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Write a 2 page single spaced paper with 11 font and could include graphs about the safety of vehicles. General prompt and the sources to be used will be included. The paper is due by tonight midnight. The paper should talk about what makes vehicles unsafe and what the solutions to that could be. Focus on how cars are flammable, airbags, brakes. Could use other references if needed.

Airbag Burns: An Unfortunate Consequence of Motor
Vehicle Safety
Thousands of people are injured in motor vehicle collisions daily and the mandated installation of airbags protects
passengers but can also cause injuries from deployment including cutaneous burns. We sought to characterize the patterns
and outcomes of burns resulting from airbag deployment by performing a retrospective review of all patients evaluated by
the burn service from May 1, 2015 to April 30, 2019. Inclusion criteria were patients of all ages with burn injuries related
to airbag deployment. Demographic data, burn characteristics, and outcomes were reviewed. Seventeen patients met the
inclusion criteria: 82.4% female and 17.6% male. The average age was 40.4 years. Fifteen patients had second-degree and
two had third-degree burns. The average TBSA was 0.45%. The hands or upper extremity (88%) were most often injured,
but there were two chest, one neck, and one anterior thigh burns. Eight patients suffered multiple burns. Burn etiology
(chemical vs thermal) was often not specified. No patients required hospitalization or surgical intervention, and all
wounds healed with wound care. The average time to re-epithelialization was 11 days. Although airbags prevent mortality
and serious injury, the exothermic chemical reaction that inflates the airbag is responsible for deployment-related burns.
Since there is a chemical and thermal component, all airbag-related burns should undergo chemical decontamination
on the initial presentation. Burns related to airbag deployment tend to be small and do not require grafting; however,
patients suffer from associated pain, scarring, and burn management can be a financial and time burden to the patient.
More than 7500 Americans are injured in motor vehicle
collisions every day, with an estimated economic burden
of $240 billion annually.1,2 All vehicles manufactured since
1997 are mandated to have dual airbags installed. The
National Highway Traffic Safety Administration estimated
that 2790 lives in people aged 13 and older were saved by
frontal airbag deployment in 2017. The use of airbags in
combination with seat belts has been shown to have an even
greater effect than either used alone.3 However, the ubiquity
of airbags in automobiles has shown a surge in airbag-related
injuries including abrasions (63.6%), contusions (37.8%),
lacerations (18.2%), burns (7.8%), fractures (3.2%), and retinal detachment (1.8%).4,5
It is important to note the mechanism of an airbag deployment, as it is intrinsically responsible for airbag-related burns.
Rapid vehicle deceleration is detected by sensors which set off
a chain reaction resulting in airbag inflation on frontal collision. The reaction starts with the ignition of sodium azide,
culminating in the release of nitrogen gas, carbon dioxide,
carbon monoxide, ammonia, and alkaline aerosol.6 The alkaline aerosol is a mixture of sodium hydroxide, sodium carbonate, and metallic oxides. The chemical reaction triggers
From the *Department of Surgery, Division of Plastic Surgery, University
of Rochester Medical Center, New York; †Jacobs School of Medicine and
Biomedical Sciences at the University of Buffalo, New York
Conflict of interest statement. None.
Disclosure of funding received: None.
Address correspondence to Kathryn E. H. Skibba, MD, Department of Surgery,
Division of Plastic Surgery, University of Rochester Medical Center, 601
Elmwood Ave, Box SURG, Rochester, NY 14642. Email: Kathryn_skibba@
© The Author(s) 2020. Published by Oxford University Press on behalf of the
American Burn Association. All rights reserved. For permissions, please e-mail:
the rapid inflation of the airbag. As an exothermic reaction,
significant heat up to 500°C is generated.7 There are vents at
the base of the airbag which prevent airbag explosion during
rapid inflation as well as allowing for cushioning of the passenger during the collision. The existence of the airbag vents
allows the heat and alkaline gases produced in the exothermic
deployment reaction to be released into the vehicle.8,9
The release of hot gas and alkaline substance into the cabin
puts patients at risk for both thermal and chemical burns.
The corrosive nature of the alkaline aerosol, particularly, the
sodium hydroxide, has been implicated in alkaline chemical
keratitis and cutaneous burns.10 The proximity of the ventilation system to the driver’s arms and hands makes this a likely
location for direct thermal burns caused by contact with the
stream of gas. Indirect thermal burns may occur by melting
or ignition of passenger clothing from contact with hot gases
released from the airbag.
There are numerous case reports of airbag-related burn
injuries but there is a paucity of literature that analyzes a cohort of these patients. The literature describes burns to the
hands or face and the etiology is split between chemical alkali
and thermal causes.6,10–13 At our institution, we have seen numerous burn injuries related to airbag deployment. This study
aims to further characterize the size, location, severity, and
consequences of burns from airbag deployment.
A retrospective review of all patients evaluated by the burn
service was performed from May 1, 2015 to April 30, 2019.
Inclusion criteria include patients of all ages with cutaneous
burn injuries resulting from motor vehicle airbag deployment.
Demographic data and burn characteristics including TBSA,
burn severity, anatomic areas involved, and surgical treatment
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Kathryn E. H. Skibba, MD,* Chelsea N. Cleveland, BS,† and Derek E. Bell, MD*
Journal of Burn Care & Research
January/February 2021
72  Skibba et al
were reviewed. The determination of burn etiology was based
on burn characteristic and patient report. Diffuse and even
depth burns over exposed skin were deemed to be chemical
burns. Patients recalling heat exposure at the site of injury
were categorized as thermal burns.
Table 1. Burn wound characteristics
Average TBSA
Burn severity
Second degree only
Third degree only
Second and third degree
Burn location
Forearm/upper arm
n (%)
6 (35.3)
3 (17.6)
8 (47.1)
15 (88.2)
1 (5.9)
1 (5.9)
10 (58.8)
6 (35.3)
2 (10.0)
1 (5.0)
1 (5.0)
While airbags are an effective method to prevent serious injury and death from motor vehicle accidents, the literature
recognizes that airbag deployment can cause cutaneous burns.
The burns are a result of either contact with alkali chemicals
and/or thermal injury created by the exothermic reaction of
deployment. The clinical manifestation of the burn may not
distinguish whether it was a chemical or thermal etiology, as
seen in 47% of our patients. We recommend that all burns related to airbag deployment should be considered as partially
chemical etiology and standard decontamination with irrigation of the wound should be performed at initial evaluation.
Interestingly, 82.4% of the patients are female. This is an
unusual finding among burn cohorts, as most are either split
equally between men and women or have a male dominant
cohort. There is no obvious explanation for this finding.
Possibilities are that women may be more likely to seek
treatment, have more exposed skin vulnerable for injury, or
that they may be more likely to be involved in a motor vehicle accident.
Notably, the average wound size was small at 0.45% TBSA.
Most wounds were determined to be second degree; only
12% of patients had full-thickness burns. This is important to
consider because all patients healed their wounds without the
need for skin grafting or other surgical intervention.
The anatomic locations of burns were consistent with current literature, mostly (88%) occurring on the hands or upper
extremity. There were no patients with facial burns, but one
patient did suffer from a burn to the neck. Interestingly, two
patients had burns to the chest and one patient had an anterior thigh burn. This indicates that airbag-related burns can
occur throughout the body. Patients presenting after airbag
deployment should prompt a thorough examination for cutaneous injury on initial evaluation.
Although the wounds may be considered minor, burn
injuries should not be an expected complication of airbag
deployment. Patients who experience injuries secondary to
airbag deployment are inconvenienced by burn management
including the emergency department and clinic visits, as well
as wound care. Hand burns are especially burdensome as open
wounds may prevent patients from returning to work until
re-epithelialization. Patients may suffer permanent scars from
the burns which primarily occur in cosmetically sensitive areas
such as the hands, neck, and face. Therefore, airbag-related
burns can burden patients with pain, scars, need for wound
care, cost of healthcare, and missed wages.
There is no question about the efficacy of airbags in preventing
death and serious injury. However, the current mechanism of
an airbag deployment is intrinsically hazardous to passengers
as it releases hot gases and alkaline substances into the cabin.
Thermal and chemical burns may be avoided with alteration in airbag deployment location or mechanism design.
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A total of 17 patients met the inclusion criteria: 82.4% female and 17.6% male. The average age at the time of burn
evaluation was 40.4 ± 23.7 years (range 20–82 years). Most
patients self-identified as Caucasian (82.4%). Comorbidities
were found in seven patients, which included obesity (35.3%),
psychiatric illness (23.5%), smoking (11.8%), hypertension
requiring medication (11.8%), and diabetes mellitus (6.5%).
The characteristics of the burns are summarized in Table 1.
There were 15 patients with exclusively second-degree burns
with an average TBSA of 0.47% ± 0.40%; one patient suffered
an exclusively third-degree burn with a TBSA of 0.16%; and
one patient had both second- and third-degree burns with a
total TBSA of 0.51%. The average TBSA for all patients was
0.45%. Ten patients (58.8%) had burns on the hand or wrist, six
(35.3%) had forearm or upper arm burns, two had chest burns,
one had a neck burn, and one had an anterior thigh burn.
In eight patients (47%), the burn etiology (thermal or
chemical) could not be clinically distinguished. Six patients
(35%) had chemical burns and three patients (18%) recalled
heat exposure and were categorized as thermal injury. One of
the three thermal patients had ignition of his clothing, causing
a flame burn injury.
Three patients required hospitalization for concurrent
injuries sustained in the motor vehicle collision. The burn
service monitored and treated the burn injuries while inpatient, but no patients required admission solely to receive
burn care. The average length of stay was 12.7 days. No
patients required intensive care or intubation. No patients
died on admission.
