Changeset 1074 for branches/maintenance/3.0/Modelica/Mechanics/MultiBody/package.mo

Timestamp:

02/26/08 09:25:23 (9 months ago)

Author:

dietmarw

Message:

updated the copyright notices to 2008 and corrected some Umlauts to HTML entities when present in the HTML environment

Files:

• branches/maintenance/3.0/Modelica/Mechanics/MultiBody/package.mo (modified) (43 diffs)

branches/maintenance/3.0/Modelica/Mechanics/MultiBody/package.mo

@@ -16 +16 @@
 3-dimensional mechanical components to model in a convenient way
 <b>mechanical systems</b>, such as robots, mechanisms, vehicles.
-Typical animations generated with this library are shown 
+Typical animations generated with this library are shown
 in the next figure:
 </p>
@@ -33 +33 @@
 
 <p>
-Copyright &copy; 1998-2007, Modelica Association and DLR.
+Copyright &copy; 1998-2008, Modelica Association and DLR.
 </p>
 <p>
 <i>This Modelica package is <b>free</b> software; it can be redistributed and/or modified
 under the terms of the <b>Modelica license</b>, see the license conditions
-and the accompanying <b>disclaimer</b> 
+and the accompanying <b>disclaimer</b>
 <a href=\"Modelica://Modelica.UsersGuide.ModelicaLicense\">here</a>.</i>
 </p><br>
@@ -98 +98 @@
 </li>
 <li>
-<a href=\"Modelica://Modelica.Mechanics.MultiBody.UsersGuide.Tutorial.FirstExample\">A first example</a> 
+<a href=\"Modelica://Modelica.Mechanics.MultiBody.UsersGuide.Tutorial.FirstExample\">A first example</a>
   describes in detail all the steps to build a simple pendulum model.
 </li>
@@ -334 +334 @@
 <p>
 In principal, now special action is needed, if
-loop structures occur (contrary to the ModelicaAdditions.MultiBody library). 
-An example is presented in the figure below. It is available as 
+loop structures occur (contrary to the ModelicaAdditions.MultiBody library).
+An example is presented in the figure below. It is available as
 <a href=\"Modelica://Modelica.Mechanics.MultiBody.Examples.Loops.Fourbar1\">
 MultiBody.Examples.Loops.Fourbar1</a>
@@ -343 +343 @@
 </p>
 <p>
-This mechanism consists of 6 revolute joints, 1 prismatic joint and forms a 
-kinematical loop. 
-It has has one degree of freedom. In the next figure the default animation 
-is shown. Note, that the axes of the revolute joints are represented by the 
-red cylinders and that the axis of the prismatic joint is represented by the 
-red box on the lower right side. 
+This mechanism consists of 6 revolute joints, 1 prismatic joint and forms a
+kinematical loop.
+It has has one degree of freedom. In the next figure the default animation
+is shown. Note, that the axes of the revolute joints are represented by the
+red cylinders and that the axis of the prismatic joint is represented by the
+red box on the lower right side.
 </p>
 <p align=\"center\">
@@ -354 +354 @@
 </p>
 <p>
-Whenever loop structures occur, non-linear algebraic 
-equations are present on \"position level\". It is then usually not possible by 
-structural analysis to select states during translation (which is possible for 
-non-loop structures). In the example above, Dymola detects a non-linear 
-algebraic loop of 57 equations and reduces this to a system of 7 coupled 
-algebraic equations. Note, that this is performed without using any 
-\"cut-joints\" as it is usually done in multi-body programs, but by just 
-appropriate symbolic equation manipulation. Via the dynamic dummy derivative 
-method the generalized coordinates on position and velocity level from one of 
-the 7 joints are dynamically selected as states during simulation. Whenever, 
-these two states are no longer appropriate, states from one of the other 
+Whenever loop structures occur, non-linear algebraic
+equations are present on \"position level\". It is then usually not possible by
+structural analysis to select states during translation (which is possible for
+non-loop structures). In the example above, Dymola detects a non-linear
+algebraic loop of 57 equations and reduces this to a system of 7 coupled
+algebraic equations. Note, that this is performed without using any
+\"cut-joints\" as it is usually done in multi-body programs, but by just
+appropriate symbolic equation manipulation. Via the dynamic dummy derivative
