Measurement Error Due to a Bent Ball Bar
Where:
L = ball bar length
DeflG = deflection due to gravity
Fmin = minimum force, applied at bar midpoint, to permanently bend the bar
DeflB = deflection at bar midpoint to permanently bend the bar
Assumptions:
tube OD = 0.75 in, tube ID = 0.62 in, steel density = 0.289 lb/in3, steel modulus (E) = 30,000,000 psi, steel yield stress (Sy) = 60,000 psi
Formulas:
DeflG = 5 * density * L ^4 / (24 * E * (OD^2 + ID^2)) (1)
Fmin = Sy * pi * (OD^4-ID^4)/(8 * OD * L) (2)
DeflB = Sy * L^2 / (6 * OD * E) (3)
Max measurement error due to a bent ball bar measured at different rotational (about the balls’ common axis) positions:
dx = 20 * density * L^3 * dy / (9 * E * (OD^2 + ID^2)) (4)
Where:
dx = Max measurement error due to a bent ball bar at different rotational positions
dy = permanent bow measured at midpoint of ball bar
For an 80” ball bar, with 0.1” of permanent bow, and the assumptions stated above:
dx = 20 * 0.289 * 80^3 * 0.1 / (9 * 30000000 * (0.75^2 + 0.62^2))
dx = 0.001158 inches