Maths and Physics.
Stiffness
Many years ago I remember reading an audio magazine which tested the rigidity of the ET2 bearing. It may have been Martin Colloms, but I can't be sure. This was done, again from memory, where accelerometrs were used and a sweep frequency was applied to the spindle. The result showed a bearing that was stiff at audio frequencies.
This is explained by the design of the bearing (it's self centering characteristics) and its extremely high resonant frequency. Many times higher than the audio spectrum. Although the bearing uses air which we know to be compliant, at the frequencies of interest, the bearing medium is stiff.
I also show here a quote from an industrial air bearing manufacturer. While these a big load bearing devices, their design is virtually identical to the ET2
"Outstanding stiffness for small deflections Most engineers visualize an air bearing as being like a hovercraft, and they erroneously conclude that a bearing which floats on air cannot be very stiff. Actually these gas bearings are many times stiffer than a ball or roller bearing. Sapphire orifices within the bearing gap control the pressure in a film of air which is only 0.0003 inches thick. As a load is applied to displace the bearing rotor or slider, the gap decreases very slightly on one side, reducing the flow of air through the adjacent sapphire orifice. This results in a pressure increase in the gap on this side which pushes the rotor back to its original position. In essence, the air bearing is a servomechanism with closed loop control, and maintains a uniform gap in spite of external forces that may be applied. This results in bearing stiffness of millions of pounds per inch for small deflections. Stiffness is linear and does not change with temperature. In contrast, ball or roller bearings have almost no stiffness unless heavily preloaded. The stiffness of a ball bearing is not linear, and varies considerably with temperature."
Amplitude
A few weeks back I posted a transmissibility graph showing the effect of excitation frequencies at various multiples of the resonant frequency. This graph can be used to show relative resultant amplitudes for known resonant and excitation frequencies.
For a standard ET2 using in my case a Shelter Harmony, we get a resonant frequency of 8.4 hz. On my heavy arm, this frequency drops to 5.3 hz. If we take the lowest frequency of interest to be 20hz we get multipliers of res freq of 2.4 and 3.8 respectively.
By applying these multipliers to the graph we can see that the system which resonates at 8.4 hz shows a small rise in amplitude about 15%. If we now compare this with the 5.3 hz example we see a much smaller rise around 5%. We have to extrapolate this answer, since it is off the scale of the graph. In other words at audio frequencies the heavy arm produces less bass boost.
You can also see that the damping applied has very little effect on the resultant gain as the lines are trending together. This means that even if we factor in a higher resonant amplitude for the heavy arm, we can see that while it alters things slightly, it has minimal effect.
There is some merit in a discussion of what happens at sub sonic frequencies but the arm with the lower multiplier (lighter arm) will face problems sooner as we decend below audible frequencies.
As before these are all first principle discussions. It is what it sounds like that matters.
Stiffness
Many years ago I remember reading an audio magazine which tested the rigidity of the ET2 bearing. It may have been Martin Colloms, but I can't be sure. This was done, again from memory, where accelerometrs were used and a sweep frequency was applied to the spindle. The result showed a bearing that was stiff at audio frequencies.
This is explained by the design of the bearing (it's self centering characteristics) and its extremely high resonant frequency. Many times higher than the audio spectrum. Although the bearing uses air which we know to be compliant, at the frequencies of interest, the bearing medium is stiff.
I also show here a quote from an industrial air bearing manufacturer. While these a big load bearing devices, their design is virtually identical to the ET2
"Outstanding stiffness for small deflections Most engineers visualize an air bearing as being like a hovercraft, and they erroneously conclude that a bearing which floats on air cannot be very stiff. Actually these gas bearings are many times stiffer than a ball or roller bearing. Sapphire orifices within the bearing gap control the pressure in a film of air which is only 0.0003 inches thick. As a load is applied to displace the bearing rotor or slider, the gap decreases very slightly on one side, reducing the flow of air through the adjacent sapphire orifice. This results in a pressure increase in the gap on this side which pushes the rotor back to its original position. In essence, the air bearing is a servomechanism with closed loop control, and maintains a uniform gap in spite of external forces that may be applied. This results in bearing stiffness of millions of pounds per inch for small deflections. Stiffness is linear and does not change with temperature. In contrast, ball or roller bearings have almost no stiffness unless heavily preloaded. The stiffness of a ball bearing is not linear, and varies considerably with temperature."
Amplitude
A few weeks back I posted a transmissibility graph showing the effect of excitation frequencies at various multiples of the resonant frequency. This graph can be used to show relative resultant amplitudes for known resonant and excitation frequencies.
For a standard ET2 using in my case a Shelter Harmony, we get a resonant frequency of 8.4 hz. On my heavy arm, this frequency drops to 5.3 hz. If we take the lowest frequency of interest to be 20hz we get multipliers of res freq of 2.4 and 3.8 respectively.
By applying these multipliers to the graph we can see that the system which resonates at 8.4 hz shows a small rise in amplitude about 15%. If we now compare this with the 5.3 hz example we see a much smaller rise around 5%. We have to extrapolate this answer, since it is off the scale of the graph. In other words at audio frequencies the heavy arm produces less bass boost.
You can also see that the damping applied has very little effect on the resultant gain as the lines are trending together. This means that even if we factor in a higher resonant amplitude for the heavy arm, we can see that while it alters things slightly, it has minimal effect.
There is some merit in a discussion of what happens at sub sonic frequencies but the arm with the lower multiplier (lighter arm) will face problems sooner as we decend below audible frequencies.
As before these are all first principle discussions. It is what it sounds like that matters.