Tonearm moment of inertia

Tonearm moment of inertia

The main criteria for designing, producing and choosing to buy tonearm are often wiring (internal and external), connections, and effective mass and geometric dimensions of tonearm which is clearly very important. However, there are some other criteria, such as tonearm moment of inertia (I_t) and acoustic resonance characteristics, which are often left out while they should be considered as well.

 I_t is the sum of cartridge moment of inertia (I_c) and counterweight moment of inertia (I_w).

 

I_t = I_c + I_w

 

Even with smallest LP twist or miscentering, I_t would affect VTA (Vertical Tracking Angle), SRA (Stylus Rake Angle) and VTF (Vertical Tracking Force) of tonearm. I_t influence on VTA, SRA and VTF causes distortions that are subjectively heard as detonation. The frequency of distortions varies from 0.5 Hz to 20 Hz and can be up to 0.5 %.

To reduce these distortions, usually damping systems are used. Both mechanical and magnetic damping systems are complicated and quite expensive. However we suggest reducing I_t, which is the main reason of distortions, first.

Moment of inertia depends on mass and square of distance formula.

Research shows that I_w can be reduced by increasing the mass of counterweight and pushing it towards the bearings. The mass of counterweight can not be highly increased because the bearings will get overloaded and that will cause friction problems. In order to minimize friction sapphire bearings with tungsten carbide points are used. These bearings rest sphere on sphere (both spheres having different radius). Clearance of bearings is not possible because it will cause azimuth and VTA errors and make additional distortions. For this reason constantly loaded bearings should be used.

 I_c can be decreased by:

1) Reducing effective mass.  Effective mass of tonearm can be reduced by changing the material of armtube. Tonearm should be matched with cartridge so that Tonearm Resonant Frequency is 10 +/- 2 Hz. Therefore, effective mass of tonearm must be within certain limits.

2) Reducing effective length of tonearm. Decreasing the length of tonearm to less than 9” increases tracking error distortions. On the other hand, if the length of tonearm is more than 12”, the distortions caused by I_t become more significant.

Statics of tonearm (static position) can be expressed by the following equations:

 

 

where

m_c – cartridge weight,

m_ch – head shell weight,

m_at – armtube weight,

l_ef – effective length,

l_w – counterweight section length,

m_w – counterweight weight,

m_wh – counterweight holder weight.

Because I_w should be (and usually is) at least 10 times smaller than I_c, for approximate calculations we should examine only I_c:

 

 

where

m_c – cartridge weight,

m_ef – effective mass,

l_ef – effective length,

 I_c should be less than 20 000 g×cm².