Real World Problem!

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RotorPro

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So in short I am trying to write an equation to determine the maximum weight of a passenger I can carry given a set of weights and locations inside the aircraft as to not exceed the center of gravity limits prescribed by the manufacturer.

Weight * Arm = Moment ( arm is the location within the aircraft relative to a specified point )

y= resultant center of gravity
x = passenger weight
a = passenger arm
b = aircraft weight
c = aircraft current center of gravity

given that:

Y = (AX+BC)/X+B

trying to solve for X

y(b+x)=ax+bc
xy+by=ax+bc
xy-ax=bc-by
x(y-a)=bc-by
x=(bc-by)/(y-a)
x=b(c-y)/(y-a)

This is the solution I came up with in an attempt to solve for X. Unfortunately when plugging in known values the results return incorrect.
Where did I go wrong in the attempt to solve for X?
 
Assuming caps and small letters are meant as the same thing, given that
Y = (AX+BC)/X+B
we have either

  1. y = \(\displaystyle \frac{a\space x\space +\space b}{x\space +\space b}\) or
  2. y = \(\displaystyle \frac{a\space x\space+\space b}{x}\space +\space b\)

Now you say (1) didn't work [I agreed with the final value you got for x], so try (2). If that doesn't work, I would say something is wrong with the problem as stated.
 
RotorPro said:
So in short I am trying to write an equation to determine the maximum weight of a passenger I can carry given a set of weights and locations inside the aircraft as to not exceed the center of gravity limits prescribed by the manufacturer.

Weight * Arm = Moment ( arm is the location within the aircraft relative to a specified point )

y= resultant center of gravity
x = passenger weight
a = passenger arm
b = aircraft weight
c = aircraft current center of gravity

given that:

Y = (AX+BC)/X+B

trying to solve for X

y(b+x)=ax+bc
xy+by=ax+bc
xy-ax=bc-by
x(y-a)=bc-by
x=(bc-by)/(y-a)
x=b(c-y)/(y-a)

This is the solution I came up with in an attempt to solve for X. Unfortunately when plugging in known values the results return incorrect.
Where did I go wrong in the attempt to solve for X?
Click to expand...

All moment arms are measured from the aircraft datum point. The aircraft cg is determined by adding the moments and dividing that total by the sum of the weights. Your formula is a simplified version of that, which is not breaking down aircraft, pilot, fuel, and cargo loads.

See a simple example here: http://en.wikipedia.org/wiki/Center_of_gravity_of_an_aircraft

Apparently, you are considering fuel, cargo, pilot, and aircraft all as one known weight and cg???

Assuming those simplifications, your algebra looks okay. You haven't shown us the actual numbers you plugged in and why you think your answer is "incorrect."

I'm also wondering why you are trying to solve for passenger weight. Are you trying to find out how heavy a passenger you can carry, or something like that?
 
Ishuda
Assuming caps and small letters are meant as the same thing, given that
Y = (AX+BC)/X+B
we have either


  1. y = [FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]b[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]b[/FONT] or
  2. y = [FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]b[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]b[/FONT]


Now you say (1) didn't work [I agreed with the final value you got for x], so try (2). If that doesn't work, I would say something is wrong with the problem as stated.​
Click to expand...

the intention was:
y=(ax+bc)/(x+b)



Apparently, you are considering fuel, cargo, pilot, and aircraft all as one known weight and cg???

Assuming those simplifications, your algebra looks okay. You haven't shown us the actual numbers you plugged in and why you think your answer is "incorrect."

I'm also wondering why you are trying to solve for passenger weight. Are you trying to find out how heavy a passenger you can carry, or something like that?
Click to expand...

The end result is a chart which I can quickly reference. The chart will have accounted for myself, my medical crew, the aircraft basic empty weight, medical interior modifications, and medical equipment as fixed values for that day. The chart does then offer various combinations of fuel vs patient weight. For example, when I have a patient who weighs X I can carry Y pounds of fuel and vice versa. What I am attempting to accomplish now is a separate chart which will offer me the maximum patient weight that can be carried given a specified fuel quantity before the aircraft will either exceed maximum gross weight limitations or center of gravity limitations.

Originally I believed that I had solved for X incorrectly, but after some sleep and reading the comments I have come to realize that the function itself is faulty. And by faulty I mean it is not returning a value which is actually the greatest amount I can carry before reaching the limitations.

Best way I can think of to determine where the function has failed is to provide as MUCH information as possible!!

These values are constant and do not change:
1.Maximum gross weight is 5225 lbs.
2.The patients station (ARM) is 75 inches.
3.The equation for determining the center of gravity (C.G.) limit is:
Y=0.0029412*X+111.83 Y cannot be less than 127.2
The C.G. Limit is exceeded when the C.G. goes below the limit
4.weight * arm = moment
Add all moments together and divide by total weight to attain the C.G.

