G‐Force Tolerance   Sample Solution 

G‐Force Tolerance Sample Solution Introduction
When different forces are applied to an object, G-Force is a term used to describe the resulting acceleration, and
is in relation to acceleration due to gravity(g). Human tolerances depend on the magnitude of the g-force, the
length of time it is applied, the direction it acts, the location of application, and the posture of the body. The
human body is flexible and deformable, particularly the softer tissues. A hard slap on the face may briefly
impose hundreds of g locally but not produce any real damage; a constant 16 g for a minute, however, may be
deadly. This portfolio will examine the tolerance humans have to both horizontal and vertical G-force. I will
create a function to model the behavior of the data and discuss the apparent implications on time and G-force.
Data and Graph
The following table and graph 1 illustrate the tolerance of human being to horizontal G-force.
Graph 1: Tolerated G-force(+Gx) vs Time
+Gx (g)
Time (min)
+Gx (g)
0.01
35
0.03
28
30
0.1
20
25
0.3
15
20
1
11
15
3
9
10
10
6
5
30
4.5
35
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
T(time in minutes)
It is obvious that T > 0 and that +Gx > 0. T, time in minutes, is the independent variable and +G(x), measured in g, is
the dependent variable. There is a definite inverse relationship between time and G-Force, +Gx. The more time a human
is exposed to g forces, the smaller the amount of gees they are capable of sustaining. This would imply that a rational,
power, or exponential function would best fit the data. In looking at the graph, there appears to be both a horizontal and
vertical asymptote, meaning that there is a certain amount of g’s that a human can tolerate indefinitely (horizontal
asymptote) and an unlimited amount of g’s that a human can withstand at an infinitesimal small amount of time. Let’s
look at both a rational function and a power function.
Rational Function
A rational function has the basic equation, y =
a
+ c , with parameters a, b, and c. The parameter c represents the
x−b
horizontal asymptote. In looking at the graph it appears that a horizontal asymptote exists at y = 4, or somewhere less
than 4. This would imply that humans can withstand 4 +G(x) indefinitely. There is no data on the maximum amount of
indefinitely sustained g forces, but the force of gravity when you are still (for example, when you sit, stand or lie down) is
considered 1 G. (reference : http://www.reference.com/browse/g+force) I will then let c = 1. The parameter b represents
the vertical asymptote, which appears to be 0. This would imply that there is no upper limit to the G-force a human can
tolerate given an extremely short amount of time. My equation then becomes y =
The parameter a will be a positive value.
Increasing a will move the graph further away
from the origin. I algebraically will solve for a
using data points.
Example 1: using data point (.01, 35)
35 =
Graph 2: The effect of changing parameter b
+Gx (g)
35
f(x)=1/x+1
f(x)=.34/x+1
f(x)=4.2/x+1
f(x)=105/x+1
f(x)=10/x+1
f(x)=24/x+1
data points
30
25
a
.34
+ 1 , a = .34 ⇒ y =
+1
.01
x
20
15
Example 2: using data point (3.9)
9=
a
+1.
x
a
24
+ 1 , a = 24 ⇒ y =
+1
3
x
10
5
2
Graph 2 shows the effects of using different
data points to solve for a. Notice some values
of a bring the graph below the data points and
other values bring the graph above.
4
6
8
10
12
14
16
18
20
22
24
26
28
30
T(time in minutes)
.34 + .81 + 1.9 + 4.2 + 10 + 24 + 50 + 105
= 24 .53
8
I averaged all of the a values from the 8 data points:
And used 24.53 as my value of a. This gives the equation, + Gx =
24.5
+ 1 to use to model the data.
T
Graph 3: Model created by hand
+Gx (g)
In looking at graph 3, the overall shape is
good, but the graph does not seem to come
close to the y-axis soon enough. The
horizontal axis could be changed to x = -1
or x = -.5, but this does not make sense in
context because T > 0.
35
f(x)=(24.53/x)+1
data points
30
25
20
15
10
5
2
I then considered a power function.
4
6
8
10
12
14
16
18
20
22
24
26
28
30
T(time in minutes)
Power Function
A power function has the basic equation, y = ax b , where a and b are the parameters. As stated before, a horizontal
asymptote at y = 1 (indefinitely sustaining 1 +G(x)) would be beneficial and change my equation to y = ax b + 1 .
Graph 4: Effects of varying the exponent in power functions
The parameter b greatly effects the graph. In looking
at graph 4, a power function takes on different shapes
depending if b is positive, negative, integer, or decimal.
