How to verify a Residual Ratio Limit.

Question:

How do I verify the residual ratio calculation using the RAH output? The charts show that the heeling area does not intersect the righting area but the RA report is showing stability assessment results.

Answer:

The residual ratio limit compares the residual area from the starting angle to equilibrium to the area from equilibrium to the terminating angle. The area to the left is technically negative, but in this ratio the absolute values are used. The area column in the RAH table displays accumulated residual area so computing the residual ratio limit from the tabular data requires only minor arithmetic.

The RAH table on Page 1 of ResidualRatio.PDF indicates that the area between the starting angle and equilibrium (angle range 12.85p to 10.81s) totals -47.14 ft-deg. The area from equilibrium to the terminating angle is the area at the terminating angle minus the area at equilibrium. For example, limit 4 has a terminating angle of RA0, so the angle range is 10.81s to 39.50s. The cumulative area at RA0 is -23.61 so subtracting the area at EQU from the area at RA0 yields, -23.61-(-47.14)=23.53. The residual ratio is then 23.53/47.14 or 0.499.

The second graph in the RAH output plots the absolute heel area and absolute righting area. In order for the righting area to intersect the heeling area, the Residual Ratio needs to be at least one. The heeling arm does intersect the righting arm, so the angles needed to evaluate the limit: ROLL, EQU, and third angle (FLD, RA0 or ABS 40), are all defined.

Normally, the three residual ratio limits from the report would be combined into two limits with RA0 appearing in both as follows:
Limit(1) Residual Ratio from Roll to RA0 or FLD
Limit(2) Residual Ratio from Roll to RA0 or ABS 40
Including RA0 in both limits prevents negative area accumulation between RA0 and the terminating angle as in limit 3. For more information, see QA076.