{
 "title": "Two identical elastic bars colliding (explicit contact)",
 "kind": "theory",
 "source": "One-dimensional elastic impact of equal bars: contact for 2L/c, interface force ρcAv₀/2, exchange of velocities, no loss of energy; Graff, Wave Motion in Elastic Solids, Dover 1991, §2.1–2.3; Johnson, Impact Strength of Materials, Arnold 1972, §1.6",
 "quantities": [
  {
   "id": "duration",
   "quantity": "time of the last contact (the bars touch from t = 0) against 2L/c = 0.2",
   "measure": {
    "from": "solve",
    "pointer": "/contacts/0/last_contact"
   },
   "value": 0.2,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "the penalty springs at the interface delay the release by a few element transit times (about 1% of 2L/c with 200 elements per bar)"
  },
  {
   "id": "force",
   "quantity": "the largest force across the interface against ρcAv₀/2 = 0.05",
   "measure": {
    "from": "solve",
    "pointer": "/contacts/0/peak_force"
   },
   "value": 0.05,
   "tolerance": {
    "relative": 0.04
   },
   "reason": "the plateau is ρcAv₀/2 to 0.01%; the peak is the plateau plus the ringing behind the discrete wave front, a few percent"
  },
  {
   "id": "v_a",
   "quantity": "velocity of the struck bar's far end after the impact (0: the striker stops)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Impact",
     "at": [
      [
       0,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/velocity/0"
   },
   "value": 0.0,
   "tolerance": {
    "absolute": 0.005
   },
   "reason": "the striker stops on average (its kinetic energy is under 0.01% of the start's, see ke); its far end node keeps the ringing of the discrete release front, a few percent of v₀"
  },
  {
   "id": "v_b",
   "quantity": "velocity of the struck bar's far end after the impact (v₀ = 0.1)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Impact",
     "at": [
      [
       20.0,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/velocity/0"
   },
   "value": 0.1,
   "tolerance": {
    "relative": 0.05
   },
   "reason": "the struck bar leaves at v₀ on average; its far end node keeps the ringing of the discrete wave fronts, a few percent"
  },
  {
   "id": "ke",
   "quantity": "kinetic energy after the impact over ½mv₀² (frictionless elastic impact)",
   "measure": {
    "from": "solve",
    "pointer": "/energy/kinetic",
    "scale": 2000.0
   },
   "value": 1.0,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "an elastic impact keeps the kinetic energy; the quadratic bulk viscosity at the compression fronts and the vibration left in the bars take a fraction of a percent"
  },
  {
   "id": "contact_energy",
   "quantity": "energy left in the contact after the separation, over ½mv₀² (its drift)",
   "measure": {
    "from": "solve",
    "pointer": "/energy/contact",
    "scale": 2000.0
   },
   "value": 0.0,
   "tolerance": {
    "absolute": 0.001
   },
   "reason": "the penalty springs give back what they took; the trapezoidal work of a spring that engages between increments drifts by far less than 0.1%"
  }
 ]
}