In [14]:
%matplotlib notebook

import pandas as pd
import numpy as np
import matplotlib

from matplotlib import pyplot as plt
import seaborn as sns
Out[14]:
<matplotlib.legend.Legend at 0x7f64ec77fb00>
In [50]:
from IPython.display import Math as latexmath
from IPython.display import IFrame
  • 6 Tip mounted Turbojets + 1 turboshaft in tail to power tail rotor
  • 3 rotors with diameter of 120m
  • Empty weight 200,000kg
  • Gross weight 453,000kg
  • 60 rpm (according to Jay Spensers book Vertical Challenge. The Hiller Aircraft Story from Jay P. Spenser. ISBN 0-295-97203-3 and also the Air Tug video)
  • Circumference = 383.02m so tips are 383m/s sea level speed of sound is 344 m/s Tips are supersonic...
$$ Takeoff\; Gross\; Weight = W_0 \\ Blade\; Position(Azimuth\; Angle)\; = \psi \\ Rotor\; Tip\; Speed\; (Hovering\; Flight)\; = V_{tip(HF)}\\ Rotor\; Tip\; Speed\; (Forward\; Flight)\; = V_{tip(FF)}\\ Freestream\; Air\; Speed\; = V_{\infty}\\ Rotor\; Rotation\; Frequency\; = \Omega\\ Rotor\; Radius\; = R_R\\ Rotor\; Hub\; Radius\; = R_H\\ Rotor\; Thrust\; = T\\ Rotor\; Thrust\; Coefficient\; = C_T\\ Rotor\; Disk\; Area\; = A\\ Air\; Density\; = \rho\\ Rotor\; Disk\; Power\; Loading\; = PL\\ Rotor\; Disk\; Loading\; = DL\\ $$
In [87]:
latexmath('V_{tip(HF)} = R_R\Omega')
Out[87]:
$$V_{tip(HF)} = R_R\Omega$$
In [ ]:
def hover_flight_tip_speed(rotor_radius, rotor_frequency):
    return rotor_radius*rotor_frequency
In [86]:
latexmath('V_{tip(FF)} = R_R\Omega + V_{\infty} \sin \psi')
Out[86]:
$$V_{tip(FF)} = R_R\Omega + V_{\infty} \sin \psi$$
In [89]:
def forward_flight_tip_speed(freestream_airspeed,blade_azimuth_angle,rotor_radius, rotor_frequency):
    return (rotor_radius*rotor_frequency)+(freestream_airspeed*np.sin(blade_azimuth_angle))
In [85]:
latexmath('A = \pi R_R^2')
Out[85]:
$$A = \pi R_R^2$$
In [ ]:
def rotor_disk_area(rotor_radius):
    return np.pi*rotor_radius**2
In [84]:
latexmath(r'C_T = \frac{T}{A R_R^2 \Omega^2 \rho}')
Out[84]:
$$C_T = \frac{T}{A R_R^2 \Omega^2 \rho}$$
In [ ]:
def rotor_thrust_coefficent(rotor_thrust,rotor_area, rotor_radius, rotor_frequency, air_density):
    return rotor_thrust/(air_density*rotor_area*(rotor_frequency**2)*(rotor_radius**2))
In [58]:
IFrame('https://upload.wikimedia.org/wikipedia/commons/e/e5/VTOL_DiscLoad-LiftEfficiency.svg',width=450,height=450)
Out[58]:
In [96]:
airspeed = np.linspace(0, 150, 151)
rotor_frequency = 1
rotor_radius = 60
blade_azimuth_angles = np.linspace(0, 2*np.pi, 20)
tip_speed_matrix = [forward_flight_tip_speed(a ,blade_azimuth_angles,rotor_radius, rotor_frequency) for a in airspeed]



print(tip_speed_matrix)
[array([ 60.,  60.,  60.,  60.,  60.,  60.,  60.,  60.,  60.,  60.,  60.,
        60.,  60.,  60.,  60.,  60.,  60.,  60.,  60.,  60.]), array([ 60.        ,  60.32469947,  60.61421271,  60.83716648,
        60.96940027,  60.99658449,  60.91577333,  60.73572391,
        60.47594739,  60.16459459,  59.83540541,  59.52405261,
        59.26427609,  59.08422667,  59.00341551,  59.03059973,
        59.16283352,  59.38578729,  59.67530053,  60.        ]), array([ 60.        ,  60.64939894,  61.22842543,  61.67433296,
        61.93880053,  61.99316899,  61.83154665,  61.47144782,
        60.95189479,  60.32918918,  59.67081082,  59.04810521,
        58.52855218,  58.16845335,  58.00683101,  58.06119947,
        58.32566704,  58.77157457,  59.35060106,  60.        ]), array([ 60.        ,  60.97409841,  61.84263814,  62.51149943,
