6 The Math¶

6.1 Depth Factors¶

_images/countersink.jpg

Where:

D = Screw head diameter
θ = Countersink included angle
n = Depth of countersink recess

Find the value of $x$ where $n = Dx$

Solving:

$x = \frac{n}{D}$

$x = \frac{\frac{n}{(\frac{D}{2})}{\frac{D}{2}}}{D}$

$x = \frac{(cot{\frac{\theta}{2})}(\frac{D}{2})}{D}$

$x = \frac{\frac{cot{\frac{\theta}{2}}}{2}D}{D}$

$x = \frac{cot{\frac{\theta}{2}}}{2}$

Application:

The depth factor for a countersink with an 82° included angle,

$x = \frac{cot{\frac{{82}}{2}}}{2}$

$x = \frac{cot{41}}{2}$

$x = \frac{1.15036}{2}$

$x = .57518$

6.2 Gauge Properties¶

_images/Gauge_calc.jpg

Where:

D = Gauge hole diameter
θ = Countersink included angle
T = Gauge thickness

Find the value of $x$ where $x = \frac{D}{T}$

$x = \frac{2\frac{D}{2}}{T}$

$x = 2tan{\frac{\theta}{2}}$

Application:

The hole size in a 1/8” thick gauge for an 82° countersink,

$\frac{D}{T} = 2tan{\frac{\theta}{2}}$

$D = 2tan{\frac{\theta}{2}}T$

$D = 2(tan{41})T$

$D = 2(.8693)(.125)$

$D = .217$