Module solids 

Solids of revolution: volume, centroid, moments of inertia and surface areas of a profile swept about its axis, either filled or as a wall of constant thickness normal to the outer surface.

Integrals. With outer radius y(x), inner radius r_i(x) (zero when filled) and x measured aft of the forward end over [0, L], per unit density:

V    = π ∫ (y² − r_i²) dx                     volume
x̄    = π ∫ x (y² − r_i²) dx / V               centroid, aft of the forward end
J_a  = (π/2) ∫ (y⁴ − r_i⁴) dx                 moment about the axis
J_t  = π ∫ [(y⁴ − r_i⁴)/4 + x² (y² − r_i²)] dx − V x̄²    moment about a transverse axis
                                                         through the centroid
S    = 2π ∫ y √(1 + y′²) dx                   outer (wetted) area, ends excluded
A_p  = 2 ∫ y dx,   x_p = ∫ x y dx / ∫ y dx    planform (side-view) area and its centroid

Each slice is a thin annulus of radii r_i < y and thickness dx, with dI_a = (π/2)(y⁴ − r_i⁴) dx about the axis and dI_t = (π/4)(y⁴ − r_i⁴) dx about its own diameter (Meriam and Kraige, appendix B), moved to the reference plane by the parallel-axis theorem.

Walls. A wall of thickness t is the set of points of the solid within t of the outer surface, so the thickness is measured normal to the surface, as a molded or laid-up shell is made. Its inner radius at station x is the lower envelope of circles of radius t centered on the profile:

r_i(x) = max(0, min_{|s − x| ≤ t} [ y(s) − √(t² − (x − s)²) ])

A point below the profile is at least t from every surface point exactly when it lies below all those circles: if it lay above the lower half of the circle about some surface point, the continuous profile would cross the point’s height closer than t, so the point would be in the wall anyway. The surface is the profile over its own length, ends included, with no extension past a cut end. At an end where the surface meets the end plane at an obtuse angle inside the wall, the wall’s inner corner is rounded by the rim’s circle instead of cut square. That removes t² (tan φ − φ)/2 of section per unit rim length, with φ the surface’s angle to the axis there: 3.1e-4 t² at 7°, and it grows without bound only as the end turns vertical, where a square cut would close the end with a disc. The integration is split where r_i reaches zero and where the nearest surface point moves between the lateral surface and a rim.

The integrals use hpr_core::quadrature::integrate on integrands scaled to order one. See docs/physics/mass.md.

Structs

RevolvedGeometry

Volume, centroid, moments and areas of a solid of revolution per unit density; multiply the volume and moments by a density to get mass and inertia.

Enums

Wall

Whether a solid of revolution is filled or a wall.

Functions

revolve

Computes the volume, moments and areas of profile swept about its axis with wall.