{k/m},\quad f_n=\omega_n/2\pi

$m \in \left[0.1, 80\right]$

$m = 0.1$ $\omega_n = 22.51$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\omega_n}{\mathrm{d}m}

$m \in \left[0.1, 80\right]$

$m = 0.603$ $\omega_n^{\prime} = -15.66$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-31 — Natural frequency ω=√(k/m)) is the form associated with Undamped SDOF. Working symbols: $k$ , $m$ $\rightarrow$ $\omega_n$ , $f_n$ . Newton mẍ + kx = 0. The eigenvalue of the conservative oscillator is √(k/m).

Purpose

Purpose: compute

\omega_n

f_n

from

k

m

in Mechanical via

\omega_n=\sqrt{k/m},\quad f_n=\omega_n/2\pi

ωn = √(k/m), fn = ωn/2π. One-mass, one-spring, no damper. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k = 2000.000\,\mathrm{N/m}

m = 4.000\,\mathrm{kg}

, the governing relation

\omega_n=\sqrt{k/m},\quad f_n=\omega_n/2\pi

yields

\omega_n = 22.36\,\mathrm{rad/s}

f_n = 3.559\,\mathrm{Hz}

. A mass on a spring, a period bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Stiffness k2000 N/mMass m4 kg

Outputs

Natural ω \omega_n22.36 rad/s
Natural frequency f_n3.559 Hz

Reading speed

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Narration of this film

A mass on a spring, a period bar.

Newton mẍ + kx = 0. The eigenvalue of the conservative oscillator is √(k/m).

Reading speed

Watch on YouTube