Skip to content
Resources / Vibration theory

From physical motion to engineering judgment

Understand the motion.
Interpret the test.

What makes a structure amplify an input? Why does damping matter? Start with one mass, one spring and one damper, then connect the response to real measurement questions.

Explore resonance ↓

Compare equal-RMS random vibration inputs ↓

Change a parameter. Follow the response.

Resonance explorer

Harmonic base excitation · SDOF
Natural frequency—
Frequency ratio—
Response / base—
Response phase—

Use the controls to explore the response of a linear spring–mass–damper system.

The model behind the motion

One mode. A useful starting point.

The base displacement is y(t); the mass displacement in a fixed reference frame is x(t). The spring and damper react to their relative motion.

mx¨+c(x˙−y˙)+k(x−y)=0m\ddot{x}+c(\dot{x}-\dot{y})+k(x-y)=0
fn=12πkm,ζ=c2km,r=ffnf_n=\frac{1}{2\pi}\sqrt{\frac{k}{m}},\quad \zeta=\frac{c}{2\sqrt{km}},\quad r=\frac{f}{f_n}
XY=1+i 2ζr1−r2+i 2ζr\frac{X}{Y}=\frac{1+i\,2\zeta r}{1-r^2+i\,2\zeta r}

The chart shows the magnitude of this complex ratio; phase describes the timing of the response relative to the base. For a single harmonic frequency, the same ratio applies to absolute acceleration amplitudes.

Same input RMS. Different structural response.

Random-vibration response explorer

Both inputs contain the same total acceleration variance. Spectrum A concentrates energy near 80 Hz; spectrum B near 250 Hz. Move the structural mode between them and compare the predicted absolute acceleration response.

How the comparison is calculated
Sa,out(f)=∣H(f)∣2Sa,in(f),arms=∫20500Sa(f) dfS_{a,\mathrm{out}}(f)=|H(f)|^2S_{a,\mathrm{in}}(f),\qquad a_{\mathrm{rms}}=\sqrt{\int_{20}^{500} S_a(f)\,df}

H is the absolute-acceleration transfer function of the same linear, viscously damped, base-excited SDOF model used above. These are stationary, zero-mean inputs with one-sided PSDs. The two spectra are alternative tests, not simultaneous inputs.

Each synthetic profile has a small flat background plus a Gaussian-shaped emphasis: A is centred at 80 Hz with a 20 Hz width parameter; B at 250 Hz with a 45 Hz width parameter. Both are normalized to the selected RMS over 20–500 Hz and are zero outside that band. The input profiles stay fixed when you move the mode.

Response variance is integrated in linear frequency using trapezoidal integration at 0.1 Hz spacing. RMS is the square root of variance; the area under a curve on logarithmic screen coordinates is not the variance.

This predicts acceleration, not stress, fatigue damage or a qualification verdict. Multiple modes, nonlinear behaviour and other response quantities require their own models.

Reference: Tom Irvine, An Introduction to Shock and Vibration Response Spectra. Explore Tom Irvine’s random vibration and PSD reference page (external website).

Use the PSD / RMS calculator →Read the test-evaluation study →
Build the physical picture

A short path through vibration theory

01 / Free vibration

What sets the natural frequency?

Displace a mass attached to a spring and release it. Mass stores kinetic energy; the spring stores elastic energy. Their balance sets the natural frequency. More stiffness raises it; more mass lowers it. With damping, the oscillation decays.

Explore damping relations →
02 / Forced vibration

What changes when the base keeps moving?

A shaker supplies continuous excitation. The steady-state response depends on excitation frequency relative to the natural frequency, as well as damping. Near resonance, even a modest base motion can produce a much larger response.

Try the resonance explorer →
03 / Damping & isolation

Is more damping always better?

Damping dissipates energy and reduces the resonant peak. For harmonic base excitation, it can also increase transmitted motion well above resonance. In this linear model, absolute-motion isolation starts above a frequency ratio of √2.

Connect damping, Q and bandwidth →
04 / Random vibration

Can equal RMS levels produce different responses?

Yes. Overall RMS describes the integrated input spectrum, but not where its energy lies in frequency. Response depends on the input PSD weighted by the squared magnitude of the relevant transfer function. Energy close to a lightly damped mode can contribute strongly.

Calculate input PSD and RMS →

Compare equal-RMS inputs interactively →

05 / Shock response

Why is peak acceleration not the whole story?

A transient excites systems differently depending on pulse shape, duration and natural frequency. A shock response spectrum reports peak responses of a family of oscillators at a stated damping. It does not uniquely reconstruct the original time history.

Read the shock and SRS relations →
Theory in the portfolio

A peak is a starting point for investigation.

An FFT peak identifies signal content. Calling it a structural mode requires more evidence about the excitation and the system response. Likewise, a shift between resonance searches calls for checking setup repeatability and environmental conditions before drawing a conclusion about structural change.

See the vibration-test evaluation study →
Explore the signal-analysis example →

Continue reading

Check measurement quality before interpreting a response →

Explore the cantilever modal-analysis study → to connect beam frequencies, spatial mode shapes and synthetic modal correlation.

The SDOF response, transmissibility and shock explanations follow the analytical framework in Tom Irvine’s An Introduction to Shock and Vibration Response Spectra. Visit Tom Irvine’s Vibrationdata reference library (external website) for further reading.

For the connection between structural response and test control, see NASA’s Force Limited Vibration Testing Monograph — NASA publication record (external website).

Use the practical formula guide →