obsidian_backup/力学书籍/FASTKinematics/auto/FASTKinematics_content_list.json

[
    {
        "type": "text",
        "text": "There are several points on a 2-bladed turbine: Z (platform reference), Y (platform mass center), T (tower node), O (tower-top / base-plate / yaw bearing mass center), U (nacelle mass center), V (arbitrary point on rotor-furl axis), W (arbitrary point on tail-furl axis), D (center of mass of structure that furls with the rotor [not including rotor]), IMU (nacelle inertial measurement unit), P (teeter pin), SG [shaft strain gage location: i.e., a point on the shaft a distance ShftGagL  towards the nacelle from point P (or point Q for a 3-blader since point P does not exist)], Q (apex of coning angle), C (hub mass center), S1 (blade node for blade 1), S2 (blade node for blade 2), I (tail boom mass center), J (tail fin mass center), and K (tail fin center-of-pressure).  There are also several reference frames: E (earth / inertial), X (platform / tower base), F (tower element body), B (tower-top / base plate), N (nacelle), R (structure that furls with the rotor—generator housing, etc…), L (low speed shaft on rotor end of LSS-compliance), H (hub / rotor), M1 (blade 1 element body), M2 (blade 2 element body), G (fixed in the high speed shaft / generator), and A (tail).  The following are derivations of the position vectors, angular velocities, linear velocities, partial angular velocities, partial linear velocities, angular accelerations, and linear accelerations of all these points on the 2-bladed turbine (point SG’s velocities and accelerations are not derived since they wont be used in the ensuing analysis).  The velocities and accelerations of points on a 3-bladed turbine are very similar. ",
        "page_idx": 0
    },
    {
        "type": "image",
        "img_path": "images/aea47d364c3db886f4957a49550f0ee56aef18eef7d0840dede2a171aa848949.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 0
    },
    {
        "type": "text",
        "text": "ZO =[qTFA1+qTFA2]a1 + Ptfm Re f+TowerHt 21S1T1F+A(S1TT1SwSr(FTlwerxFLl) eqxT2LFA) 1qT+2SSS12T2F+AS(2TT2SSw(rTFlwerxFLl) exqT2LF) A2qT2+SS 22S+1T2F2AS(1T2TSSw(rTFwlerxFLle) xqLTF) A1qTqSTSF1Aq2TSS 2 u +[qTSS1+qTSS 2]a3 ",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "OU =NacCMxnd1+NacCMznd2−NacCMynd3 rVD=(RFrlCMxn−RFrlPntxn)rf1+(RFrlCMzn−RFrlPntzn)rf2−(RFrlCMyn−RFrlPntyn)rf3 rVIMU= $\\big(N c I M U x n-R F r l P n t x n\\big)r f_{I}+\\big(N c I M U z n-R F r l P n t z n\\big)r f_{J}-\\big(N c I M U y n-R F r l P n t y n+R F r l P n t z n\\big)$ )rf3 rVP= −RFrlPntxnrf1+(Twr2Shft−RFrlPntzn)rf2−(Yaw2Shft−RFrlPntyn)rf3+OverHangc1 ",
        "page_idx": 1
    },
    {
        "type": "image",
        "img_path": "images/f0aa977f81135ff9ada5ba5bbff16d69e441e535ce42fa7471ef27b93ae21afb.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "PSG = ShftGagLc ",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "PQ= −UndSlingg rQC=HubCMg ",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "",
        "page_idx": 1
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{r l}&{\\Bigr|\\int_{I}^{B I}+\\Bigr[\\psi_{I}^{B I}\\left(r\\right)q_{B I F I}+\\psi_{2}^{B I}\\left(r\\right)q_{B I F2}+\\psi_{3}^{B I}\\left(r\\right)q_{B I E I}\\Bigr]\\dot{j}_{2}^{B I}}\\\\ &{\\Bigr.}\\\\ &{\\Bigr.\\Bigr.\\Bigr.\\Bigr.\\qquad^{2}+S_{33}^{B I}\\left(r\\right)q_{B I E I}^{2}+2S_{I2}^{B I}\\left(r\\right)q_{B I F I}q_{B I F2}+2S_{23}^{B I}\\left(r\\right)q_{B I F2}q_{B I E I}+2S_{I3}^{B I}\\left(r\\right)q_{B I F I}q_{B I E I}\\Bigr]\\Biggr\\}.}\\end{array}\n$$$$\n\\begin{array}{l}{{\\displaystyle7.8I F I+\\phi_{2}^{B I}\\left(r\\right)q_{B I F2}+\\phi_{3}^{B I}\\left(r\\right)q_{B I E I}\\biggr]\\dot{J}_{I}^{B I}+\\left[\\psi_{I}^{B I}\\left(r\\right)q_{B I F I}+\\psi_{2}^{B I}\\left(r\\right)q_{B I F I2}+\\psi_{3}^{B I}\\left(r\\right)q_{B I E I}\\right]\\dot{J}_{I}^{B I}+\\cdots}}\\\\ {{\\displaystyle:R a d-\\frac{I}{2}\\Bigl[S_{I I}^{B I}\\left(r\\right)q_{B I F I}^{2}+S_{22}^{B I}\\left(r\\right)q_{B I F2}^{2}+S_{33}^{B I}\\left(r\\right)q_{B I E I}^{2}+2S_{I2}^{B I}\\left(r\\right)q_{B I F I}q_{B I F2}+2S_{23}^{B I}\\left(r\\right)q_{B I F}^{2}\\Bigr]\\dot{J}_{I}^{B I}\\,,}}\\end{array}\n$$",