No patients received antibiotic therapy and there were no
burn wound infections. All patients were able to heal their
burns with daily wound care; therefore, no patients required
surgical intervention. The average number of wound care days
performed before burn wound re-epithelialization was 11 ± 5.
Journal of Burn Care & Research
Volume 42, Number 1
Skibba et al  73
Airbag-related burns should not be minimized as they cause
pain, scarring, and a financial and time burden to the patient.
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Washington (DC): National Highway Traffic Safety Administration;
2015. Report No.: DOT HS 812 013.
3. National Center for Statistics and Analysis. Lives saved in 2017 by restraint use and minimum-drinking-age laws. Traffic Safety Facts Crash
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Stats. Washington (DC): National Highway Traffic Safety Administration;
2019. Report No.: DOT HS 812 683.
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Simulation-Based Assessment of Vehicle Safety Behavior
under Hazardous Driving Conditions
Suren Chen, P.E., M.ASCE1; and Feng Chen, S.M.ASCE2
Abstract: Future sustained economic growth of the nation very much depends on the reliability and efficiency of its highway infrastructure system. Some vehicles, such as trucks, emergency vehicles, and sport utility vehicles, often experience increasing risks of singlevehicle accidents under hazardous driving conditions, such as inclement weather and/or complicated topographical conditions. An
advanced simulation-based single-vehicle accident assessment model is developed considering the coupling effects between vehicles and
hazardous driving conditions, including wind gust, snow-covered or icy road surface, and/or curving. Compared to existing simulation
models, the new model focuses on characterizing the transient process of accidents, introducing new critical variables on assessing the
accident risks under more comprehensive hazardous driving conditions and establishing more realistic accident criteria. As a holistic
deterministic model, it can be used to provide useful assessment and prevention information for traffic and emergency management. For
example, it can be used to define appropriate safe driving speed limits for vulnerable vehicles under normal and extreme conditions and
predict potential crash and injury risk of vulnerable drivers. Moreover, the new deterministic vehicle safety behavior simulation model
lays a critical basis for future reliability-based studies of single-vehicle accident risks of vulnerable vehicles under hazardous conditions.
After the model is introduced, numerical analyses on a typical truck under several representative hazardous scenarios will be conducted
for demonstration purposes.
DOI: 10.1061/å…±ASCEå…²TE.1943-5436.0000093
CE Database subject headings: Traffic accidents; Hazards; Vehicles; Simulation.
Author keywords: Accident; Hazardous; Traffic; Conditions; Simulation.
In the United States as well as other developed countries, road
accidents are causing more injuries and casualties than any other
natural or man-made hazard. Large commercial trucks, high-sided
sport utility vehicles å…±SUVså…², and emergency vehicles å…±e.g., fire
trucks and emergency medical service å…±EMSå…² vehicleså…² are especially vulnerable to single-vehicle crashes å…±e.g., rollover, sideslipå…²
under hazardous driving environments on rural highways. The
hazardous driving environments may include inclement weather
å…±e.g., strong crosswind gusts, snow, rain, or iceå…² and/or complicated terrain å…±e.g., steep slopes or sharp curveså…² å…±The National
Academies 2006; USDOT 2005å…². In 2005, single-vehicle accidents were responsible for 57.8% of accident fatalities å…±USDOT
2005å…². Each year in the United States, adverse weather alone is
associated with more than 1.5 million vehicular crashes, which
result in 800,000 injuries and 7,000 fatalities å…±The National Academies 2006å…². Among various causes of crashes in rural areas, it
has been found that the dominant causes are excessive speeds and
adverse environments å…±The Road Information Program 2005å…². In
Assistant Professor, Dept. of Civil and Environmental Engineering,
Colorado State Univ., Fort Collins, CO 80523 å…±corresponding authorå…².
E-mail: suren.chen@colostate.edu
Graduate Research Assistant, Dept. of Civil and Environmental
Engineering, Colorado State Univ., Fort Collins, CO 80523. E-mail:
Note. This manuscript was submitted on October 14, 2008; approved
on August 13, 2009; published online on August 15, 2009. Discussion
period open until September 1, 2010; separate discussions must be submitted for individual papers. This paper is part of the Journal of Transportation Engineering, Vol. 136, No. 4, April 1, 2010. ©ASCE, ISSN
addition to direct safety threats, frequent single-vehicle accidents
will also cause serious congestions, affecting the functionality of
the whole highway network in normal situations, as well as under
emergency. Therefore, for trucking industries, transportation, and
emergency management agencies, it is critical to accurately predict the crash risk and further advise appropriate driving speeds
under complicated adverse driving environments.
Different from multivehicle crashes, single-vehicle crashes
under adverse or hazardous environments were found to be
closely related to the coupling among vehicle, infrastructure, and
environment å…±Baker 1991; Guo and Xu 2006; Chen and Cai 2004;
Chen et al. 2009å…². As a result of this unique coupling, observations solely from historical crash data in one place can hardly be
translated into accurate risk prediction in different places or under
driving environments which were not covered by the actual crash
data. Therefore, in addition to analyzing actual historical crash
data gathered after the crashes, investigations on single-vehicle
crashes also require a reasonable simulation model which can be
used more than for an after-the-fact reconstruction of the crash
å…±TRB 2007å…², but more important, to reasonably predict the potential risk of crashes under comprehensive scenarios including those
which may not be covered by historical crash data.
In automobile engineering, significant efforts have been put
forth on simulating vehicle dynamics and accidents with engineering simulation models, from the simple rigid body model, the
bicycle model, to the complicated spring-mass multiple-degreeof-freedom model å…±Thomas 1992å…². Despite extensive works in
these fields å…±e.g., Winkler and Ervin 1999; Gaspar et al. 2004,
2005; Sampson 2000å…², research on vehicle accident risks, which
considers the coupling among the vehicle dynamic model, inclement weather, and topographical condition, is still very limited. Baker å…±1986, 1987, 1991, 1994å…² was the first researcher who
tried to investigate the high-sided vehicle accident risks under
strong crosswind. In his studies, vehicle accident risks were assessed through solving several static equilibrium equations with
some predefined accident criteria. Based on Baker’s work, several
reliability-based accident assessments were recently conducted
共Sigbjörnsson and Snæbjörnsson 1998; Snæbjörnsson et al. 2007兲.
Chen and Cai å…±2004å…² improved the accident risk assessment by
introducing a general dynamic interaction model, based on which
the vehicle accident assessment was conducted by considering
excitations from the supporting structure å…±e.g., bridgeå…². Guo and
Xu å…±2006å…² introduced an integrated vehicle safety assessment
model on bridges. In the model, the dynamic bridge-vehicle-wind
interaction analysis as well as the safety assessment was carried
out at the same time based on the same accident criteria by Baker
å…±1991å…². In most existing studies, however, only situations that
vehicles are driven on straight routes with only crosswind excitation were considered. In the present study, a general vehicle
safety behavior simulation model is introduced to consider the
coupling effects with more realistic hazardous environments, including combinations of both inclement weather and complicated
topographical conditions. Improved transient dynamic equations,
accident criteria, and new critical variables will also be incorporated into the model.
Theoretical Formulation
Fig. 1. Addition of the velocity vectors
ar = longitudinal distances from the center of sprung mass to the
front and the rear axles, respectively.
Crosswind Forces
Crosswind velocity can be obtained from actual measurements or
from numerical simulations based on existing wind velocity spectra å…±Baker 1991; Chen and Cai 2004å…². Typically, quasistatic assumptions are applied to simulate the wind loadings acting on
moving vehicles å…±Baker 1987, 1994; Coleman and Baker 1994å…².
The crosswind-induced quasistatic forces and moment acting on
the vehicle body on x-, y-, and z-directions are defined as follows
å…±Baker 1994å…²:
Fx = 0.5␳CFxAVre
drag force
lift force
Fy = 0.5␳CFyAVre
The general accident simulation model is introduced in this
section: after the primary forces acting on a vehicle are introduced, series dynamic models are developed to simulate the dynamic response under different stages of the transient process of
Fz = 0.5␳CFzAVre
side force
M x = 0.5␳C MxAVre
hre rolling moment
Primary Forces Acting on Vehicles
M y = 0.5␳C MyAVre
hre yawing moment
M z = 0.5␳C MzAVre
hre pitching moment
Tire Forces
When a vehicle is cornering, the lateral tire forces perpendicular
to the direction of the driving velocity applied at the contact
patches of the wheels are approximated to be proportional to
the tire slip angle. The lateral tire forces of the front and the
rear tires are defined in Eqs. å…±1å…² and å…±2å…² as follows, respectively
å…±Gaspar et al. 2004, 2005å…²:
Fy,f = ␮c f ␣ f
Fy,r = ␮cr␣r
where ci 共i = f or r兲 = tire cornering stiffness and ␣i 共i = f or r兲
= tire side slip angle associated with the front and the rear axles,
respectively; ␮ = road adhesion coefficient; and subscripts y, f,
and r denote the lateral direction å…±y-directionå…², front and rear
wheels, respectively.
The classic equations for the tire slip angles of the front 共␣ f 兲
and the rear 共␣r兲 wheels can be defined as 共Gaspar et al. 2004,
␣ f = − ␤ + ␦ − a f ␺˙ /V
␣r = − ␤ − ar␺˙ /V
where ␤, ␦, and ␺˙ = sideslip angle, steer angle, and yaw
rate, respectively; V = driving speed of the vehicle; and a f and
where ␳ = density of air; A = reference area; hre = reference arm;
CFx, CFx, and CFz = wind force coefficients; and C Mx, C My, and
C Mz = wind moment coefficients in å…±aboutå…² x-, y-, and z-directions,
respectively. These wind coefficients, which are typically obtained from wind tunnel testing å…±Baker 1994å…², are related to the
profile of a specific vehicle and are functions of attack angle ␽.
