G Force Tolerance Sample Solution

Transcription

1 G Force Tolerance Sample Solution Introduction When different forces are applied to an oject, G-Force is a term used to descrie 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 ody. The human ody is fleile and deformale, particularly the softer tissues. A hard slap on the face may riefly impose hundreds of g locally ut not produce any real damage; a constant 16 g for a minute, however, may e deadly. This portfolio will eamine the tolerance humans have to oth horizontal and vertical G-force. I will create a function to model the ehavior of the data and discuss the apparent implications on time and G-force. Data and Graph The following tale and graph 1 illustrate the tolerance of human eing to horizontal G-force. Time (min) Graph 1: Tolerated G-force(+G) vs Time It is ovious that T > 0 and that +G > 0. T, time in minutes, is the independent variale and +G(), measured in g, is the dependent variale. There is a definite inverse relationship etween time and G-Force, +G. The more time a human is eposed to g forces, the smaller the amount of gees they are capale of sustaining. This would imply that a rational, power, or eponential function would est fit the data. In looking at the graph, there appears to e oth 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 oth a rational function and a power function. Rational Function a A rational function has the asic equation, y = + c, with parameters a,, and c. The parameter c represents the horizontal asymptote. In looking at the graph it appears that a horizontal asymptote eists at y = 4, or somewhere less than 4. This would imply that humans can withstand 4 +G() indefinitely. There is no data on the maimum amount of indefinitely sustained g forces, ut the force of gravity when you are still (for eample, when you sit, stand or lie down) is

2 considered 1 G. (reference : I will then let c = 1. The parameter represents the vertical asymptote, which appears to e 0. This would imply that there is no upper limit to the G-force a human can a tolerate given an etremely short amount of time. My equation then ecomes y =. The parameter a will e a positive value. Increasing a will move the graph further away from the origin. I algeraically will solve for a using. Eample 1: using data point (, ) = a.34, a =.34 y = Eample 2: using data point (3.9) 9 = a 24, a = 24 y = 3 Graph 2: The effect of changing parameter f()=1/+1 f()=.34/+1 f()=4.2/+1 f()=/+1 f()=/+1 f()=24/+1 Graph 2 shows the effects of using different to solve for a. Notice some values of a ring the graph elow the and other values ring the graph aove I averaged all of the a values from the 8 : = And used 24.3 as my value of a. This gives the equation, + G = to use to model the data. T In looking at graph 3, the overall shape is good, ut the graph does not seem to come close to the y-ais soon enough. The horizontal ais could e changed to = -1 or = -., ut this does not make sense in contet ecause T > 0. Graph 3: Model created y hand f()=(24.3/) I then considered a power function.

3 Power Function A power function has the asic equation, y = a, where a and are the parameters. As stated efore, a horizontal asymptote at y = 1 (indefinitely sustaining 1 +G()) would e eneficial and change my equation to y = a +1. Graph 4: Effects of varying the eponent in power functions y f()= f()=^2 f()=^ f()=^(-1) f()=^2.3 f()=^-2.3 f()=^(-.8) The parameter greatly effects the graph. In looking at graph 4, a power function takes on different shapes depending if is positive, negative, integer, or decimal. A negative value for will give an inverse relationship etween the variales. The parameter a will e a positive value. Increasing a will move the graph further away from the origin. I will algeraically solve for a and y choosing two points. 3 Eample 1: using points (1, 11) and (.1, ) G = at = a(1) = a(.1) = a sustitute 19 = (.1) 1.9 =.1 ln(1.9) = ln(.1) = ln(1.9) ln(.1) = G = T.278 Graph : Power equations f()=^(-.278)+1 f()=8.89^(-.2914) Eample 2: using points (, ) and (, 4.) = a() 34 = a () + G = at 34 = a, sustitute () + G = 8.89T = a() 3. =.097 = = () () ln(.097) ln() ln() =.2914 These functions can e seen in graph to the left. Both fit very well and are a etter match to the data than the rational equation. The inverse relationship etween time and +G is clear and the equations hit almost every data point. I will choose G = T to model the data for its simplicity.

4 Technology Another equation to fit this data can e found using technology. A power model, + G = 11. proved to e the est fit with a coefficient of determination, r 2 = The two equations look almost identical, oth proving to e an ecellent model for the data. Graph 6: By hand vs Technology y hand: ^(-.278)+1 technology:11.x^ Vertical G forces The human ody is consideraly less adept to surviving vertical g-forces. Aircraft, in particular, eert g-force along the ais aligned with the spine. This causes significant variation in lood pressure along the length of the suject's ody, which limits the maimum g-forces that can e tolerated. ( Graph 7 shows vertical G- Graph 7: Vertical G force and the original model force, +Gz and our original +Gz (g) power model. The shape of the graph and the data are original model: ^(-.278)+1 similar, ut clearly changes new model:7.08x^-.199 need to e made. The fact that humans can tolerate vertical forces less can e seen y the falling elow most of the graph. The parameter a, needs to e smaller to account for this. An equation such as, Gz = 7.082T has a high coefficient, r 2 =.9979 of determination and the parameter a has changed from to 7.08 Notice how the new equation is stretched ack towards the ais and the. The new equation still incorporates the idea of sustaining 1 G from gravity. Time (min) +Gz (g)

5 Limitations Human tolerance of g-force in the horizontal direction can e modeled y the equation + Gz = T.278 and similarly.199 vertical g-force can e modeled y + Gz = 7.082T. Both power models can interpolate values within the domain. 01 T quite accurately. But how well do the models etrapolate 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 = 0, suggesting humans can tolerate any etremely high amount of gees, ut I don t elieve this is true. Humans can survive up to aout to g instantaneously (for a very short period of time). Any eposure to around 0 g or more, even if momentary, is likely to e lethal, although the record is g. ( Another limitation is where the data originated not knowing who this data is aout, or how the +G were calculated. Prolems could arise in using the model to predict if the suject or circumstances are different. To some degree, G - tolerance can e trainale, and there is also considerale variation in innate aility etween individuals. Overall a power model fits the data well and would e an ecellent source for interpolation.