        "text_format": "latex",
        "page_idx": 1
    },
    {
        "type": "equation",
        "text": "",
        "text_format": "latex",
        "page_idx": 1
    },
    {
        "type": "text",
        "text": "where, ",
        "page_idx": 1
    },
    {
        "type": "image",
        "img_path": "images/ef4f76a950c8f1ce0bea0dba255bdc0dc45aee29c9c4dd7e27c2905433d614c8.jpg",
        "img_caption": [],
        "img_footnote": [
            "The equation for $r^{\\varrho s2}\\left(r\\right)$ is similar. "
        ],
        "page_idx": 2
    },
    {
        "type": "text",
        "text": "Note limit on r : 0≤r≤TipRad−HubRad=BldFlexL ",
        "page_idx": 2
    },
    {
        "type": "equation",
        "text": "$$\n\\pmb{r}^{o w}=T F r l P n t x n\\pmb{d}_{\\jmath}+T F r l P n t z n\\pmb{d}_{\\jmath}-T F r l P n t y n\\pmb{d}_{\\jmath}\n$$",
        "text_format": "latex",
        "page_idx": 2
    },
    {
        "type": "text",
        "text": "$r^{\\prime\\prime}=\\left(B o o m C M x n-T F r l P n t x n\\right)t f_{I}+\\left(B o o m C M z n-T F r l P n t z n\\right)t f_{J}-\\left(B o o m C M y n-T F r u n t z n\\right)t f_{I}=0.$ TFrlPntyn)tf3 ",
        "page_idx": 2
    },
    {
        "type": "equation",
        "text": "$$\nr^{\\mu\\nu}=\\left(T F i n C M x n-T F r l P n t x n\\right)t f_{I}+\\left(T F i n C M z n-T F r[P n t z n)t f_{J}-\\left(T F i n C M y n-T F i n C M z n\\right)t f_{I}\\right).\n$$",
        "text_format": "latex",
        "page_idx": 2
    },
    {
        "type": "equation",
        "text": "$$\nr^{W K}=\\left(T F i n C P x n-T F r l P n t x n\\right)t f_{I}+\\left(T F i n C P z n-T F r l P n t z n\\right)t f_{2}-\\left(T F i n C P y n-T F r u n t z n\\right)t f_{I}.\n$$",
        "text_format": "latex",
        "page_idx": 2
    },
    {
        "type": "text",
        "text": "Angular Velocities: ",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "${\\}\\pmb{\\omega}^{X}=\\dot{q}_{R}\\pmb{\\Sigma}_{I}+\\dot{q}_{Y}\\pmb{\\Sigma}_{2}-\\dot{q}_{P}\\pmb{\\Sigma}_{3}$ ",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "EωF( h)=EωX dφ1TSS ( h) qTSS1 + dφ2TSS ( h) qTSS 2 a dφ1 ( h) qTFA1 + dφ2TFA ( h) qTFA2 dh dh dh dh ",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "dφ1TSS( h) dφ2TSS( h) dφ1TFA( h) dφ2TFA( h) ω ω qTSS1+ qTSS 2 a1 qTFA1+ qTFA2 a3 dh dh dh dh h =TwrFlexL h =TwrFlexL h =TwrFlexL h =TwrFlexL ",
        "page_idx": 3
    },
    {
        "type": "equation",
        "text": "$$\n{}^{E}{\\pmb\\omega}^{N}={}^{E}{\\pmb\\omega}^{B}+\\dot{q}_{\\mathrm{{}}\\scriptscriptstyle{Y a w}}{\\pmb d}_{z}\n$$",
        "text_format": "latex",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "${}^{E}{\\pmb\\omega}^{R}={}^{E}{\\pmb\\omega}^{N}+\\dot{q}_{R F r l}{\\pmb r}{\\pmb f}{\\pmb\\dot{a}}$ where, $r f\\!\\!a=\\!c o s\\left(R F r l S k e w\\right)\\!c o s\\left(R F r l T i l t\\right)\\!d_{I}+s i n\\left(R F r l T i l t\\right)\\!d_{J}-s i n\\left(R F r l T i l t\\right)\\!d_{I}-s i n\\left(R F r l T i l t\\right)\\!d_{I}+s i n\\left(R F r l T i l t\\right)\\!d_{I}.$ lSkew) cos (RFrlTilt)d3 ",
        "page_idx": 3
    },
    {
        "type": "equation",
        "text": "$$\n{}^{E}{\\pmb\\omega}^{L}={}^{E}{\\pmb\\omega}^{R}+\\dot{q}_{D r T r}{\\pmb c}_{I}+\\dot{q}_{G e A z}{\\pmb c}_{I}\n$$",