Due to the lack of wind tunnel testing results of vehicles during
the process of accident-related motions, it is assumed in the
present study that the wind loadings acting on the vehicle remain
the same during the process of rollover or sideslip. Vre is the wind
velocity relative to the vehicle, which is defined as å…±Fig. 1å…²
Vre = 冑V2 + 关U + u共t兲兴2 + 2V关U + u共t兲兴 · cos ␸
where U = mean wind velocity and uå…±tå…² = turbulent component of
wind velocity in the alongwind direction. Wind turbulent velocity
can be obtained from actual wind measurements or from simulations based on wind velocity spectrums 共Chen and Cai 2004兲. ␸ is
the wind direction å…±Fig. 1å…².
Forces due to Topology
In typical highway designs, there will be an appropriate roadway
superelevation on any curved path to provide centripetal acceleration which acts toward the center of the curvature å…±AASHTO
2004兲. So it is necessary to consider the corresponding superelevation ␪ in the model to replicate the real situation when a vehicle moves through a curved path. In the following numerical
Fig. 2. Single-body vehicle model: å…±aå…² elevation view; å…±bå…² top view
results, ␪ is defined based on typical design values suggested by
AASHTO å…±2004å…², which are dependent on the road design speed
and radius of curvature.
Basic Vehicle Dynamic Model—Wheels Are Not Lifted
up or Sideslip
rFy,r = − mu,rV共hu,r − r兲共␤˙ + ␺˙ 兲 + mu,rg共hu,r − r兲␾t,r
+ mu,rg共hu,r − r兲␪ − arollIx⬘x⬘mr/m − mu,ray共hu,r − r兲
˙ −␾
Ë™ t,rå…² + ur
+ kt,r␾t,r − kr共␾ − ␾t,r兲 − lr共␾
The vehicle model is shown with the coordinate system fixed on
the vehicle in Fig. 2. In the following model, pitching and bouncing motions are not considered because they typically have insignificant impacts on the rolling and lateral movements of the
vehicle å…±Sampson 2000å…². The sprung mass rotates about the roll
center which is dependent on the kinematical properties of the
suspensions. The unsprung masses can also rotate, combined with
the effect of the vertical compliance of the tires. The vehicle
motion equations are developed according to the change of the
momentum and the sum of external forces based on the model
introduced by Sampson å…±2000å…². The suspension parameters such
as damping coefficients are assumed to be constant.
As a general model which considers wind load, road superelevation, curvature, and excitations from supporting structures
å…±e.g., vibration induced by pavement roughness or bridge/vehicle
interactionså…², five force and moment equilibrium equations of
vehicle motions of sprung mass and suspensions in y- and zdirections are defined in Eqs. 共12兲–共16兲, respectively
where Fw,y, M x, and M z = lateral wind force, wind-induced roll
moment, and wind-induced yaw moment, respectively; ␪ = road
superelevation; ay and aroll = accelerations in y-direction and rolling direction of the supporting infrastructures å…±e.g., pavement or
bridgeå…², respectively; m, ms, and mu = total mass, sprung mass,
and unsprung mass, respectively; h = height of the center of
sprung mass, measured upwards from the roll center; r and
hu = heights of rolling center and unsprung mass center, measured
upwards from ground, respectively; Fy,f and Fy,r = lateral forces of
front and rear tires, respectively; Ix⬘x⬘, Ix⬘z⬘, and Iz⬘z⬘ = roll
moment, yaw-roll product, and yaw moment of inertia of sprung
mass, respectively; k, kt, and l = suspension roll stiffness, tire roll
stiffness, and suspension roll damping rate, respectively; ␾
and ␾t = absolute roll angle of sprung mass and unsprung mass,
respectively; ␤ and ␺ = sideslip angle and heading angle; and
u = active roll torque. A full list of all variables can be found in the
The above equations can be expressed using a state-space representation, which is suitable for numerical integrations
¨ = mV共␤˙ + ␺˙ 兲 − Fy,f − Fy,r + Fw,y − mg␪ + may
ẋ = Ax + B0u + B1␦ + C
− Ix⬘z⬘␾¨ + Iz⬘z⬘␺¨ = Fy,f a f + Fy,rar + M z
˙ ␾t,f ␾t,r 兴T
x = 关␤ ␺˙ ␾ ␾
u = å…³u f ur å…´T
Ix⬘x⬘␾¨ − Ix⬘z⬘␺¨ = msgh␾ + msVh共␤˙ + ␺˙ 兲 + M x − msgh␪ + msgay
Ë™ t,f å…²
+ Fw,yhw − k f 共␾ − ␾t,f 兲 − l f 共␾˙ − ␾
+ u f − kr共␾ − ␾t,r兲 − lr共␾˙ − ␾˙ t,r兲 + ur
rFy,f = − mu,f V共hu,f − r兲共␤˙ + ␺˙ 兲 + mu,f g共hu,f − r兲␾t,f
B0 = E−1
+ mu,f g共hu,f − r兲␪ − arollIx⬘x⬘m f /m − mu,f ay共hu,f − r兲
Ë™ t,f å…² + u f
+ kt,f ␾t,f − k f 共␾ − ␾t,f 兲 − l f 共␾˙ − ␾
0 0 1 1 0 0
0 0 1 0 1 0
B1 = E−1关␮Y ␦ − ␮N␦ 0 r␮Y ␦,f 0 0 兴T
A = E−1
␮Y ␤
␮Y ␺˙ + mV
− ␮N␤
− ␮N␺˙
r␮Y ␤,f
msgh − k f − kr − l f − lr
r␮Y ␺˙ ,f − mu,f V共hu,f − r兲
− lf
k f + kt,f + mu,f g共hu,f − r兲
− kf
r␮Y ␤,r r␮Y ␺˙ ,r − mu,rV共hu,r − r兲
C = E−1
− kr
− lr
k f + kt,r + mu,rg共hu,r − r兲
Fw,y − mg␪ + may,M z,M x − msgh␪ + msayh + Fw,yhw,mu,f g共hu,f − r兲␪ − mu,f ay共hu,f − r兲,
mu,rg共hu,r − r兲␪ − mu,ray共hu,r − r兲,0
− mV
I z⬘z⬘
m sh
0 − I x⬘z⬘
− I x⬘z⬘ 0
− msVh
mu,f V共hu,f − r兲
mu,rV共hu,r − r兲
N and Y terms in 关Eqs. 共21兲 and 共22兲兴 are partial derivatives of net
tire yaw moment or lateral force, and the detailed definitions can
be found in the nomenclature. The Runge-Kutta method will be
used to solve dynamic equations in time domain with a time step
of dt = 0.001 s.
Criteria of Wheel Being Lifted up or Sideslip
Taking the summation of moment about the point on the ground
plane at the midtrack position, one can get the weight transfer
ratio between the left and right wheels
Wtrans = 兵关mV共␤˙ + ␺˙ 兲 + may + mg共␾ − ␪兲兴 ⫻ hcm + Fw,y共hw + r兲
+ M x + arollIx⬘x⬘其/d
Wheel Being Lifted up
When the weight transferred between the left and right wheels is
larger than a half of the vehicle weight minus a half of the vertical
wind force å…±lift forceå…², there is no reaction force existing on one
side of wheels. In addition, the roll angle between the sprung
mass and the suspension system typically cannot exceed 6 or 7°
due to the mechanical restraints of the suspension movements
å…±Sampson 2000å…². Thus if either of the following two criteria is
satisfied, the wheel is believed to be lifted up
Wtrans ⬎ mg/2 − Fw,z/2
艌 ␾cri or ␾i − ␾t,r
艌 ␾cri
␾i − ␾t,f
where ␾cri = maximum allowable relative roll-over angle due to
the mechanical restraints 共e.g., 7°兲.
The front or the rear wheel will start to sideslip when the actual
lateral tire forces Fy,f or Fy,r quantified with 关Eqs. 共1兲 and 共2兲兴
exceeds the corresponding sideslip critical friction forces, respectively
I x⬘x⬘
− l f − lr
− lf 0
0 − lr
Fy,f ⬎ Fla,f
= ␮Fz,f
= ␮Fz,r
Fy,r ⬎ Fla,r
where Fz,f and Fz,r = vertical reaction forces on the front and rear
axles, respectively; Fla,f
and Fla,r
= sideslip critical friction forces
of the front and the rear wheels, respectively; and ␮ = static lateral
friction coefficient. The longitudinal rolling resistance of the tires
in the driving direction, which is related to vehicle driving speed
and tire condition å…±temperature, inflation pressure, and so onå…², is
relatively insignificant to the vehicle stability compared to the
side friction force. Therefore, the longitudinal rolling resistance is
not considered in this model.
Two sets of criteria, as shown in 关Eqs. 共26兲–共29兲兴, will be
checked at each time step to identify whether any wheel will be
lifted up or will start to sideslip, under either of which, the corresponding new dynamic equations as introduced below will be
used to continue the simulation.
Updated Vehicle Dynamic Model
After Wheels Being Lifted up
After wheels on one side of the vehicle are lifted up, the suspension system of a vehicle cannot generate resistant moment anymore and the roll center moves toward the wheels which are not
yet lifted up. Accordingly, in 关Eqs. 共12兲–共16兲兴, Ix⬘x⬘, Iz⬘z⬘, and Ix⬘z⬘
will be changed to I⬘x x , Iz⬘ z , and Ix⬘ z , which are moments of
⬘⬘ ⬘⬘
inertia about the wheels remaining on the ground in three directions, respectively. kt,i␾t,i in Eqs. 共12兲–共16兲 will be changed to
kt,i␾ⴱt,i, where ␾ⴱt,i is the value of ␾t,i when the wheels are just
lifted up. In addition, in Eqs. 共12兲–共16兲, all the moment reference
arms are changed to the distances to the wheels remaining on the
ground from, originally, to the suspension roll center of the vehicle due to the fact that the vehicle starts to rotate about the
contact points of the wheels remaining on the ground once the
wheels on one side are lifted up.
