Topics related to collision & Trajectory analysis formerly on our 'Registrants only' area however which we get asked about frequently so believe shoud be in the open forum too
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The following is a quick list of some papers on the Critical Speed formula.
- Links are provided to SAE and the papers are available for approx $20 each.
Most SAE papers are also available at your local college or university library for review. Go to the library!
They are listed with most recent publications first. This is merely listed to provide a source for critical speed publications.
Please let us know of any additions at firstname.lastname@example.org
- A Comparison of Bayesian Speed Estimates from Rollover and Critical Speed Methods,SAE Technical Paper 2015-01-1434, Davis, G.,
Uncertainty in Calculations Using Lambourn's Critical Speed Procedure,SAE Technical Paper 2013-01-0779, Wach, W.,
- Abstract: Martinez and Schlueter  described a three-phase model for reconstructing tripped rollover crashes, where the vehicle's path is divided into pre-trip, trip, and post-trip phases. Brach and Brach  also described this model and noted that the trajectory segmentation method for the pre-trip phase needed further validation. When a vehicle leaves a measurable yaw mark at the start of its pre-trip phase it might be possible to compare estimates from the three-phase model to those obtained using the critical speed method, and this paper describes Bayesian reconstruction of two such cases. For the first, the 95 percent confidence interval for the case vehicle's initial speed, estimated using the critical speed method, was (64 mph, 81 mph) while the 95 percent confidence interval via the three-phase model was (66 mph, 79 mph). For the second case an initial inconsistency between the two estimates suggested a modification to the three-phase model, leading to confidence intervals of (72 mph, 91 mph) and (79 mph, 92 mph) for the critical speed method and three-phase model, respectively.
Analysis of Critical Speed Yaw Scuffs Using Spiral Curves, SAE Technical Paper 2012-01-0606, Daily, J.,
- Abstract: Critical Speed Formula (CSF) belongs to the canon of tools used in reconstruction of vehicle accidents. It is used to calculate vehicle speed at the beginning of tire yaw marks and, together with the entire methodology of processing the information contained in the marks into the data, is often referred to as the Critical Speed Method (CSM). Its great practical importance as well as recurring doubts as to the reliability make it one of the best experimentally and theoretically studied methods. Although the CSF applies in fact to a point mass, it is used with reference to a vehicle, i.e., an increasingly complicated multi-body system. Accident reconstruction experts point out the particular usefulness of Lambourn's research concerning the CSM in respect to a passenger car. Because his method by virtue of solid research basis is extensively applied in practice, the paper is focused specifically on Lambourn's methodology in terms of analysis of uncertainty and influence of vehicle properties. Regarding the first problem, sensitivity and uncertainty due to the parameters of the CSM was analyzed. Regarding the latter one, simulation tests using programs of different degrees of complexity were performed, taking into account criteria which were not considered in experimental tests. It has also been shown that in the case of unknown braking deceleration and large curvature of yaw mark, it will be desirable to shorten the chord c during the measurement and adjust the m/c (where m denotes the middle coordinate) ratio to the range from 0.01 to 0.03. This will ensure the reduction of calculation uncertainty to a level not exceeding ± 10%, regardless of the torques applied to the wheels. The article presents the theoretical arguments supporting and extending Lambourn's thesis concerning error no greater than ± 10%, subject to the guidelines prescribed by Lambourn.
A Critique of Critical Speed Yaw Mark Research, SAE Technical Paper 2012-01-0600, Richardson, S., Orton, T., Josevski, N., Pok, W. et al.,
- Abstract: Total station equipment, triangulation, or some other mapping technique can generate x-y coordinates describing curved tire marks on the pavement. These marks may result from a critical speed maneuver. Traditionally, these marks are assumed to follow a circular arc and a radius can be determined for use in the critical speed yaw formula. However, critical speed yaw marks typically have a decreasing radius in the direction of travel and a spiral is a more precise fit to the data. In this paper, a total least squares fitting approach is presented to fit the parameters of three types of spiral curves to coordinate data. These are a clothoid spiral, a logarithmic spiral, and an Archimedean spiral which are evaluated and compared for usability in a critical speed yaw analysis. A spreadsheet implementation is presented that makes use of the Microsoft Excel Solver Add-in to perform the minimization of the total least squares fit for the spirals. The goodness of fit was evaluated and showed an average deviation of close to 2 cm over a 44 meter arc length. An example is presented to demonstrate feasibility and application. Once the best fit spiral geometry was determined, the radius of curvature was evaluated and it was found the clothoid spiral may overestimate the radius of curvature at the beginning of the mark. A particle dynamics solution including normal and tangential components of friction is developed to show how speeds evolve while the vehicle travels in a spiral path. It was found that the logarithmic spiral matched the speed and acceleration data from the test example the closest. From the fitted parameters and a friction model, the velocity and acceleration of the vehicle can be determined. It can be concluded that spiral geometry better represents the actual path of travel and analysis based on spiral geometry has the potential to render accurate results over the length of the arc when using coordinate data for critical speed yaw analysis.
