DIMENSIONAL LAYOUT ERRORS

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DIMENSIONAL LAYOUT ERRORS

By

James R. Howe

 

When measuring the distance between two bores of equal size a pin to pin method is usually employed when the layout is performed on a three dimensional table with digital readout. This method is implemented by inserting a cylinder pin, of the appropriate diameter, into the bores. Then a measurement is taken between the surfaces of the two pins (figure 1) by finding top dead center (TDC) of pin 1 with the indicator (blue arrow). Once TDC is discovered for Pin 1 both the indicator and the probe arm are zeroed at that point. The probe and the indicator are then directed to pin 2 and adjusted until TDC is found on that pin. The indicator is then returned to zero by adjusting the probe arm. This procedure will yield the dimension from the surface of Pin 1 to the surface of Pin 2 (red arrow). Since the radii of the two pins (black arrows) are equal they cancel each other out and the probes digital read out is the distance between the centers of the two pins (green arrow).

 

However, when measuring the distance between two bores of unequal size we must consider not only the radii of the pins but also the relative position of the pins to each other.  First let us consider the relative positioning of the pins in figures 2 and 3.

 

To find the dimension from the center of Pin 1 to the center of Pin 2 (green arrow) the radii of both pins (black arrows) must be accounted for.  Whether the radii are added or subtracted depends on the relative position of the pins to each other.  Indicating on Pin 1 (as a reference pin) to locate top dead center (TDC) yields the following.

 

In FIGURE 2: Pin 1 is above Pin 2 so the surface to surface dimension (red arrow) falls beyond the center of Pin 1 and short of the center of Pin 2. The only way to obtain the center to center dimension is to subtract the Pin 1 radius from the red arrow and add the Pin 2 radius to the red arrow.

 

In FIGURE 3: just the opposite is true, Pin 1 is below Pin 2 so Pin 1 falls short of center and Pin 2 falls beyond center so we must add the Pin 1 radius and subtract the Pin 2 radius from the red arrow.

 

Failure to properly understand these relationships can lead to gross dimensional errors because a radius might be added when it should be subtracted.

 

Using Figure 4 as an example,  when the indicator (blue arrow) approaches top dead center the dimension is read from the surface of Pin-1 to the surface of Pin-2 (red arrow). To obtain the dimension from the center of Pin-1 to the center of Pin-2 (green arrow) the radius of Pin-1 (black arrow = 2.010) must be subtracted from the surface dimension (red arrow) because it is beyond the center of Pin-1. Then  the radius of Pin-2 (black arrow =.9955)  must be added to the surface dimension (red arrow) because it is short of the center of Pin 2. The result of this manipulation is the true dimension (green arrow) from pin center to pin center.

 

 

Not withstanding any mathmatical errors that might occur  there exist yet another source of errors that must be taken into account. Lets examine a close up of pin 1 shown in FIGURE 5.

 

 

When indicating to find top dead center (TDC) an error can be introduced if the indicator approach is not a straight line. For example, when the indicator approach is on a steep angle (red Indicator), do to surrounding metal obstructing a direct 180 degree straight line approach (green indicator), an error results because TDC is not true. It could be as much as 10 degrees away from TDC, depending on operator technique, degree of obstruction, and available indicators (size of ball). If it is not at true TDC the resulting error is a function of the cosine when indicating to the side of the pin and a function of the sine when indicating to the top of the pin.

 

Cosine is defined as the adjacent side divided by the hypotenuse. Sine is defined as the opposite side divided by the hypotenuse.

 

cosine = adjacent side / hypotenuse or adjacent side = cosine x  hypotenuse

sine = opposite side / hypotenuse or opposite side = sine x hypotenuse

 

When indicating to the side of the pin (cosine) let the red arrow (radius) be the hypotenuse. As it sweeps through the quadrant the green arrow travels along the blue line and defines the adjacent side as the point of intersection on the blue line to the center of the pin. The blue line approaches the radius as the indicator approaches top dead center on the side.

