船舶原理1.1 Simpson's Rule
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• A curve that is irregular and not easily defined mathematically
It is an APPROXIMATION of the true integration
Given an integral in the following form:
Simpson’s 1st Rule is the one we use here since it gives an EVEN number of divisions
Here’s how it’s put to use in this course:
Waterplane Area, Awp Awp = 2 y(x) dx
Simpson’s 2nd Rule
Area = 3/8 ∆x [yo + 3y1 + 3y2 + 2y3 + 3y4 +3y5 + 2y6 +… + 3y n-1 + yn]
where: - n is an EVEN number of stations - Repeats in a pattern 1,3,3,2,3,3,2,3,3,2,……2,3,3,1
Vsubmerged =
Asect(x) dx
Vsub = 1/3 ∆x [Ao + 4A1 + 2A2+…2A n-2 + 4A n-1 + An]
Vertical Center of Buoyancy, KB
This is similar to the LCF in that it is a CENTROID, but where LCF is the centroid of the Awp, KB is the centroid of the submerged volume of the ship measured from the keel… z y Awp
LCF =
x δA/Awp
2/Awp
x y(x)δx δ
*see below!
Substituting into Simpson's Eq., you’ll get the following: LCF = 2/Awp x 1/3 ∆x [(1) (xo) (yo) + 4 (x1) (y1) + 2 (x2) (y2) +… + (xn) (yn) ] Note that the blue terms are what we have to add to make Simpson work for LCF. Remember to include them in your calculations!
Awp = 2 x 1/3 ∆x [yo + 4y1 + 2y2+…2y n-2 + 4y n-1 + yn]
The “2” is needed because the data you’ll have is for a half-section… Section Area, Asect Asect= 2 y(z) dz
Longitudinal Center of Flotation, LCF, (cont’d)
- Since our sectional areas are done in half-sections this needs to be multiplied by 2 - Remember, δA = y(x)δx, so we can substitute for δA - Awp is constant, so it moves left δA
And FINALLY,… Longitudinal Center of Buoyancy, LCB
Simpson’s Rule
Simpson’s Rule is used when a standard integration technique is too involved or not easily performed. • A curve that is not defined mathematically
where: - n is an ODD number of stations - ∆x is the distance between stations - yn is the value of y at the given station along x - Repeats in a pattern of 1,4,2,4,2,4,2……2,4,1
Asection and Awp are examples of how Simpson’s rule is used to find area… … The next slides show how it can be used to find the First Moment of Area, that is, finding the centroid of a given area. The only difference in the procedure is the addition of another term, the distance of the individual area segments from the y-axis…the value of x. The values of x will be the progressive sum of the ∆x… if ∆x is the width of the sections, say10, then x0=0, x1=10, x2=20,x3=30… and so on.
y y = f(x)
∆x
…and summing the individual areas.
y
y = f(x)
x
Notice that:
∆x
• The curve area is the same • Spacing is constant along x (the δx in the integral is the ∆x here) • The value of y (the height) depends on the location on x (y is a function of x, aka y= f(x)) • The area of the series of “rectangles” is more easily summed up
- ∆x here is 81.6 ft - Awp would be given - “2” because you’re dealing with a half-breadth section
Remember, this gives only part of the equation! You still need the “2/Awp x 1/3 ∆x” part!
y
δA
y(x)
x FP
∆x
AP
LCF =
x δA/Awp
That is, LCF is the sum of all the areas, δA, and their distances from the y-axis, divided by the total area of the water plane…
Asect = 2 x 1/3 ∆x [yo + 4y1 + 2y2+…2y n-2 + 4y n-1 + yn]
Note: You will always know the value of y for the stations (x or z)! It will be presented in the Table of Offsets or readily measured…
Longitudinal Center of Flotation, LCF
-This is the CENTROID of the Awp of the ship, or the 1st Moment of Area of the Awp. -For this reason you now need to introduce the distance, x, of the section ∆x from the y-axis
*This is actually a “moment balance” equation! Awp LCF = Σ (x δA) See the .ppt presentation for further discussion!
It’s often easier to put all the information in tabular form on an Excel spreadsheet:
Simpson’s Rule breaks the curve into these sections and then sums them up for total area under the curve
Simpson’s 1st Rule Area = 1/3 ∆x [yo + 4y1 + 2y2+…2y n-2 + 4y n-1 + yn]
We can now move onto the next dimension, VOLUMES!
Volume, Submerged, Vsubmerged
- It gets a little trickier here… remember, since you are now dealing with a VOLUME, the y term previous now becomes an AREA term for that station section because you are summing the areas:
Station Dist from FP (x value) 0 1 2 3 4 0.0 81.6 163.2 244.8 326.4 HalfBreadth (y value) 0.39 12.92 20.97 21.71 12.58 Moment xy 0.0 1054.3 3422.3 5314.6 4106.1 Simpson Multiplier 1 4 2 4 1 Product of Moment x Multiplier 0.0 4217.1 6844.6 21258.4 4106.1 36426.2
You can now put this into Simpson’s Equation:
Байду номын сангаас
KB =
zAwp(z) δz
KB =Awp x 1/3 ∆z [(1) (zo) (Awpo) + 4 (z1) (Awp1) + 2 (z2) (Awp2) +… + (zn) (Awpn) ]
Remember that the blue terms are what we have to add to make Simpson work for KB. Don’t forget to include them in your calculations!
y y = f(x)
y(x) dx
x
Where y is a function of x, that is, y is the dependent variable defined by x, the integral can be approximated by dividing the area under the curve into equally spaced sections, ∆x, …
KB
x
KB =
zAwp(z) δz
where: - Awp is the area of the waterplane at the distance z from the keel - z is the distance of the Awp section from the x-axis - δz is the spacing between the Awp sections, or ∆z in Simpson’s Eq.
