April 9th 1997

What is Vas?

Problem:

Measuring spring constant, or compliance, as an equivalent volume of air is part of the standard Thiele-Small parameters. Vas is a quantity of air, that when compressed by the driver, exhibits a restoring force on the cone that is equal to that supplied by the driver suspension. A simple relationship between Vas and mechanical spring constant of the driver suspension (Km) needs to be found.

Solution:

I have shown that air does not compress linearly. But we can make an excellent approximation by assuming it is. Let B be the adiabatic bulk modulus of air.

     B =  P = 1.40 * 101.3 * 10^3 = 1.42 * 10^5 N/m^2     (1)

Consider a volume of air (V) that is enclosed in a box that a piston is attached to it so that the volume can be changed. By changing the volume (V) of that air we know the net resultant pressure (P) it exerts on all enclosing surfaces; P = -B V/V. The surface area of the piston (A) and the amount the piston is displaced determines V = A x. The net force on the piston due to the pressure in the box is F = P A. Therefore

      F       A x                 2  1
     --- = -B ---    =>   F = -B A  --- x                   (2)
      A        V                     V

The piston displacement results in a force in the opposite direction; the air in the box acts like a spring. We ask, what volume (V) would be required if (V2) obeyed just like a mechanical spring; F = - K x ?  Set them equal to each other:

                  2  1                  B   2
     - K x = - B A  --- x    =>    V= --- A                (3)
                     V                 K

If this piston is a driver then V has a special designation: Vas. The mechanical spring is the suspension of the driver; K = Km. Therefore,

             B   2
      Vas = --- A                                           (4)
            Km

Vas is a function of two parameters, area of the cone (A) and suspension spring constant (Km). Obviously, having one parameter (Vas) to describe two parameters has its advantages and disadvantages. Although deriving solutions is more intuitive for me using spring constants and area, I will stick to the convention of using Vas in conclusions when I can.