-
+
HW 8
due 11/13/2024 before midnight via Learning Suite
@@ -153,7 +153,7 @@
HW 8
-The fuel tanks and rocket body are made of aluminum. Note that a stacked configuration is used and that about 10 meters of extra length is required below the tanks for the engines. As shown on the right, it is easiest to estimate the required tank size by assuming that the tanks are cylinders and then add on about 3 meters per tank to account for the fact that the round end caps will need to be longer than a pure cylinder and that there will be some ullage volume (that means the tanks won’t be 100% filled). For simplicity, I’d treat the diameter of the tanks as the same as that of the rocket body (it’s a really small difference). Note that the cylindrical tanks are meant to have the same volume as the rounded tanks (see right-most figure). So for a fixed volume they will be shorter (which is why we add in the extra length), but the surface area should be roughly the same. In other words, the added 3m length per tank should only affect the total length calculation of the stage, it should not affect the structural mass calculation.
+The fuel tanks and rocket body are made of aluminum. Note that a stacked configuration is used and that about 10 meters of extra length is required below the tanks for the engines. As shown on the right, it is easiest to estimate the required tank size by assuming that the tanks are cylinders and then add on about 3 meters per tank to account for the fact that the round end caps will need to be longer than a pure cylinder and that there will be some ullage volume (that means the tanks won’t be 100% filled). For simplicity, I’d treat the diameter of the tanks as the same as that of the rocket body (it’s a really small difference).
Because the structural mass depends on the propellant mass, and the propellant mass depends on the structural mass, an iterative process is required (i.e., root finding). If you’re struggling to know if you’re on the right track, because this is an actual rocket, you could start with the known propellant mass and work your way through the equations checking your numbers against the actual rocket, then fine tune from there.
diff --git a/me415/schedule/hw9/index.html b/me415/schedule/hw9/index.html
index f720831..9589987 100644
--- a/me415/schedule/hw9/index.html
+++ b/me415/schedule/hw9/index.html
@@ -11,7 +11,7 @@
- HW 9 · ME 415
+ HW · ME 415
@@ -87,84 +87,56 @@
-
- HW 9: Rockets
-
-due 12/6/2017 before midnight via Learning Suite
-50 possible points
-
-All of these exercises will be done in class (though some of you may need a little extra time outside of class to finish up). I’d like to give you practice with rocket analyses, while still preserving your out of class time for finishing up your plane and report.
-
-
- -
-
Basic Sizing of a Rocket Engine. Some parameters for a liquid-fueled rocket engine are given below:
-
-
-
-
- \(\gamma\)
- ratio of specific heats for combustion gas mixture
- 1.2
-
-
- \(M_w\)
- molecular weight of combustion gas mixture
- 12 (g/mol)
-
-
- \(T_{Tc}\)
- combustion (total) temperature
- 3500 K
-
-
- \(P_{Tc}\)
- combustion (total) pressure
- 20 MPa
-
-
- \(I_{sp}\)
- required specific impulse at altitude
- 400 seconds
-
-
- \(T\)
- required thrust at altitude
- 2 MN
-
-
-
- nozzle type
- 80% bell nozzle
-
-
- \(\sigma_c\)
- max allowable stress in combustion chamber wall
- 55 MPa
-
-
-
-
- Design for peak efficiency at an altitude of 12.5 km (shortly after max-Q). The required thrust and specific impulse apply at this altitude.
-
- Determine/design the following:
-
-
- - the areas (3) at the end of the combustion chamber, throat, and exit.
- - the lengths (3) of the combustion chamber, the converging, and diverging sections of the nozzle.
- - the wall thickness (1) of the combustion chamber assuming a thin-shell cylindrical combustion chamber.
- - Draw your nozzle to scale.
-
-
- Note: These specs correspond to the Space Shuttle Main Engine so you can check some of your numbers (though don’t expect them to be exactly the same, there are important boundary layer losses and multiphase flow losses on a nozzle that we are neglecting).
