{
 "title": "Natural frequencies of a free beam: rigid-body modes and the first bending mode",
 "kind": "theory",
 "source": "Euler–Bernoulli free–free beam, β₁L = 4.7300; Timoshenko, Young and Weaver, Vibration Problems in Engineering, 4th ed., Wiley 1974, §5.3",
 "quantities": [
  {
   "id": "rigid_6",
   "quantity": "sixth frequency (the last rigid-body mode)",
   "measure": {
    "from": "modes",
    "pointer": "/modes/5/frequency"
   },
   "value": 0.0,
   "tolerance": {
    "absolute": 8.1264203095911e-05
   },
   "reason": "a free body has six zero frequencies; numerical rigid-body frequencies below 10⁻³ of the first elastic one (eigenvalues below 10⁻⁶ of it) are rounding of the shifted solve"
  },
  {
   "id": "f7",
   "quantity": "first free–free bending frequency (mode 7)",
   "measure": {
    "from": "modes",
    "pointer": "/modes/6/frequency"
   },
   "value": 0.08126420309591101,
   "tolerance": {
    "relative": 0.015
   },
   "reason": "Euler–Bernoulli neglects shear and rotary inertia, about 0.8% at L/h = 20 for the free–free mode"
  },
  {
   "id": "mass",
   "quantity": "total mass",
   "measure": {
    "from": "modes",
    "pointer": "/total_mass"
   },
   "value": 20.0,
   "tolerance": {
    "relative": 1e-12
   },
   "reason": "the mass matrix integrates the density exactly"
  }
 ]
}