{
  "title": "Illustrative uncoupled damped oscillator bank",
  "status": "Hypothetical mathematical illustration, not measured or predicted device performance",
  "equation_of_motion": "m*x'' + c*x' + k*x = F0*cos(2*pi*f*t)",
  "definitions": {
    "natural_frequency_hz": "sqrt(k/m)/(2*pi)",
    "damping_ratio": "c/(2*sqrt(k*m))",
    "static_response": "F0/k for each channel",
    "frequency_ratio": "r_i=f/fn_i"
  },
  "magnitude_formula": "H_i=1/sqrt((1-r_i^2)^2+(2*zeta*r_i)^2)",
  "aggregate_formula": "sqrt(mean(H_i^2)) across the three channels at each frequency",
  "natural_frequencies_hz": [
    250.0,
    500.0,
    1000.0
  ],
  "damping_ratios": [
    0.1,
    0.5
  ],
  "grid": {
    "spacing": "logarithmic (numpy.geomspace)",
    "start_hz": 50,
    "stop_hz": 2000,
    "points": 1601,
    "includes_endpoints": true
  },
  "csv_rows": 3202,
  "normalization": "Each displacement magnitude is divided by that channel's own static displacement F0/k; equal normalized forcing is not equal absolute displacement or voltage.",
  "assumptions": [
    "linear time-invariant steady state",
    "three independent uncoupled oscillators",
    "same damping ratio assigned to all channels in each panel",
    "harmonic forcing",
    "no noise or transients"
  ],
  "excluded": [
    "coherent sum and phase interference",
    "electrical transduction",
    "coupling",
    "microphone sensitivity",
    "acoustic loading",
    "actual source parameters",
    "actual device geometry or performance"
  ],
  "checks": {
    "all_outputs_finite": true,
    "H_at_zero_equals_one": true,
    "H_at_fn_equals_one_over_2zeta": true
  },
  "software": {
    "python": "3.12.10",
    "numpy": "2.5.3",
    "matplotlib": "3.11.2"
  },
  "reproduction": "python reproduce.py",
  "figure_alt": "Two panels compare dimensionless response versus frequency for three uncoupled channels at 250, 500 and 1000 Hz. Damping 0.1 produces narrow peaks near 5; damping 0.5 produces broad low peaks. The dashed black curve is the RMS of the three normalized magnitudes, not an electrical sum."
}