After Starting to Sideslip
When a wheel starts to sideslip, the lateral slipping friction forces
can be assumed approximately equal to the sideslip critical fricmax
tion forces Fla,f
and Fla,r
that the road can generate for the left
and right wheels, respectively. Before the vehicle hits roadside or
another object, Fy,f and Fy,r in Eqs. å…±12å…², å…±15å…², and å…±16å…² will be
changed to Fla,f
and Fla,r
, respectively. As a result, the vehicle
will laterally slip with the slipping acceleration aslip which can be
derived as
aslip = 关mV共␤˙ + ␺˙ 兲 − Fmax
f,f − F f,r + Fw,y − mg␪ + ma y − msh␾兴/m
Vehicle Accident Assessment Criteria
Vehicle Rollover
A vehicle ultimately rolls over only when the y value of the center
of gravity exceeds the y-coordinate of the wheel. Therefore, the
corresponding roll angle at the moment when the vehicle ultimately rolls over is set as the criterion to identify the occurrence
of rollover accidents
␾ ⬎ arc sin共d/2冑d2/4 + h2cm兲 + ␪
only take 2 s to go through a ramp at one specific driving speed.
If the CST for this vehicle under the specific combination of the
adverse environmental and driving conditions is larger than 2 s,
the accident may not really happen as the environmental conditions will change right after 2 s. One common situation is when
the truck suddenly experiences a change of strong wind gust load
on the vehicle å…±i.e., both imposing and removingå…² due to special
topographical conditions, such as getting into a valley from open
areas or passing a bridge tower or mountain and getting to open
According to the Green book å…±AASHTO 2004å…², the median
reaction time of drivers is 0.66 s based on the data from 321
drivers. The design reaction time is 2.5 s which exceeds 90th
percentile of reaction time for all drivers å…±AASHTO 2004å…². In the
present study, both “median reaction time” 共0.66 s兲 and “design
reaction time” 共2.5 s兲 will be checked. If the CST is larger than
the reaction time of the driver, the driver may have sufficient
time to take appropriate actions å…±e.g., reduce speedså…² to possibly
prevent the occurrence of accidents. Obviously, CDS suggests
the appropriate driving speed assuming the driver has sufficient
time to react while CST discloses the information about whether
the driver has enough time to react under a particular driving
where d = track width of the truck; hcm = height of the mass center
of the truck; and ␪ = road superelevation.
Once a vehicle starts to sideslip, driver operations such as applying steering or brakes usually have little effect on stopping the
motion before the vehicle hits an object å…±e.g., road side curbs,
other vehicleså…², which may or may not cause tripped rollover.
With the purpose of introducing the general model in this study,
the travel distance after sideslip starts will be the critical variable
to be investigated without dealing with different site-specific road
conditions å…±e.g., different distances from the center of the driving
lane to the curbå…². It is noted that any particular tripped rollover
scenario can be simulated with the proposed model as long as the
specific descriptions of the obstacle å…±e.g., locations, size, and materialå…² are available. Due to the limited scope of the present study,
different particular tripped rollover scenarios will not be discussed in this paper.
For any given hazardous condition and any specific vehicle, the
occurrence of single-vehicle accidents is significantly related
to excessive driving speeds. To maintain an appropriate driving
speed to balance the safety and efficiency is obviously critical.
Therefore, for the proposed deterministic model, the “critical
driving speed 共CDS兲” is the highest allowable driving speed without causing any type of accidents under a specific combination of
environmental and vehicular conditions. In the future reliabilitybased model, it will become the highest allowable driving speed
which results in the crash risk at the desired level.
In addition to the CDS which has been studied in some existing studies å…±e.g., Baker 1991; Chen and Cai 2004; Guo and Xu
2006; Sigbjörnsson and Snaebjörnsson 1998; Snæbjörnsson et al.
2007å…² another critical variable which has been rarely discussed is
the “critical sustained time 共CST兲.” CST is the minimum time
period required to sustain the specific combination of the adverse
environments å…±e.g., wind speed, curvatureå…² and the driving conditions å…±e.g., specific driving speedå…². For example, a vehicle may
Parametric Study
A numerical example will be conducted for demonstration purposes. Although the simulation process as introduced above can
be applied to any type of single-body vehicle, a truck model is
adopted in the parametric study because of its relatively larger
safety risks under hazardous driving conditions. Comparative
studies among different types of vehicles are beyond the scope of
the present study.
Truck Model
A singe-body truck model will be used in the numerical studies
and the same parameters from Winkler and Ervin å…±1999å…² are
adopted 共Table 1兲. In automobile engineering, the steering angle ␦
is typically expressed as ␦ = L / R 共neutral steer兲, where L is the
wheelbase of the vehicle, R is the turning radius of the curved
path, and the steering angle can be determined for each different
R. The corresponding superelevation is considered for different R
and typical speed limits according to AASHTO å…±2004å…². Although
there are some limited studies on quantifying the steering angle
due to driver behavior å…±Baker 1991; Chen and Cai 2004å…², to the
writers’ knowledge, there is not yet a well-accepted model which
can accurately relate the steering angle and the motion of vehicles
from existing literature. Besides, existing studies also showed
limited impact of driver behavior of steering on single-vehicle
accidents å…±Chen and Cai 2004å…². Therefore, in the present study,
impacts of driving behavior on steering angles will not be considered. However, it is noted that the present model can easily incorporate the driver behavior model on steering angles when a
reliable one becomes available in the future.
Adverse/Hazardous Driving Environments
It is well known that the same vehicle experiences different accident risks under different driving environments. For large trucks,
some driving environments can be hazardous which often cause
rollover or sideslip accidents. These adverse driving environments
Table 1. Parameters of the Single-Body Truck Model
mu,f , mu,r
a f , ar
hw å…±hreå…²
u f , ur
23,000 lb
1,202 lb, 4603 lb
14.8 ft, ⫺5.22 ft
2.12 ft
3.98 ft
5.46 ft
2.42 ft
6 ft
0 in. lb, 0 in. lb
C f , Cr
k f , kr
l f , lr
kt,f , kt,r
hu,f , hu,r
714.6 lb/deg, 2,544 lb/deg
24,119 in. lb/deg, 245,826 in. lb/deg
393 lb/deg, 938 lb/deg
31,8491 in. lb/deg, 274,583 in. lb/deg
1.67 ft, 1.67 ft
66,132 in. lb s s
31,799 in. lb s s
465,180 in. lb s s
107.6 ft2
typically include strong crosswind gust, slippery road surface
which is covered by snow or ice, a curved path, or with dynamic
excitations from the supporting structures å…±e.g., roughness of
pavement and/or bridge structureå…². These adverse driving environments may work individually or integrally to significantly increase the crash risk of trucks. In the present study, three different
road surface conditions å…±dry, snow-covered, and icyå…² and the situation with excitations from supporting structures will be considered along with different wind gust conditions, on both straight
and curved paths. To capture the most critical scenarios, wind is
assumed to be perpendicular to the driving direction of the vehicle all the time, including on both straight and curved paths.
Given the randomness of actual wind directions in nature, this
assumption will lead to slightly more conservative results than the
reality, which is usually preferred in engineering fields. However,
the present model can easily consider varying wind directions
during curving when one specific initial wind direction is given.
In the following sections, CST, CDS, and transient accidentrelated response in time domain will be investigated for various
Critical Sustained Time of Accidents
Straight Road
When the truck is driven on a straight road with dry road surface,
rollover accidents are found to occur first. Fig. 3 shows the relationship between CST of rollover, the wind speed U, and the
driving speed V. The lowest V of each curve also suggests the
CDS under which the accident may happen. There are two hori-
Fig. 3. CST of rollover on a straight and dry road
zontal lines shown in the figure which suggest the median å…±0.66 så…²
and design reaction time å…±2.5 så…², respectively. It can be found
from Fig. 3 that the CST of rollover decreases when U or V
increases. When the wind speed is relatively low å…±lower than 45
mphå…², the required CST to roll over a truck quickly drops when
the vehicle driving speed increases. For instance, when the truck
moves in a speed of 70 mph and the wind speed is 35 mph, it
requires about 1.7 s to roll over the truck. Depending on the
driving experience and how fast an individual driver senses and
reacts to the danger, a rollover accident may or may not actually
happen. When the wind speed is more than 50 mph, it only takes
around 0.6 s to roll over the truck with about 25-mph driving
speed and there is no significant difference for the CST of rollover when the wind speed keeps increasing. As compared to the
median reaction time å…±0.66 så…², 0.6 s is usually too short for most
drivers to react and a rollover accident is very likely to happen in
this scenario. An accurate accident risk assessment based on the
CST relies on a reliability-based risk assessment model considering the uncertainties of reaction times among different drivers. It
will be the future task for the writers based on the proposed
deterministic model. It is also noted that none of the scenarios in
Fig. 3 can satisfy the design reaction time requirement å…±2.5 så…² as
specified in AASHTO, which is known to be very conservative.