-
- -
-
Basic Sizing/Performance of a Rocket. The worksheet is available online. I’ve already laid out the methodology so you don’t need to repeat it here. Just report the four critical values, and include a brief discussion on any lessons learned.
-
-
-
-
-
-
+
+ HW 9
+
+due 11/20/2024 before midnight via Learning Suite
+15 possible points
+
+
+
+This problem uses the same rocket from the previous homework. In this case we are interested in the flight trajectory. The Saturn V does not fly at a straight angle during the first stage and so it would be difficult to provide an accurate estimate using the closed-form rocket equation. Instead, we need to use numerical integration. Though we will still ignore drag in this analysis.
+
+As mentioned the heading angle changes significantly throughout the flight. I fit a curve to postflight trajectory data and computed the heading angle as a function of time during stage 1. Note that \(\theta = 0\) corresponds to vertical flight:
+
+
+
+\(\theta = p_1 \arctan\left(p_2 t^{p_3}\right)\)
+where
+\(p_1 = 0.866, p_2 = 2.665 \times 10^{-5}, p_3 = 2.378\)
+
+Other parameters you will need:
+
+
+
+
+ thrust
+ 35.1 MN
+
+
+ total rocket mass
+ \(2.97 \times 10^6\) kg
+
+
+ specific impulse
+ 283 s
+
+
+
+
+You could solve this using any ODE solver (e.g. scipy.integrate.solve_ivp in python), or you could write a basic forward Euler method. This means that you setup a time vector, and a starting point for \(V, m, z, x\), and then execute a for loop. At iteration (\(i\)) you update those four values using data from the previous iteration (\(i - 1\)). For example, using the last ODE (and setting \(\Delta t = t^{(i)} - t^{(i-1)}\)):
+
+\[x^{(i)} = x^{(i-1)} + V^{(i-1)}\sin\theta^{(i-1)} \Delta t\]
+
+Report the following. Be sure to clearly show your work and assumptions.
+
+
+
+
+ - Plot the trajectory with forward distance on the \(x\)-axis and altitude on the \(y\)-axis. (I tabulated the actual flight data if you are interested in comparing, first column is the forward distance, and the second column is the altitude).
+ - Your final velocity, altitude, and forward distance.
+
+
diff --git a/me415/schedule/index.html b/me415/schedule/index.html
index 06cc447..22295af 100644
--- a/me415/schedule/index.html
+++ b/me415/schedule/index.html
@@ -394,7 +394,7 @@ Schedule
W Nov 20
Rocket HW
- HW 9
+ HW 9
F Nov 22
diff --git a/me415/schedule/tanks.png b/me415/schedule/tanks.png
index b19b904..373d83b 100644
Binary files a/me415/schedule/tanks.png and b/me415/schedule/tanks.png differ
diff --git a/me415/schedule/traj.dat b/me415/schedule/traj.dat
new file mode 100644
index 0000000..e80be0a
--- /dev/null
+++ b/me415/schedule/traj.dat
@@ -0,0 +1,168 @@
+0 64
+0 65
+0 66
+0 67
+0 71
+0 77
+0 85
+1 96
+1 109
+1 125
+1 144
+2 164
+1 187
+1 213
+1 242
+1 274
+0 307
+0 345
+-1 386
+-1 429
+-2 475
+-2 524
+-2 576
+-2 633
+-2 692
+-1 755
+0 822
+2 892
+4 967
+6 1044
+10 1126
+14 1212
+18 1302
+24 1396
+31 1494
+38 1597
+47 1705
+57 1815
+69 1931
+82 2052
+97 2177
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+153 2582
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+4082 11493
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+4605 12226
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+5481 13373
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+6833 14988
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+8843 17156
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+12866 20994
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+18034 25288
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+31439 34553
+32528 35227
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+34787 36591
+35957 37292
+37155 37994
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+39638 39419
+40923 40141
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+47802 43860
+49267 44322
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+52271 46155
+53813 46928
+55381 47704
+56976 48484
+58599 49269
+60249 50055
+61927 50846
+63633 51641
+65368 52440
+67132 53242
+68915 54049
+70749 54859
+72602 55674
+74486 56492
+76402 57315
+78348 58141
+80327 58972
+82337 59808
+84381 60647
+86458 61491
+88568 62339
+90713 63192
+92903 64054
+94272 64588
+95111 64915
\ No newline at end of file
HW 8
due 11/13/2024 before midnight via Learning Suite @@ -153,7 +153,7 @@
HW 8
The fuel tanks and rocket body are made of aluminum. Note that a stacked configuration is used and that about 10 meters of extra length is required below the tanks for the engines. As shown on the right, it is easiest to estimate the required tank size by assuming that the tanks are cylinders and then add on about 3 meters per tank to account for the fact that the round end caps will need to be longer than a pure cylinder and that there will be some ullage volume (that means the tanks won’t be 100% filled). For simplicity, I’d treat the diameter of the tanks as the same as that of the rocket body (it’s a really small difference). Note that the cylindrical tanks are meant to have the same volume as the rounded tanks (see right-most figure). So for a fixed volume they will be shorter (which is why we add in the extra length), but the surface area should be roughly the same. In other words, the added 3m length per tank should only affect the total length calculation of the stage, it should not affect the structural mass calculation.
