Anyone want to help vet this?
https://docs.myhi.fi/tools/tonearm-lf-mechanics.html
It's built off the physics of LD's "loafer" spreadsheet. I added the contact force/wear multiplier plot.
Technical (AI) explanation:
The system is modeled as a single degree of freedom damped harmonic oscillator with base excitation. The mass is the combined tonearm effective mass and cartridge/fixings mass. The spring is the cartridge cantilever compliance. Damping comes from two sources: the cartridge suspension (derived from the ratio of static to dynamic compliance at a known frequency) and the tonearm (bearings, fluid dampers, or other mechanical damping).
The undamped natural frequency follows from Hooke's law: fₙ = √(k/m) / 2π, where k = 1000/compliance and m is total effective mass. The damped resonant peak frequency depends on the system's overall damping and may differ slightly from the undamped natural frequency.
The frequency response uses the cartridge output transfer function, which represents the relative motion between stylus and arm body — this is what generates the cartridge's electrical signal. It has a high-pass characteristic: zero output at DC (the arm tracks the groove perfectly), a resonant peak where relative motion is amplified, and unity gain at high frequencies where the arm can no longer follow the groove modulation.
The contact force plot uses the base-excited displacement transmissibility: T(f) = √((1 + (2ζr)²) / ((1-r²)² + (2ζr)²)), where r is the frequency ratio and ζ is the overall damping ratio. This approximates how the mechanical resonance modulates the contact force between the stylus and the groove wall. At resonance, the cantilever deflection is amplified, swinging the contact force above and below VTF. Above resonance, the arm's inertia prevents it from responding, and the contact force returns to VTF. The contact force envelope shows VTF ± (k × excitation × T) at each frequency. This is an approximation — the transmissibility captures the qualitative behavior correctly but does not precisely model the cantilever force at all frequencies.
The wear multiplier applies Archard's wear law with a Hertzian contact exponent of 1.5. Because wear rate is proportional to force^1.5, the cycle-averaged wear from an oscillating contact force is always greater than the wear from steady force at the same mean value. The multiplier is computed by numerically integrating force^1.5 over one oscillation cycle and normalizing to the steady-state wear at VTF.
Cartridge damping ratio (Tc) is derived from the difference between static and dynamic compliance: Tc = (k × √((Z/k)² - 1)) / (2π × f_dynamic) / c_critical, where Z = 1000/dynamic_compliance and c_critical = 2mωₙ. This captures the viscoelastic loss in the cantilever suspension. Tonearm damping (Ta) is either entered directly or computed from the logarithmic decrement method using two successive peak amplitudes of a decaying oscillation.
The trackability analysis runs a time-domain simulation of forced vibration at a specified frequency and amplitude. The simulation uses only the cartridge damping in the equation of motion, consistent with the original Luckydog spreadsheet model. This is valid for the typical trackability test frequencies (100 Hz–1 kHz) where the modulation frequency is far above the arm-cartridge resonance and arm damping has negligible effect. The minimum VTF required is the peak dynamic force perturbation during the simulation, representing the tracking force needed to prevent stylus liftoff at that modulation.