Sine bar principle, setting and error

(a) Principle

A sine bar is a hardened steel bar with two equal precision rollers fixed at a known centre distance $L$ (100, 200 or 300 mm). When one roller rests on slip gauges of height $h$ and the other on the surface plate, the bar is inclined at angle $\theta$ to the plate:

        ________  <- sine bar
      /   L    /(roller)
    /  theta  /|
   /_________/_| h  (slip gauges)
   surface plate

$$\sin\theta = \frac{h}{L}$$

Used to set a component at a known angle, or to measure an unknown angle by adjusting $h$ until the work surface is level (checked by a dial indicator).

Limitations

• Accuracy falls at large angles (above $45^\circ$) because $\cos\theta$ becomes small (see (c)); not suitable above $60^\circ$.
• Needs a surface plate and a good set of slip gauges; cannot measure steep tapers directly.
• Errors in centre distance, roller diameters and parallelism of rollers directly give angle error.
• Only conical or flat surfaces that can be set against the bar can be checked; tedious for large work.

(b) Stack height

$$h = L\sin\theta = 200\sin(28.5^\circ) = 200 \times 0.47716 = 95.432\ \text{mm}$$

Answer: slip gauge height = 95.432 mm.

(c) Error

Differentiating $h = L\sin\theta$ gives $\delta h = L\cos\theta\,\delta\theta$, so

$$\delta\theta = \frac{\delta h}{L\cos\theta}$$

The same stack error causes a larger angular error at steeper angles. Hence the sine bar is best used below $45^\circ$, and for large angles the complement is measured.