        62.9082008 ,  62.98975348,  62.74731998,  62.20717173,
        61.42784218,  60.49378377,  59.50621623,  58.57215782,
        57.79282827,  57.25268002,  57.01024652,  57.0917992 ,
        57.48850057,  58.15736186,  59.02590159,  60.        ]), array([ 60.        ,  61.29879788,  62.45685085,  63.34866591,
        63.87760106,  63.98633797,  63.66309331,  62.94289564,
        61.90378957,  60.65837836,  59.34162164,  58.09621043,
        57.05710436,  56.33690669,  56.01366203,  56.12239894,
        56.65133409,  57.54314915,  58.70120212,  60.        ]), array([ 60.        ,  61.62349735,  63.07106356,  64.18583239,
        64.84700133,  64.98292247,  64.57886663,  63.67861955,
        62.37973697,  60.82297295,  59.17702705,  57.62026303,
        56.32138045,  55.42113337,  55.01707753,  55.15299867,
        55.81416761,  56.92893644,  58.37650265,  60.        ]), array([ 60.        ,  61.94819682,  63.68527628,  65.02299887,
        65.8164016 ,  65.97950696,  65.49463996,  64.41434346,
        62.85568436,  60.98756754,  59.01243246,  57.14431564,
        55.58565654,  54.50536004,  54.02049304,  54.1835984 ,
        54.97700113,  56.31472372,  58.05180318,  60.        ]), array([ 60.        ,  62.27289628,  64.29948899,  65.86016535,
        66.78580186,  66.97609145,  66.41041329,  65.15006737,
        63.33163175,  61.15216213,  58.84783787,  56.66836825,
        54.84993263,  53.58958671,  53.02390855,  53.21419814,
        54.13983465,  55.70051101,  57.72710372,  60.        ]), array([ 60.        ,  62.59759575,  64.9137017 ,  66.69733183,
        67.75520213,  67.97267594,  67.32618661,  65.88579129,
        63.80757914,  61.31675672,  58.68324328,  56.19242086,
        54.11420871,  52.67381339,  52.02732406,  52.24479787,
        53.30266817,  55.0862983 ,  57.40240425,  60.        ]), array([ 60.        ,  62.92229522,  65.52791441,  67.5344983 ,
        68.72460239,  68.96926044,  68.24195994,  66.6215152 ,
        64.28352654,  61.48135131,  58.51864869,  55.71647346,
        53.3784848 ,  51.75804006,  51.03073956,  51.27539761,
        52.4655017 ,  54.47208559,  57.07770478,  60.        ]), array([ 60.        ,  63.24699469,  66.14212713,  68.37166478,
        69.69400266,  69.96584493,  69.15773327,  67.35723911,
        64.75947393,  61.6459459 ,  58.3540541 ,  55.24052607,
        52.64276089,  50.84226673,  50.03415507,  50.30599734,
        51.62833522,  53.85787287,  56.75300531,  60.        ]), array([ 60.        ,  63.57169416,  66.75633984,  69.20883126,
        70.66340293,  70.96242942,  70.07350659,  68.09296302,
        65.23542132,  61.81054049,  58.18945951,  54.76457868,
        51.90703698,  49.92649341,  49.03757058,  49.33659707,
        50.79116874,  53.24366016,  56.42830584,  60.        ]), array([ 60.        ,  63.89639363,  67.37055255,  70.04599774,
        71.63280319,  71.95901392,  70.98927992,  68.82868693,
        65.71136872,  61.97513508,  58.02486492,  54.28863128,
        51.17131307,  49.01072008,  48.04098608,  48.36719681,
        49.95400226,  52.62944745,  56.10360637,  60.        ]), array([ 60.        ,  64.2210931 ,  67.98476526,  70.88316422,
        72.60220346,  72.95559841,  71.90505325,  69.56441084,
        66.18731611,  62.13972967,  57.86027033,  53.81268389,
        50.43558916,  48.09494675,  47.04440159,  47.39779654,
        49.11683578,  52.01523474,  55.7789069 ,  60.        ]), array([ 60.        ,  64.54579257,  68.59897798,  71.7203307 ,
        73.57160372,  73.9521829 ,  72.82082657,  70.30013475,
