measuring straightness by wire

Описание: This is a quick demonstration of using a wire and a microscope to measure the straightness of the ways of a lathe.

calculation of the wire sag

The (German) wikipedia article
https://de.wikipedia.org/wiki/Seilstatik
gives the following formula for the sag of a rope/wire under evenly distributed load

with
y = vertical position of the wire
x = horizontal position

x0 / y0 = position of left wire end
L / yL = position of right wire end

q0 = vertical force caused by the distributed load of the wire (the weight of the wire)
H = horizontal tensioning force


y(x) = q0 / ( 2 H ) * x^2 ( ( yL - y0 ) / L -q0 L / ( 2 H ) ) * x + y0

this can be reduced

y0 = yL = 0;
x0 = 0;

L = length of the machine bed (in this case 1.2 m)

this gives

y(x) = q0 / ( 2 H ) x^2 ( 0 - q0 / ( 2 H ) ) x + 0

y(x) = q0 / ( 2 h ) ( x^2 - L x )

plausibility check at the ends

y(x=0) = q0 / ( 2 H ) ( 0^2 - L 0 ) = 0 (ok)
y(x=L) = q0 / ( 2 H ) ( L^2 - L L ) = 0 (ok)


Now we need the forces q0 and H

The tensioning force is calculated from the mass of the tensioning weight and the gravitational constant g

H = W * g


The distributed force caused by the weight per meter of the wire M is

q0 = M * g

The wire mass per meter is the product of the cross section A and the density D

M = A * D


with the radius of the wire r

M = pi r^2 * D

this gives the factor

q0/(2H) = ( pi r^2 * D g ) / ( W g )

g cancels out

q0/(2H) = ( pi r^2 * D ) / W

inserting the actual values

with D = 1140 kg/m^3 as the density for (probably) Nylon

q0/(2H) = ( pi * ( 0.04 10^-3 m ) ^2 * 1140 kg/m^3 ) / ( 2 * 0.5 kg )
q0/(2H) = 5.7303 10^-6 / m


the maximal sag is in the center at x = 0.6 m

this results in

y(x=0.6m) = q0/(2H) * ( 0.6m ^2 - 1.2m * 0.6m )

= 5.7303 10^-6 / m * -0.36 m^2 = -2.0629e-06 m
~= -2 µm