        "text_format": "latex",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "EωM1(r )=EωH dψ1B1(r ) dr qB1F1 +dψ2B1(r ) dr qB1F 2 dψ3B1(r ) dr qB1E1 j1B1 dφ1B1(r )qB1F1+dφ2B1(r ) qB1F 2 +dφ3B1(r ) dr qB1E1 2 ",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "The equation for $^{E}\\pmb{\\omega}^{M2}\\left(r\\right)$ is similar. ",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "Since the generator is attached to the high speed shaft which may or may not rotate in the opposite direction of the low speed shaft and since $q_{G e A z}$ represents the position of the low speed shaft near the entrance of the gearbox, ",
        "page_idx": 3
    },
    {
        "type": "equation",
        "text": "$$\n{}^{E}{\\pmb\\omega}^{G}={}^{E}{\\pmb\\omega}^{R}+G e n D i r\\cdot G B R a t i o\\cdot{\\dot{q}}_{G e d z}{\\pmb{c}}_{I}\n$$",
        "text_format": "latex",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "where, $G e n D i r=\\binom{-I}{I}\\quad f o r\\,\\,\\,\\,G B\\,R e\\,\\nu e r s e=T r u e}\\\\ {\\,\\,\\,\\,G\\d{p}\\,\\,\\,\\,G B\\,R e\\,\\nu e r s e=F a l s e}$ ",
        "page_idx": 3
    },
    {
        "type": "text",
        "text": "${}^{E}{\\pmb\\omega}^{A}={}^{E}{\\pmb\\omega}^{N}+\\dot{q}_{{}_{T F r l}}t{\\pmb f}\\dot{\\pmb u}$ where, tfa=cos (TFrlSkew) cos (TFrlTilt)d1+sin (TFrlTilt)d2−sin (TFrlSkew) cos (TFrlTilt)d3 ",
        "text_level": 1,
        "page_idx": 4
    },
    {
        "type": "text",
        "text": "Linear Velocities: ${}^{E}{\\pmb{\\nu}}^{Z}=\\dot{q}_{S g}\\bar{\\sf z}_{I}+\\dot{q}_{H\\nu}\\bar{\\sf z}_{2}-\\dot{q}_{S w}\\bar{\\sf z}_{3}$ ",
        "text_level": 1,
        "page_idx": 5
    },
    {
        "type": "equation",
        "text": "$$\n{}^{E}{\\pmb{\\nu}}^{Y}={}^{E}{\\pmb{\\nu}}^{Z}+{}^{E}{\\pmb{\\omega}}^{X}\\times{\\pmb{r}}^{Z Y}\n$$",
        "text_format": "latex",
        "page_idx": 5
    },
    {
        "type": "equation",
        "text": "$$\n{}^{E}{\\nu}^{T}\\left(h\\right)={}^{E}{\\nu}^{Z}+{}^{X}{\\nu}^{T}\\left(h\\right)+{}^{E}{\\omega}^{X}\\times{r}^{Z T}\\left(h\\right)\n$$",
        "text_format": "latex",
        "page_idx": 5
    },
    {
        "type": "text",
        "text": "where, ",
        "page_idx": 5
    },
    {
        "type": "text",
        "text": "XvT( h)=φ1TFA( h) qTFA1+φ2TFA( h) qTFA2a1 S1T1FA( h) qTFA1qTFA1+S2T2FA( h) qTFA2qTFA2+S1T2FA( h)(qTFA1qTFA2+qTFA1qTFA2) +S1T1SS( h) qTSS1qTSS1+S2T2SS( h) qTSS 2qTSS 2+S1T2SS( h)(qTSS1qTSS 2+qTSS1qTSS 2) +φ1TSS( h) qTSS1+φ2TSS( h) qTSS 2a3 ",
        "page_idx": 5
    },
    {
        "type": "text",
        "text": "${}^{E}{\\nu}^{o}={}^{E}{\\nu}^{Z}+{}^{X}{\\nu}^{o}+{}^{E}{\\omega}^{X}\\times{r}^{Z O}$ where, ",
        "page_idx": 5
    },
    {
        "type": "text",
        "text": "XvO=[qTFA1+qTFA2]a1 S1T1FA(TwrFlexL) qTFA1qTFA1+S2T2FA(TwrFlexL) qTFA2qTFA2+S1T2FA(TwrFlexL)(qTFA1qTFA2+qTFA1qTFA2) +S1T1SS(TwrFlexL) qTSS1qTSS1+S2T2SS(TwrFlexL) qTSS 2qTSS 2+S1T2SS(TwrFlexL)(qTSS1qTSS 2+qTSS1qTSS 2) +[qTSS1+qTSS 2]a3 ",
        "page_idx": 5
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l}{{{^E}_{\\nu}{^U}={^E}_{\\nu}{^o}+{^E}_{\\omega}{^N}\\times r^{o U}}}\\\\ {{{}}}\\\\ {{{^E}_{\\nu}{^V}={^E}_{\\nu}{^o}+{^E}_{\\omega}{^N}\\times r^{o V}}}\\\\ {{{}}}\\\\ {{{^E}_{\\nu}{^D}={^E}_{\\nu}{^V}+{^E}_{\\omega}{^R}\\times r^{V D}}}\\end{array}\n$$",
        "text_format": "latex",
        "page_idx": 5
    },
    {
        "type": "text",
        "text": "The equation for $E_{\\nu}^{\\phantom{\\mu\\nu}}{}^{m U}$ is similar. ",
        "page_idx": 5
    },
    {
        "type": "equation",
        "text": "$$\n{}^{E}{\\pmb{\\nu}}^{P}={}^{E}{\\pmb{\\nu}}^{V}+{}^{E}{\\pmb{\\omega}}^{R}\\times{\\pmb{r}}^{V P}\n$$",