+The fuel tanks and rocket body are made of aluminum. Note that a stacked configuration is used and that about 10 meters of extra length is required below the tanks for the engines. As shown on the right, it is easiest to estimate the required tank size by assuming that the tanks are cylinders and then add on about 3 meters per tank to account for the fact that the round end caps will need to be longer than a pure cylinder and that there will be some ullage volume (that means the tanks won’t be 100% filled). For simplicity, I’d treat the diameter of the tanks as the same as that of the rocket body (it’s a really small difference).
Because the structural mass depends on the propellant mass, and the propellant mass depends on the structural mass, an iterative process is required (i.e., root finding). If you’re struggling to know if you’re on the right track, because this is an actual rocket, you could start with the known propellant mass and work your way through the equations checking your numbers against the actual rocket, then fine tune from there.
diff --git a/me415/schedule/hw9/index.html b/me415/schedule/hw9/index.html index f720831..9589987 100644 --- a/me415/schedule/hw9/index.html +++ b/me415/schedule/hw9/index.html @@ -11,7 +11,7 @@
-
- HW 9: Rockets
-
-due 12/6/2017 before midnight via Learning Suite
-50 possible points
-
-All of these exercises will be done in class (though some of you may need a little extra time outside of class to finish up). I’d like to give you practice with rocket analyses, while still preserving your out of class time for finishing up your plane and report.
-
-
- -
-
Basic Sizing of a Rocket Engine. Some parameters for a liquid-fueled rocket engine are given below:
-
-
-
-
- \(\gamma\)
- ratio of specific heats for combustion gas mixture
- 1.2
-
-
- \(M_w\)
- molecular weight of combustion gas mixture
- 12 (g/mol)
-
-
- \(T_{Tc}\)
- combustion (total) temperature
- 3500 K
-
-
- \(P_{Tc}\)
- combustion (total) pressure
- 20 MPa
-
-
- \(I_{sp}\)
- required specific impulse at altitude
- 400 seconds
-
-
- \(T\)
- required thrust at altitude
- 2 MN
-
-
-
- nozzle type
- 80% bell nozzle
-
-
- \(\sigma_c\)
- max allowable stress in combustion chamber wall
- 55 MPa
-
-
-
-
- Design for peak efficiency at an altitude of 12.5 km (shortly after max-Q). The required thrust and specific impulse apply at this altitude.
-
- Determine/design the following:
-
-
- - the areas (3) at the end of the combustion chamber, throat, and exit.
- - the lengths (3) of the combustion chamber, the converging, and diverging sections of the nozzle.
- - the wall thickness (1) of the combustion chamber assuming a thin-shell cylindrical combustion chamber.
- - Draw your nozzle to scale.
-
-
- Note: These specs correspond to the Space Shuttle Main Engine so you can check some of your numbers (though don’t expect them to be exactly the same, there are important boundary layer losses and multiphase flow losses on a nozzle that we are neglecting).
