48 Collision www.collisionpublishing.com

Basic Integral Calculusfor Crash Reconstruction

Sean HaightTexas Christian University

College of Engineering

Collision Safety Institute ∫

ollision

e International Compendium for Crash Research

C

www.collisionpublishing.com Collision 49

There is an obvious link between Newtonian physics and crash re-construction; however, the link to mathematics is not as intuitive. Among other things, basic math is commonly used when adding vectors, evaluating crush, and converting between imperial and metric units. This article examines how fundamental calculus tech-niques can help define reconstruction variables, specifically delta-V (velocity change). As a refresher for some and first exposure in the application for others, this is a look at calculus and the definite integral in the context of crash analysis.

But why? As crash data becomes more common and more auto manufacturers (OEs) make their stored data available, we will most assuredly see that the data storage schemes will continue to vary between manufacturers and parts suppliers. As part of the differ-ences between OEs, we already see how some record delta-V differ-ently. Some OEs store delta-V while others store acceleration. For example, General Motors vehicles typically store delta-V in 10ms “blocks” for varying crash pulse periods and Ford vehicles store acceleration which is then used by the Bosch Crash Data Retrieval (CDR) software to calculate delta-V. But, how is that calculation done? What’s necessary to take a set of acceleration data points and convert that to some quantity of delta-V? When would I want to even do that sort of analysis?

Starting with the last question first, say you have a set of accelera-tion data from a crash test or imagine you download crash data from a car and it offers a crash pulse but not the delta-V. What are you going to do with the acceleration data? Simply put, you can calculate the “area under the curve” and find the delta-V. That delta-V is then used, of course, with other calculations to find clos-ing velocity and, in some cases, impact velocity. Before any of that; however, we have to work on getting from acceleration to delta-V and, to get there, we’ll cover some basics first.

We can start by reviewing the fundamentals of an acceleration pulse. Figure 1 is an example of a crash pulse captured during a frontal crash test. The horizontal axis is time in seconds, and the vertical axis is acceleration in g’s. There are several data points that are important in this plot. For example, the measured max accel-eration is around 14.5g, and the crash duration is about 96 milli-seconds. However, cumulative delta-V can’t be found just by look-ing at the graph and considering a maximum value so this is where calculus comes into play and where we contemplate the “definite integral.” For our purposes, the integral will be a number that represents the area enclosed by a time based curve, the time axis, and two time points.

ollision

e International Compendium for Crash Research

C

www.collisionpublishing.com

Collision Publishing, LLC PO Box 3383

Kirkland, WA 98083-3383 Phone: (800) 280-7940

Fax: (866) 966-7033 email: editor@collisionpublishing.com

Annual Subscription Form

Instructions Please print the PDF form, complete the information and mail or fax to Collision Publishing LLC. If you have any difficulty with this form or need assistance, please contact Collision Publishing at (800) 280-7940. Billing Information First Name: Last Name:

Billing Address:

City: State: Zip:

Company:

Email: Phone:

How did you hear about “Collision”? (Print clearly):

Shipping Information First Name: Last Name:

Shipping Address:

City: State: Zip:

Company:

Email: Phone:

Subscription Fees (one year) $139 per year – Only $119 per year for Collision Safety Institute Students! USE COUPON CODE “CSI” when placing your order online. Payment Information

Type of Card (AMEX, VISA, MC): Credit Card #:

Name on Card: Expiration Date (MM/YY):

□ Check (Payable to Collision Publishing LLC) I certify all the above information is true, correct and complete to the best of my knowledge. I authorize Collision Publishing, LLC to verify the above information and charge my credit card (if paying by credit card).

________________Signature

______________Date

MapScenes Forensic CAD, MapScenes Capture and MapScenes PointCloud provide professional investigators with the tools needed to visually communicate the output via accurate and compelling diagrams and animations.

Law enforcement and accident investigators use MapScenes Forensic CAD.

MapScenes Forensic CAD 2010 is the most precise desktop forensic mapping software available in the world. This powerful and feature-rich application provides law enforcement professionals and accident reconstruction specialists, like you, with confidence that the evidence they present is precise, compelling, and irrefutable.

Turn the page for more information about our power-ful animation and point cloud components.

Three desktop mapping & animation software packages to ensure your exact needs are met.

ACCIDENT RECONSTRUCTION AND FORENSIC MAPPING SOLUTIONS DESKTOP MAPPING & ANIMATION SOFTWARE | EVIDENCE COLLECTION SOFTWARE & HARDWARE | TRAINING

1.800.668.3312www.mapscenes.comMapScenes and MicroSurvey are Registered in the U.S. Patent and Trademark OfficeAll other brand names, product names or trademarks are property of their respective owners.®

PointCloud 2010PointCloud 2010

CDR Interface ModuleBosch ACM Cable

Air Bag ControlModule (ACM)

Computer running Bosch CDR Software

Power SupplyCord

USB-Serial Adapter (opt.)

Module to Computer Connection

The CDR System is a tool that provides access to vehicle crash and pre-crash data that may be stored in late model passenger cars, light trucks and/or SUVs.

For more information, quotes, or purchases, contactCrash Data Group

www.cdr-system.com(800) 280-7940