In our paper "

**CRASH-97 - Refinement of the Trajectory Solution Procedure**" SAE Paper 97-0949, there is a discussion of approximations for drag factors.

- As we point out in the paper, with the astounding power and performance of common PC computers the use of a simulation to approximate the spinout of a vehicle to rest is a relatively easy task. Any user of the CRASH3/EDCRASH or other CRASH based program has the trajectory solution procedure from SMAC included. The CRASH3 trajectory simulation procedure is basically the SMAC trajectory solution procedure.

**APPENDIX 1: DISCUSSION OF SPIN2**

- The SPIN2 procedure of the original CRASH program uses as a starting point the relationships developed by Marquard [1].

The Marquard procedure takes into account the fact that the linear and angular (i.e., yaw rotation) displacements of a four wheeled vehicle subsequent to a collision each occur under conditions of intermittent deceleration when the wheels are free to rotate. By approximating the linear and angular deceleration rates of a vehicle with either- (1) all wheels freely rotating or

(2) all wheels locked during different phases of spinout motion,

In the CRASH program [2], the SPIN2 routine was developed to extend the relatively simple Marquard relationships to include the cases of partial braking and/or damage-locked individual wheels.- Evaluations of the resulting, modified relationships by means of trial applications to spinout trajectories generated with SMAC [3] revealed several shortcomings of the initial SPIN2 relationships. First, a residual linear velocity frequently exists at the end of the rotational (i.e., yawing) motion. Next, the general shapes of plots of linear and angular velocity vs. time changed substantially as functions of the ratio of linear and angular velocity at separation from the collision. Finally, the transitions between the different deceleration rates of linear and angular motions were found to occur gradually rather than abruptly. Slope changes in the plots of linear and angular velocity vs. time were found to generally occur in the form of rounded "corners" in the curves.

To improve the accuracy of approximations of separation velocities, provisions for the introduction of a residual linear velocity at the end of the rotational motion and the development of empirical coefficients, in the form of polynomial functions of the ratio of linear to angular velocity at separation, were incorporated in the SPIN2 analytical relationships of the CRASH program. Since the separation velocity ratio is initially unknown, a solution procedure was developed whereby several trial values of the ratio, based on an approximate equation, were used to test multiple solutions.

- Polynomial functions to generate empirical coefficients were developed, on the basis of 18 single-vehicle SMAC runs with relatively high linear and angular velocities for starting (i.e., separation) conditions. In the more common, real-life accident case, a relatively small rotation (i.e., yawing) velocity may exist at separation. In such a case the initial direction of the velocity vector with respect to the longitudinal axis of the vehicle will obviously affect the sequence and the duration of the linear and angular deceleration rates of the vehicle.

- A representative sample of actual accident cases was selected from the NCSS [5] files for use in the study. A total of 50 cases were selected and then reconstructed with the SMAC computer program. For each of the SMAC reconstructions, separation information was used to formulate a basis for a refinement of the SPIN2 empirical coefficients.

A careful examination of the time-history plots of linear and angular velocities for all of the cases in the sample revealed a significant number of cases in which the SMAC-predicted behaviour deviated from the analytical assumptions upon which the SPIN2 routine is based. Attempts were undertaken within the research project to discriminate characteristics of separation conditions. Unfortunately, only partial success was achieved in the attempts to accommodate deviations by means of the use of logic and discriminators.

*"To achieve a general improvement in the reliability and accuracy of approximations of the angular and linear velocities at separation, a step-by-step time history form of trajectory solution should be implemented."*

- Any proposed refinements of the SPIN empirical coefficients and any reconstruction techniques which are based on the refinements of the SPIN empirical approach will ultimately fail in some applications to individual case reconstructions due to the possibililty that the particular case being investigated may be characteristic of a "scatter" point.

**The research cited in this paper strongly supports the conclusion from 1981 that implementation of a trajectory solution procedure should utilize an iterative time-history simulation.**

- (1) all wheels freely rotating or

- 1)Marquard, E.,“Progress in the calculations of Vehicle Collisions” Automobiletechnische Zeitschrift, Jarq. 68, Heft 3, 1966

2)McHenry, R.R., “The CRASH Program - A simplified Collision Reconstruction Program” Proceedings of the Motor Vehicle Collision Investigation Symposium, Calspan, 1975

3)McHenry, R.R.,"A Computer Program for Reconstruction of Highway Accidents", SAE Paper 73-0980, Proceedings of the 17th Stapp Car Conference, November 1973

4)McHenry, R.R., McHenry, B.G., "National Crash Severity Study - Quality Contol, Task V: Analysis to Refine Spinout Aspects of CRASH", Calspan Field Services, Inc. ZP-6003-V-4; DOT-HS-6-01442, January 1981

5)Kahane, C.J, et al, "The National Crash Severity Study", Sixth International Technical Conference on Experimental Safety Vehicles (1976) 495-516

6)Fonda, A.G., Metz, L.D., "Post-Impact Spin, 1968-1993", SAE Paper 93-0653

7)Fonda, A.G., "Energy and Major Diversion in Accident Reconstruction", SAE Paper 96-0888