        "text_format": "latex",
        "page_idx": 5
    },
    {
        "type": "text",
        "text": "EvQ=EvP+EωH×r ",
        "text_level": 1,
        "page_idx": 6
    },
    {
        "type": "text",
        "text": "× ",
        "page_idx": 6
    },
    {
        "type": "text",
        "text": "EvS1(r )=EvQ+HvS1(r )+EωH×rQS1(r ) ",
        "page_idx": 6
    },
    {
        "type": "text",
        "text": "where, ",
        "page_idx": 6
    },
    {
        "type": "text",
        "text": "HvS1(r )=φ1B1(r ) qB1F1+φ2B1(r ) qB1F 2+φ3B1(r ) qB1E1j1B1+ψ1B1(r ) qB1F1+ψ2B1(r ) qB1F 2+ψ3B1(r ) qB1E1j2B1 S1B11(r ) qB1F1qB1F1+S2B21(r ) qB1F 2qB1F 2+S3B31(r ) qB1E1qB1E1 +S1B21(r )(qB1F1qB1F 2+qB1F1qB1F 2)+S2B31(r )(qB1F 2qB1E1+qB1F2qB1E1)+S1B31(r )(qB1F1qB1E1+qB1F1qB1E1) ",
        "page_idx": 6
    },
    {
        "type": "text",
        "text": "The equation for $\\boldsymbol{\\varepsilon}_{\\boldsymbol{\\nu}}^{s_{2}}\\left(\\boldsymbol{r}\\right)$ is similar. ",
        "page_idx": 6
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{r l}&{^{E}_{\\nu}\\psi^{\\nu}=^{E}_{\\nu}\\!\\!\\!\\phantom{^{(0)}}^{\\!\\!E}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \n$$",
        "text_format": "latex",
        "page_idx": 6
    },
    {
        "type": "text",
        "text": "Partial Angular Velocities: ",
        "page_idx": 7
    },
    {
        "type": "text",
        "text": "Recall that: $^{E}\\!\\omega^{N_{i}}\\left(\\dot{q},q,t\\right)\\!=\\!\\!\\left(\\sum_{r=l}^{22}\\d^{E}\\omega_{r}^{N_{i}}\\left(q,t\\right)\\!\\dot{q}_{r}\\right)\\!+\\!\\;^{E}\\!\\omega_{t}^{N_{i}}\\left(q,t\\right)$ for each rigid body $N_{i}$ in the system.  Note that all of the ${}^{E}{\\pmb{\\omega}}_{t}^{N_{i}}$ terms are zero as will be shown. ",
        "page_idx": 7
    },
    {
        "type": "text",
        "text": "z1 for r=R z3 for r=P EωX= 7 for r=Y 0 otherwise EωtX = 0 ",
        "page_idx": 7
    },
    {
        "type": "table",
        "img_path": "images/2277d15a0b56afcd7524f8a378e1de9a8ca839f48c68db2aebeacb21a1ed8e85.jpg",
        "table_caption": [],
        "table_footnote": [],
        "table_body": "\n\n<html><body><table><tr><td rowspan=\"4\">E (h) の + E</td><td>dpi TFA h a3 for r =TFAl</td></tr><tr><td>dh dpi TSS h a1 for r =TSS1</td></tr><tr><td>dh dΦ2 TFA (h) a3 for r=TFA2</td></tr><tr><td>dh dΦ2 TSS h a for r =TSS2 dh otherwise</td></tr><tr><td colspan=\"2\">0 0F (h) =0</td></tr></table></body></html>\n\n",
        "page_idx": 7
    },
    {
        "type": "table",
        "img_path": "images/a593b4bb09ff3bcc0aa64f9fe4cad48bc1ea831df66f0ed67ed5622ce3ff0a5e.jpg",
        "table_caption": [],
        "table_footnote": [],
        "table_body": "\n\n<html><body><table><tr><td rowspan=\"4\">Q</td><td>dpIFA TFA (h) dh</td><td>a3 for r =TFAl h=TwrFlexL</td></tr><tr><td>d$iss T'SS (h) dh</td><td>for r = TSS1 h=TwrFlexL</td></tr><tr><td>dΦ2 TFA (h dh TSS</td><td>a3 for r=TFA2 h=TwrFlexL</td></tr><tr><td>dp? h dh</td><td>a for r =TSS2 h=TwrFlexL</td></tr><tr><td>E =0</td><td>0</td><td>otherwise</td></tr></table></body></html>\n\n",
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "d for r=Yaw   \nE ω ω 0 otherwise   \nE ω ",
        "page_idx": 8
    },
    {
        "type": "text",
        "text": "rfa for r=RFrlω ω0 otherwiseE ω R =0",
        "page_idx": 8
    },
    {
        "type": "image",
        "img_path": "images/3398e4ae47d978c05d1ac6d8435467b2a54017a3ad095973c8560076ec510976.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 8
    },
    {
        "type": "image",
        "img_path": "images/6312d612c59210b4b88ca58006215d23618c8c2ae4a8405023d27e07a9523168.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 9
    },
    {
        "type": "image",
