{
 "title": "Natural frequencies of a cantilever (Euler–Bernoulli)",
 "kind": "theory",
 "source": "Euler–Bernoulli beam: f_n = (β_n L)²/(2π L²)·√(EI/ρA), β₁L = 1.8751, β₂L = 4.6941; Timoshenko, Young and Weaver, Vibration Problems in Engineering, 4th ed., Wiley 1974, §5.3",
 "quantities": [
  {
   "id": "f1",
   "quantity": "first bending frequency (modes 1 and 2 are the y and z pair)",
   "measure": {
    "from": "modes",
    "pointer": "/modes/0/frequency"
   },
   "value": 0.012770863695053732,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "Euler–Bernoulli neglects shear and rotary inertia, which lower f1 by about 0.2% at L/h = 20"
  },
  {
   "id": "f1_pair",
   "quantity": "the other first bending frequency",
   "measure": {
    "from": "modes",
    "pointer": "/modes/1/frequency"
   },
   "value": 0.012770863695053732,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "as f1: a square section bends alike in y and z"
  },
  {
   "id": "f2",
   "quantity": "second bending frequency",
   "measure": {
    "from": "modes",
    "pointer": "/modes/2/frequency"
   },
   "value": 0.08003363662360005,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "shear and rotary inertia lower the second mode by about 1% at L/h = 20 (Timoshenko)"
  }
 ]
}