Validation of the Circular Trajectory Assumption in Critical Speed,SAE Technical Paper 2005-01-1189, Masory, O., Delmas, S., Wright, B., and Bartlett, W.,
- Abstract: Critical speed yaw marks are commonly used in collision reconstruction to estimate vehicle speed. Research and laboratory testing have demonstrated that critical speed calculations can be used to accurately estimate vehicle speed. Thus, the principles supporting critical speed yaw analysis are fundamentally and theoretically valid and are not being challenged in this study. However, there are observed and documented limitations with respect to the appropriate application and execution of critical speed yaw analysis. This paper reviews the published research to-date and identifies limitations of critical speed yaw analysis. Examples of collision scenes are provided which quantify the inaccuracies associated with the misuse of critical speed yaw calculations. Areas for further research are identified and detailed.
Challenging the Critical Speed Formula In Light Of the Daubert Decision,SAE Technical Paper 2005-01-3141, Fischer, W.,
- Abstract: The critical speed model is a tool used by accident reconstructionists to determine vehicle speeds. One assumption implicit in the model is that when in a critical speed yaw, the vehicle's center of mass travels in a circular arc. The validity of this assumption was investigated by comparing the results obtained by manually measuring the tire marks, assuming them parallel to the center of mass path, and fitting a polynomial. The results indicate that the assumption of a circular path is reasonably accurate.
Critical Speed Yaw - Special Situations, 2008 Pennsylvania State Police, Reconstruction Seminar (PSP 2008), Powerpoint presentation (includes a literature review), Ofc. Wade Bartlett, PE, Mechanical Forensics Engineering, Handout - 42 pages, 84 slides
- Abstract: Yaw marks are often observed at the scene of an accident when a car spins out of control on a curve. Police investigators measure the radius of these marks and conduct skid tests at the scene to determine the friction, or drag factor, on that road surface. The radius and friction values are then plugged into the Critical Speed Formula, or CSF, to compute the car's speed. Such estimates may later form the basis of the officer's expert testimony for speed related prosecutions, including criminally negligent homicide. The CSF is derived from a more generalized equation that balances cornering force with inertia. But the CSF is too simplistic to account for all the variables that affect a cornering vehicle, and the methods normally used to employ it are not valid. By default, the braking friction value from the police cruiser is substituted into the equation for the cornering force limit of the crashed car, because cornering friction is either too dangerous or impractical to obtain at the scene. The car that caused the yaw marks may be too damaged to drive - the best evidence destroyed - and the roadway not wide enough to conduct a skid pad test. The operant hypothesis is that the braking friction limit of every test car is equal to the cornering friction limit of any car on the road.
- Analysis of independent, real world data shows the potential for error using such methods exceeds 41%
Yaw Testing of an Instrumented Vehicle with and without Braking,
- Abstract: A review of literature on CSY and presentation of the most recent testing involving ABS, stability control systems, braking, and throttle on pavement, gravel, and grass surfaces.
Cliff, W., Lawrence, J., Heinrichs, B., and Fricker, T., SAE Technical Paper 2004-01-1187
An Analytical Assessment of the Critical Speed Formula,SAE Technical Paper 970957, Brach, R.
- Abstract: Two methods for calculating speed from curved tire marks were investigated. The commonly used critical speed formula and a computer simulation program were evaluated based on their ability to reproduce the results of full-scale yaw tests. The effects of vehicle braking and friction coefficient were studied. Twenty-two yaw tests were conducted at speeds between 70 and 120 km/h. For half of the tests, about 30% braking was applied. Using the measured sliding coefficient of friction, both the critical speed formula and the computer simulations under-predicted the actual speed of the vehicle. Using the measured peak coefficient of friction, both methods over-estimated the actual speed. There was less variance in the computer simulation results. Braking tended to increase the speeds calculated by the critical speed formula.