        "img_path": "images/fa88774625bec40d9835cefc14b9dab265ecad758ce4f53a290b13815e357f12.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "The equations for $^{E}\\pmb{\\omega}_{r}^{M2}\\left(r\\right)$ and $^{E}\\pmb{\\omega}_{t}^{M2}\\left(r\\right)$ are similar. ",
        "page_idx": 9
    },
    {
        "type": "image",
        "img_path": "images/e31134e876cef5a881092ca70c43cba1fd09da2d674dd9f5d4cac0be697c2b35.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 9
    },
    {
        "type": "text",
        "text": "Partial Linear Velocities: ",
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "Recall that: $^{E}\\nu^{X_{i}}\\left(\\dot{q},q,t\\right)=\\left(\\sum_{r=I}^{22}\\varepsilon_{r}^{\\phantom{R}}\\left(q,t\\right)\\dot{q}_{r}\\right)+^{E}\\nu_{t}^{X_{i}}\\left(q,t\\right)$ for each point $X_{i}$ in the system.  Note that all of the $\\ensuremath{\\boldsymbol{\\varepsilon}}_{\\ensuremath{\\boldsymbol{\\nu}}_{t}^{X_{i}}}$ terms are zero as will be shown. ",
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "7 for r=Sg z3 for r=Sw for r=Hv 0 otherwise v = 0 ",
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "E ωr X ×r ZY for r=4,5,6 0 otherwise = 0 ",
        "page_idx": 10
    },
    {
        "type": "text",
        "text": "EvrT( h)=EvrZ+ $\\begin{array}{r l r}&{\\left\\{\\begin{array}{l l}{\\varepsilon_{\\omega}^{x}\\times r^{z T}\\left(h\\right)}&{f o r}&{r=d,5,6}\\\\ {\\phi_{l}^{T R L}\\left(h\\right)a_{l}-\\left[S_{l l}^{T R L}\\left(h\\right)q_{T E A l}+S_{l2}^{T R L}\\left(h\\right)q_{T E A l}\\right]a_{2}}&{f o r}&{r=T F A l}\\end{array}\\right.}\\\\ &{\\left\\{\\begin{array}{l l}{\\phi_{l}^{T S S}\\left(h\\right)a_{3}-\\left[S_{l l}^{T S S}\\left(h\\right)q_{T S S l}+S_{l2}^{T S S}\\left(h\\right)q_{T S S2}\\right]a_{2}}&{f o r}&{r=T S S I}\\\\ {\\phi_{2}^{T R L}\\left(h\\right)a_{l}-\\left[S_{22}^{T R L}\\left(h\\right)q_{T E A2}+S_{l2}^{T R L}\\left(h\\right)q_{T E A l}\\right]a_{2}}&{f o r}&{r=T F A2}\\end{array}\\right.}\\\\ &{\\left\\}\\phi_{2}^{T R L}\\left(h\\right)a_{l}-\\left[S_{22}^{T R L}\\left(h\\right)q_{T E A2}+S_{l2}^{T R L}\\left(h\\right)q_{T E A l}\\right]a_{2}}&{f o r}&{r=T F A2}\\\\ &{\\phi_{2}^{T S S}\\left(h\\right)a_{3}-\\left[S_{22}^{T S S}\\left(h\\right)q_{T S S2}+S_{l2}^{T S S}\\left(h\\right)q_{T S S l}\\right]a_{2}}&{f o r}&{r=T S S2}\\\\ {0}&{o t h e r w i s e}\\end{array}\\right.}\\end{array}$ $\\boldsymbol{\\varepsilon}_{\\boldsymbol{\\nu}_{t}^{T}}(h)\\!=\\!\\boldsymbol{\\ O}$ ",
        "page_idx": 10
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{r}{\\left[\\begin{array}{l l l}{\\varepsilon_{\\omega}^{\\scriptscriptstyle X}\\times r^{2\\theta}}&{f o r}&{r=\\mathscr{I},\\mathscr{S},\\theta}\\\\ {a_{I}-\\Big[S_{I I}^{\\scriptscriptstyle T\\scriptscriptstyle F\\scriptscriptstyle L}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle A I}+S_{I2}^{\\scriptscriptstyle T\\scriptscriptstyle L}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle A2}\\Big]a_{2}}&{f o r}&{r=T F A I}\\\\ {a_{3}-\\Big[S_{I I}^{\\scriptscriptstyle T\\scriptscriptstyle S\\scriptscriptstyle S}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle S\\scriptscriptstyle I}+S_{I2}^{\\scriptscriptstyle T\\scriptscriptstyle S\\scriptscriptstyle S}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle S\\scriptscriptstyle S2}\\Big]a_{2}}&{f o r}&{r=T S S I}\\\\ {a_{I}-\\Big[S_{22}^{\\scriptscriptstyle T\\scriptscriptstyle E\\scriptscriptstyle A}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle A2}+S_{I2}^{\\scriptscriptstyle T\\scriptscriptstyle E\\scriptscriptstyle A}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle A I}\\Big]a_{2}}&{f o r}&{r=T F A2}\\\\ {a_{3}-\\Big[S_{22}^{\\scriptscriptstyle T\\scriptscriptstyle S\\scriptscriptstyle S}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle A2}+S_{I2}^{\\scriptscriptstyle T\\scriptscriptstyle S\\scriptscriptstyle S}\\big(T w r F l e x L\\big)q_{T r\\scriptscriptstyle A2}}&{f o r}&{r=T S S2}\\\\ {\\theta}&{o t h e r w i s e}\\end{array}\\right]}\\end{array}\n$$",