When road surface is covered by snow or ice, sideslip accidents usually happen first. Fig. 4 shows the relationship between
the CST of sideslip and the driving speed V when the truck moves
straight on a snow-covered and ice-covered road surfaces, along
with different wind speeds. For any higher driving speeds beyond
the x-coordinate of each curve in Fig. 4, rollover accidents will
Fig. 4. CST of sideslip accident on a straight road
Fig. 5. CST of rollover accident on a dry curved road
Fig. 7. CST of sideslip accident on ice-covered curved roads
happen first, as marked in the figure with the text “rollover.”
Under the same wind condition, driving faster å…±e.g., higher Vå…² will
require a shorter duration of the sustained hazardous condition
å…±i.e., smaller CSTå…² to finally make the accident happen.
When the truck is on icy road surface, the CST remains relatively constant when the driving speed is above 20 mph å…±Fig. 4å…².
The CST values are all smaller than the median reaction time
when the wind speed is more than 40 mph, which suggest sideslip
accidents are very difficult to be avoided by majority of drivers.
When the truck is on icy road surface with 40-mph wind speed, it
is also found that the CST actually slightly increases with the
driving speed and this trend continues until the driving speed
reaches 30 mph when the CST becomes nearly constant despite
further increase of the driving speed. This is different from the
observation under the snow-covered situation, and it is probably
related to the unique vehicle movement manner on ice when the
lateral friction is very small. As shown in Fig. 4, when wind is
strong å…±U = 40 mphå…², the truck will experience rollover accidents
first when the driving speed is over 57.5 mph on both snowcovered and icy road surfaces.
can be found that the respective CST considerably decreases
under the same curving situation when the wind speed increases.
For example, when the curving radius R is 130 ft å…±a typical value
of many highway rampså…², the driving speed of 42.5 mph and
higher may cause rollover accidents with the CST about 3 s when
there is no wind. When the wind speed is 20 mph, the driving
speed of 37.5 mph or higher will cause rollover accidents with the
CST of about 2 s on the same path. When the wind speed is
further increased to 40 mph, the truck will roll over with the
driving speed of 30 mph and the CST is about 0.6 s. In reality,
wind gust with 20–40 mph wind speed is pretty common on
highways and 0.6 s is typically not enough for most drivers to
react. As we often observe on highways, curving operations of
large trucks in windy weather, especially under a sharp curve
å…±e.g., ramp or in mountain areaså…², are much more vulnerable than
the situation without strong wind. By comparing the differences
of results for R = 130 and 260 ft under various wind speeds, it is
found that the different radiuses affect the CST significantly when
wind is not strong. While wind is strong å…±e.g., U = 40 mphå…², different radiuses only affect the CST slightly, which suggests that
the dominant impact shifts from the geometric condition å…±curvatureå…² to environmental condition å…±windå…².
Fig. 6 gives the results of the CST of sideslip under different
combinations of driving speeds and curvature radiuses when the
wind speed is 0 or 20 mph while the road surface is covered by
snow. It is found that when the radius R is about 130 ft, 60-mph
driving speed without existence of wind, or 50-mph driving speed
with 20-mph wind will all cause the CST to be lower than the
median reaction time.
Fig. 7 shows the CDS under different radiuses when the road
surface is covered by ice. Similar to the results for the snowcovered road on curves and icy surface on a straight road, sideslip
accidents will dominate and the truck may experience sideslip
accidents when it is driven in a speed of 45 mph on a curve with
a radius of 260 ft or the driving speed of 25 mph on a curve with
a radius of 130 ft when there is no wind. When wind speed
increases to 20 mph, the CDS will be changed to 17.5 mph on a
curve with a radius of 260 ft or 15 mph on a curve with a radius
of 130 ft. For all these cases, the CST is generally between the
median and design reaction time.
Curved Roads
Fig. 5 shows the relationship between the CST of rollover and the
driving speed V under different radiuses of curvature R and wind
conditions. Two representative curve radiuses å…±130 and 260 ftå…²
and three wind conditions å…±U = 0, 20, and 40 mphå…² are studied. It
Fig. 6. CST of sideslip accident on snow-covered curved roads
Excitations from Supporting Infrastructures
When a vehicle moves on roadways, vehicles will be excited to
vibrate in several directions by the surface roughness on the roadway å…±Xu and Guo 2003; Chen and Cai 2004å…². When a vehicle
Fig. 8. CST of rollover on a supporting structure
moves on a bridge, dynamic interactions between the bridge and
the vehicle will cause the vehicle to vibrate more significantly
å…±Cai and Chen 2004å…². In either case å…±i.e., on pavement or on
bridgeså…², the vehicle will experience additional accelerations as a
type of base excitations. In the present vehicle accident assessment model, safety behavior of the truck will be evaluated
through a general consideration of excitations from supporting
infrastructures by defining accelerations in the lateral direction
ay and that in rolling direction aroll as base excitations. The relationship between rollover critical time and ay as well as aroll
å…±U = 40 mph, V = 40 mphå…² is demonstrated in Fig. 8. When aroll
⬎ 0.4 rad/ s2, rollover accidents will occur even when the wind
speed and vehicle velocity are both not very high. It is found that
the rolling acceleration caused by interaction with supporting
structures is pretty critical to the truck safety and will increase
the chance of having accidents when all other conditions are the
same. Since considerable rolling excitations may exist on some
bridges, it suggests that vehicles are more vulnerable to rollover
accidents on a vibrating bridge, which has also been observed in
existing studies å…±Guo and Xu 2006; Chen and Cai 2004å…².
Critical Driving Speeds of Accidents
Straight Roads
Assuming that the CST of the specific environmental conditions
is satisfied å…±i.e., the actual time duration of a specific set of conditions is longer than the required CSTå…², Fig. 9 shows the CDS of
the truck under different wind conditions on dry 关Fig. 9共a兲兴, snowcovered 关Fig. 9共b兲兴, and icy road surfaces 关Fig. 9共c兲兴.
With the increase of the wind speed, the CDS generally decreases. It can be found from Fig. 9å…±aå…² that a sideslip accident will
not occur before a rollover accident does first when the truck
moves on a dry straight path. Generally speaking, depending on
the driving speed of the truck, it is found that there exist various
levels of rollover risk when the wind speed exceeds 35 mph.
When the wind speed is more than 55 mph, even the truck in still
å…±V = 0 mphå…² will have the risk of being blown over. Fig. 9å…±bå…²
shows the critical U or V under which at least one type of accident may happen when the truck is driven on a snow-covered
road surface. It is easy to find that when U and V are not high, no
rollover or sideslip accidents will happen. If the wind speed is
moderate, sideslip accident will likely happen when the vehicle
driving speed is more than 20 mph. When the wind speed is more
Fig. 9. CDSs on a straight road with various surface conditions: å…±aå…²
dry road surface; å…±bå…² snow-covered road surface; and å…±cå…² icy road
than 50 mph, rollover accidents instead of sideslip will happen
first. Fig. 9å…±cå…² shows the critical U or V when the truck moves on
an icy road surface assuming the CST of sideslip is satisfied. By
comparing Figs. 9共a–c兲, it is obvious that sideslip accidents will
be more prone to occur first than rollover accidents when the road
kinetic friction coefficient decreases. Sideslip accidents can happen even when the wind speed is below 20 mph and the vehicle
Fig. 10. CDSs on dry roads with different radius and wind conditions
driving speed is 25 mph on icy roads. This observation is consistent with the frequent sideslip accidents observed in cold regions.
Curved Roads
Fig. 10 gives the results of CDS under different wind speeds U
and radiuses R when the road surface is dry. It can be found
that with the increase of wind speeds or the decrease of radius,
the CDS decreases dramatically. When wind is very weak
共U ⬍ 10 mph兲, any radius lower than 330 ft will impose considerable safety threats to the truck with a driving speed about
65 mph or higher. A further decrease of the radius to 165 and
100 ft leads to a dramatic decrease of the CDS to around 50 and
37.5 mph, respectively. With the increase of wind speed, the CDS
under the same radius will also significantly decrease compared to
the case only with breeze. For example, when the wind speed
increases from 10 to 40 mph, the CDS for a radius of 360 ft will
decrease from 70 to 35 mph.
Fig. 11 gives the CDS results under different curvature radiuses when the road surface is covered by snow. It can be found
that sideslip will be the only accident type which will happen first
å…±if there is an accidentå…². It is found that depending on the driving
speeds, the curvature radius of 590 ft and lower along with 20mph wind speed will possibly cause accidents. With a radius of
130 ft, 30 mph will be the CDS for the truck in the present study
if there is no wind. Due to the high number of possible combinations of wind, driving speed, and curvature radius, a full parametric study of all possible scenarios will not be discussed here. By
comparing Fig. 11å…±aå…² with Fig. 11å…±bå…², it can be found that if the
driving speed is more than 35 mph and the radius of the curved
road is more than 130 ft, the possibility of sideslip increases dramatically when the wind speed changes from 0 to 20 mph.
Fig. 12 shows the CDS under different radiuses when the road
is covered by ice. Two different wind speeds å…±0 and 20 mphå…² are
studied. Similar to the results for snow-covered curved roads and
icy straight roads, sideslip accidents will dominate and the truck
may experience sideslip accidents when it is driven in a speed of
60 mph on the curved road with a radius of 330 ft or in a driving
speed of 25 mph on the curve with a radius of 130 ft when there
is no wind. We can find that the truck with the driving speed more
than 20 mph is prone to sideslip accidents when the radius is
more than 330 ft and the wind speed is 20 mph. Comparing Figs.
12å…±a and bå…², people can find that even very moderate wind can
affect the stability of the truck significantly on curved roads covered by ice.
