Shock response spectrum (SRS): Visual walkthrough

We assume you know only what a push, a spring, and a wobble are. Everything else we construct.

Step 1 — One tiny wobbler on a shaking floor

WHAT. We name three properties of the wobbler:

$m$ = mass of the block (how much stuff, in kg). Big $m$ = hard to shove.
$k$ = stiffness of the spring (how hard it pushes back per metre stretched, N/m). Big $k$ = snappy spring.
$c$ = damping of the piston (how hard it resists speed, N per m/s). Big $c$ = the wobble dies fast.

WHY these three. They are the only ingredients a bell needs to ring and then go quiet: a mass to carry momentum, a spring to pull it back (that's the ringing), a damper to bleed energy (that's the fading). Every real bracket, board, or tank mount reduces to this near its main resonance.

PICTURE. In the figure, the floor carries a jagged shock. The block lags behind the floor — that lag is the thing we must track.

Why relative? Because the spring and damper only feel the stretch between block and floor — they don't care about the whole room, only about how far apart their two ends are.

Step 2 — Newton on the shaking floor gives the equation of motion

WHAT. Add up every force on the block. Newton says mass × acceleration = sum of forces.

Spring force $= - k z$ — pulls back, opposite to the stretch (hence minus).
Damper force $= - c \overset{z}{˙}$ — resists motion, opposite to the speed.
Inertial (fictitious) force $= - m a (t)$ — the backward shove from riding the accelerating floor. Here $a (t)$ is the floor's acceleration (the shock we were handed).

WHY the minus on $ma (t)$ . When the floor accelerates one way, the block feels thrown the other way — exactly the car example. Look at the pink arrow in the figure pointing opposite the floor's motion.

Now divide every term by $m$ so the mass drops out and only ratios survive:

Recall Why

$ω_{n} = k / m$ and not $k / m$ ? Where does $k / m$ come from ::: Set damping and forcing to zero; $\overset{z}{¨} = - (k / m) z$ is solved by a sine of frequency $k / m$ — the classic "acceleration proportional to minus position" swing.

See Quality factor Q and damping: engineers often quote $Q = \frac{1}{2 ζ}$ , so $ζ = 0.05$ means $Q = 10$ .

Step 3 — What a single kick does (the building block)

WHAT. Feed the wobbler one instantaneous kick (an impulse) at time zero. It responds with a decaying sine: it swings, and each swing is smaller than the last.

WHY a decaying sine. The spring makes it swing (sine); the damper makes each swing shrink (a shrinking multiplier). Two effects, two factors — multiply them.

Why the tool "exponential × sine" and not, say, a polynomial? Because it is the only shape whose acceleration equals a fixed combination of itself and its own speed — which is exactly what the EOM demands. Nothing else fits.

Step 4 — Add up all the kicks: Duhamel's integral

WHAT. Slide a kick's response so it starts at time $τ$ (its "age" is $t - τ$ ), scale it by that kick's strength $- a (τ) d τ$ , and integrate from the start ( $0$ ) to now ( $t$ ).

PICTURE. In the figure, three sample kicks each launch their own shrinking sine; the black curve is their running sum — that sum is $z (t)$ .

Step 5 — From relative wobble to the acceleration that actually breaks things

WHAT. True (absolute) acceleration = what the floor did + what the block did relative to the floor: $a_{abs} (t) = a (t) + \overset{z}{¨} (t)$

Now use the EOM from Step 2 to erase the ugly $\overset{z}{¨}$ . Rearranging $\overset{z}{¨} + 2 ζ ω_{n} \overset{z}{˙} + ω_{n}^{2} z = - a (t)$ gives $\overset{z}{¨} = - a (t) - 2 ζ ω_{n} \overset{z}{˙} - ω_{n}^{2} z$ . Substitute:

WHY it cancelled. The floor's shove enters the block only through the spring and damper — there is no other path. So once we know the stretch and speed, we already know everything the floor did to the block.

Step 6 — Take the worst moment: that single number is the SRS point

Step 7 — Sweep every frequency: the spectrum is born

WHAT — and all the cases you must never be surprised by:

Very low $f_{n}$ (slow wobbler). Its period is far longer than the pulse; the pulse is done before the block has moved much. The block barely responds → SRS is small, rising like $f_{n}^{2}$ (that's $+ 40$ dB/decade). See the left rising branch.
The knee, near $f_{n} \approx \frac{1}{π τ}$ (with $τ$ = pulse duration). Here the wobbler completes about one swing during the pulse — perfect timing to "ring up." SRS peaks. A shorter pulse pushes this knee to a higher frequency.
Very high $f_{n}$ (fast wobbler). It swings so fast it simply tracks the floor peak; no amplification is left to gain. SRS flattens toward the raw peak acceleration ( $0$ dB/octave).

The one-picture summary

The whole chain: shock $a (t)$ → chop into kicks → Duhamel sum → subtract to get true acceleration → grab the peak → sweep $f_{n}$ → spectrum.

Recall Feynman retelling — say it back in plain words

A bolt explodes and the floor of the spacecraft lurches in a jagged, milliseconds-long jerk. I want to know: for a little gadget that likes to wobble at some speed, how hard does it get slammed? So I imagine that gadget as one block on one spring with a shock-absorber. I climb onto the shaking floor to watch it, which invents a fake backward shove — that shove is the shock feeding in. I figure out how the block answers a single instantaneous flick: it swings and the swings shrink. Since flicks don't interfere, I chop the whole messy shock into a rain of flicks and add up every answer — that's Duhamel's integral, giving me the block's motion relative to the floor. But damage cares about the real acceleration in the room, so I add back the floor's motion; magically the spring stretch and the damper speed alone tell me that real acceleration. I scan the whole ring-down and keep the single biggest slam — that's one dot. Then I do it all again for a slow wobbler, a medium one, a fast one — hundreds of them — and connect the dots. Slow ones barely feel it (rising branch), the one tuned to about one swing per pulse gets rung up the most (the knee peak), and fast ones just ride the floor's peak (the flat top). That connected curve is the shock response spectrum — one picture that tells every component whether it lives or dies.

Related: Pyroshock environments · Shock testing methods · MIL-STD-810 Method 516 · Modal analysis · Mechanical impedance · Force limiting in shock testing · Vibration power spectral density (PSD)