This is the Sommerfeld effect in action, it's a non-linear change in a system with multiple degrees of freedom, described by Armold Sommerfeld in 1902, where the work of a "non-ideal" motor is converted into vibrational energy rather than torque near a resonance frequency, leading to a capture of performance until increasing power leads to escape. I did some background research and sketched out a hypothesis with the help of Gemini AI yesterday.
Q: Can there be a Sommerfeld effect in a system with two jointed rods, rod A driven and rod B fixed at the other end?
A: Yes, a Sommerfeld effect can occur in a system of two jointed rods where one is driven and the other is fixed at its far end. This phenomenon is a hallmark of non-ideal vibration systems, where the energy source (the motor) and the mechanical structure it drives are mutually coupled.
How the Effect Manifests in This System
In your specific setup (rod A driven, rod B fixed at the other end), the Sommerfeld effect would appear during the "coast-up" phase as the motor attempts to increase the reciprocating speed of rod A.
Resonance Capture: As the driving frequency approaches the natural frequency of the two-rod assembly, energy from the motor is diverted into increasing the vibration amplitude of the rods rather than increasing the motor's speed.
Speed Stagnation: Even if you increase the power supply to the motor, the rotational speed of rod A may remain "stuck" or caught near the resonance point for a significant range of input power.
Jump Phenomenon: Once the input power exceeds a critical threshold, the system suddenly "escapes" resonance. This results in an abrupt jump in motor speed and a simultaneous, sharp decrease in the vibration amplitude of the rods.
Key Requirements for the Effect
For this effect to be observable in your two-rod system, three conditions must typically be met:
Non-Ideal Source: The motor must have limited power. An "ideal" source (infinitely powerful) would simply force the system through resonance without slowing down.
Coupling: There must be a physical interaction where the rods' motion creates a back-torque on the motor, especially near resonance.
Flexibility/Mounting: The fixed end of rod B or the foundation of the motor must have some degree of flexibility (like being mounted on elastic supports) to allow for the structural vibrations that consume the energy.
Similar effects have been studied in reciprocating mechanisms like slider-cranks and double-Cardan joint drivelines, which share the jointed-linkage characteristics of your two-rod system.
Q: Compare the effect when the resonant frequency of rod A is higher than rod B and then when the resonant frequency of rod B is higher than rod A.
A: In a system of two jointed rods, the order of their resonant frequencies significantly alters the power requirements and vibration severity during a coast-up. Because rod A is directly driven and rod B is "downstream," the system behaves as a multi-degree-of-freedom (MDOF) oscillator where the driven rod acts as the primary transmission path.
Case 1: Resonant Frequency of Rod A > Rod B
In this scenario, the system's "weakest" point (rod B) resonates first as the motor speeds up.
Early Capture: The motor speed will likely get "captured" at a lower RPM. Because rod B is fixed at one end, its resonance creates a significant reactive torque that rod A must overcome.
High Vibration in Rod B: Rod B will exhibit large-amplitude oscillations
while rod A may remain relatively stable (my italics).
Easier Escape: Since the driving rod (A) is not yet in its own resonance, it can more efficiently transfer torque to push the system past rod B's critical speed, provided the motor has a small power reserve.
Case 2: Resonant Frequency of Rod B > Rod A
This is generally considered more "dangerous" or difficult for the motor to navigate.
Direct Path Blockage: As the motor approaches rod A’s resonant frequency, the very component doing the "driving" begins to vibrate violently.
Severe Power Saturation: The energy from the motor is "sucked" into rod A's vibrations before it even reaches rod B. This often leads to a more stubborn speed stagnation where the motor cannot provide enough torque to "jump" to the next speed level.
Sequential Resonance: If the motor successfully escapes rod A’s resonance, it immediately faces a second Sommerfeld "trap" when it hits rod B’s higher frequency. This can lead to a double-jump phenomenon.
Summary Comparison Table
Feature / Rod A Frequency > Rod B / Rod B Frequency > Rod A
First Trap / Occurs at lower RPM (Rod B resonance) / Occurs at lower RPM (Rod A resonance)
Capture Severity / Moderate; Rod A can still "push" / High; the drive path itself is vibrating
Jump Magnitude / Usually a single, smaller jump / Potentially larger or multiple jumps
Primary Risk / Fatigue at the fixed end of Rod B / Motor stalling or "stuck" at low RPM
Q: Can you apply the Sommerfeld effect to a phono cartridge with a cantilever (rod A) and tie-wire (Rod B; tension wire), where the resonant frequency of the tie-wire is lower than the cantilever, as it is driven by a frequency sweep on a test LP?
A: Applying the "Two-Rod" Model to a Phono Cartridge
Rod A (Driven): The cantilever assembly. It is driven by the stylus tracing the high-frequency modulations of a test LP.
The "motor" is non-ideal, because the stylus can escape VTF and friction by riding up the grooves or jumping (my insertion in italics).
Rod B (Fixed at Other End): The tie wire (tension wire). It is anchored at the cartridge body and provides the restoring "spring" force for the cantilever.
Behavior When Cantilever Resonance (A) > Tie Wire Resonance (B)
In modern high-performance cartridges (especially those using boron or sapphire cantilevers), the cantilever is designed to be extremely stiff, pushing its own natural resonance (Rod A) into the ultrasonic range (25kHz–50kHz).
The tie-wire and its surrounding elastomer damper (Rod B) have a much lower fundamental resonance, typically in the mid-band or low-treble (approx. 1kHz–9kHz).
When a frequency sweep passes through these points:
Low-Frequency "Capture" (Rod B): As the sweep reaches the tie wire/damper resonance (e.g., ~9kHz), the system enters a "transitional phase"
*. In a Sommerfeld context, energy from the groove is diverted into vibrating the tie-wire and elastomer rather than purely generating electrical output
(clearly wrong).
Increased Mechanical Load: At this point, the stylus experiences higher "drag" because it is fighting the mechanical resonance of the tie wire. If the turntable motor were weak (a "non-ideal source"), you might even see a microscopic slowing of the platter—the classic Sommerfeld speed stagnation—though modern heavy platters usually "brute force" through this.
No, the stylus would escape.
Damping as the "Escape" Mechanism: Because the cantilever (Rod A) is still rigid and well above its own resonance, it effectively "pushes" the system through the tie wire's resonance. High-quality dampers (like the "donut" rings used by Audio-Technica) prevent the "jump phenomenon" from becoming an audible distortion spike.
Summary
In a sine wave sweep, there is a capture of the stylus/cantilever energy as it reaches the resonant frequency of the tie-wire (9-12 kHz in the recorded sweeps,
*Gemini cites my post here for the 9 kHz figure, so there is a risk of circular reasoning). The increasing groove velocity (with increasing frequency) gets stuck in driving the amplitude of the stylus and volume (the 1-2 dB peak), instead of the frequency of oscillation and playback frequency. As the energy from the grooves increases, there is an escape from the resonant frequency of the tie-wire and a transient drop in amplitude (the dip). The size of the jump needs to be limited (tie-wire tension, elastomer, electromechanical damping) so that it is not sufficient to cause mistracking.
At least that's the hypothesis. There are some experiments that might help to test it. I've posted one piece of experimental data on the Fun With Vinyl Measurements thread
here.