        66.6632635 ,  62.30432426,  57.69567574,  53.3367365 ,
        49.69986525,  47.17917343,  46.0478171 ,  46.42839628,
        48.2796693 ,  51.40102202,  55.45420743,  60.        ]), array([ 60.        ,  64.87049204,  69.21319069,  72.55749717,
        74.54100399,  74.9487674 ,  73.7365999 ,  71.03585866,
        67.1392109 ,  62.46891885,  57.53108115,  52.8607891 ,
        48.96414134,  46.2634001 ,  45.0512326 ,  45.45899601,
        47.44250283,  50.78680931,  55.12950796,  60.        ]), array([ 60.        ,  65.19519151,  69.8274034 ,  73.39466365,
        75.51040426,  75.94535189,  74.65237323,  71.77158257,
        67.61515829,  62.63351344,  57.36648656,  52.38484171,
        48.22841743,  45.34762677,  44.05464811,  44.48959574,
        46.60533635,  50.1725966 ,  54.80480849,  60.        ]), array([ 60.        ,  65.51989098,  70.44161612,  74.23183013,
        76.47980452,  76.94193638,  75.56814655,  72.50730648,
        68.09110568,  62.79810803,  57.20189197,  51.90889432,
        47.49269352,  44.43185345,  43.05806362,  43.52019548,
        45.76816987,  49.55838388,  54.48010902,  60.        ]), array([ 60.        ,  65.84459045,  71.05582883,  75.06899661,
        77.44920479,  77.93852087,  76.48391988,  73.24303039,
        68.56705307,  62.96270263,  57.03729737,  51.43294693,
        46.75696961,  43.51608012,  42.06147913,  42.55079521,
        44.93100339,  48.94417117,  54.15540955,  60.        ]), array([ 60.        ,  66.16928991,  71.67004154,  75.90616309,
        78.41860505,  78.93510537,  77.39969321,  73.9787543 ,
        69.04300047,  63.12729722,  56.87270278,  50.95699953,
        46.0212457 ,  42.60030679,  41.06489463,  41.58139495,
        44.09383691,  48.32995846,  53.83071009,  60.        ]), array([ 60.        ,  66.49398938,  72.28425425,  76.74332957,
        79.38800532,  79.93168986,  78.31546653,  74.71447821,
        69.51894786,  63.29189181,  56.70810819,  50.48105214,
        45.28552179,  41.68453347,  40.06831014,  40.61199468,
        43.25667043,  47.71574575,  53.50601062,  60.        ]), array([ 60.        ,  66.81868885,  72.89846697,  77.58049604,
        80.35740558,  80.92827435,  79.23123986,  75.45020212,
        69.99489525,  63.4564864 ,  56.5435136 ,  50.00510475,
        44.54979788,  40.76876014,  39.07172565,  39.64259442,
        42.41950396,  47.10153303,  53.18131115,  60.        ]), array([ 60.        ,  67.14338832,  73.51267968,  78.41766252,
        81.32680585,  81.92485885,  80.14701319,  76.18592603,
        70.47084265,  63.62108099,  56.37891901,  49.52915735,
        43.81407397,  39.85298681,  38.07514115,  38.67319415,
        41.58233748,  46.48732032,  52.85661168,  60.        ]), array([ 60.        ,  67.46808779,  74.12689239,  79.254829  ,
        82.29620612,  82.92144334,  81.06278651,  76.92164995,
        70.94679004,  63.78567558,  56.21432442,  49.05320996,
        43.07835005,  38.93721349,  37.07855666,  37.70379388,
        40.745171  ,  45.87310761,  52.53191221,  60.        ]), array([ 60.        ,  67.79278726,  74.7411051 ,  80.09199548,
        83.26560638,  83.91802783,  81.97855984,  77.65737386,
        71.42273743,  63.95027017,  56.04972983,  48.57726257,
        42.34262614,  38.02144016,  36.08197217,  36.73439362,
        39.90800452,  45.2588949 ,  52.20721274,  60.        ]), array([ 60.        ,  68.11748673,  75.35531782,  80.92916196,
        84.23500665,  84.91461233,  82.89433317,  78.39309777,