        "text_format": "latex",
        "page_idx": 11
    },
    {
        "type": "text",
        "text": "ω × OU for r=4,5,,11 otherwise ",
        "page_idx": 11
    },
    {
        "type": "text",
        "text": "OV for r=4,5,,11 ω × otherwise EvV =0 ",
        "page_idx": 11
    },
    {
        "type": "image",
        "img_path": "images/e3d6a23ac82c553a6109e1bbd076a589d53454144682502bb72c0663d0760822.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 11
    },
    {
        "type": "text",
        "text": "The equations for $E_{\\nu_{r}}^{\\,,\\,\\,\\,\\,\\mu\\nu}$ and EvIMU are similar. ",
        "page_idx": 11
    },
    {
        "type": "image",
        "img_path": "images/afa45461db2a2b95476a24dfe5f2800411c600e2ef02e1410f267618deae6c23.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 11
    },
    {
        "type": "image",
        "img_path": "images/200f40a4ff42db1bfd8e7b1ddc2568d911234a08e251bb70cdd737ad06440482.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "$\\begin{array}{r l r}&{}&{f o r\\ \\ r}\\\\ &{}&{\\left\\{\\phi_{l}^{B I}\\left(r\\right)j_{l}^{B I}+\\psi_{l}^{B I}\\left(r\\right)j_{2}^{B I}-\\left[S_{1l}^{B I}\\left(r\\right)q_{B I F I}+S_{12}^{B I}\\left(r\\right)q_{B I F I}+S_{13}^{B I}\\left(r\\right)q_{B I E I}\\right]j_{3}^{B I}\\ \\ \\ f o r\\ \\ r}\\\\ &{}&{\\left\\{\\phi_{3}^{B I}\\left(r\\right)j_{l}^{B I}+\\psi_{3}^{B I}\\left(r\\right)j_{2}^{B I}-\\left[S_{33}^{B I}\\left(r\\right)q_{B I E I}+S_{23}^{B I}\\left(r\\right)q_{B I F I}+S_{13}^{B I}\\left(r\\right)q_{B I F I}\\right]j_{3}^{B I}\\ \\ \\ f o r\\ \\ r}\\\\ &{}&{\\left\\{\\phi_{2}^{B I}\\left(r\\right)j_{l}^{B I}+\\psi_{2}^{B I}\\left(r\\right)j_{2}^{B I}-\\left[S_{22}^{B I}\\left(r\\right)q_{B I F I}+S_{12}^{B I}\\left(r\\right)q_{B I F I}+S_{23}^{B I}\\left(r\\right)q_{B I E I}\\right]j_{3}^{B I}\\ \\ \\ f o r\\ \\ r}\\\\ &{}&{\\left\\{\\varepsilon_{O H}^{B I}\\times r^{Q S I}\\left(r\\right)\\right.}\\\\ &{}&{\\left.\\rho t\\rightarrow r^{r}\\left(r\\right)}\\\\ &{}&{\\left.\\rho t h e r w i\\right.}\\end{array}$ =4,5,,14 =B1F1 =B1E1 =B1F2 =Teet ise $\\boldsymbol{\\varepsilon}_{\\boldsymbol{\\nu}_{t}^{S I}}\\left(\\boldsymbol{r}\\right)=\\boldsymbol{\\ O}$ ",
        "page_idx": 12
    },
    {
        "type": "equation",
        "text": "",
        "text_format": "latex",
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "The equations for $^E_{\\nu_{r}^{S2}}\\left(r\\right)$ and $^E_{\\nu_{t}^{S2}}\\left(r\\right)$ are similar. ",
        "page_idx": 12
    },
    {
        "type": "text",
        "text": "ω ×r OW for r=4,5,,11 0 otherwise E = 0 ",
        "page_idx": 12
    },
    {
        "type": "image",
        "img_path": "images/e86624538ba5b2865e2e19027cff55cc7241673860f69c31ddc8f7297c122302.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 13
    },
    {
        "type": "text",
        "text": "Angular Accelerations: ",
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "Recall that: $^{E}\\pmb{\\alpha}^{N_{i}}\\left(\\ddot{q},\\dot{q},q,t\\right)\\!=\\!\\left(\\sum_{r=l}^{22}^{E}\\pmb{\\omega}_{r}^{N_{i}}\\left(q,t\\right)\\ddot{q}_{r}\\right)\\!+\\!\\left[\\sum_{r=l}^{22}\\frac{d}{d t}\\!\\left(^{E}\\pmb{\\omega}_{r}^{N_{i}}\\left(q,t\\right)\\right)\\!\\dot{q}_{r}\\right]\\!+\\!\\frac{d}{d t}\\!\\left(^{E}\\pmb{\\omega}_{t}^{N_{i}}\\left(q,t\\right)\\right)\\!\\left(^{E}\\pmb{\\omega}_{p}^{N_{i}}\\left(q,t\\right)\\right),$ for each rigid body $N_{i}$ in the system.  Note that the $\\frac{d}{d t}\\Big(^{E}\\omega_{r}^{N_{i}}\\Big)$ terms are all vector functions of $\\left({\\dot{q}},q,t\\right)$ and that all of the $\\frac{d}{d t}\\Big(^{E}\\omega_{t}^{N_{i}}\\Big)$ terms are zero as will be shown. ",