Vehicle Critical Speed Formula - Values for the Coefficient of Friction - A Review - , SAE Technical Paper 971148, Sledge, N. and Marshek, K.,
- Abstract: The Critical Speed Formula is used in the field of accident reconstruction for the estimation of the speed of a vehicle that has been given a sudden unidirectional steer maneuver by the driver and when the tires develop a high enough sideslip to leave curved visible marks on the pavement. This and other uses of the formula are investigated in this paper. Reconstructions are done using computerized dynamic simulations of a turn maneuver for 3 different, driver forward control modes: braking, coasting and accelerating. The experimental results of Shelton ( Accident Reconstruction Journal , 1995 ) are analyzed statistically and are compared to the results of the simulations. Results show that the Critical Speed Formula can give reasonably accurate results but that the accuracy varies with several factors. One is where along the trajectory measurements are made to estimate the tire mark curvature. Another factor is the forward control mode; the accuracy is the highest when the vehicle accelerates through the turn and is the lowest for braking. The experimental data is also used to determine the statistical uncertainty of speed estimation.
Formulas for Estimating Vehicle Critical Speed From Yaw Marks - A Review, SAE Technical Paper 971147, Sledge, N. and Marshek, K.,
- Abstract: This paper covers briefly the theory of tire-road friction, coefficient of friction measurement techniques, and the vagaries of tire-road friction as they relate to critical speed estimation. A literature review of tire-road friction studies was conducted to identify the primary factors effecting the tire-road coefficient of friction. Background information is presented covering general definitions and the connection between the basic critical speed formulas and the coefficient of friction. The primary components of tire-road friction, adhesion and hysteresis, are discussed along with minor effects such as tearing, wear, waves, and roll formation. Common coefficient of friction field measuring techniques are described, including the skid-to-stop test and drag sled. Influential factors such as tire characteristics, tire inflation pressure, road conditions, and dynamic factors are reviewed. Important dynamic factors are listed and the connection between longitudinal and lateral friction is discussed. Overwhelmingly, the literature indicates that the coefficient of friction is the function of many variables and that it is the most ubiquitous factor affecting speed estimates when the critical speed formula is used. Unfortunately, there appears to be no consensus regarding the appropriate value or measurement technique for the tire-road coefficient of friction used to estimate critical speed.
Evaluation of Vehicle Velocity Predictions Using the Critical Speed Formula,SAE Technical Paper 950137, Dickerson, C., Arndt, M., Arndt, S., and Mowry, G.,
- Abstract: This paper provides an exposition of the basic and some refined inertial critical speed estimation formulas. A literature review of existing inertial formulas for estimating critical cornering speed were identified for the ultimate purpose of developing a useful, compact, and more accurate speed estimation formula. Background information is presented covering the general definitions and utility of critical speed formulas. First, as a point of reference, the basic critical speed formulas are derived. Included is a list of the key assumptions on which the basic formulas are based. It is shown that the basic formulas are founded on the fundamental principles of physics and engineering mechanics; namely, Newton's Second Law and centrifugal force. Then refined formulas are presented which account for the effects of many important kinematic and dynamic factors ignored in the basic formulas such as: road grade, vehicle weight distribution, vehicle side-slip angle, axle and tire slip angles, superelevation, lateral and longitudinal drag factors, wheelbase, front steering angle, cornering stiffnesses, lateral load shift, friction dependency on load, aerodynamic forces, and anti-lock brake effectiveness.
Braking and Cornering Effects with and without Anti-Lock Brakes," SAE Technical Paper 940723, Lambourn, R.
- Abstract: Tire marks left by the vehicle prior to impact, rollover, or other event, are important forensic evidence reconstruction of motor vehicle accidents. Often these tire marks have some curvature that is measured and used to calculate the speed of vehicles prior to the event. This calculation is based on the coefficient of friction of the tire/road interface and the radius of curvature of the vehicle center of gravity (c.g.) path. There is controversy about the validity of this approach. To explore this theory, a test vehicle was driven through a series of maneuvers that produced yaw marks for direct comparison of actual vehicle velocity to the velocity calculated by the critical speed formula. Test results show the critical speed formula is inaccurate for most circumstances and does not correctly describe vehicle limit performance behavior.
- Abstract: The results of three inquiries of relevance to accident reconstruction are given. In the first the difference between anti-lock and locked-wheel braking on a dry surface is examined, with the finding that average decelerations with anti-lock are about 12% higher. In the second the effect of anti-lock operation on the curved yaw-mark method of speed calculation is explored; the finding is that the method is still valid, although it is possible that there is a greater tendency towards underestimation. The third inquiry is into the appearance and enhancement of tyre marks from anti-lock braking, where it is found that marks are infrequently made; they are more likely to be found on uneven surfaces, and while faintly visible marks can sometimes be seen more easily with polarised light, no way has been found of developing invisible or latent marks.