Fig. 11. CDSs on snow-covered roads with various radiuses: å…±aå…² U
= 0 mph; å…±bå…² U = 20 mph
Transient Accident-Related Responses
Fig. 13 shows the time-history results of course angle and lateral
displacement of the truck on snow-covered and icy road surfaces,
respectively, when V = 32.5 mph and U = 47.5 mph. Fig. 14 displays the corresponding time history of lateral friction force.
It can be found that when wind gust is applied on the truck moving on the snow-covered surface, after a slight lateral displacement about 0.6 ft in Fig. 13å…±bå…², the joint effect of wind-induced
lateral force and moment will change the vehicle course angle
关Fig. 13共a兲兴 and bring the driving direction of the truck opposite
to the wind direction until the truck moves laterally about 1.8 ft,
when the lateral friction force of the rear tire reaches the sideslip
critical friction force å…±Fig. 14å…². So at 0.8 s after wind gust is
applied on the truck, the truck starts to sideslip after it has traveled laterally about 1.8 ft from its original path. As shown in
Fig. 13共a兲, the course angle is lower than 2° when sideslip just
happens. But 0.6 s after the vehicle starts to sideslip, the course
angle is about 11°, which suggests that a strong rotational movement of the truck has occurred after the tires start to sideslip.
When the road is covered by ice, as shown in Fig. 13å…±aå…², the
course angle is lower than ⫺2° when sideslip happens. But 0.6 s
after the truck starts to sideslip, the course angle is about 4°,
which means that strong rotational motion of the truck has happened under the strong wind load after the tires start to sideslip.
Fig. 13å…±bå…² suggests that the lateral displacement of the truck on
icy roads is pretty straightforward and gradually increasing along
the wind direction, which is different from that observed on the
Fig. 13. Time histories of vehicle course angle and lateral distance:
å…±aå…² course angle; å…±bå…² lateral displacement
Fig. 12. CDSs on icy roads with various radiuses: å…±aå…² U = 0 mph; å…±bå…²
U = 20 mph
snow-covered road. It is found that the lateral friction force of the
rear tire increases quickly over time and will exceed the critical
friction forces and start to sideslip at about 0.5 s å…±Fig. 14å…². While
the same truck is driven in the same speed on a snow-covered
road, it requires 0.8 s to start sideslip 关Fig. 13共a兲兴. As discussed
earlier, once sideslip starts, the driver usually can do very little to
regain the control of the vehicle.
Compared to existing simulation models, the new model has the
following improvements: å…±1å…² adopting a series of dynamic equations to simulate the transient process of accidents; å…±2å…² for the first
time, combining crosswind, different road surfaces, curving, and
excitations from supporting structures in one single model which
can be used to consider more realistic scenarios; å…±3å…² introducing a
new and important variable CST of each specific combination of
adverse environmental and driving conditions in addition to the
CDS which has been adopted in existing studies. Such a new
variable will be helpful on characterizing the accident risks
more realistically; and å…±4å…² as a holistic deterministic model, the
present study can be used directly to provide useful information
for traffic and emergency management as well as accident preventions. Moreover, the developed model also lays a critical basis
for future reliability-based vehicle safety studies under hazardous
Several assumptions have been made in the proposed model
due to the lack of more detailed information: å…±1å…² driver behavior
uncertainties on steering angle is not considered due to the lack of
a reliable model. Possible solutions include adopting CST to
study driver behavior and consider uncertainties using the reliability theory and å…±2å…² wind loads on a truck during the rollover
process are assumed to be constant. A preliminary sensitive study
conducted by the writers showed the impact from such an assumption is insignificant. If necessary, this could be further improved by conducting more extensive wind tunnel tests or
applying the reliability theory to appropriately simulate the distributions of wind force coefficients during the rollover process.
More comprehensive parametric studies and site-specific analyses
can easily be conducted based on the model developed in the
present study, which will be reported by the writers later.
An integrated vehicle safety behavior simulation model was developed which adopts more realistic dynamic equations and accident criteria to characterize the transient process of accidents.
Numerical analyses on one type of typical trucks under several
representative scenarios were conducted. Major findings from the
numerical studies are summarized as follows:
Fig. 14. Time history of tire lateral force
The new model can be used to predict the safety performance
of vulnerable vehicles under various hazardous weather, topographic, and road surface conditions by using the variables
CST and CDS. The rigorous validation of the new simulation
model depends on the availability of comprehensive experimental data, which is beyond the scope of the present study;
For both straight and curved roads, rollover accidents usually
happen first when the road surface is dry. When the wind
speed is low, the difference of curvature has noteworthy impacts on CST and CDS. With the increase of the wind speed,
wind will gradually replace the curvature to dominate the
impacts on CST and CDS;
Sideslip accidents usually happen first on curved roads when
the road surface is covered by either snow or ice. Both CST
and CDS usually decrease with the increase of the driving
speed or the wind speed significantly. When wind is weak,
the decrease of the curvature radius will cause the CST and
CDS dramatically decrease under the same driving speed.
When wind is strong, the CST and CDS will only slightly
decrease for smaller curvature radiuses. It was found that the
truck is very vulnerable to accidents on curved roads covered
by ice even with the existence of very moderate wind;
On straight roads, the dominant accident type exhibited a
relatively complicated pattern when the road surface is covered by snow or ice. When the wind speed is moderate å…±U is
not more than 50 mphå…², sideslip accidents may happen first
on snow-covered roads depending on the specific combination of wind and driving speeds. On icy roads, sideslip accidents usually happen first when the wind speed is not very
high å…±less than 50 mphå…². When the wind speed exceeds 50
mph, rollover accidents usually will happen first for both
snow-covered and icy road surfaces;
It was found that the road surface condition, wind speed, and
the curvature all play vital roles on the accident risks integrally. Accurately predicting the safety risk under adverse
driving conditions requires a detailed simulation with the developed model on a case by case basis; and
CST was found to be a critical variable which can be used to
conduct a more accurate and personalized risk analysis by
considering the site-specific environmental conditions as
well as reaction time of individual drivers. This will be incorporated into the reliability-based accident model based on
the present study in the future.
This publication was partially supported by the Colorado Department of Transportation and the CDC NIOSH Mountain and Plains
Education and Research Center å…±Grant No. 1T42OH009229-01å…².
Its contents are solely the responsibility of the writers and do not
necessarily represent the official views of CDOT or CDC NIOSH
and MAP ERC.
The following symbols are used in this paper:
a ⫽ longitudinal distance to axle, measured forward
from center of sprung mass;
ay ⫽ lateral acceleration caused by the movement of
c1 , c2 ⫽ tire cornering stiffness coefficients, in Fy / ␣ = c1Fz
+ c2Fz2;
c␣ ⫽ tire cornering stiffness, measured at rated vertical
tire load;
d ⫽ track width;
Fw,y ⫽ lateral wind force;
Fw,z ⫽ vertical wind force;
Fy ⫽ lateral tire force;
Fz ⫽ vertical tire force;
g ⫽ acceleration due to gravity;
h ⫽ height of center of sprung mass, measured upwards
from roll center;
hcm ⫽ height of center of mass for whole truck, measured
upwards from ground;
hs ⫽ height of center of sprung mass, measured upwards
from ground;
hu ⫽ height of center of unsprung mass, measured
upwards from ground;
hw ⫽ height of lateral wind load Fw,y, measured upwards
from roll center;
Ixx ⫽ roll moment of inertia of sprung mass, measured
about sprung center of mass;
Ix⬘x⬘ ⫽ roll moment of inertia of sprung mass, measured
about origin of å…±x0 ; y 0 ; z0å…² coordinate system;
Ixz ⫽ yaw-roll product of inertia of sprung mass,
measured about sprung mass center;
Ix⬘z⬘ ⫽ yaw-roll product of inertia of sprung mass,
measured about origin of å…±x0 ; y 0 ; z0å…² coordinate system;
Iyy ⫽ pitch moment of inertia of sprung mass, measured
about sprung mass center;
Izz ⫽ yaw moment of inertia of sprung mass, measured
about sprung mass center;
Iz⬘z⬘ ⫽ yaw moment of inertia of total mass, measured
about origin of å…±x0 ; y 0 ; z0å…² coordinate system;
k ⫽ suspension roll stiffness;
kt ⫽ tire roll stiffness;
L ⫽ wheelbase;
l ⫽ suspension roll damping rate;
M x ⫽ wind-induced roll moment;
M z ⫽ wind-induced yaw moment;
m ⫽ total mass;
ms ⫽ sprung mass;
mu ⫽ unsprung mass;
N␤ ⫽ ⳵ M z / ⳵␤ = 兺 ja⬘j c␣,j, partial derivative of net tire yaw
moment with respect to sideslip angle;
N␦ ⫽ ⳵ M z / ⳵␦ = −a⬘1c␣,1, partial derivative of net tire yaw
moment with respect to steer angle;
N␺˙ ⫽ ⳵ M z / ⳵␺˙ = 兺 ja⬘2c␣,j / U, partial derivative of net tire
yaw moment with respect to yaw rate;
r ⫽ height of roll axis, measured upwards from ground;
U ⫽ forward speed;
u ⫽ active roll torque;
Y ␤ ⫽ ⳵Fy / ⳵␤ = 兺 jc␣,j partial derivative of net tire lateral
force with respect to sideslip angle;
Y ␦ ⫽ ⳵Fy / ⳵␦ = −c␣,1 partial derivative of net tire lateral
force with respect to steer angle;
Y ␺˙ ⫽ ⳵Fy / ⳵␺˙ = 兺 ja⬘c␣,j / U partial derivative of net tire
lateral force with respect to yaw rate;
␣ ⫽ tire slip angle;
␤ ⫽ sideslip angle;
␦ ⫽ steer angle;
␪ ⫽ road superelevation;
␾ ⫽ absolute roll angle of sprung mass;
␾t ⫽ absolute roll angle of unsprung mass;
␾ⴱt ⫽ roll angle of unsprung mass when one wheel lift
⫽ critical roll angle of unsprung mass;
␺ ⫽ heading angle; and
␺˙ ⫽ yaw rate.
f ⫽ front;
j ⫽ jth axle, counted from front; and
r ⫽ rear.
