Torsten Mayer-Gürr Institute of Geodesy, NAWI Graz Technische Universität Graz
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1 Insttute of Geodes GGOS and Reference Sstems Euler-ouvlle equaton Torsten Maer-Gürr Insttute of Geodes, NWI Gra Technsche Unverstät Gra Torsten Maer-Gürr 1
2 Insttute of Geodes Reference sstem Need of a global, precse, long tme stable reference sstems! ctuall, need of at least two reference sstems - n Earth fed, rotatng reference sstem Terrestral Reference Frame TRS - Terrestral observatons - Geophscal processes - Poston of observaton statons - space fed, quas-ntertal reference sstem Celestal Reference sstem CRS - Satellte moton - Poston of planetar obects - Quas-nertal sstem Torsten Maer-Gürr
3 Insttute of Geodes Rotatng reference sstems Torsten Maer-Gürr 3
4 Insttute of Geodes Torsten Maer-Gürr 4 Rotatng coordnate sstems e e I e I e e e e dt d e e e e e e Poston Veloct D dt d Rotaton of the reference sstem Moton n the rotatng sstem D dt d Vald for all vectors Dervaton operator
5 Insttute of Geodes Equaton of moton Poston r R Equaton of moton f r I e e r e R R d dt d R dt I e Dervaton n sstem I Torsten Maer-Gürr 5
6 Insttute of Geodes Torsten Maer-Gürr 6 Equaton of moton dt d dt d dt d D dt d D D D D D D D D D D Dervaton operator D dt d D dt d
7 Insttute of Geodes Equaton of moton Inertal sstem Equaton of moton: d r f dt Newton sstem / Quas nertal sstem Equaton of moton: D f R Galle sstem rotatng D Equaton of moton: f R ω ω ω ω D Centrfugal force Euler force Corols force Torsten Maer-Gürr 7
8 Insttute of Geodes Corols force Vdeo from the Unverst of Illnos WW1 Proect. Torsten Maer-Gürr 8
9 Insttute of Geodes Euler s equaton of moton Torsten Maer-Gürr 9
10 Insttute of Geodes near and angular momentum Change of momentum needs a force p F near momentum p mr Change of angular momentum needs a torque r p r F ngular momentum M r p mr r Force Torque F M r F F r Torsten Maer-Gürr 1
11 Insttute of Geodes ngular momentum balance Total torque M r F Partcle sstem Total angular momentum r p Change of angular momentum needs a torque M Torsten Maer-Gürr 11
12 Insttute of Geodes ngular momentum balance Total torque M r F Partcle sstem Total angular momentum r p r m r r m m r Dr r r Momentum p mr Dervaton operator d dt D Fed partcle sstem Dr Torsten Maer-Gürr 1
13 Insttute of Geodes Torsten Maer-Gürr 13 ngular momentum balance Total angular momentum m r r Partcle sstem In bod fed coordnates r Total angular momentum n coordnates m m m
14 Insttute of Geodes ngular momentum balance Total angular momentum m r r In bod fed coordnates r Partcle sstem Total angular momentum n coordnates m m m m m m m m m T Torsten Maer-Gürr 14
15 Insttute of Geodes Torsten Maer-Gürr 15 ngular momentum balance Total angular momentum Rgd bod In bod fed coordnates r Total angular momentum n coordnates dm dm dm dm dm dm dm dm dm T dm r r
16 Insttute of Geodes near moton vs. Rotaton near moton Change of momentum needs a force p F wth the lnear momentum p mr T Rotaton Change of angular momentum needs a torque M wth the angular momentum Inertal mass Veloct Inerta tensor ngular veloctes Torsten Maer-Gürr 16
17 Insttute of Geodes Torsten Maer-Gürr 17 Inerta tensor ngular momentum T C The nerta tensor s a 33 matr Eample Rotaton about the -as: The moment of nerta C s large
18 Insttute of Geodes Torsten Maer-Gürr 18 Inerta tensor ngular momentum T C The nerta tensor s a 33 matr Eample Rotaton about the -as: The moment of nerta s small
19 Insttute of Geodes Rotaton of the coordnate sstem ngular momentum T R T RTR R RTR ' T' ' T T R R New coordnate sstem ' R ' R Rotar matr R T T T' RTR R I Rgd bod Choosng a coordnate sstem n whch the nerta tensor becomes dagonal Egen value decomposton T' C Prncpal coordnate sstem Hauptachsensstem Torsten Maer-Gürr 19