-
- -
-
Basic Sizing/Performance of a Rocket. The worksheet is available online. I’ve already laid out the methodology so you don’t need to repeat it here. Just report the four critical values, and include a brief discussion on any lessons learned.
-
-
-
-
-
-
+
+ HW 9
+
+due 11/20/2024 before midnight via Learning Suite
+15 possible points
+
+
+
+This problem uses the same rocket from the previous homework. In this case we are interested in the flight trajectory. The Saturn V does not fly at a straight angle during the first stage and so it would be difficult to provide an accurate estimate using the closed-form rocket equation. Instead, we need to use numerical integration. Though we will still ignore drag in this analysis.
+
+As mentioned the heading angle changes significantly throughout the flight. I fit a curve to postflight trajectory data and computed the heading angle as a function of time during stage 1. Note that \(\theta = 0\) corresponds to vertical flight:
+
+
+
+\(\theta = p_1 \arctan\left(p_2 t^{p_3}\right)\)
+where
+\(p_1 = 0.866, p_2 = 2.665 \times 10^{-5}, p_3 = 2.378\)
+
+Other parameters you will need:
+
+
+
+
+ thrust
+ 35.1 MN
+
+
+ total rocket mass
+ \(2.97 \times 10^6\) kg
+
+
+ specific impulse
+ 283 s
+
+
+
+
+You could solve this using any ODE solver (e.g. scipy.integrate.solve_ivp in python), or you could write a basic forward Euler method. This means that you setup a time vector, and a starting point for \(V, m, z, x\), and then execute a for loop. At iteration (\(i\)) you update those four values using data from the previous iteration (\(i - 1\)). For example, using the last ODE (and setting \(\Delta t = t^{(i)} - t^{(i-1)}\)):
+
+\[x^{(i)} = x^{(i-1)} + V^{(i-1)}\sin\theta^{(i-1)} \Delta t\]
+
+Report the following. Be sure to clearly show your work and assumptions.
+
+
+
+
+ - Plot the trajectory with forward distance on the \(x\)-axis and altitude on the \(y\)-axis. (I tabulated the actual flight data if you are interested in comparing, first column is the forward distance, and the second column is the altitude).
+ - Your final velocity, altitude, and forward distance.
+
+
diff --git a/me415/schedule/index.html b/me415/schedule/index.html
index 06cc447..22295af 100644
--- a/me415/schedule/index.html
+++ b/me415/schedule/index.html
@@ -394,7 +394,7 @@ Schedule
W Nov 20
Rocket HW
- HW 9
+ HW 9
F Nov 22
diff --git a/me415/schedule/tanks.png b/me415/schedule/tanks.png
index b19b904..373d83b 100644
Binary files a/me415/schedule/tanks.png and b/me415/schedule/tanks.png differ
diff --git a/me415/schedule/traj.dat b/me415/schedule/traj.dat
new file mode 100644
index 0000000..e80be0a
--- /dev/null
+++ b/me415/schedule/traj.dat
@@ -0,0 +1,168 @@
+0 64
+0 65
+0 66
+0 67
+0 71
+0 77
+0 85
+1 96
+1 109
+1 125
+1 144
+2 164
+1 187
+1 213
+1 242
+1 274
+0 307
+0 345
+-1 386
+-1 429
+-2 475
+-2 524
+-2 576
+-2 633
+-2 692
+-1 755
+0 822
+2 892
+4 967
+6 1044
+10 1126
+14 1212
+18 1302
+24 1396
+31 1494
+38 1597
+47 1705
+57 1815
+69 1931
+82 2052
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+153 2582
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+202 2878
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+261 3194
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+333 3532
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+418 3892
+466 4081
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+575 4475
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+772 5111
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+30377 33886
+31439 34553
+32528 35227
+33644 35909
+34787 36591
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+37155 37994
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+39638 39419
+40923 40141
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+46365 43103
+46591 43224
+47802 43860
+49267 44322
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+52271 46155
+53813 46928
+55381 47704
+56976 48484
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+60249 50055
+61927 50846
+63633 51641
+65368 52440
+67132 53242
+68915 54049
+70749 54859
+72602 55674
+74486 56492
+76402 57315
+78348 58141
+80327 58972
+82337 59808
+84381 60647
+86458 61491
+88568 62339
+90713 63192
+92903 64054
+94272 64588
+95111 64915
\ No newline at end of file
HW 9: Rockets
- -due 12/6/2017 before midnight via Learning Suite -50 possible points
- -All of these exercises will be done in class (though some of you may need a little extra time outside of class to finish up). I’d like to give you practice with rocket analyses, while still preserving your out of class time for finishing up your plane and report.
