Sir Isaac Takes the Wheel

From a pared-down perspective, vehicle dynamics mathematically describe the movement of a vehicle on a road surface. More specifically, it describes any change in direction, elevation, or speed that a vehicle can exhibit while it is in motion. The equation that expresses that movement derives from Newton's second motion law: force equals mass multiplied by acceleration (F = m x a). Ultimately, it is acceleration that sets objects in motion. Recalling Newton's first law that mass needs an outside force to compel its motion, one realizes that there would be no accidents to investigate without “a” because “m” would remain at rest. A host of mathematical expressions develop from that initial equation to describe and model the actions and interactions of automotive systems and operations. Thus, vehicle dynamics are the ultimate analytical and design framework for engineers.

The advent of microprocessor technology has revolutionized the application of vehicle dynamic methodologies in designing, manufacturing, and testing vehicles. Sure there are still crash test dummies, proving grounds, and wind tunnels to run real-world tests, but they are extensions of virtual computer laboratories that can model vehicles and their systems well before assembly-line implementation.

There are substantial bottom-line benefits to these CAD systems that optimize mathematical variables so easily. Features like ABS and electronic stability control were once considered luxuries. Nowadays, they appear more quickly as standard features, making improved handling and performance affordable at virtually all price points.

History Needs Math Props

Even as vehicle dynamics point the way to the future of automotive design and functionality, the discipline's tools are useful beyond the automaker's realm. Vehicle dynamic applications can track history for automotive insurance professionals. For instance, what happened when Car A slammed into Car B? Was tire failure the cause — or the result — of the accident? Was a driver speeding? Any number of questions can be answered by applying mathematical analyses to the data gathered at the scene of an accident. In essence, vehicle dynamic equations can function as a litmus test for driver and witness statements. Using the scenario sketched above, a simple speed-from-skids analysis could show that Driver A was speeding when Cars A and B collided, perhaps contradicting Driver A's statement that he was traveling at the posted speed when the accident occurred. Many lower-dollar-value claims can be resolved by using simplified vehicle dynamic equations to examine fault issues.

Typically, though, accident investigations require a nuanced approach that reflects an in-depth understanding of broader vehicle dynamic issues. The most common situations calling for vehicle dynamics are those accidents with unresolved issues of tire, braking, and cornering performance as well questions pertaining to steering and acceleration. However, it is crucial to understand that the analytical challenges facing investigating engineers differ significantly from those involved in manufacturing.

Engineers working in CAD environments track immense volumes of data and can assign values to equation variables with relative ease based on controlled testing. Those investigating accident sites must compensate for evidence that is damaged, elusive, and, frequently, uncollectible. As a result, compiling the data or the “history” from the accident scene to perform a mathematical analysis is as much an art as it is a science. The art aspect comes from experiential knowledge of vehicle dynamics rather than a creative whim. The engineers most adept at forensic investigations are those who couple intellectual depth with hands-on experience of vehicle dynamics as competition drivers, kit-car hobbyists, motorcyclists, truckers, or even test engineers and the like. Ultimately, that real-world know-how amplifies the skills needed to sift through roadway debris, measure tire skid marks, and probe crumpled steel to assign numeric values that can then be plugged into vehicle dynamic equations.

Also important to reconstructing automotive accidents are the databases of crash studies and vehicle/component specifications developed by the automotive industry and its satellite consumer groups, governmental and regulatory agencies, and testing laboratories. When conducting a tire analysis, for example, an engineer will correlate the tire data gathered at the accident scene with tire performance attributes collected from manufacturers and the automotive press. The engineer also may use tables specifying roadway friction coefficients or crush measurements compiled by researchers from controlled testing. Such resources bridge the gap between the known and unknown accident data and help the investigating engineer test the rigor and validity of the report's conclusions. Above all, the accident reconstruction performed by an automotive engineer must be based on solid reasoning supported by mathematical proofs.

When Cars Collide

A forensics engineer usually has a rough idea of an accident's parameters when starting an investigation. Chances are, he will suspect that certain elements influenced the outcome; however, he may not have the means of measuring their impact on the accident. By sketching out the worst and least likely scenarios and compensating for the unknowns, the engineer can typically find an answer that satisfies unresolved questions.

For example, in a fatal collision of two vehicles, the outstanding question was determining the traveling speeds of the vehicles prior to the accident, based on the events preceding and following the crash. The engineer relied on police reports and photographs; vehicle inspection reports; witness interviews; preliminary hearing summaries; and a report filed by the opposing side's expert to reconstruct the accident mathematically.

Vehicle 1 was turning left from a northbound road to a westbound road when Vehicle 2 struck his car broadside on the passenger side. Following the impact, both cars moved in a southerly direction and ended up in proximity to each other, about 50 feet from the point of impact. Vehicle 1 rotated clockwise about 120 degrees from its initial westerly direction as it slid and came to rest. Vehicle 2 continued in its southerly direction of travel as it too slid and came to rest. There were two deaths and serious injuries among the passengers of Vehicle 1. Three independent witnesses gave statements, but two had prior knowledge of the driver of Vehicle 2, and the third witness had legal issues and did not want to impeach himself.

Armed with the facts, the engineer performed a speed and force analysis based on the principles of conservation of momentum and of energy, two fundamental principles at work in a vehicle dynamics analysis. He input data derived from the police report to help calculate the velocities and angular velocity components of the two vehicles. Balancing the known data and weighting the unknown information, the engineer determined (with momentum analysis) that Vehicle 2 was traveling between 78 and 115 miles per hour prior to impact. Subsequently, using calculations following conservation of energy principles, he concluded that Vehicle 2 was traveling at 78 miles per hour at the moment of the collision. He performed the two analyses and used the information corroboratively to determine the probable speed. His results, when rendered mathematically, satisfied all outstanding issues and assigned fault to the driver of Vehicle 2, a conclusion that highlighted the errors in the opposing side's expert assumptions and calculations. The claim was resolved in favor of the driver of Vehicle 1.

Saving Claim Dollars

While vehicle dynamics analysis is not for everyone, its advantages — especially when applied to automotive accident investigations — are readily applicable for clarifying the factual issues affecting the resolution of complex claims. A vehicle dynamics analysis conducted by a skilled engineer will provide a fact-based, dispassionate perspective while answering outstanding questions. It can therefore be considered a resource for those unraveling a claim with high-dollar value exposures.

While a forensics engineer's input is useful at any point in the claim resolution process, the key to maximizing the benefits of this input is to engage an engineer early in the claim cycle before unresolved issues can snowball into expensive legal actions. Determining causation and facts in the initial stages increases settlement rates and equips the insured's counsel with solid data to support arguments when litigation is the probable outcome. Although a brilliant thinker beyond his time, Newton never conceptualized the internal combustion engine that powers today's automobiles. Moreover, he certainly never imagined that his mathematical processes would illuminate the fault issues when “m's” collide.

Peter R. Thom is principal of Peter R. Thom & Associates Inc., a national firm of consulting automotive engineers. Vernell M. Hance is a 21-year associate. Contributor Donald L. Margolis, Ph.D., is an 18-year associate and a professor of mechanical engineering at the University of California. They may be reached at 800-874-1664; www.prtassoc.com.