Bar Talk: What's Your Sine?

Bar Talk: What's Your Sine?
George Schuetz, Mahr Federal Inc.

 A few months ago this column described the use of surface plates, observing that a flat surface is the basis for most dimensional measurements.  Many workpieces, of course, are neither flat nor straight.  In order to measure the angular accuracy or straightness of an angled surface using a surface plate, a sine instrument comes into play.  By placing the workpiece on the sine instrument and raising one end of the instrument to the proper height, it is possible to orient the workpiece parallel to the surface plate.  Straightforward measurements can then be performed with a test indicator and height stand.

 Sine bars are relatively narrow (up to about 1" wide) fine-ground or lapped steel bars, with precision cylinders at each end resting against stops machined into the bottom surface.  Some sine bars are completely ambidextrous; others have "upper" and "lower" ends as well as distinct top and bottom surfaces.  Holes machined in the instrument enable the placement of stops or clamps to hold workpieces in place.  Close cousins to sine bars include sine blocks, which are simply wider sine bars; and sine plates (the most popular version), in which the bottom cylinder is actually part of a hinge connected to an attached base.  While most sine instruments serve primarily as measuring instruments, some sine plates are rugged enough to serve as fixturing devices for machining operations.  The principle of operation is identical for all versions, so for the sake of simplicity, we'll refer only to sine plates in the following discussion.

 To set the sine plate at the proper angle, one simply selects a gage block or gage block stack of the appropriate height and places it under the upper cylinder.  An imaginary right triangular prism is thus created, the vertical face of which passes through the gage block and terminates at the axis of the upper cylinder.  The horizontal base of the prism is above and parallel to the surface plate, terminating at the axis of the lower cylinder (hinge), while the ends of the hypotenuse are defined by the axes of the two cylinders.  The top surface of the sine plate is therefore parallel to the hypotenuse.  The use of cylinders as contact points ensures that the length of the hypotenuse remains the same regardless of its angle.  Most sine instruments set the cylinders at a fixed distance between centers that is easy to manipulate mathematically, 5" and 10" being the most common lengths.

 To calculate the required height of gage blocks, use the following formula:
   Sin B = b/a where:
   B = required angle
   b = height of triangle
   a = length of hypotenuse.

 For example, say we want to measure the straightness of a surface on a workpiece that's angled 27° from a reference surface on the same workpiece.  Let's also say our sine plate is 5" between cylinder centers.  To obtain the sine value, we either refer to a table of trig values, or simply punch it up on a $10 scientific calculator.

   Sin 27° = b/5"
   0.453,990,499 = b/5"
   b = 2.270"

 So we wring up a stack of gage blocks, place it beneath the upper cylinder of the sine plate, and voilá, we have a surface that's 27° out-of-parallel with the surface plate.  We stage the workpiece on the sine plate facing the opposite direction and, if all is well, the workpiece surface should be parallel to the surface plate.  We can then measure it for flatness, straightness, and angular accuracy, using conventional surface plate methods.  To measure straightness, an indicator held by a test stand is drawn across the surface, and comparative height measurements are taken at regular intervals.  By drawing a best-fit line through the data, it is possible to break out the flatness of the surface and to calculate the accuracy of the machined angle.

 Because sine values change rapidly between angular measurements of low numerical values, and change very little between higher numerical value angles, the accuracy that can be achieved using a sine instrument varies considerably: precision is much better for shallow angles than for steep ones.  Therefore, where the surface of a workpiece is angled at greater than 45°, it is often advisable to use the complement of the angle (90° - x) in the calculation.

 Even so, the accuracy that can be achieved with sine instruments is somewhat limited, because there are so many separate mechanical elements to the set-up (the surface plate, the gage block(s), the test indicator, the test stand, and the sine instrument itself), each of which imposes a certain degree of uncertainty.  We'll look at methods to perform higher-precision angular measurements in a future column.