It changes the shell's mass and stiffness, so it moves the shell's modes.
Depth and rib width decide by how much.
Fig. 4
Machining removes mass to a plan.Machining removes mass to a plan.
A Kumiko shell starts as a 10-ply maple shell. The isogrid is cut into the outside: triangular pockets through some of the outer plies, with ribs left standing between them.
Every pocket takes mass out of the wall. Every rib left behind keeps some stiffness.
Fig. 4 / Cutting an isogrid into 10-ply maple14 × 6.5 · pattern covers 80% of the shell
Shell
Mass
Stiffness
Oval mode
n=3
n=4
n=5
Plain shell7.2 mm, 10 plies
930 g
100%bending
95 Hz
268 Hz
515 Hz
833 Hz
Machined5 of 10 plies cut
703 g-24%
49%
77 Hz-19%
217 Hz-19%
416 Hz-19%
673 Hz-19%
Same weight, thinner5.4 mm solid
703 g-24%
43%
72 Hz-24%
203 Hz-24%
389 Hz-24%
630 Hz-24%
Where the shell modes sit against the heads
Where the heads are busiest
100 Hz200 Hz500 Hz1k Hz
Dashed: plain shell. Orange: machined shell. Shell modes n = 2 to 5.
3.6 mm
4.0 mm
Pockets take mass out of the outer plies and leave ribs running three ways. Every extra millimeter of depth makes the wall lighter and less stiff. The first cuts remove the outer plies, which do most of the work in bending, so stiffness falls faster than mass and the shell's modes drop toward the heads. Past about half the wall, the ribs carry much of the bending: the wall keeps getting lighter and softer, but the two fall together and the modes level off. Depth decides where the shell's modes land. Simplified smeared-rib model; real shells add bearing edges, lugs and glue lines.
Mass and stiffness move the shell's modes.Mass and stiffness move the shell's modes.
A shell's modes go up with stiffness and down with mass. Take out mass alone and they would rise. Take out stiffness alone and they would fall.
A pocket takes out both. The deeper the pocket, the lighter and less stiff the wall. What depth changes is the balance between the two, and that balance is the design lever.
The first cuts cost the most stiffness. The outer plies do most of the work in bending, so stiffness falls faster than mass and the shell's modes drop toward the heads.
Deeper cuts keep making the wall lighter and less stiff, but the ribs now carry much of the bending. Stiffness and mass fall together, and the modes level off.
Why a triangle grid.Why a triangle grid.
Ribs in three directions, 60° apart, resist bending equally in every direction. The shell stays even under the hoop and the lugs.
Square or one-way ribs would make the shell stiffer one way than the other. Aerospace uses the same triangle grid to stiffen panels for the same reason.
It has to survive the drum.It has to survive the drum.
A machined shell has to keep material where it carries load: at the bearing edges, and under the lugs and throwoff.
Fig. 4 assumes the pattern covers about 80% of the shell, with full-thickness bands left for those jobs.
Kumiko Series in Honey Wheat: the isogrid, cut into maple.
Lighter is a choice.Lighter is a choice.
A machined shell is mechanically rewritten. The pattern you see is the change you hear.
That is the idea behind the Kumiko, Hive and Asanoha shells: geometry that looks like what it does.
Bibliography
Selected sources.Selected sources.
Primary papers and references used throughout the essay.