What is the minimum amount of weight I can add to the 75 inch arm within this configuration so that either the weight does not exceed 5225 lbs or the C.G. does not go below the C.G. Limit within these 3 scenarios? (Run On Scentence, good thing this is a Math Forum)

The aircraft weighs 4308 lbs
current C.G. 129.9 inches

The aircraft weighs 4674 lbs
current C.G. is 130.5 inches

The Aircraft weighs 5135 lbs
current C.G. is 131.0 inches

Thank you for the help all of you have provided so far!
 
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And this would be a great answer to all of those students who say, "Why should I pay attention? I'm never going to use this stuff anyways!"
 
RotorPro said:
... For example, when I have a patient who weighs X I can carry Y pounds of fuel and vice versa....
3.The equation for determining the center of gravity (C.G.) limit is:
Y=0.0029412*X+111.83 Y cannot be less than 127.2 ...


...The aircraft weighs 4308 lbs
current C.G. 129.9 inches

The aircraft weighs 4674 lbs
current C.G. is 130.5 inches

The Aircraft weighs 5135 lbs
current C.G. is 131.0 inches...
Click to expand...

The data provided seem inconstant [edit: see I never make misteaks] to me or I am min-interpreting [edit: what?]:

In the first quoted section above
y=0.0029412*x+111.83 \(\displaystyle \ge\) 127.2
would mean you had to have a passenger who weighed at least 5225.8 pounds.


In the second quoted section, either the aircraft weights are the 'empty weight' (including your self, your medical crew, the aircraft basic empty weight, medical interior modifications, and medical equipment as fixed values for that day) or it is an 'empty weight' plus something. If it were the 'empty weight' and the aircraft weights could be treated as a single entity, then the distance from the datum point (the arm) would be equal to the center of gravity. But that would mean the heavier the aircraft was, the closer to the datum point you would need to get which is not the case here. So something is missing. [If the something missing were the weight of the passenger, the resulting three equations may be over determined. That is, you could end up with two equations but three unknowns.]
 
(It appears as if the images quality was greatly reduced when transferring to the site. I have fixed the issue and the links should provide good resolution images)

I am hoping the following screen shots of the tables I am working on may answer the questions being asked a bit better.

Med Equipment.jpg This page is where the equipment that is not attached to the aircraft is entered.

Information Page.jpg The pilot uses this page to enter the weight of himself, the med crew, and other variables.

TableExplanation.jpg This is the loading table which will be taken on a flight to assist in rapidly determining either how much fuel or how much patient can be carried.
 
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In an attempt to answer the questions arising, I'm hoping these screen shots of the spread sheet I'm working on will help...


Information Page -- http://tinypic.com/r/x425hk/8 <-- This page is used by the pilot to enter values such as crew weight, pilot weight, and other variables

Equipment Loading -- http://tinypic.com/r/2zthagh/8 <-- This page is used to enter the weight and location in the A/C of non permanent equipment

Loading Table -- http://tinypic.com/r/2d7vggh/8 <-- This page is a loading table used for quick reference in the field to determine at what fuel load can what weight of patient be carried. My goal is to insert a new row here to display the maximum patient weight that can be carried at a specified fuel load. The limiting factors for possible patient weight are C.G. limits and Max Gross limits.
 
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Cant seem to view the next posts I have made...

*edit

problem solved, cannot hyperlink images.
 
In an attempt to answer the questions arising, I'm hoping these screen shots of the spread sheet I'm working on will help...


http://tinypic.com/r/x425hk/8 <-- This page is used by the pilot to enter values such as crew weight, pilot weight, and other variables

http://tinypic.com/r/2zthagh/8 <-- This page is used to enter the weight and location in the A/C of non permanent equipment

http://tinypic.com/r/2d7vggh/8 <-- This page is a loading table used for quick reference in the field to determine at what fuel load can what weight of patient be carried. My goal is to insert a new row here to display the maximum patient weight that can be carried at a specified fuel load. The limiting factors for possible patient weight are C.G. limits and Max Gross limits.
 
RotorPro said:
In an attempt to answer the questions arising, I'm hoping these screen shots of the spread sheet I'm working on will help...


http://tinypic.com/r/x425hk/8 <-- This page is used by the pilot to enter values such as crew weight, pilot weight, and other variables

http://tinypic.com/r/2zthagh/8 <-- This page is used to enter the weight and location in the A/C of non permanent equipment

http://tinypic.com/r/2d7vggh/8 <-- This page is a loading table used for quick reference in the field to determine at what fuel load can what weight of patient be carried. My goal is to insert a new row here to display the maximum patient weight that can be carried at a specified fuel load. The limiting factors for possible patient weight are C.G. limits and Max Gross limits.
Click to expand...
Let
W = total weight of everything except fuel and passenger, assumed fixed for the day
S = sum of arm * weight for everything included in W, assumed fixed for the day.
FW = weight of Fuel
FA = arm of Fuel
PW = weight of passenger
PA = arm of passenger
y = new center of gravity

y = (S + FA * FW + PA * PW) / (W + FW + PW)

Part of your problem was that B changed with the change in fuel weight. Also you shouldn't have been using the old center of gravity as the arm for the new fuel weight.

Note: This assumes that the only balance needed is along the axis from which the arm is measured.
 
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