source: PSPA/madxPSPA/doc/usrguide/Introduction/global_system.html @ 430

Last change on this file since 430 was 430, checked in by touze, 11 years ago

import madx-5.01.00
File size: 4.3 KB
Line 
1<head>
2<title>Global Reference System</title>
3<!-- Changed by: Chris ISELIN, 17-Jul-1997 -->
4<!-- Changed by: Hans Grote, 10-Jun-2002 -->
5</head>
6
7<body bgcolor="#ffffff">
8
9<center>
10EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
11<IMG SRC="http://cern.ch/madx/icons/mx7_25.gif" align=right>
12<h2>Global Reference System</h2>
13</center>
14
15The <a href="#global">global reference orbit</a> of the accelerator
16is uniquely defined by the sequence of physical elements.
17The local reference system (<i>x</i>, <i>y</i>, <i>s</i>)
18may thus be referred to a global Cartesian coordinate system
19(<i>X</i>, <i>Y</i>, <i>Z</i>) (see <a href="#global">Figure 1</a>).
20The positions between beam elements are numbered 0,...,i,...n.
21The local reference system
22(<i>x<sub>i</sub>, y<sub>i</sub>, s<sub>i</sub></i>)
23at position <i>i</i>,
24i.e. the displacement and direction of the reference orbit
25with respect to the system (<i>X</i>, <i>Y</i>, <i>Z</i>)
26are defined by three displacements
27(<i>X<sub>i</sub></i>, <i>Y<sub>i</sub></i>, <i>Z<sub>i</sub></i>)
28and three angles
29(<i>Theta<sub>i</sub></i>, <i>Phi<sub>i</sub></i>, <i>Psi<sub>i</sub></i>)
30The above quantities are defined more precisely as follows:
31<ul>
32<li>X:
33Displacement of the local origin in <i>X</i>-direction.
34<li>Y:
35Displacement of the local origin in <i>Y</i>-direction.
36<li>Z:
37Displacement of the local origin in <i>Z</i>-direction.
38<li><a name=theta>THETA</a>:
39Angle of rotation (azimuth) about the global <i>Y</i>-axis,
40between the global <i>Z</i>-axis and the projection
41of the reference orbit onto the (<i>Z</i>, <i>X</i>)-plane.
42A positive angle THETA forms a right-hand screw with the <i>Y</i>-axis.
43<li><a name=phi>PHI</a>:
44Elevation angle, i.e. the angle between the reference orbit and its projection
45onto the (<i>Z</i>, <i>X</i>)-plane.
46A positive angle PHI correspond to increasing <i>Y</i>.
47If only horizontal bends are present,
48the reference orbit remains in the (<i>Z</i>, <i>X</i>)-plane.
49In this case PHI is always zero.
50<li><a name=psi>PSI</a>:
51Roll angle about the local <i>s</i>-axis,
52i.e. the angle between the intersection (<i>x</i>, <i>y</i>)- and
53(<i>Z</i>, <i>X</i>)-planes and the local <i>x</i>-axis.
54A positive angle PSI forms a right-hand screw with the <i>s</i>-axis.
55</ul>
56The angles (THETA, PHI, PSI) are <b>not</b> the Euler angles.
57The reference orbit starts at the origin and points by default
58in the direction of the positive <i>Z</i>-axis.
59The initial local axes (<i>x</i>, <i>y</i>, <i>s</i>)
60coincide with the global axes (<i>X</i>, <i>Y</i>, <i>Z</i>) in this order.
61The six quantities (X<sub>0</sub>, Y<sub>0</sub>, Z<sub>0</sub>,
62THETA<sub>0</sub>, PHI<sub>0</sub>, PSI<sub>0</sub>)
63thus all have zero initial values by default.
64The program user may however specify different initial conditions.
65<p>
66Internally the displacement is described by a vector <i>V</i>
67and the orientation by a unitary matrix <i>W</i>.
68The column vectors of <i>W</i> are the unit vectors spanning
69the local coordinate axes in the order (<i>x, y, s</i>).
70<i>V</i> and <i>W</i> have the values:
71<p>
72<img align=bottom src="../equations/VW.gif">
73<p>
74where
75<p>
76<img align=bottom src="../equations/PhiThetaPsi.gif">
77<p>
78The reference orbit should be closed and it should not be twisted.
79This means that the displacement of the local reference system
80must be periodic with the revolution frequency of the accelerator,
81while the position angles must be periodic modulo(2 pi)
82with the revolution frequency.
83If PSI is not periodic module(2 pi), coupling effects are introduced.
84When advancing through a beam element,
85MAD computes <i>V<sub>i</sub></i> and <i>W<sub>i</sub></i>
86by the recurrence relations
87<p>
88<i>V<sub>i</sub> = W<sub>i-1</sub>R<sub>i</sub> + V<sub>i-1</sub></i>,
89<i>W<sub>i</sub> = w<sub>i-1</sub>S<sub>i</sub></i>.
90<p>
91The vector <i>R<sub>i</sub></i> is the displacement and the matrix
92<i>S<sub>i</sub></i> is the rotation of the local reference system
93at the exit of the element <i>i</i> with respect to the entrance
94of the same element.
95The values of <i>R<sub>i</sub></i> and <i>S<sub>i</sub></i>
96are listed in the:
97
98<a href="local_system.html#straight">straight reference system</a>
99for each physical element type.
100<center>
101<a name=global><img align=bottom src="../figures/global.gif"></a>
102<p>
103<b>Figure 1:</b> Global Reference System
104</center>
105<p>
106<address>
107<a href="http://www.cern.ch/Hans.Grote/hansg_sign.html">hansg</a>,
108January 24, 1997
109</address>
110
111</body>
Note: See TracBrowser for help on using the repository browser.