AASHTO. å…±2004å…². A policy on geometric design of highways and streets,
AASHTO, Washington, D.C.
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Published 03/28/2017
Copyright © 2017 SAE International
A Comprehensive Validation Method with Surface-Surface Comparison for
Vehicle Safety Applications
Junqi Yang, Zhenfei Zhan, Ling Zheng, and Gang Guo
Chongqing University
Changsheng Wang
Tongji University
Computer Aided Engineering (CAE) models have proven themselves to be efficient surrogates of real-world systems in automotive
industries and academia. To successfully integrate the CAE models into analysis process, model validation is necessarily required to
assess the models’ predictive capabilities regarding their intended usage. In the context of model validation, quantitative comparison
which considers specific measurements in real-world systems and corresponding simulations serves as a principal step in the
assessment process. For applications such as side impact analysis, surface deformation is frequently regarded as a critical factor to be
measured for the validation of CAE models. However, recent approaches for such application are commonly based on graphical
comparison, while researches on the quantitative metric for surface-surface comparison are rarely found. To deal with this problem, a
validation metric, which combines the discrepancies measurements in magnitude and shape, is proposed to evaluate the inconsistence
between two deformed surfaces. For magnitude error, an exploited 2-Dimensional Dynamic Time Warping (2D-DTW) method is
applied to address the mismatch in surface features between two surfaces. Geometric features, say mean curvatures of surfaces, are
extracted for shape comparison. For decision making, the original assessments are then transformed into scores through a linear
regression method. An analytical case is employed to verify the employed algorithms in the proposed method. Furthermore, the method
is implemented on a real-world case involving surface comparison to show its potential in vehicle safety applications.
CITATION: Yang, J., Zhan, Z., Zheng, L., Guo, G. et al., “A Comprehensive Validation Method with Surface-Surface Comparison for
Vehicle Safety Applications,” SAE Int. J. Trans. Safety 5(1):2017, doi:10.4271/2017-01-0221.
As a crucial step in validation activity, metric development for
quantitative comparison between test and corresponding simulation
needs to consider various features of responses. For vehicle safety
applications, considerable researchers have paid their efforts to
develop numerous potential metrics with various case studies. Cheng
[2] utilized a wavelet decomposition based method to extract the
features of crash pulses, and introduced a metric in the form of rating
score. Yang [3] reviewed several popular metrics in safety
applications, and proposed a metric to address the bias introduced by
Subject Matter Expert and quantitative metric itself. More recently,
Xi et al. [4] and Xu et al. [5] paid efforts on the metrics development
involving uncertainty consideration. Moreover, several previously
introduced metrics, say CORA [6], EARTH [7] and EEARTH [8]
become the foundations of the published ISO standards “Road
vehicles – Objective rating metrics for dynamic systems”. Though
great progress has been achieved in the field of curve-curve
comparison based validation, the development of quantitative metric
for surface-surface comparison has not been well studied.
With the ever growing concerns about occupant protection,
automotive manufacturers have to meet several increasingly strict
vehicle safety regulations and laws, such as the mandatory Federal
Motor Vehicle Safety Standards FMVSS and the market-driven New
Car Assessment Program (NCAP). Traditional approach to assess
whether these requirements are satisfied is commonly based on costly
and time-consuming physical tests. To shorten the development time,
Computer Aided Engineering (CAE) models based virtual prototype
tests to evaluate the vehicle crashworthiness are on the rise. To
maximize the use of these models, the validity and predictive
capabilities of these models need to be assessed objectively and
quantitatively. This calls for the process of CAE model validation.
The fundamental concept of model validation has been introduced
mainly by several professional societies and institutions, including the
U.S. Department of Energy (DOE), the American Institute of
Aeronautics and Astronautics (AIAA), the U.S. Department of Defense
(DOD), and the American Society of Mechanical Engineers Standards
Committee (ASME). It is commonly defined as the process of
determining the degree to which a model is an accurate representation
of the real world from the perspective of its intended uses.
For many vehicle safety applications, surface deformation is
frequently regarded as a critical factor to be measured for validating
the CAE models. For instance, to ensure occupant protection, besides
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
time histories responses, modeler would also concern the deformed
door surfaces of vehicle in side impact test and accident. Currently,
validation methods for these applications are commonly based on
graphical comparison, which is neither objective nor accurate. Ideal
metric should provide quantitative indication for the degree of
agreement between computational and experimental data. For such
purpose, a comprehensive validation metric for surface-surface
comparison is proposed in this paper. The proposed metric for
surface-surface comparison combines the measurements of errors in
magnitude and shape. For magnitude error, a 2-Dimensional Dynamic
Time Warping (2D-DTW) method is proposed to address the effect of
shape features on magnitude. As for shape error, curvature based
measurement is introduced to characterize the geometric features of
surfaces. For convenient decision making, the two error measurements
are transformed into scores through a linear regression method.
The rest of this paper is organized as follows. The proposed surface
validation method is introduced in the following section. The third
section demonstrates the validity of the techniques involved in this
metric through an intuitive case. Then the proposed method is applied
to a case to show its potential in real-world engineering application.
Let DR = [xR, yR, zR] and DC = [xC, yC, zC] be the coordinates of the
nodes on real-world surface and corresponding CAE model based
surface. For the convenience of formulation, the original coordinates
data are projected onto respective principal planes, so that to
represent the original data by the height information of the
transformed nodes [9]. The other two dimensions are transformed as
the mesh indexes of the surfaces. Therefore, the following validation
activities are performed based on the projected and meshed CAE and
test data, say ZR and ZC, which are both m × n matrices.
The proposed surface-surface validation process contains two main
parallel assessment procedures for magnitude error and shape error,
respectively. For magnitude error, an exploited 2D-DTW method is
applied to address the mismatch regarding surface features (i.e. peaks
and valleys) between two surfaces. Then the pre-processed surfaces
are submitted to the following Euclidean distance calculation. In
terms of shape error, geometric feature, say mean curvatures of
surfaces, are extracted for topology comparison. Since the ranges of
two errors are quite different, the original assessments are
transformed into scores through a linear regression method.
Magnitude Error Measurement
Given two surfaces data to be compared, the most direct way to
assess the discrepancy is computing Euclidean distance, which forms
the foundation of distance measure [10]. However, Euclidean
distance is quite sensitive to even small mismatches between two
data. For example, slightly misaligned phase of time histories or
shifted shape features (i.e. peaks and valleys) of surfaces would lead
to unreasonably large Euclidean distance measurement [11]. To
address such problem, a 2D-DTW is exploited to separate the
interaction between the features of shape and magnitude.
2D-DTW based Euclidean distance measurement
Dynamic Time Warping (DTW) is originally an algorithm for
measuring similarity between two temporal sequences which may
vary in speed and was initially used in context with speech
recognition [12, 13]. It shows remarkable potential in various
applications, e.g., data mining [14], medical imaging [15], and
geological monitoring [16]. The key idea of DTW is that any point of
a time history can be (forward and/or backward) aligned with
multiple points of the other time history that lie in different temporal
positions, so as to compensate the temporal shifts [17]. With such
advantage, DTW has also proven itself to be an exceptionally useful
tool in the field of model validation to map the samples of two time
history curves [18]. However, for surface-surface comparison, the
matching process needs to be performed in a higher dimensionality
space. For such purpose, a 2D-DTW method is developed to
minimize the effect of mismatched surfaces features on the
magnitude error measurement.
In the context of DTW for one dimensional problems, the warping
path that minimize the cost to match two curves needs to be
calculated. Similarly, the proposed 2D-DTW evaluates the warping
paths in two dimensionalities, say horizontal and vertical , as
shown in Figure. 1.
Figure 1. Dimensionalities definition for surfaces to be compared.
For the two directions, the distance matrices of the two surfaces are
calculated as:
where ||•||2 indicates L2-norm operation.
imply that the
original matrices ZR and ZC are both sliced as n-column vectors,
imply ZR and ZC are partitioned as m-row vectors.
For each dimensionality, respective distance matrix can be obtained.
The alignments of the two surfaces are optimized in the sense that it
minimizes the cumulative distance. To optimize the warping paths
“vertically” and “horizontally”, the matrices that minimize the
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
cumulative distances to match two surfaces are required to be
calculated based on dh and dv. For each warping step, the cumulative
distance is expressed as:
Dh and Dv indicate the cumulative distances regarding the two
dimensionalities. The optimal warping paths to match the two
surfaces can be determined based on Dh and Dv.
In practice, backtracking is needed to identify the indexes for each
step. Taking the horizontal warping process as example, backtracking
along the minimum cumulative index pairs (ih, jh) starting from the
top-right corner indexes (n, n) of Dh yields the DTW results. In case
that the original sequences of matrices indexes are not optimal,
certain sample indexes of
will be repeated for several
times to make the two surfaces matched, which will results in the
prolongation of the original matrix. Assume the obtained matrices be
in the size of m′ × n′ (m′ ≥ m, n′ ≥ n), the following magnitude
comparison is conducted based on the warped surfaces data Z′R and
Z′C. Then the discrepancy in magnitude can be calculated based on
Euclidean distance, as shown in Equation (3).
Shape Error Measurement
In the context of differential geometry, curvature is a critical attribute
to describe the shape of a surface [19]. In this study, mean curvature
is employed to characterize the surfaces to be compared.
Geometric Features Extraction
At any point on a surface, its normal planes that contain the normal
vector would certainly intersect with the surface. The intersection
forms a curve called a normal section. Different normal sections
correspond to different curvatures at the evaluated point, and the
maximum and minimum values of these curvatures are called the
principal curvatures k1 and k2, respectively. As expressed in
Equation (4), the mean curvature is defined as the average of the
two principal curvatures:
Mean curvature is employed in this study as it is a physically
meaningful quantity. In the theory of differential geometry, the
implication of M related to geometry is provided in Table 1.
Table 1. Implication of mean curvature related to geometry.
Let S (DR, DC ⊆ S), which belongs to 3-dimensional Euclidean space,
represent the surface data to be evaluated, the mean curvature matrix
of the surface can be calculated as:
is the gradient of S , while
is the Hessian matrix of S .
Trace (•) denotes the trace of the matrix correspondingly. Details can
be referred to ref. [20, 21].
Regression Based Rating
As the ranges of these two errors, say D and M, are nondeterministic
and quite different, it is difficult for engineers to interpret how good
or how bad a CAE model is based on these raw error data. To provide
more intuitive rating based on the original measurements, a rating
score method is employed (Zhan et al., 2011b). The regression based
method translates the original errors into one score between 0 and
100%, so that it can provide an intuitive rating score. In this study,
Equation (6) is used to calculate the magnitude score Sm,
where D* is the maximum allowable magnitude error, km defines the
order of the regression. In this way, the best magnitude score is
100%. If the magnitude is equal to or greater than the maximum
allowable threshold M*, then the score is 0%. In between, the score is
calculated by regression method. Equation (7) provides the similar
rating rule for shape error.
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
In the proposed method, the validity of 2D-DTW would largely affect
the calculated magnitude score, while the capability of curvature
extraction algorithm is the core of shape score calculation. In this
section, the validity of 2D-DTW and curvature calculation algorithms
is verified through a demonstrative case study.
Two free surfaces in the size of 64 × 64 are provided in this case, as
shown in Figure 2. These two surfaces are globally similar in terms of
the surface features. However, it is observed that several geometric
characteristics cannot be well matched. The proposed 2D-DTW is
utilized to address such problem. Based on the 2D-DTW algorithm,
the optimum paths in directions and are obtained, as shown in
Figure 3. The dash lines indicate the reference paths in two
directions, while the dotted lines are the optimal warping paths. It is
noted that when the matrices to be matched are consistent, the
optimal warping path will be identical with the diagonal line of the
chessboard. Under such condition, sequentially matching the samples
can result in the minimum distance. As the warping paths are not
identical with the reference ones, the two surfaces need to be
expanded to make the global features matched. Through the warping
process, the sizes of surface matrices Z1 and Z2 are automatically
changed into 71 × 69. Figure 4 shows the resultant surfaces.
To obviously show the effectiveness of the warping process, contours
of surfaces are presented in Figure 5. Figure 5(a) is the original
surface contours, while Figure 5(b) shows the resultant contours. In
these figures, “P” indicates “peak”, and “V” means “valley”. As
shown in Figure 5(a), the locations of
on Z1 are different
from that of
on Z2. Through warping, the surfaces are
modified and the geometric features of peak and valley can be
maximally matched, as shown in Figure 5(b). Therefore, it can be
concluded that the proposed 2D-DTW method has the capability to
minimize the effect of shape mismatch on magnitude evaluation.
Figure 3. Warping paths of surfaces in two directions.
Figure 4. Matched surfaces based on 2D-DTW.
Figure 5. Contours of the (a) original surfaces, and (b) warped surfaces.
Figure 2. Surfaces to be compared.
In terms of shape error, the ability to extract surface features correctly
is critical to the resultant measurements. Curvatures are able to
represent the geometric characteristics of a surface. For each point on
the surface, mean curvature can be evaluated to show whether it is a
convex or concave point according to the sign of the evaluated value.
In this case study, the mean curvatures of nodes on surfaces Z1 and
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
Z2 are calculated, as provided in Figure 6. According to the
implications of mean curvature, positive mean curvature indicates
convex region, while negative value corresponds to concave region.
Figure 8. Deformed door surfaces to be compared.
Error Measurement
Figure 6. Extracted mean curvatures for the two surfaces.
The evaluated results in Figure 6 are consistent with the intuitive
observations. Based on the reliable evaluations, differences between
the two surfaces regarding mean curvature can be calculated.
As shown in Figure 8, ZR and ZC are globally similar according to
visual comparison, except for several slight differences in local
details. For the purpose of quantitative evaluation, 2D-DTW is firstly
performed for ZR and ZC to match the two surfaces. As shown in
Figure 9, in total 17 shifting steps are needed in the matching process.
The two warped surfaces are provided in Figure 10. The mismatches
in geometric features have been globally compensated. Then,
magnitude errors at each points of the surfaces can be calculated.
In this section, a real-word test based deformed surface and
corresponding FE surface are employed to demonstrate the method
for model fidelity quantification. The left-front door after crash and
corresponding FE surface are presented in Figure 7. The real-world
deformed door is scanned by laser scanning facility and represented
as a set of coordinates. Meanwhile, the nodes coordinates are
extracted through post-processing the FE model based crash results.
After transforming the two surfaces into the same space, the
validation activity can be conducted. The surfaces and in the size of
64×64 are presented in Figure 8.
Figure 9. Warping paths of the two deformed door surfaces.
In terms of shape error, the extracted curvatures of ZR and ZC are
presented in Figure 11. It is observed that the features of “convex”
and “concave” can be correctly represented by the evaluated
curvatures. Similarly, shape errors can be calculated according to the
discrepancies in curvatures evaluated for ZR and ZC.
Figure 7. Deformed real-world door and simulated door.
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
Sensitivity Study
Figure 10. Feature-matched door surfaces for magnitude comparison.
It is known that only one parameter, say the meshing size of the
surface, needs to be determined by engineers in the validation
process. The meshing size, which can be regarded as the resolution of
the evaluation, will certainly affect the precision of assessment
results. Therefore, it is necessary to perform sensitivity study on the
resolution of the surface. As introduced before, the results in Table 2
are obtained based on the surfaces which are meshed in the size of 64
× 64. In this subsection, the matrix size is modified for comparison.
The data are shown in Table 3, while corresponding results are
provided in Figure 12 to show the tendency. It is observed in Figure
12(a) that magnitude score is insensitive to the resolution of surfaces.
In terms of shape assessment, as resolution increases, the score
initially shows its uptrend, then keeps in stable value when the
resolution becomes higher than 64 × 64. It is reasonable to choose a
high resolution for accurate assessment. However, the increase of
entries in matrix would lead to the increase of computational cost. To
ensure robustness with relatively high computational efficiency, the
size of 64 × 64 is preferable for this case. It is suggested that, when
dealing with practical validation problem, performing sensitivity
study in advance is able to specify a resolution correspondingly that
helps obtain robust evaluation results.
Table 3. Magnitude and shape scores evaluations evaluated based on different
Figure 11. Extracted mean curvatures on (a) real-world door surface, and (b)
FE-based door surface.
Based on the evaluations for magnitude and shape discrepancies,
corresponding error measurements are obtained by specifying D* and
M*. D* and M* are defined as the discrepancies between the
maximum and minimum values of D and M, respectively. The values
of the thresholds and the calculated scores are provided in Table 2.
Table 2. Defined thresholds values and corresponding assessed scores.
Figure 12. Scores evaluated based on different resolutions: (a) Magnitude
score, (b) Shape score.
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
Figure 12 (cont). Scores evaluated based on different resolutions: (a)
Magnitude score, (b) Shape score.
For safety applications, such as side impact and rollover analysis,
surface deformation is a critical factor to be measured in tests and
corresponding simulations. Model validation activities for such
applications need to compare the deformed surfaces obtained in
simulations with that of real-world tests. However, how to assess the
discrepancy between two surfaces quantitatively is still an
outstanding issue in the field of model validation. This paper presents
a comprehensive validation metric for models involving surfacesurface comparison. The metric is composed of 2D-DTW based
magnitude error measurement and geometric feature based shape
measurement. An illustrative case study is to show the validity of the
algorithms employed in the validation procedure. Furthermore, the
proposed method is applied to the validation an impact model.
Sensitivity study is performed on the resolution parameter of the
proposed method for this case. It is suggested that with appropriate
resolution, robust assessment results can be obtained with no increase
in computational cost.
The scope of this paper is to introduce a metric for surface-surface
comparison to be integrated in existing validation framework. Further
efforts will be paid to integrate the proposed metric with the
well-established curve-curve comparison based metrics to formulate
a more comprehensive validation framework for safety applications
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Corresponding author:
Zhenfei Zhan, Ph.D.
Research Professor
State Key Laboratory of Mechanical Transmission, Chongqing
University, Chongqing, PR China
This work was supported by the National Natural Science Foundation
of China under Grant NO. 51405041, and the Ford project
“Development of a Validation Method for 3D Surface Comparison in
Vehicle Simulations”.
Yang et al / SAE Int. J. Trans. Safety / Volume 5, Issue 1 (April 2017)
D* – Threshold of magnitude error.
DR, DC – Coordinates data of surfaces
dh, dv – Distances matrices in 2 directions
Dh, Dv – Optimal cumulative distances in 2 directions
M* – Threshold of shape error.
D, D – Euclidean based magnitude error.
Sm – Magnitude score.
M, M – Mean curvature of surface.
Ss – Shape score.
ZR, ZC – Surfaces Matrices
k1 , k2 – Principal curvatures
Z′R, Z′C – Warped surfaces matrices.
km, ks – Regression order
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