ABSTRACT
In essence, impact mechanics involves the study of colliding billiard balls rolling on a frictionless flat pool table and applies the results to vehicle crashes. The impact phase begins at first contact and ends at separation—for billiard balls, a short time indeed. Times and displacements are so short that they are often taken to be infinitesimal for billiard ball collisions, but that restriction does not apply to vehicle crashes; indeed, it is not necessary. In fact, the impact duration is mostly independent of collision speed, depending only on the masses and stiffnesses of the colliding bodies, as shown in Chapter 17.
In coplanar impact mechanics, each vehicle has three degrees of freedom: X and Y position, and yaw angle. The reconstruction task is: given the X and Y velocities and yaw rate for each vehicle at separation, find the translational and rotational velocities for each crash partner at impact—a total of six unknowns for the system of two vehicles.
Chapter 18 introduces the quantities mass, moment of inertia, impulse, and momentum, so that the general principles of impulse–momentum based impact mechanics can be discussed. The concept of effective mass is introduced so that eccentric collisions can be addressed. Using particle mass analysis for eccentric collisions is discussed, as is impulse–momentum using each body as a system. The result is an equation set with six unknown velocity quantities (three for each vehicle), plus the two unknown components of impulse, for a total of eight. However, we still have only six equations. We need two more.
One approach to this conundrum is Brach’s Planar Impact Mechanics. One equation (a constraint equation) is obtained by treating the restitution coefficient as a known quantity, thus relating closing velocity to separation velocity. A second constraint equation comes from introducing a quantity called the impulse ratio coefficient, and assuming that it is known. The equation system is then inverted (solved) to find the six unknown impact velocities.
In the Collision Safety Engineering approach, the constraint equations come from regulating the velocity components at the impulse center, which is specified by the user, such that no restitution occurs. The system of equations cannot be inverted, so they necessarily pose a forward-looking (initial value) problem. Use of this approach requires that the initial velocity estimates be adjusted, and the calculations repeated, until the separation velocities are reasonably close to those obtained from the run-out analysis. Adjustment of the impulse center may also be required.
Another approach is to treat the two vehicles as a single system, and apply the principle of energy conservation as well as momentum conservation. This approach requires the calculation of crush energy for both collision partners, which in turn involves comparing the crash damage incurred in the field accident to that incurred in crash tests. Such a comparison is a desirable (some might say necessary) check on the validity of the results. This approach of using both energy and momentum conservation is discussed further in Chapters 19–27.
