Physics examples
These examples are classic problems of physics and engineering mechanics, each with a known answer to check against. Many are eigenvalue, elasticity or flow problems. For these, the geometry is drawn in planex2d, netgen meshes it, NGSolve solves it, and the result is compared with a formula, a table or a published value, from Euler’s buckling load to the stress intensity factor of a crack. Others need only the geometry, as the waterline of a heeled boat at constant displacement, rays reflected
by conic mirrors or a bead racing down a ramp, an arc and a cycloid.
In this section
Isospectral drums Two drums with the same eigenvalues, so the shape cannot be heard. The Helmholtz resonator The note of a bottle from an acoustic eigenvalue solve, against the formula with its end corrections. Chladni figures The modes of a free square plate against the tabulated frequencies, and the nodal lines where sand gathers. Tuning a marimba bar The arch under a bar that puts its first overtone two octaves up, from its bending modes. The tuning fork Two tines swinging against each other leave the stem at rest; the clang mode and the length for 440 Hz. Runs in the browser Boat stability The waterline at constant displacement and the righting lever curve. Flow through sand Stokes flow through packed grains against the Kozeny-Carman law. Solar shading The shaded share of a roof panel over a day, every quarter of an hour, at the solstices and the equinox. The catenary arch Hooke's inverted chain in compression, a semicircle opening at its haunches, and the thrust w a. Euler buckling The critical load of a column for the four end conditions, against Euler and Engesser's shear correction. Cantilever deflection A beam as a plane solid against Euler–Bernoulli and Timoshenko, from 1 to 20 times as long as high. Stress in a thick-walled pipe Lamé's hoop stress, which no wall thickness brings below the pressure, and a shrink-fitted ring that lowers it. The stress intensity factor of a crack A central crack from the energy it releases, against Griffith and Feddersen. Runs in the browser Mirrors from conic sections A whispering gallery and a Cassegrain telescope from exact rational B-splines, traced ray by ray. The fastest slide A ramp, an arc and the cycloid raced from rest, and the same time from any height. Runs in the browser Transition curves The Euler spiral between a straight and an arc: its curvature, the shift of the circle and the ride. See Examples