It is an APPROXIMATION of the true integration
Given an integral in the following form:
Simpson’s 1st Rule is the one we use here since it gives an EVEN number of divisions
Here’s how it’s put to use in this course:
Waterplane Area, Awp Awp = 2 y(x) dx
Simpson’s 2nd Rule
Area = 3/8 ∆x [yo + 3y1 + 3y2 + 2y3 + 3y4 +3y5 + 2y6 +… + 3y n-1 + yn]
where: - n is an EVEN number of stations - Repeats in a pattern 1,3,3,2,3,3,2,3,3,2,……2,3,3,1
Vsubmerged =
Asect(x) dx
Vsub = 1/3 ∆x [Ao + 4A1 + 2A2+…2A n-2 + 4A n-1 + An]
Vertical Center of Buoyancy, KB
This is similar to the LCF in that it is a CENTROID, but where LCF is the centroid of the Awp, KB is the centroid of the submerged volume of the ship measured from the keel… z y Awp
LCF =
x δA/Awp
2/Awp
x y(x)δx δ
*see below!
Substituting into Simpson's Eq., you’ll get the following: LCF = 2/Awp x 1/3 ∆x [(1) (xo) (yo) + 4 (x1) (y1) + 2 (x2) (y2) +… + (xn) (yn) ] Note that the blue terms are what we have to add to make Simpson work for LCF. Remember to include them in your calculations!
Awp = 2 x 1/3 ∆x [yo + 4y1 + 2y2+…2y n-2 + 4y n-1 + yn]
The “2” is needed because the data you’ll have is for a half-section… Section Area, Asect Asect= 2 y(z) dz
Longitudinal Center of Flotation, LCF, (cont’d)
- Since our sectional areas are done in half-sections this needs to be multiplied by 2 - Remember, δA = y(x)δx, so we can substitute for δA - Awp is constant, so it moves left δA
And FINALLY,… Longitudinal Center of Buoyancy, LCB
Simpson’s Rule
Simpson’s Rule is used when a standard integration technique is too involved or not easily performed. • A curve that is not defined mathematically
where: - n is an ODD number of stations - ∆x is the distance between stations - yn is the value of y at the given station along x - Repeats in a pattern of 1,4,2,4,2,4,2……2,4,1
Asection and Awp are examples of how Simpson’s rule is used to find area… … The next slides show how it can be used to find the First Moment of Area, that is, finding the centroid of a given area. The only difference in the procedure is the addition of another term, the distance of the individual area segments from the y-axis…the value of x. The values of x will be the progressive sum of the ∆x… if ∆x is the width of the sections, say10, then x0=0, x1=10, x2=20,x3=30… and so on.
y y = f(x)
∆x
…and summing the individual areas.
y
y = f(x)
x
Notice that:
∆x
• The curve area is the same • Spacing is constant along x (the δx in the integral is the ∆x here) • The value of y (the height) depends on the location on x (y is a function of x, aka y= f(x)) • The area of the series of “rectangles” is more easily summed up
- ∆x here is 81.6 ft - Awp would be given - “2” because you’re dealing with a half-breadth section
Remember, this gives only part of the equation! You still need the “2/Awp x 1/3 ∆x” part!
y
δA
y(x)
x FP
∆x
AP
LCF =
x δA/Awp
That is, LCF is the sum of all the areas, δA, and their distances from the y-axis, divided by the total area of the water plane…
Asect = 2 x 1/3 ∆x [yo + 4y1 + 2y2+…2y n-2 + 4y n-1 + yn]
Note: You will always know the value of y for the stations (x or z)! It will be presented in the Table of Offsets or readily measured…
Longitudinal Center of Flotation, LCF
-This is the CENTROID of the Awp of the ship, or the 1st Moment of Area of the Awp. -For this reason you now need to introduce the distance, x, of the section ∆x from the y-axis
*This is actually a “moment balance” equation! Awp LCF = Σ (x δA) See the .ppt presentation for further discussion!
It’s often easier to put all the information in tabular form on an Excel spreadsheet:
Simpson’s Rule breaks the curve into these sections and then sums them up for total area under the curve
Simpson’s 1st Rule Area = 1/3 ∆x [yo + 4y1 + 2y2+…2y n-2 + 4y n-1 + yn]
We can now move onto the next dimension, VOLUMES!
Volume, Submerged, Vsubmerged
- It gets a little trickier here… remember, since you are now dealing with a VOLUME, the y term previous now becomes an AREA term for that station section because you are summing the areas:
Station Dist from FP (x value) 0 1 2 3 4 0.0 81.6 163.2 244.8 326.4 HalfBreadth (y value) 0.39 12.92 20.97 21.71 12.58 Moment xy 0.0 1054.3 3422.3 5314.6 4106.1 Simpson Multiplier 1 4 2 4 1 Product of Moment x Multiplier 0.0 4217.1 6844.6 21258.4 4106.1 36426.2
You can now put this into Simpson’s Equation:
Байду номын сангаас
KB =
zAwp(z) δz
KB =Awp x 1/3 ∆z [(1) (zo) (Awpo) + 4 (z1) (Awp1) + 2 (z2) (Awp2) +… + (zn) (Awpn) ]
Remember that the blue terms are what we have to add to make Simpson work for KB. Don’t forget to include them in your calculations!
y y = f(x)
y(x) dx
x
Where y is a function of x, that is, y is the dependent variable defined by x, the integral can be approximated by dividing the area under the curve into equally spaced sections, ∆x, …
KB
x
KB =
zAwp(z) δz
where: - Awp is the area of the waterplane at the distance z from the keel - z is the distance of the Awp section from the x-axis - δz is the spacing between the Awp sections, or ∆z in Simpson’s Eq.