        71.89868483,  64.11486476,  55.88513524,  48.10131517,
        41.60690223,  37.10566683,  35.08538767,  35.76499335,
        39.07083804,  44.64468218,  51.88251327,  60.        ]), array([ 60.        ,  68.4421862 ,  75.96953053,  81.76632843,
        85.20440691,  85.91119682,  83.81010649,  79.12882168,
        72.37463222,  64.27945935,  55.72054065,  47.62536778,
        40.87117832,  36.18989351,  34.08880318,  34.79559309,
        38.23367157,  44.03046947,  51.5578138 ,  60.        ]), array([ 60.        ,  68.76688567,  76.58374324,  82.60349491,
        86.17380718,  86.90778131,  84.72587982,  79.86454559,
        72.85057961,  64.44405394,  55.55594606,  47.14942039,
        40.13545441,  35.27412018,  33.09221869,  33.82619282,
        37.39650509,  43.41625676,  51.23311433,  60.        ]), array([ 60.        ,  69.09158514,  77.19795596,  83.44066139,
        87.14320745,  87.9043658 ,  85.64165315,  80.6002695 ,
        73.32652701,  64.60864853,  55.39135147,  46.67347299,
        39.3997305 ,  34.35834685,  32.0956342 ,  32.85679255,
        36.55933861,  42.80204404,  50.90841486,  60.        ]), array([ 60.        ,  69.41628461,  77.81216867,  84.27782787,
        88.11260771,  88.9009503 ,  86.55742647,  81.33599341,
        73.8024744 ,  64.77324312,  55.22675688,  46.1975256 ,
        38.66400659,  33.44257353,  31.0990497 ,  31.88739229,
        35.72217213,  42.18783133,  50.58371539,  60.        ]), array([ 60.        ,  69.74098408,  78.42638138,  85.11499435,
        89.08200798,  89.89753479,  87.4731998 ,  82.07171732,
        74.27842179,  64.93783771,  55.06216229,  45.72157821,
        37.92828268,  32.5268002 ,  30.10246521,  30.91799202,
        34.88500565,  41.57361862,  50.25901592,  60.        ]), array([ 60.        ,  70.06568355,  79.04059409,  85.95216083,
        90.05140824,  90.89411928,  88.38897313,  82.80744123,
        74.75436918,  65.1024323 ,  54.8975677 ,  45.24563082,
        37.19255877,  31.61102687,  29.10588072,  29.94859176,
        34.04783917,  40.95940591,  49.93431645,  60.        ]), array([ 60.        ,  70.39038301,  79.65480681,  86.7893273 ,
        91.02080851,  91.89070378,  89.30474645,  83.54316514,
        75.23031658,  65.26702689,  54.73297311,  44.76968342,
        36.45683486,  30.69525355,  28.10929622,  28.97919149,
        33.2106727 ,  40.34519319,  49.60961699,  60.        ]), array([ 60.        ,  70.71508248,  80.26901952,  87.62649378,
        91.99020878,  92.88728827,  90.22051978,  84.27888905,
        75.70626397,  65.43162148,  54.56837852,  44.29373603,
        35.72111095,  29.77948022,  27.11271173,  28.00979122,
        32.37350622,  39.73098048,  49.28491752,  60.        ]), array([ 60.        ,  71.03978195,  80.88323223,  88.46366026,
        92.95960904,  93.88387276,  91.13629311,  85.01461296,
        76.18221136,  65.59621607,  54.40378393,  43.81778864,
        34.98538704,  28.86370689,  26.11612724,  27.04039096,
        31.53633974,  39.11676777,  48.96021805,  60.        ]), array([ 60.        ,  71.36448142,  81.49744494,  89.30082674,
        93.92900931,  94.88045726,  92.05206643,  85.75033687,
        76.65815876,  65.76081066,  54.23918934,  43.34184124,
        34.24966313,  27.94793357,  25.11954274,  26.07099069,
        30.69917326,  38.50255506,  48.63551858,  60.        ]), array([ 60.        ,  71.68918089,  82.11165766,  90.13799322,
        94.89840957,  95.87704175,  92.96783976,  86.48606078,
