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Mass and Spring Resonance
Begin by noting that the general equation for a mass and spring resonator is given by equation 1a and 1b. In the case of an
enclosed compressible volume of air having a vent, the spring and masses become equations relating the box volume to port
dimensions. Using radians per second rather than cycles per second may be a little easier to follow, so these equations
are also given (1 radian = 2*PI cycles).
1a) W = sqrt(K/M) - frequency in radians/second
1b) F = sqrt(K/M)/2*PI - frequency in cycles/second
K=DensityOfAir*C*C*Sv*Sv/Vb // Air spring for area Sv and volume Vb
M=DensityOfAir*Sv*Lv // Air mass contained by vent
Helmholtz Resonators
Substituting the equations for K and M into equation 1a or 1b derives the Helmholtz resonator equation:
2a) Wh = C*sqrt(Sv/(Lv*Vb)) - radians/second
2b) Fh = C*sqrt(Sv/(Lv*Vb))/2*PI - cycles/second
Wh = Helmholtz frequency in radians per second
Fh = Helmholtz frequency in cycles per second
C = Speed of sound
Sv = Vent area
Lv = Vent length
Vb = Box Volume
Box Resonance - Effective Vent Length
The effective length of a vent is however somewhat longer than its physical length. This is because some volume of air in
front or behind the vent becomes coupled to the air mass inside the vent and moves sympathetically with the mass inside the
vent (mass loading). Literature on this topic varies, but the general consensus is that for a flush-fitted port, Lv increases
by 1.463*Rv where Rv=vent radius.
Lv = Lv_cut + 1.463*Rv Reversing the equation for Helmholtz resonance, the vent length for a particular vent area and box volume can now be
calculated. The radians per second equations are used here to temporarily hide the value of '2*PI' making the equations
easier to follow.
2a) Wh = C*sqrt(Sv/(Lv*Vb))
Lv = C*C*Sv/(Wh*Wh*Vb)
Sustituting Sv=Rv*Rv*PI
Lv = C*C*Rv*Rv*PI/(Wh*Wh*Vb) As mentioned previously, the vent' physical cut length is found by compensating for the front and back mass coupled weight.
Finally, since radian frequency Wh was used, this is converted to cycles per second using the relationship Wh=2*PI*Fh.
Lv_cut = Lv - 1.463*Rv
Lv_cut = C*C*Rv*Rv*PI/(Wh*Wh*Vb) -1.463*Rv
Lv_cut = C*C*Rv*Rv/(4*PI*Fh*Fh*Vb) -1.463*Rv
Notes:
Various literature sources use constants that may create confusion. The fudge factor that relates the physical port length to the
effective port length is 1.463 in the equations above, or in some cases 16/(3*PI)=1.70. Since this constant will vary with the
physical properties of air as well as the box and port configuration it comes down to an opinion. This is why measuring the inbox
electrical characteristics and comparing them to the simulation is so important.
Another common practice is to assume a constant value for the speed of sound. The result is that a constant can now be put in the
equation for the value C*C/4*PI. Further clouding the point is that the units of C can be either metric (meters) or English
(inches), and the box volume may not even be in the same units. Even worse, Keeles pocket calculator method computes an interim
'alpha' value using the vent area and C. Needless to say, this results in many different constants and equations for the same
problem.
The following equations are pulled from various references and summarized. Interestingly, the equation for Svmin, the minimum
vent area needed for the port to be somewhat linear, suffers from essentially the same condition, constants that are not actually
constant, units of measure and matters of opinion. In the end, the final consensus of a number of experts has been that the vent
air velocity should not exceed 5 to 10% of the speed of sound. These days, this is much more easily computed within the simulator
than it was in the days that these equations were first derived. And, in the final analysis, the original equations were
quite good.
Dickason's Loudspeaker Design Cook Book
Lv_cut = (1.463E7*Rv^2/(Fh^2*Vb)) - 1.463*Rv Rv in^2, Vb in^3, Lv inches
Keeles Pocket Calculator Method
Svmin = 0.02*Fh*Vd Vd volume displacement in in^3
alpha = Vb*(2*PI*Fh/C)^2 ~= 3.7E-4*Vb*Fh^2 Vb box volume in ft^3
Lv_cut = (Sv/alpha) - 0.83*sqrt(Sv) Sv vent area in sq_inches, Lv in inches
As can be seen from the examples above, these equations look quite different. However, they are in fact derived from a common
source. Furthermore, at least the constant relating the speed of sound can now be fully derived. The uncertain part of the
equation then becomes the fudge factor relating the physical and effective vent length.
From Derivation Above:
Lv_cut = C*C*Rv*Rv/(4*PI*Fh*Fh*Vb) -1.463*Rv Constant from Dickason
Or, reversing the equations from Michael Lamptons program, a new constant is found
Lv_cut = C*C*Rv*Rv/(4*PI*Fh*Fh*Vb) -(16/3*PI)*Rv Constant from Lampton
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