A negative value for b will give an inverse relationship
between the variables.
y
9
f(x)=x
f(x)=x^2
f(x)=x^5
f(x)=x^(-1)
f(x)=x^2.3
f(x)=x^-2.3
f(x)=x^(-5.8)
8
7
6
5
The parameter a will be a positive value. Increasing a
will move the graph further away from the origin. I
will algebraically solve for a and b by choosing two
points.
4
3
Example 1: using points (1, 11) and (.1, 20)
2
1
x
1
2
3
4
5
6
7
8
9
+ Gx = aT b + 1
20 = a (.1) b + 1
11 = a(1) b + 1
10 = a substitute ⇒
19 = 10(.1) b
1.9 = .1b
ln(1.9) = b ln(.1)
b=
ln(1.9 )
ln(.1)
= −.278
+ Gx = 10T −.278 + 1
Graph 5: Power equations
Example 2: using points (.01, 35) and (30, 4.5)
+Gx (g)
35
30
+ Gx = aT b + 1
4.5 = a (30) b + 1
35 = a(.01) b + 1
f(x)=10x^(-.278)+1
f(x)=8.89x^(-.2914)+1
data points
25
3.5 =
34 = a (.01) b
34
(.01) b
20
= a, substitute ⇒
.097 =
b=
15
34
(.01) b
30b
.01b
(30) b
.01b
ln(.097 )
ln( 30 ) − ln(.01)
= −.2914
+ Gx = 8.89T −.2914 + 1
10
5
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
T(time in minutes)
These functions can be seen in graph 5 to the left.
Both fit very well and are a better match to the data
than the rational equation. The inverse relationship
between time and +Gx is clear and the equations hit
almost every data point. I will choose
+ Gx = 10T −.278 + 1 to model the data for its
simplicity.
Technology
Another equation to fit this data can be found using technology. A power model, + Gx = 11.2555 x −.24988 proved
to be the best fit with a coefficient of determination, r2 = .9978. The two equations look almost identical, both
proving to be an excellent model for the data.
Graph 6: By hand vs Technology
+Gx (g)
35
30
by hand: 10x^(-.278)+1
technology:11.255X^-.24988
data points
25
20
15
10
5
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
T(time in minutes)
Vertical G‐forces The human body is considerably less adept to surviving vertical g-forces. Aircraft, in particular, exert g-force along
the axis aligned with the spine. This causes significant variation in blood pressure along the length of the
subject's body, which limits the maximum g-forces that can be tolerated. (http://www.reference.com/browse/g+force)
Graph 7 shows vertical Gforce, +Gz and our original
power model. The shape of
the graph and the data are
similar, but clearly changes
need to be made. The fact
that humans can tolerate
vertical forces less can be seen
by the data points falling
below most of the graph. The
parameter a, needs to be
smaller to account for this.
An equation such as,
+ Gz = 7.082T −.199 has a high
coefficient, r2 = .9979 of
determination and the
parameter a has changed
from10 to 7.08
Graph 7: Vertical G force and the original model
+Gz (g)
35
30
original model: 10x^(-.278)+1
new model:7.08X^-.199
data points
25
20
Time (min)
+Gz (g)
0.01
0.03
0.1
0.3
1
3
10
30
35
28
20
15
11
9
6
4.5
15
10
5
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
T(time in minutes)
Notice how the new equation is stretched back towards the axis and the data points. The new equation still incorporates
the idea of sustaining 1 G from gravity.
Limitations
Human tolerance of g-force in the horizontal direction can be modeled by the equation + Gz = 10T −.278
and similarly
−.199
vertical g-force can be modeled by + Gz = 7.082T
. Both power models can interpolate values within the domain
.01 ≤ T ≤ 30 quite accurately. But how well do the models extrapolate information? Both models suggest that humans
can tolerate 1g indefinitely. Is this realistic or can humans tolerate a higher amount of g?
The more serious implication concerns the vertical asymptote at x = 0, suggesting humans can tolerate any extremely high
amount of gees, but I don’t believe this is true. Humans can survive up to about 20 to 35 g instantaneously (for a very
short period of time). Any exposure to around 100 g or more, even if momentary, is likely to be lethal, although the record
is 179.8 g . (http://www.reference.com/browse/g+force)
Another limitation is where the data originated – not knowing who this data is about, or how the +Gx were calculated.
Problems could arise in using the model to predict if the subject or circumstances are different. To some degree, G tolerance can be trainable, and there is also considerable variation in innate ability between individuals.
Overall a power model fits the data well and would be an excellent source for interpolation.