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Answer #1: The four fitted constants are calculated only once, before the start of an application run. When a given radial spring starts to unload, the residual crush for that spring is calculated from equations (17) and (24), or the equivalent equation (9). Equation (8) was presented only for the purpose of showing that the value of K_{1} is independent of the extent of restored energy and, thereby, of the restitution coefficient.
Answer #2: The derivation of equation (8) is presented in appendix 2 in the form of equation (26). While the manner of derivation is different, the result is equivalent to equation (10) in SAE #910119.
Answer #3: In figure 6A, the listed values of A, B, (/)_{1, }()_{1}, and ()_{1} were used to calculate the value for K_{v} (by application of either equations (2) through (5) or equation (8)). In Figures 7, 8 and 9, the same A and B values as in Figure 6A were applied with different inputs for the restitution properties:
Fig.6A |
Fig. 7, 8 & 9 |
||
A |
317 |
317 |
LB/IN |
B |
56 |
56 |
LB/IN^{2} |
(_{m})_{1} |
30 |
30 |
IN |
()_{1} |
0.2000 |
0.1500 |
- |
(/)_{1} |
0.8000 |
0.8775 |
- |
K_{1} |
54.74 |
63.7 |
LB/IN^{2} |
K_{2} |
54.74 |
95.5 |
LB/IN^{2} |
Note that a major point in the paper is the fact that a given set of values for A and B can serve to represent a wide range of restitution behavior.
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Answer#4: Clarification #7 refers specifically to the CRASH (EDCRASH) computer program in which the implied elastic deformation range, in terms of full dimensional recovery is equal to A/B. In prior publications (e.g., SAE papers 910119, 920607, etc.) "A" has been referred to as the "zero residual crush force" or the "stiffness coefficient which represents the maximum force per unit width of the contact area which produces no crush." Therefore, the quantity A/B must be recognized as effectively being the elastic deformation range in terms of full dimensional recovery.
With the revised form of restitution modeling presented in SAE #970960, the corresponding elastic deformation range, in terms of full dimensional recovery, is given by equation (26) with set to zero. In that case, the maximum deformation for elastic behavior, in terms of full dimensional recovery, is defined as:
For the A, B values of the first Figure A3, which were fitted by Monk and Guenther (Monk, M.W., Guenther, D.A., "Update of CRASH II Computer Model Damage Tables, Vol. I", DOT HS-806446, March 1983), the following results can be obtained:
While 12.93 inches of elastic deformation, in terms of full dimensional
recovery, also seems large, it is certainly more believable than the 27.5
inches of elastic deformation in terms of full dimensional recovery in
CRASH (EDCRASH) for the same (questionable) fitted values of A and B.
The purpose for inclusion of Figure A3 was to demonstrate that (1) unusual A, B fits such as Category 4, Rear, can be accommodated with crush properties that appear to be reasonable and (2) force equilibrium is possible in collisions between vehicles with widely different values for A and B.
As indicated in the paper, progress towards a rigorous and complete validation is data-limited at the present time.
Answer#5: In Figure 8 of SAE #970960 (revised SMAC) the
maximum deformation corresponding to a zero residual deformation, from
equation (9):
For = 5.308 Inches,
the coefficient of restitution, from equation (1):
Since the calculated value for is >
1.00, is set equal
to the maximum value of 1.0.
Thus, extension of the slope of the V
line in Figure 8 will include a change in the slope of the
V
line at that value of residual deflection where the calculated
equals
1.0.
From equation (1):
=6.410 Inches
From equation (9) the corresponding value of residual deflection is:
The intercept is then obtained using equation (19) and the coefficient
of restitution (1.0):
= (2.00)(6.74)
= 13.48 MPH
This means that, for the given fitted properties, the residual deformation would be zero in a 6.74 MPH SAE barrier crash. The corresponding V would be 13.48 MPH.
Another item to note is that because of the selected form of restitution control in the original NHTSA SMAC (EDSMAC), the V line has an intercept through (0,0). In particular, in original SMAC the residual deflection approaches ~95% of the maximum deflection at small deflections (see Figure 6 of the paper). In real-life, the residual deflection should become a very small portion of the maximum deflection at small deflections (i.e, the ratio of should approach 0.0).