+method the generalized coordinates on position and velocity level from one of
+the 7 joints are dynamically selected as states during simulation. Whenever,
+these two states are no longer appropriate, states from one of the other
 joints are selected during simulation.
 </p>
 <p>
-The efficiency of loop structures can usually be 
-enhanced, if states are statically fixed at translation time. For this 
-mechanism, the generalized coordinates of joint j1 (i.e., the 
-rotation angle of the revolute joint and its derivative) can always be used as 
-states. This can be stated by setting parameter \"enforceStates = <b>true</b>\" 
-in the \"Advanced\" menu of the desired joint. This flag sets the attribute 
-stateSelect of the generalized coordinates of the coresponding joint to 
-\"StateSelect.always\". When setting this flag to <b>true</b> for joint j1 in 
-the four bar mechanism, Dymola detects a non-linear algebraic loop of 40 
-equations and reduces this to a system of 5 coupled non-linear algebraic 
+The efficiency of loop structures can usually be
+enhanced, if states are statically fixed at translation time. For this
+mechanism, the generalized coordinates of joint j1 (i.e., the
+rotation angle of the revolute joint and its derivative) can always be used as
+states. This can be stated by setting parameter \"enforceStates = <b>true</b>\"
+in the \"Advanced\" menu of the desired joint. This flag sets the attribute
+stateSelect of the generalized coordinates of the coresponding joint to
+\"StateSelect.always\". When setting this flag to <b>true</b> for joint j1 in
+the four bar mechanism, Dymola detects a non-linear algebraic loop of 40
+equations and reduces this to a system of 5 coupled non-linear algebraic
 equations.
 </p>
@@ -384 +384 @@
 equations analytically. For a certain class of systems this can
 be performed also with the MultiBody library. This technique
-is described in section 
+is described in section
 \"<a href=\"AnalyticLoopHandling\">Analytic loop handling</a>\".
 </p>
@@ -394 +394 @@
         annotation (Documentation(info="<HTML>
 <p>
-In the figure below, the model of a V6 engine is 
-shown that has a simple combustion model. It is available as 
+In the figure below, the model of a V6 engine is
+shown that has a simple combustion model. It is available as
 <a href=\"Modelica://Modelica.Mechanics.MultiBody.Examples.Loops.EngineV6\">
-MultiBody.Examples.Loops.EngineV6</a>. 
+MultiBody.Examples.Loops.EngineV6</a>.
 </p>
 <p align=\"center\">
@@ -403 +403 @@
 </p>
 <p>
-The Modelica schematic of one cylinder 
-is given in the figure below. Connecting 6 instances of this 
+The Modelica schematic of one cylinder
+is given in the figure below. Connecting 6 instances of this
 cylinder appropriately together results in the engine schematic displayed
 above.
@@ -412 +412 @@
 </p>
 <p>
-In the next figure the animation of the engine is 
-shown. Every cylinder consists essentially of 1 prismatic and 2 revolute 
-joints that form a planar loop, since the axes of the two revolute joints are 
-parallel to each other and the axis of the prismatic joint is orthogonal to 
-the revolute joint axes. All 6 cylinders together form a coupled set of 6 
-loops that have together 1 degree of freedom. 
+In the next figure the animation of the engine is
+shown. Every cylinder consists essentially of 1 prismatic and 2 revolute
+joints that form a planar loop, since the axes of the two revolute joints are
+parallel to each other and the axis of the prismatic joint is orthogonal to
+the revolute joint axes. All 6 cylinders together form a coupled set of 6
+loops that have together 1 degree of freedom.
 </p>
 <p align=\"center\">
@@ -423 +423 @@
 </p>
 <p>
-All planar loops, and especially the engine, result in 
+All planar loops, and especially the engine, result in
 a DAE (= Differential-Algebraic Equation system)
-that does not have a unique solution. The reason is that, e.g., the cut 
-forces in direction of the axes of the revolute joints cannot be uniquely 
-computed. Any value fulfills the DAE equations. This is a structural property 
-that is determined by the symbolic algorithms. Since they detect that the DAE is 