20 Insttute of Geodes Egen value decomposton Rotaton of the nerta tensor T RTR T' dag,, C Matr T s 33, smmetrc, and postve defnte Tv Tv Tv v T 1 3 v v v v k k 3 The egenvectors v are orthonormal T T v 1, v, v v , v, v v, v, v v, v v , TR T R T T' Rotar matr R T R T v 1, v, v R I T RTR T' => The rotar matr R contans the egenvectors => The matr T contans the egen values Torsten Maer-Gürr
21 Insttute of Geodes Rotaton of the coordnate sstem ngular momentum T ' T' ' Rgd bod Rotaton n a new coordnate sstem ' R ' R Wth the rotar matr T T' RTR R T R I Choosng a coordnate sstem n whch the nerta tensor becomes dagonal Egen value decomposton T' C Prncpal coordnate sstem Hauptachsensstem Torsten Maer-Gürr 1
22 Insttute of Geodes near moton vs. Rotaton near moton Change of momentum needs a force p F wth the lnear momentum p mr T Rotaton Change of angular momentum needs a torque M wth the angular momentum Inertal mass Veloct Inerta tensor ngular veloctes Newton-Euler equaton of moton m const m r F Epressed n coordnates mr mr mr F F F Euler's rotaton equatons for rgd bodes T C M C T M Epressed n prncple coordnates C M M Torsten Maer-Gürr
23 Insttute of Geodes Torsten Maer-Gürr 3 Euler's equaton of moton alance equaton M Dervaton operator D dt d In the rotatng sstem M D ngular momentum T M T T D Rgd bod T D M T T Inerta tensor n the prncple coordnate sstem C T Euler's equaton of moton M C M C M C
24 Insttute of Geodes near moton vs. Rotaton near moton Change of momentum needs a force p F wth the lnear momentum p mr T Rotaton Change of angular momentum needs a torque M wth the angular momentum Inertal mass Veloct Inerta tensor ngular veloctes Newton-Euler equaton of moton m const m r F Epressed n coordnates mr mr mr F F F Euler's rotaton equatons for rgd bodes T C M C T M Epressed n prncple coordnates C M M Torsten Maer-Gürr 4
25 Insttute of Geodes Torsten Maer-Gürr 5 Inerta moton Euler's equaton of moton M C M C M C Smplfed case: M - Free of torques nerta moton C C C T C - Ellpsod of rotaton Flattened Earth
26 Insttute of Geodes Inerta moton Euler's equaton of moton smplfed C C bbrevaton General soluton C c const const Flattened Earth acos t t asn t t c wth a const, c const, t C const const Torsten Maer-Gürr 6
27 Insttute of Geodes Inerta moton Torsten Maer-Gürr 7
28 Insttute of Geodes Torsten Maer-Gürr 8 Inerta moton ll equatons n the bod fed coordnate sstem : od as 1 e Rotar vector c t t a t t a sn cos ngular momentum Cc t t a t t a sn cos T 1 e const c a const c C a const c C a Cc cos e e Scalar products const c a c cos e e const c a c C a cos Scalar trple products e ll vectors are n a plane
29 Insttute of Geodes ngular momentum as od as Rotar as Torsten Maer-Gürr 9
30 Insttute of Geodes alance equaton M od as ngular momentum as Rotar as ngular momentum vector s fed n space const Torsten Maer-Gürr 3
31 Insttute of Geodes Rotaton of the Earth Rotar vector acos t t C asn t t wth const c Propertes of the Earth Mass M 5, kg Equator radus R ,6 m Moment of nerta,39618 MR Moment of nerta, MR Moment of nerta C,3377 MR Dal Rotaton 7, rad/s Results n the Euler perod of C 35 sdereal das Torsten Maer-Gürr 31
32 Insttute of Geodes Rotaton of the Earth ~35 das Torsten Maer-Gürr 3
33 Insttute of Geodes Rotaton of the Earth Rotar vector acos t t C asn t t wth const c Propertes of the Earth Mass M 5, kg Equator radus R ,6 m Moment of nerta,39618 MR Moment of nerta, MR Moment of nerta C,3377 MR Dal Rotaton 7, rad/s Results n the Euler perod of C 35 sdereal das Seth Carlo Chandler ut: Observed s the Chandler perod of ~43 sdereal das Torsten Maer-Gürr 33
34 Insttute of Geodes Can the torques of sun and moon eplan the Chandler perod? Torsten Maer-Gürr 34