- --
-
-
-
Basic Sizing of a Rocket Engine. Some parameters for a liquid-fueled rocket engine are given below:
- -- -
- -- -\(\gamma\) -ratio of specific heats for combustion gas mixture -1.2 -- -\(M_w\) -molecular weight of combustion gas mixture -12 (g/mol) -- -\(T_{Tc}\) -combustion (total) temperature -3500 K -- -\(P_{Tc}\) -combustion (total) pressure -20 MPa -- -\(I_{sp}\) -required specific impulse at altitude -400 seconds -- -\(T\) -required thrust at altitude -2 MN -- -nozzle type -80% bell nozzle -- - -\(\sigma_c\) -max allowable stress in combustion chamber wall -55 MPa -Design for peak efficiency at an altitude of 12.5 km (shortly after max-Q). The required thrust and specific impulse apply at this altitude.
- -Determine/design the following:
- --
-
- the areas (3) at the end of the combustion chamber, throat, and exit. -
- the lengths (3) of the combustion chamber, the converging, and diverging sections of the nozzle. -
- the wall thickness (1) of the combustion chamber assuming a thin-shell cylindrical combustion chamber. -
- Draw your nozzle to scale. -
Note: These specs correspond to the Space Shuttle Main Engine so you can check some of your numbers (though don’t expect them to be exactly the same, there are important boundary layer losses and multiphase flow losses on a nozzle that we are neglecting).
-
- -
-
Basic Sizing/Performance of a Rocket. The worksheet is available online. I’ve already laid out the methodology so you don’t need to repeat it here. Just report the four critical values, and include a brief discussion on any lessons learned.
-
-
HW 9
+ +due 11/20/2024 before midnight via Learning Suite +15 possible points
+ ++ +
This problem uses the same rocket from the previous homework. In this case we are interested in the flight trajectory. The Saturn V does not fly at a straight angle during the first stage and so it would be difficult to provide an accurate estimate using the closed-form rocket equation. Instead, we need to use numerical integration. Though we will still ignore drag in this analysis.
+ +As mentioned the heading angle changes significantly throughout the flight. I fit a curve to postflight trajectory data and computed the heading angle as a function of time during stage 1. Note that \(\theta = 0\) corresponds to vertical flight:
+ + + +\(\theta = p_1 \arctan\left(p_2 t^{p_3}\right)\) +where +\(p_1 = 0.866, p_2 = 2.665 \times 10^{-5}, p_3 = 2.378\)
+ +Other parameters you will need:
+ +| thrust | +35.1 MN | +
| total rocket mass | +\(2.97 \times 10^6\) kg | +
| specific impulse | +283 s | +
You could solve this using any ODE solver (e.g. scipy.integrate.solve_ivp in python), or you could write a basic forward Euler method. This means that you setup a time vector, and a starting point for \(V, m, z, x\), and then execute a for loop. At iteration (\(i\)) you update those four values using data from the previous iteration (\(i - 1\)). For example, using the last ODE (and setting \(\Delta t = t^{(i)} - t^{(i-1)}\)):
Report the following. Be sure to clearly show your work and assumptions.
+ + + +-
+
- Plot the trajectory with forward distance on the \(x\)-axis and altitude on the \(y\)-axis. (I tabulated the actual flight data if you are interested in comparing, first column is the forward distance, and the second column is the altitude). +
- Your final velocity, altitude, and forward distance. +