Foundations — Shock response spectrum (SRS)

This page builds every symbol the parent note throws at you, starting from a child who has never seen a spring, an integral, or a Greek letter. Read it top to bottom — each idea is the floor the next one stands on.

1. The picture behind everything: a mass on a spring

Before any symbol, meet the machine the whole topic is about.

Look at the figure. A block (the component — a circuit board, a bracket) sits on a spring and a dashpot (a piston that pushes back when you move it fast). The bottom of the spring is glued to the base — the wall of the spacecraft. When the spacecraft gets kicked, the base moves, the spring stretches, and the block jiggles.

Why start here? Because a real spacecraft has millions of wiggling parts, but near any one resonance each part behaves like this one simple bell (this is the promise of Modal analysis). Master one bell, and you can describe them all.

2. Position, and the two things that ride on top of it

Everything below is measured along the single up-down line of that block.

The parent note writes these with dots on top. A dot means "rate of change of," borrowed from Newton himself.

Why do we need acceleration specifically? Because acceleration is what breaks things. A gentle drift across a room never hurt anyone; being slammed (huge acceleration) shatters a solder joint. The whole SRS measures peak acceleration for exactly this reason. See Acceleration response in structures.

3. Base motion vs. relative motion — the trickiest picture

Here is the confusion that sinks most beginners. There are two motions happening at once.

• The base moves (the spacecraft wall jerks). Call its acceleration $a(t)$ — the shock we are handed.
• The block moves too, dragged along by the spring.
• What the spring actually feels is the difference between them: how much the block lags behind the base.

4. The Greek letters: naming the bell's personality

Every bell has a pitch and a ring-time. Two Greek letters capture them.

Why two versions of the same thing? $f_n$ is friendly for humans ("this board rings at 200 Hz"). ${\omega }_n$ is friendly for the equations, because the maths of circles and springs speaks in radians. They are the same fact in two costumes.

The figure shows three blocks plucked and released: one barely damped (long ring), one moderately, one heavily. The envelope — the fading dotted line — is set by $\zeta$; the wiggle rate is set by ${\omega }_n$.

5. The ingredients that make ${\omega }_n$ and $\zeta$

Where do pitch and ring-time come from? From three physical properties of the block-spring-damper.

These three combine into the personality letters:

Why the square root in ${\omega }_n$? Because the restoring push grows with stretch while the sluggishness grows with mass, and the wiggle-rate turns out to depend on their ratio's square root — the same maths that makes a pendulum's period depend on $\sqrt{L/g}$. You don't need to prove it here; just trust the shape: pitch $\propto \sqrt{k/m}$.

6. The law that governs the block: $F=ma$ in disguise

Now assemble the pieces into the parent's central equation.

Newton says: (mass)(acceleration) = (sum of forces). On our block, the forces are the spring pull, the damper drag, and the inertial kick from the shaking base.

Term by term, in words:

• $\ddot{z}$ — the block's own jiggle-acceleration.
• $2\zeta {\omega }_n\dot{z}$ — the damper fighting the speed.
• ${\omega }_n^{2}z$ — the spring fighting the stretch.
• $-a(t)$ — the base's kick, felt as a push in the opposite direction (the "pressed into your seat" feeling in an accelerating car). See Mechanical impedance for the force-vs-motion view.

Why does this one line matter? Because if you can solve it — find $z(t)$ from a given kick $a(t)$ — you can compute the block's slamming, and therefore its SRS.

7. The tools that solve it: sine, exponential, integral

The parent's solution (Duhamel's integral) uses three maths tools. Meet them plainly.

8. From one bell to the whole spectrum

One bell (one $f_n$) gives one number: its worst slam.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.