35 Insttute of Geodes Torsten Maer-Gürr 35 Torques of sun and moon C C r GM t M 3 5 Torques wth the poston of sun and moon t r
36 Insttute of Geodes Rotaton of the Earth ~35 das Torsten Maer-Gürr 36
37 Insttute of Geodes Rotaton of the Earth wth torque of the moon ~14 das ~35 das Torsten Maer-Gürr 37
38 Insttute of Geodes Rotaton of the Earth wth torque of the sun ~18 das Torsten Maer-Gürr 38
39 Insttute of Geodes Torsten Maer-Gürr 39 Torques of sun and moon C C r GM t M 3 5 Torques wth the poston of sun and moon t r
40 Insttute of Geodes Torsten Maer-Gürr 4 Torques of sun and moon C C r GM t M 3 5 Torques wth the poston of sun and moon t r
41 Insttute of Geodes Torques do not change the Euler perod to the Chandler perod. Torsten Maer-Gürr 41
42 Insttute of Geodes Rotaton vector of the Earth Sgrd öhm Torsten Maer-Gürr 4
43 Insttute of Geodes Rotaton vector of the Earth Torsten Maer-Gürr 43
44 Insttute of Geodes ength of da Torsten Maer-Gürr 44
45 Insttute of Geodes The Earth s not a rgd bod mass transports n the atmosphere ocean crosphere Hdrosphere Earth tde Plate tectoncs Post glacal rebound Core Earth quakes Dnamc Sstem Earth MIT/Hastack Torsten Maer-Gürr 45
46 Insttute of Geodes Euler-ouvlle equaton eonhard Euler Joseph ouvlle Torsten Maer-Gürr 46
47 Insttute of Geodes ngular momentum balance Total angular momentum: r m r Partcle sstem Torsten Maer-Gürr 47
48 Insttute of Geodes ngular momentum balance Total angular momentum: r r dm D dm dm Dervaton operator d D dt D dm Sold deformable bod Mass term T Moton term h Relatve angular momentum Total angular momentum: T h Torsten Maer-Gürr 48
49 Insttute of Geodes Torsten Maer-Gürr 49 ngular momentum balance alance equaton Dervaton operator D dt d M In the rotatng sstem M D ngular momentum h T M h T h T D Euler-ouvlle equaton M h h T T T D D M h T h T T D D T D
50 Insttute of Geodes Rotaton deformaton pole tde Euler-ouvlle equaton T T DT h Dh M Torsten Maer-Gürr 5
51 Insttute of Geodes Rotaton deformaton pole tde Euler-ouvlle equaton T T DT h Dh M Torsten Maer-Gürr 51
52 Insttute of Geodes Rotaton deformaton pole tde Euler-ouvlle equaton T T DT h Dh M The Rotaton deformaton pole tdes changes the nerta tensor and slows down the Euler perod to the Chandler perod Torsten Maer-Gürr 5
53 Insttute of Geodes Rotaton deformaton pole tde Torsten Maer-Gürr 53
54 Insttute of Geodes Mass varatons n the atmosphere Torsten Maer-Gürr 54
55 Insttute of Geodes Torsten Maer-Gürr 55 tmospherc model NCEP Moton term Floran Set 4: tmospherc and Oceanc Influences on Earth rotaton t t t t h
56 Insttute of Geodes Inerta tensor T D E D F E F C tmospherc model NCEP E F C Floran Set 4: tmospherc and Oceanc Influences on Earth rotaton Torsten Maer-Gürr 56
57 Insttute of Geodes sun, moon, planets sold Earth tdes ocean tdes The dnamc sstem Earth ocean currents atmospherc tdes atmosphere currents gravt feld of the Earth ground water postglacal rebound plate tectonc vulkansm geometr of the Earth ocean loadng ocean angular momentum atmosphere loadng atmos. angular momentum mass dstrbuton rotaton deformaton Earth quakes vegetaton core, mantel... torque orentaton of the Earth nerta tensor Torsten Maer-Gürr 57
58 Insttute of Geodes sun, moon, planets sold Earth tdes ocean tdes The dnamc sstem Earth ocean currents atmospherc tdes atmosphere currents gravt feld of the Earth ground water postglacal rebound plate tectonc vulkansm geometr of the Earth ocean loadng ocean angular momentum atmosphere loadng atmos. angular momentum mass dstrbuton rotaton deformaton Earth quakes vegetaton core, mantel... torque orentaton of the Earth nerta tensor Torsten Maer-Gürr 58
59 Insttute of Geodes Earth rotaton cannot predcted precsel, but must be observed dal Torsten Maer-Gürr 59
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