        "page_idx": 14
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l}{\\displaystyle\\frac{d}{d t}\\Big(\\sp\\varepsilon\\omega_{r}^{X}\\Big)=O}\\\\ {\\displaystyle\\frac{d}{d t}\\Big(\\sp\\varepsilon\\omega_{{r}}^{X}\\Big)=O}\\end{array}\n$$",
        "text_format": "latex",
        "page_idx": 14
    },
    {
        "type": "text",
        "text": "$\\begin{array}{l l}{\\displaystyle\\frac{d}{d t}\\Big[^{\\varepsilon}\\pmb{\\omega}_{r}^{F}\\left(h\\right)\\Big]\\!=\\!\\left\\{\\!\\!\\begin{array}{l l}{\\displaystyle\\varepsilon_{\\pmb{\\omega}}^{\\varepsilon}\\!\\times^{\\varepsilon}\\!\\omega_{r}^{F}\\left(h\\right)}&{\\displaystyle f o r\\;\\;r=7,\\delta,...,l O}\\\\ {\\displaystyle\\theta}&{o t h e r w i s e}\\end{array}\\!\\right.}\\\\ {\\displaystyle\\frac{d}{d t}\\Big[^{\\varepsilon}\\pmb{\\omega}_{\\prime}^{F}\\left(h\\right)\\Big]=O}\\end{array}$   \n$\\begin{array}{l}{\\displaystyle\\frac{d}{d t}\\Big({}^{E}{\\omega}_{r}^{B}\\Big)=\\left\\{{\\!\\!\\begin{array}{l l}{\\displaystyle E_{{\\pm}}{\\omega}^{X}\\times{}^{E}{\\omega}_{r}^{B}}&{\\displaystyle f o r\\ \\ r=7,\\mathrm{}\\mathrm{\\it{\\mathscr{S}}},\\ldots,{\\it{\\it{10}}}}\\\\ {\\displaystyle O}&{\\ o t h e r w i s e}\\end{array}}\\right.}\\\\ {\\displaystyle\\frac{d}{d t}\\Big({}^{E}{\\omega}_{{\\iota}}^{B}\\Big)=\\cal{O}}\\end{array}$ E E N for r=Yaw   \nd E ω Yaw   \nω ω   \ndt d otherwise   \nd E   \nωt =0   \ndt ",
        "page_idx": 14
    },
    {
        "type": "image",
        "img_path": "images/9b38d49d69832767897ff11cc45ca36c4165b88b914426af13a479d09b6d4510.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 15
    },
    {
        "type": "text",
        "text": "The equations for ${\\frac{d}{d t}}\\Big[^{\\,E}\\pmb{\\omega}_{r}^{{\\scriptscriptstyle M}2}\\,\\big(r\\big)\\Big]$  and d Eω $\\frac{d}{d t}\\Big[^{\\,E}\\pmb{\\omega}_{t}^{{M}2}\\left(r\\right)\\Big]$ are similar. ",
        "page_idx": 15
    },
    {
        "type": "image",
        "img_path": "images/c5881e166cabde96bd49b6631c6b951f97f8b82ca05b9ad9c40af4ae957d03ed.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 16
    },
    {
        "type": "text",
        "text": "Linear Accelerations: ",
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "Recall that: $\\varepsilon_{\\mathbf{}}\\boldsymbol{\\alpha}^{X_{i}}\\left(\\ddot{q},\\dot{q},q,t\\right)\\!=\\!\\!\\left(\\sum_{r=l}^{22}\\varepsilon_{\\nu_{r}^{X_{i}}}\\left(q,t\\right)\\ddot{q}_{r}\\right)\\!+\\!\\!\\left[\\sum_{r=l}^{22}\\!\\frac{d}{d t}\\!\\left({^{\\varepsilon}\\nu_{r}^{X_{i}}}\\left(q,t\\right)\\right)\\!\\dot{q}_{r}\\right]\\!+\\!\\frac{d}{d t}\\!\\left({^{\\varepsilon}\\nu_{{^I}}^{X_{i}}}\\left(q,t\\right)\\right)\\!\\left({^{\\varepsilon}\\nu_{{^I}}^{X_{i}}}\\left(q,t\\right)\\right)\\!+\\!\\frac{d}{d t}\\!\\left({^{\\varepsilon}\\nu_{{^I}}^{X_{i}}}\\left(q,t\\right)\\right)\\!+\\!{^{\\varepsilon}\\nu_{{^I}}^{X_{i}}}\\left(q,t\\right),$ for each point $X_{i}$ in the system.  Note that the ${\\frac{d}{d t}}{\\Big(}^{E}\\nu_{r}^{X_{i}}{\\Big)}$ terms are all vector functions of $\\left({\\dot{q}},q,t\\right)$ and that all of the ${\\frac{d}{d t}}{\\Big(}^{E}\\nu_{t}^{X_{i}}{\\Big)}$ terms are zero as will be shown. ",
        "page_idx": 17
    },
    {
        "type": "equation",
        "text": "$$\n\\begin{array}{l}{\\displaystyle\\frac{d}{d t}\\Big(\\sp\\varepsilon\\pmb{\\nu}_{r}^{z}\\Big)=O}\\\\ {\\displaystyle\\frac{d}{d t}\\Big(\\sp\\varepsilon\\pmb{\\nu}_{{\\pmb\\nu}}^{z}\\Big)=O}\\end{array}\n$$",
