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Numerical Methods in Excel

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Numerical Methods in Excel. In flight, the thrust produced by an aircraft's ... Alternately, the power available from the propulsion system is used to ... – PowerPoint PPT presentation

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Title: Numerical Methods in Excel


1
Numerical Methods in Excel
2
  • In flight, the thrust produced by an aircrafts
    propulsion system is used to overcome the
    aerodynamic drag.
  • Alternately, the power available from the
    propulsion system is used to compensate for the
    power required to overcome the aerodynamic drag.
  • If the power available is greater than the power
    required, the excess power can be used to
  • accelerate the aircraft to a higher speed
  • cause the aircraft to climb or increase its
    potential energy.

3
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4
  • This chart illustrates the variation of power
    required and power available with respect to
    aircraft speed for a Boeing 747.
  • At speeds where the power available is greater
    than the power required, the excess power can be
    used to accelerate the aircraft.
  • The speed at which the power available is equal
    to the power required is the maximum level flight
    speed for the aircraft.

5
  • The excess power can also be used to cause the
    aircraft to climb or to increase the aircrafts
    potential energy.
  • There is a particular speed at which the excess
    power and the rate of climb are a maximum.
  • Here, the maximum flight speed is indicated by
    the speed at which the rate of climb is zero,
    i.e., when the excess power is zero.
  • We will use simple numerical methods to find the
    maximum flight speed, maximum rate of climb, and
    the speed to fly for maximum rate of climb.

6
  • double click on the worksheet tab and rename the
    worksheet to calculations
  • create the table of calculation constants
  • create the table for calculation results with the
    formatting shown

7
  • enter the formula to calculate the Power Required
  • note that the calculated power is divided by 106
    so that the results are shown in MW
  • format the calculated results as shown

8
  • enter the formula to calculate the Power
    Available
  • note that the calculated power is divided by 106
    so that the results are shown in MW
  • format the calculated results as shown

9
  • enter the formula to calculate the Climb Velocity
    or Rate of Climb
  • note that the excess power is multiplied by 106
    to convert to Nm or Watts.
  • format the calculated results as shown

10
  • Plot the Power Required and Power Available
    versus Flight Speed

11
  • format the plot as shown

12
  • right-click in the chart area and select Move
    Chart from the pop-up menu

13
  • place the plot on a New Sheet named power

14
  • rearrange the plot elements to get the appearance
    shown

15
  • plot the Rate of Climb and move it to its own
    worksheet

16
  • note that the speed at which the Rate of Climb is
    zero is also the speed at which the Excess Power
    is zero
  • this is the Maximum Level Flight Speed
  • we will locate the maximum speed by finding the
    speed at which the Rate of Climb is zero

17
  • copy the table headings and first row of
    calculations to create a new table as shown
  • label the table Maximum Speed

18
  • in the Data Tools group on the Data tab, select
    Goal Seek

19
  • the Goal Seek tool will search for values of
    Flight Speed (cell G10) such that the Climb
    Speed (cell J10) is zero
  • we are using Goal Seek as a root finding tool,
    i.e., we want to find the value of V such that
    Vc(V) 0

20
  • the result is that the Climb Speed is zero when
    the flight speed is 67.40206 m/s
  • is this right?

21
  • the speed that Goal Seek found is the minimum
    speed rather than the maximum speed
  • this is a result of starting cell G10 with a
    value of 30 m/s
  • the closest solution was the minimum speed

22
  • set the initial value of the flight speed to 200
    m/s
  • restart the Goal Seek tool

23
  • now we see that the maximum level flight speed is
    310.0789 m/s

24
  • here, we can verify that we have found the
    correct root to Vc(V) 0
  • now we would like to find the maximum rate of
    climb and the speed at which it occurs
  • note that the rate of climb is a maximum when the
    slope of Vc(V) is zero

25
  • the slope of the Vc(V) curve can be written as
  • we can approximate the slope at a speed of 90 m/s
    with
  • recall that the slope is equivalent to the
    derivative of the function which is defined as
  • we are using a central difference approximation
    to the derivative at 90 m/s
  • implement this approximation as a cell formula
    and replicate it for all but the first and last
    values of velocity

26
  • plot the derivative or slope versus the flight
    speed
  • move the plot to a worksheet titled dVcdV

27
  • recall that we are looking for the speed for
    maximum rate of climb which will be the speed at
    which the slope is equal to zero
  • from the plot, the speed for maximum rate of
    climb is approximately 190 m/s
  • this is another root finding problem

28
  • create another table by copying the table
    headings and 3 rows from the original table
  • we are using a central difference approximation
    to find the derivative at the middle velocity
    value
  • make the derivative approximation more accurate
    by using a smaller DV
  • enter formulas for the velocities above and below
    the center value such that they are V 1 m/s

29
  • use the Goal Seek tool to find the value of V
    such that dVc/dV 0

30
  • the maximum rate of climb is 16.36 m/s at a
    flight speed of 193.4531 m/s

31
  • verify that these results appear to be correct
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