        77.13410615,  65.92540525,  54.07459475,  42.86589385,
        33.51393922,  27.03216024,  24.12295825,  25.10159043,
        29.86200678,  37.88834234,  48.31081911,  60.        ]), array([ 60.        ,  72.01388036,  82.72587037,  90.9751597 ,
        95.86780984,  96.87362624,  93.88361309,  87.22178469,
        77.61005354,  66.08999984,  53.91000016,  42.38994646,
        32.77821531,  26.11638691,  23.12637376,  24.13219016,
        29.0248403 ,  37.27412963,  47.98611964,  60.        ]), array([ 60.        ,  72.33857983,  83.34008308,  91.81232617,
        96.83721011,  97.87021073,  94.79938641,  87.95750861,
        78.08600094,  66.25459443,  53.74540557,  41.91399906,
        32.04249139,  25.20061359,  22.12978927,  23.16278989,
        28.18767383,  36.65991692,  47.66142017,  60.        ]), array([ 60.        ,  72.6632793 ,  83.95429579,  92.64949265,
        97.80661037,  98.86679523,  95.71515974,  88.69323252,
        78.56194833,  66.41918902,  53.58081098,  41.43805167,
        31.30676748,  24.28484026,  21.13320477,  22.19338963,
        27.35050735,  36.04570421,  47.3367207 ,  60.        ]), array([ 60.        ,  72.98797877,  84.56850851,  93.48665913,
        98.77601064,  99.86337972,  96.63093307,  89.42895643,
        79.03789572,  66.58378361,  53.41621639,  40.96210428,
        30.57104357,  23.36906693,  20.13662028,  21.22398936,
        26.51334087,  35.43149149,  47.01202123,  60.        ]), array([  60.        ,   73.31267824,   85.18272122,   94.32382561,
         99.7454109 ,  100.85996421,   97.54670639,   90.16468034,
         79.51384311,   66.7483782 ,   53.2516218 ,   40.48615689,
         29.83531966,   22.45329361,   19.14003579,   20.2545891 ,
         25.67617439,   34.81727878,   46.68732176,   60.        ]), array([  60.        ,   73.63737771,   85.79693393,   95.16099209,
        100.71481117,  101.85654871,   98.46247972,   90.90040425,
         79.98979051,   66.91297279,   53.08702721,   40.01020949,
         29.09959575,   21.53752028,   18.14345129,   19.28518883,
         24.83900791,   34.20306607,   46.36262229,   60.        ]), array([  60.        ,   73.96207718,   86.41114665,   95.99815857,
        101.68421144,  102.8531332 ,   99.37825305,   91.63612816,
         80.4657379 ,   67.07756738,   52.92243262,   39.5342621 ,
         28.36387184,   20.62174695,   17.1468668 ,   18.31578856,
         24.00184143,   33.58885335,   46.03792282,   60.        ]), array([  60.        ,   74.28677665,   87.02535936,   96.83532504,
        102.6536117 ,  103.84971769,  100.29402637,   92.37185207,
         80.94168529,   67.24216197,   52.75783803,   39.05831471,
         27.62814793,   19.70597363,   16.15028231,   17.3463883 ,
         23.16467496,   32.97464064,   45.71322335,   60.        ]), array([  60.        ,   74.61147611,   87.63957207,   97.67249152,
        103.62301197,  104.84630219,  101.2097997 ,   93.10757598,
         81.41763269,   67.40675656,   52.59324344,   38.58236731,
         26.89242402,   18.7902003 ,   15.15369781,   16.37698803,
         22.32750848,   32.36042793,   45.38852389,   60.        ]), array([  60.        ,   74.93617558,   88.25378478,   98.509658  ,
        104.59241223,  105.84288668,  102.12557303,   93.84329989,
         81.89358008,   67.57135115,   52.42864885,   38.10641992,
         26.15670011,   17.87442697,   14.15711332,   15.40758777,
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In [ ]: