Jayneel Gajjar, Madhu Kandampadath | Vehicle Crash Dynamics | December 19, 2015

Mini Sled Project GROUP 11

PAGE 1

Objective -:

Conduct a crash reconstruction of collision of bullet object with target object which is fit with a cup. Generate a

mathematical model to find the Pre Impact Velocity and Post Impact Velocity of both bullet and target object.

Compare the values obtained with the values obtained from accelerometer reading and using the crash analysis

software TEMA and give your conclusion.

Data Available -:

We need to make a calibration curve for Ecrush Vs Crush and coefficient of restitution (e) vs Crush. In order to

make the curves the data which is required is given below

Crush Length

(in.) Sled Weight (lb) Air Pressure (psi)

0.53 14.0 40

0.95 14.0 50

0.93 14.0 60

0.67 19.7 40

1.05 19.7 50

1.26 19.7 60

0.54 25.4 30

1.60 25.4 40

2.02 25.4 50

The above data is obtained by conducting a full frontal collision with rigid barrier. The cup is attached to the

barrier. In the experimental setup, the bullet is made to move using compressed air which makes piston move

which in turn pulls the bullet sled. So the test is conducted at different air pressure which will provide different

Pre Impact velocity and different crush length.

Test Condition -: The test is recorded by using a camera which can record at speed of 1000 frames per second.

The lighting used in the lab is direct current since alternating current has frequency of 60 Hz it will cause

fluctuation in brightness while video is recorded.

TEMA -:

TEMA is a crash analysis software which makes use of the video recorded to analyze the crash. The basic input

which are required for the TEMA software -:

1.) Recording Speed ( frames per second)

2.) Gamma Value

3.) Distance of the reference plane from camera

4.) Distance of setup from reference plane

5.) Scale distance between stationary points.

Upload the video of the 9 experiments conducted one by one and follow the mentioned procedure.

PAGE 2

Once the video is opened on the software we need to set T0. After that locate the bullet, target and reference

points using add point tool. Using the quadrant geometry locate the points correctly and set size for the object.

Once the points the located we need to provide distance of the reference plane from camera and Scale distance

between stationary points which help TEMA to analysis the motion. The data is -:

Distance from Camera Lens to Reference Frame (in.) 38.25

Scale Distance between Stationary Targets (in.) 3

Distance from Reference Frame to Sled Target (in.) -3.25

Select the point on the bullet to obtain the velocity vs time graph and acceleration time graph. From the velocity

time graph we obtain the Pre Impact velocity and Post Impact Velocity. Also we know the crush depth for each

case. We obtain the coefficient of restitution by taking the ratio of Post impact velocity to Pre Impact Velocity.

The corresponding values are tabulated below -:

Test Air

Pressure

(psi)

Vpre(m/s) Vpost(m/s) e Crush

Sled

Weight

(lb)

Sled

Wt.(kgs) Ecrush(J)

TEST

1 40 2.6 -0.206 0.079 0.53 14 6.3 21.301

TEST

2 50 2.38 -0.205 0.086 0.95 14 6.3 17.829

TEST

3 60 2.0923 -0.196 0.094 0.93 14 6.3 13.760

TEST

4 40 1.4 -0.1 0.071 0.67 19.7 8.9 8.701

TEST

5 50 2.25 -0.162 0.072 1.05 19.7 8.9 22.472

TEST

6 60 1.88 -0.154 0.082 1.26 19.7 8.9 15.665

TEST

7 30 2.2 -0.174 0.079 0.54 25.4 11.5 27.671

TEST

8 40 1.95 -0.126 0.065 1.60 25.4 11.5 21.785

TEST

9 50 1.48 -0.115 0.078 2.02 25.4 11.5 12.526

The only input value which we have with us is the crush depth. Therefore we make a calibration plot of

Coefficient of restitution vs Crush Depth

PAGE 3

Also we know that

Epre=Epost+Ecrush

Since we know the pre impact velocity and post impact velocity from the graphs we can obtain Ecrush for each

case. Therefore we make another calibration curve of Ecrush vs Crush Depth.

Since we have the calibration curves with us we can use the data from calibration curve to obtain Ecrush and

Coefficient of restitution for our experiment.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.00 0.50 1.00 1.50 2.00 2.50

e (C

oef

fici

ent

of

resi

tuti

on

)

Crush Depth (inch)

E Vs Crush Depth

0

5

10

15

20

25

30

0.00 0.50 1.00 1.50 2.00 2.50

Ecr

ush

(J)

Crush Depth (inch)

Ecrush(J)

PAGE 4

Actual Setup -:

Distance of reference plane from camera -: 43.5 inch

Distance between two points on reference plane -: 12 inch

Distance between reference plane and setup -: 7 inch

Using the steps shown above we setup the case in TEMA. Then we plot acceleration vs time and velocity vs time

curve for our actual experiment.

The yellow curve shows the velocity of bullet and red curve shows the velocity of target. From the graph we see

that the bullet approaches the target with velocity of around 3 m/s. We chose the point just before the velocity

starts to decrease therefore the pre impact velocity of bullet is 2.91 m/s. Whereas the target is at rest therefore the

pre impact velocity of the target is 0 m/s. The post impact velocity of bullet is 1.15 m/s and for the target it is 1.52

m/s. From the graph we see that the velocity of bullet tends to increase at time 135 msec. This might be because

vibration in the table which must have caused the reference points to move a bit causing error in reading the pixel

in TEMA. Velocity of bullet does not become negative which means that there is no change in direction of its

motion.

PAGE 5

Accelerometer

The accelerometer are fit on the bullet and target which measures the deceleration during impact. From the data

obtained we plot acceleration vs time graph for bullet and target.

For Bullet Sled

From the graph we can see that initially the accelerometer reads zero. Then when it starts to accelerate we find the

value in accelerometer changes. At around 0.1 sec the bullet reaches a constant velocity so again the

accelerometer reads zero. At around 0.18sec it impacts the targets and we see that accelerometer reads the data.

On integrating the acceleration curve we obtain the velocity curve.

From the velocity time graph we see that the bullet sled speed increases from zero to 3.25 m/s between 0.2 sec to

0.8 sec. It remains constant till around 0.18 sec and as soon as it impact it decelerates so its velocity decreases and

reaches 1.3 m/s.

-150

-100

-50

0

50

100

150

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Acc

eler

atio

n (

g's

)

Time (sec)

Acceleration vs time for bullet

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

-0.05 0.05 0.15 0.25 0.35 0.45

Vel

oci

ty (

m/s

)

Time (sec)

Velocity of Bullet sled

PAGE 6

For Target Sled

The target sled is initially at rest. There are small reading in acceleration curve from 0.02 sec to 0.05 sec because

of minor vibration in the setup. As the bullet sled impacts the target at around 0.18 sec, we see that accelerometer

shows reading. The target sleds accelerates till 0.25 secs and maintains a constant velocity.

-70.000

-50.000

-30.000

-10.000

10.000

30.000

50.000

70.000

-0.05 0.05 0.15 0.25 0.35 0.45

Acc

eler

atio

n(m

/s^

2)

Time(sec)

Acceleration vs Time graph of Target Sled

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.05 0.05 0.15 0.25 0.35 0.45

Vel

oci

ty(m

/s)

Time(secs)

Velocity v/s Time graph for Target Sled

PAGE 7

Fig : i. Before Crush ii. After Crush

Mathematical Model

Our mathematical model is based on 3 important equations. From our experiment which we conducted we get the

value of the crush depth. We measure crush depth at four different points on the cups. The original length of the

cup is 4.5 inch. The length of cup after impact at four different region are 86 mm, 92.3 mm, 91 mm and 93 mm.

We take average of the four values and we get average length of cup after crash as 90.575 mm or 3.566 inch. The

crush depth is

Crush depth = Original Length before impact – Length after impact

Crush depth = 4.5 – 3.566 = 0.934 inch

Using the calibration curve which we made of Coefficient of restitution vs crush depth and Ecrush vs crush Depth

we obtained the coefficient of restitution and Ecrush for our crush depth of 0.934 inch. The corresponding value

of coefficient of restitution (e) = 0.08 and Ecrush =20 J.

Since we have to find the pre and post impact velocity of bullet and target sled. We have 4 unknown out of which

𝑣2𝑝𝑟𝑒 = 0. Therefore we are left with 3 unknowns. We require three equations to solve

Basic governing equations for solving the Mathematical model

PAGE 8

1. Conservation of Momentum Eq:

𝑚1𝑣1𝑝𝑟𝑒 + 𝑚2𝑣2

𝑝𝑟𝑒 = 𝑚1𝑣1𝑝𝑜𝑠𝑡 + 𝑚2𝑣2

𝑝𝑜𝑠𝑡

m1 =7.67 kgs (Mass of bullet sled) m2= 7.83kgs (Mass of Target Sled)

𝑣2𝑝𝑟𝑒 = 0 (Since the target sled is in rest initially)

7.67𝑣1𝑝𝑟𝑒 = 7.67𝑣1

𝑝𝑜𝑠𝑡 + 7.83𝑣2𝑝𝑜𝑠𝑡 ----- i

2. Conservation of Energy Eq :

1

2𝑚1𝑣1

𝑝𝑟𝑒2+

1

2𝑚2𝑣2

𝑝𝑟𝑒2=

1

2𝑚1𝑣1

𝑝𝑜𝑠𝑡2+

1

2𝑚2𝑣2

𝑝𝑜𝑠𝑡2+ 𝐸2𝑐𝑟𝑢𝑠ℎ

𝑣2𝑝𝑟𝑒 = 0

3.835𝑣1𝑝𝑟𝑒2

= 3.835𝑣1𝑝𝑜𝑠𝑡2

+ 3.915𝑣2𝑝𝑜𝑠𝑡2

+ 20𝐽 ------ ii

3. Coefficient of Restitution Eq :

𝑒 = −(𝑣2

𝑝𝑜𝑠𝑡 − 𝑣1𝑝𝑜𝑠𝑡

𝑣2𝑝𝑟𝑒 − 𝑣1

𝑝𝑟𝑒)

𝑒 = 0.08

−0.08𝑣1𝑝𝑟𝑒 = −𝑣2

𝑝𝑜𝑠𝑡 + 𝑣1𝑝𝑜𝑠𝑡 ------ iii

Simplifying equations:

𝑣1𝑝𝑟𝑒 = 𝑣1

𝑝𝑜𝑠𝑡 + 1.02𝑣2𝑝𝑜𝑠𝑡

−0.08(𝑣1𝑝𝑜𝑠𝑡 + 1.02𝑣2

𝑝𝑜𝑠𝑡) − 𝑣1𝑝𝑜𝑠𝑡 + 𝑣2

𝑝𝑜𝑠𝑡 = 0

𝑣2𝑝𝑜𝑠𝑡 = 1.176𝑣1

𝑝𝑜𝑠𝑡

3.835(𝑣1𝑝𝑜𝑠𝑡 + 1.02𝑣2

𝑝𝑜𝑠𝑡)2 − 3.835𝑣1𝑝𝑜𝑠𝑡2

+ 3.915(1.176𝑣1𝑝𝑜𝑠𝑡)2 + 20𝐽

9.355𝑣1𝑝𝑜𝑠𝑡2

= 20

𝑣1𝑝𝑜𝑠𝑡 = 1.46 𝑚/𝑠

𝑣1𝑝𝑟𝑒 = 3.2 𝑚/𝑠

𝑣2𝑝𝑜𝑠𝑡 = 1.71 𝑚/𝑠

PAGE 9

Observations-:

MODE BULLET (m/s) TARGET(m/s)

V1pre V1post V2pre V2post

TEMA- VIDEO ANALYSIS 2.91 1.15 0 1.52

ACCELEROMETER 3.25 1.3 0 1.6

MATHEMATICAL MODEL 3.2 1.46 0 1.71

ERROR of Mathematical model

w.r.t Accelerometer 1.538462 -12.3077 NA -6.875

Error of Tema w.r.t

Accelerometer 10.46154 11.53846 NA 5

Possible Error sources in TEMA are

1. Anomalous output from Video analysis.

When the setup vibrate the position of the reference points in the video tend to change which affects the

tracking ability of the software. As we can see from the velocity vs time curve after collision the bullet

tends to accelerate. This cannot not be possible without any force applied on the bullet. We expect that the

table was shaking so the position of reference points changed which resulted in the error. This causes

error in taking the right Vpost values. This accounts for the maximum error in the Vpost. The Vpost value

fluctuates between 1.15 to 1.3 m/s. We took Vpost as 1.25 m/s. So the relative error in taking the Vpost

value ranges from -4.2% to 8.3%.

2. Reference Error

If the actual distance between camera and reference points and what we measure is off by some value it

can result in error. It will basically offset the values.

For the given conditions with a slight reference plane value offset of 0.25” ~6.34mm the value of the

velocity changes by 0.15m/s which gives an error up to 3%.

3. Human Reading error

This error is accountable along with the error in the output from the TEMA software. The error in reading

the values can occur like if the reading is taken 2 times then the probability of getting two different values

are bit high as the TEMA software doesn’t give that much precise curve and the variations in the curve

makes difficult to get exact values.

For a variation in getting V values form the graph is slightly off by 0.05m/s then the % error in the value

of V Is 1.36%. We can say reading error will be around 1.5%

4. Processing Error

The TEMA works by tracking the pixel as it updates the pixel matrix displacement with respect to its

previous location and the with the help of reference plane it calculates the Angle thus the variation in

angle gives the desired value of velocity(triangulation/trilateration). Due to its high velocity and low

frame capture there happens to have the offset in the pixels thus causes the variation in speed thus results

in cumulative error in the tracking.

PAGE 10

This error is reduced by using the 2 Camera tracking system which is widely used in photogrammetry

nowadays to reduce the error.

This error accounts for the error % ranging about <1%.

Possible error in Mathematical Modelling

1. Calibration Curve Fit Error.

The calibration curve is made using TEMA software which again returns about 2.4% error in the values.

Also the calibration values are plotted by the number of similar experiments. It can occur some of the

points shows the anomalous behavior thus that point doesn’t give to the prover curve fit. Which

eventually gives an error to the “ e “ and “ Ecrush “ values . Thus the error is carried and multiplied in the

Vpre and Vpost calculations.

For 2 different curve fits for Ecrush even if value of Ecrush is varies by +/- 1 Joule the error in velocity

ranges from 5 to 8%.

2. Inability of Mathematical model to process the actual scenario.

From the values of V pre and V post we observe that major error occurs in the Vpost value of the bullet. It

is because when the bullet crushes target the accelerometer reads the deceleration whereas the

mathematical model is unable to read this. This results in higher value of Vpost of the bullet. This error

ranges between 3 to 4%

3. Reading error from the Graph

There is a probability of the reading error form the graph. For e.g. from the calibration graph there is

chance of taking up value with error 1- 2.5% error.

Possible error from the Accelerometer

Accelerometer gives the best approximation of the values w.r.t to the other methods still it can carry some error

such as

1. Vibration of the system

The vibration of the system on which the accelerometer is mounted is subject to rapid acceleration and

deceleration due to crush and external force which can induce many internal vibration to the

accelerometer which adds up to the accelerometer reading of its own.

This causes and error in the graph which causes error in the velocity curve which is obtained by

integrating the acceleration graph thus the cumulative error is estimated up to 1.5% to the actual values.

2. Aliasing

Aliasing is a type of sampling error in which frequencies that do not actually exist appear in the data.

Since nearly all data acquisition devices are digital, they have a sample rate: the rate at which samples are

collected and stored. If you use the wrong sampling rate the sampled waveform will differ significantly

from original.

Thus the common error is accounted upto 1%.

PAGE 11

Error Chart in Accelerometer % Internal Error

Vibration of System Aliasing

Reading Unaccounted Error

Error Chart in Mathematical Modelling % Error w.r.t. accel

Calibration Fit Curve

Inability to process actual sc.

Human Reading error

Unaccounted Error

3. Time delay

The third biggest source of error is not accounting for the impact of the time delay between sampling the

data and the application software processing that information.

Error Sources Error Estimation -:

Considering the maximum of every error possible we have plotted the below pie chart -:

1. Accelerometer -:

Error Chart in Accelerometer

Error sources

% Internal

Error

Cumulative

Error

Vibration of System 1.5 36.31961259

Aliasing 1 24.21307506

Reading 0.33 7.99031477

Unaccounted Error 1.3 31.47699758

Total error 4.13 100

2. Mathematical Modeling -:

Error Chart in Mathematical Modelling

Error sources

% Error

w.r.t. accn

Cumulative

Error

Calibration Fit Curve 8 48.48484848

Inability to process

actual sc. 3 18.18181818

Human Reading

error 2.5 15.15151515

Unaccounted Error 3 18.18181818

Total error 16.5 100

PAGE 12

Error Chart in TEMA modelling % Error w.r.t Accel

Anomalous Output Reference Error

Reading Error Unaccounted Error

3. TEMA

Error Chart in TEMA modelling

Error sources

% Error

w.r.t Accel

Cumulative

Error

Anomalous Output 12 64.86486486

Reference Error 3 16.21621622

Reading Error 1.5 8.108108108

Unaccounted Error 2 10.81081081

Total error 18.5 100

Conclusion -:

Mini Sled project was conducted using three platform accelerometer values, TEMA and mathematical model. The

accelerometer gives the most accurate values even though there are some errors associated with it. Possible causes

for the error has been above mentioned for the data obtained. Our mathematical model gives us the second most

accurate values followed by TEMA. In mathematical model, the calibration curve fit error can be reduced by

including more sampling point and eliminating the ones with abnormal values. The calibration curve values are

obtained from TEMA. Other way to reduce the error can be done by obtaining the values of each experiment

using accelerometer rather than TEMA thereby reducing the error carried forward to mathematical model. The

second error which is the inability of mathematical model to replicate the actual scenario we can consider some

correction factor which will reduce the error. This can be taken into account by conducting several experiment

and deciding the empirical value of the correction factor.