 

When indicating at the top of the pin (sine) let the pink arrow (radius) be the hypotenuse and the green arrow becomes the side opposite and approaches the radius as the indicator approaches top dead center.

 

An error is introduced when adding and subtracting pin radii when the indicator is not on true top dead center because the indicated radius  is less than the true radius. see table 1.

 

Table 1 displays the error for the .9955 radius in red. The table reveals that as you indicate further away from top dead center the error grows larger.

 

Table 2 displays the error for the 2.01 radius. This table also reveals that as the radius of the cylinder pins increases so does the error.  Compare .015 error for .9955 radius and .030 error  for the 2.01 radius. (at 10 degrees before top dead center)

 

The tables indicate that these errors can accumulate from pin to pin and (at a position of 10 degrees before top dead center) could be as much as .045 error (.015 plus .030).

 

Conclusion:

 

If the approach of  the indicator is not  in direct line with the TDC of the pin your measurments could be in error. Do not perform the layout using the pin to pin method. Discard the pins and measure the TDC of the bores directly using a  smaller more accurate indicator such as a Starret  “Test Indicator” which should allow a more direct approach and minimize the error. If many such layouts must be done submit a request for a CMM.

 

 

Note: The trig functions change signs from one quadrant to another. Error calculations were adjusted to compensate for this.

 

 

TABLE 1    ERROR FOR .9955      
       
       
       

DEGREES

  OPPOSITE LENGTH
BTDC SINE SIDE-LENGTH ERROR
80.00 0.984808 0.980376 -0.015124
81.00 0.987688 0.983244 -0.012256
82.00 0.990268 0.985812 -0.009688
83.00 0.992546 0.988080 -0.007420
84.00 0.994522 0.990047 -0.005453
85.00 0.996195 0.991712 -0.003788
86.00 0.997564 0.993075 -0.002425
87.00 0.99863 0.994136 -0.001364
88.00 0.999391 0.994894 -0.000606
89.00 0.999848 0.995348 -0.000152
90.00 1 0.995500 0.000000
       
       
DEGREES   ADJACENT LENGTH
BTDC COSINE SIDE-LENGTH ERROR
170 -0.98481 -0.980376 0.015124
171 -0.98769 -0.983244 0.012256
172 -0.99027 -0.985812 0.009688
173 -0.99255 -0.988080 0.007420
174 -0.99452 -0.990047 0.005453
175 -0.99619 -0.991712 0.003788
176 -0.99756 -0.993075 0.002425
177 -0.99863 -0.994136 0.001364
178 -0.99939 -0.994894 0.000606
179 -0.99985 -0.995348 0.000152
180 -1 -0.995500 0.000000

 

 

 

       
       
TABLE 2 ERROR FOR 2.01      
       
       
DEGREES   OPPOSITE LENGTH
BTDC SINE SIDE-LENGTH ERROR
80.00 0.984808 1.979464 -0.030536
81.00 0.987688 1.985254 -0.024746
82.00 0.990268 1.990439 -0.019561
83.00 0.992546 1.995018 -0.014982
84.00 0.994522 1.998989 -0.011011
85.00 0.996195 2.002351 -0.007649
86.00 0.997564 2.005104 -0.004896
87.00 0.99863 2.007245 -0.002755
88.00 0.999391 2.008776 -0.001224
89.00 0.999848 2.009694 -0.000306
90.00 1 2.010000 0.000000
       
       
DEGREES   ADJACENT LENGTH
BTDC COSINE SIDE-LENGTH ERROR
170 -0.98481 -1.979464 0.030536
171 -0.98769 -1.985254 0.024746
172 -0.99027 -1.990439 0.019561
173 -0.99255 -1.995018 0.014982
174 -0.99452 -1.998989 0.011011
175 -0.99619 -2.002351 0.007649
176 -0.99756 -2.005104 0.004896
177 -0.99863 -2.007245 0.002755
178 -0.99939 -2.008776 0.001224
179 -0.99985 -2.009694 0.000306
180 -1 -2.010000 0.000000

 

Copyright :   James R. Howe

 

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