        "text_format": "latex",
        "page_idx": 17
    },
    {
        "type": "text",
        "text": "ω ω × for r=4,5,6 v dt otherwise (EvtY) =0 dt ddt $\\iota_{\\Psi_{r}^{T}}(h)\\!\\!\\!\\int_{=\\!\\!\\!\\!\\!\\!\\int_{0}^{M}}\\!\\left[\\!\\!\\!\\begin{array}{l l l}{\\varepsilon_{\\omega_{r}^{X}\\times\\left[\\!\\!\\!\\begin{array}{l}{x}{\\nu}{\\nu^{r}\\left(h\\right)+}^{\\varepsilon}\\omega^{X}\\times r^{Z r}\\left(h\\right)\\!\\!\\right]}}&{f o r}&{r=d,5,6}\\\\ {-\\left[\\!\\!\\begin{array}{l l l}{S_{11}^{T R A}\\left(h\\right)\\dot{q}_{T r A l}+S_{12}^{T R A}\\left(h\\right)\\dot{q}_{T r A l}]\\!\\!\\right]a_{2}+}&{\\varepsilon_{\\omega}^{X}\\times^{E}\\nu_{T R A l}^{T}\\left(h\\right)}&{f o r}&{r=T R A l}\\\\ {-\\left[\\!\\!\\begin{array}{l l l}{S_{11}^{T S B}\\left(h\\right)\\dot{q}_{T S S1}+S_{12}^{T S B}\\left(h\\right)\\dot{q}_{T S S2}}\\end{array}\\right]a_{2}+}&{\\varepsilon_{\\omega}^{X}\\times^{E}\\nu_{T S S1}^{T}\\left(h\\right)}&{f o r}&{r=T S S I}\\\\ {-\\left[\\!\\!\\begin{array}{l l l}{S_{22}^{T R A}\\left(h\\right)\\dot{q}_{T R A2}+S_{12}^{T R A}\\left(h\\right)\\dot{q}_{T R A l}\\right]a_{2}+}&{\\varepsilon_{\\omega}^{X}\\times^{E}\\nu_{T R A2}^{T}\\left(h\\right)}&{f o r}&{r=T F A2}\\\\ {-\\left[\\!\\!\\begin{array}{l l l}{S_{22}^{T R A}\\left(h\\right)\\dot{q}_{T S2}+S_{12}^{T R A}\\left(h\\right)\\dot{q}_{T R A l}}\\end{array}\\right]a_{2}+}&{\\varepsilon_{\\omega}^{X}\\times^{E}\\nu_{T R S2}^{T}\\left(h\\right)}&{f o r}&{r=T F A2}\\\\ {-\\left[\\!\\!\\left[S_{22}^{T S S}\\left(h\\right)\\dot{q}_{T S S2}+S_{12}^{T S S}\\left(h\\right)\\dot{q}_{T S S1}\\right]a_{2}+}&{\\varepsilon_{\\omega}^{X}\\times^{E}\\nu_{T S S2}^{T}\\left(h\\right)}&{f o r}&{r=T S S2}\\\\ {0}&{o t h e r w i s e}\\end{array}\\right]$ d[EvtT(h )]=0 ",
        "page_idx": 17
    },
    {
        "type": "image",
        "img_path": "images/7f540e8a0b94d65e0c8e85862cbfffbd9db962a124a331a6b20ff526e17acf02.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 18
    },
    {
        "type": "image",
        "img_path": "images/f5e77b445b09c605b565937c9931c544ab43aea41659c32cb20fd54ba1f400a3.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 19
    },
    {
        "type": "text",
        "text": "The equations for $\\frac{d}{d t}\\big(\\varepsilon_{\\nu_{r}^{I M U}}\\big)$ and $\\frac{d}{d t}\\big(\\boldsymbol{\\varepsilon}_{\\boldsymbol{\\nu}_{t}^{I M U}}\\big)$ are similar. ",
        "page_idx": 19
    },
    {
        "type": "image",
        "img_path": "images/3ce1abcaeeff5430134d742b3e324508a553d05c8c0f30117b673678c8672d72.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 19
    },
    {
        "type": "image",
        "img_path": "images/b46f6d5b32c278d5fa117b282e8e89c83efcfe951642463192021963be2383f9.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 20
    },
    {
        "type": "image",
        "img_path": "images/a5ccd2c04b4bb71c7215c42e6a3f7b90011a6bd8b375fadde860693f0fd076b8.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "dtEvtS1(r )=0 ",
        "text_level": 1,
        "page_idx": 20
    },
    {
        "type": "text",
        "text": "The equations for $\\frac{d}{d t}\\Big[^{\\varepsilon}\\nu_{r}^{s_{2}}(r)\\Big]$ and $\\frac{d}{d t}\\Big[^{\\varepsilon}\\nu_{t}^{s2}\\left(r\\right)\\Big]$ are similar. ",
        "page_idx": 20
    },
    {
        "type": "image",
        "img_path": "images/b425e4c4b9ef2bc7d01a78777fb711557ad6ceeba0f84bf46ae380326bc9c813.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 21
    },
    {
        "type": "image",
        "img_path": "images/2d58bc879c460dac3de92bb198a38c050563fc9038b4acdddcdd1b0c5834c1ca.jpg",
        "img_caption": [],
        "img_footnote": [],
        "page_idx": 22
    }
]