-structurally singular, a further processing is not possible. Without 
-additional information it is also impossible that the symbolic algorithms 
-could be enhanced because if the axes of rotations of the revolute joints are 
-only slightly changed such that they are no longer parallel to each other, the 
-planar loop can no longer move and has 0 degrees of freedom. Algorithms based 
-on pure structural information cannot distinguish these two cases. 
-</p>
-<p>
-The usual remedy is to remove superfluous constraints, 
-e.g., along the axis of rotation of <b>one</b> revolute joint. Since this is 
+that does not have a unique solution. The reason is that, e.g., the cut
+forces in direction of the axes of the revolute joints cannot be uniquely
+computed. Any value fulfills the DAE equations. This is a structural property
+that is determined by the symbolic algorithms. Since they detect that the DAE is
+structurally singular, a further processing is not possible. Without
+additional information it is also impossible that the symbolic algorithms
+could be enhanced because if the axes of rotations of the revolute joints are
+only slightly changed such that they are no longer parallel to each other, the
+planar loop can no longer move and has 0 degrees of freedom. Algorithms based
+on pure structural information cannot distinguish these two cases.
+</p>
+<p>
+The usual remedy is to remove superfluous constraints,
+e.g., along the axis of rotation of <b>one</b> revolute joint. Since this is
 not easy for an inexperienced modeler, the special joint:
 <a href=\"Modelica://Modelica.Mechanics.MultiBody.Joints.RevolutePlanarLoopConstraint\">
-RevolutePlanarLoopConstraint</a> is provided that removes these constraints. 
+RevolutePlanarLoopConstraint</a> is provided that removes these constraints.
 Exactly one revolute joint in a every planar loop must be replaced by this
 joint type. In the engine example, this special joinst is used for
@@ -448 +448 @@
 </p>
 <p>
-If a modeler is not aware of the problems with planar 
+If a modeler is not aware of the problems with planar
 loops and models them without special consideration, a Modelica
-translator, such as Dymola, displays an error 
-message and points out that a planar loop may be the reason and suggests to 
+translator, such as Dymola, displays an error
+message and points out that a planar loop may be the reason and suggests to
 use the RevolutePlanarLoopConstraint joint.
-This error message is due to an annotation in 
+This error message is due to an annotation in
 the Frame connector.
 </p>
@@ -463 +463 @@
 </pre>
 <p>
-If no assignment can be found for some 
-forces in a connector, the \"unassignedMessage\" is displayed. In most cases the 
-reason for this is a planar loop or two joints that constrain the same motion. 
+If no assignment can be found for some
+forces in a connector, the \"unassignedMessage\" is displayed. In most cases the
+reason for this is a planar loop or two joints that constrain the same motion.
 Both cases are discussed in the error message.
 </p>
 <p>
-Note, that the non-linear algebraic equations occurring 
-in planar loops can be solved analytically in most cases and therefore it is 
+Note, that the non-linear algebraic equations occurring
+in planar loops can be solved analytically in most cases and therefore it is
 highly recommended to use the techniques discussed in section
 \"<a href=\"AnalyticLoopHandling\">Analytic loop handling</a>\"
@@ -481 +481 @@
 
         annotation (Documentation(info="<HTML>
-It is well known that the non-linear 
-algebraic equations of most mechanical loops in technical devices can be 
-solved analytically. It is, however, difficult to perform this fully 
-automatically and therefore none of the commercial, general purpose multi-body 
-programs, such as MSC ADAMS, LMS DADS, SIMPACK, have this feature. 
-These programs solve loop structures with pure numerical methods. Multi-body 
-programs that are designed for real-time simulation of the dynamics of 
-specific vehicles, such as ve-DYNA, usually contain manual implementations 
-of a particular multi-body system (the vehicle) where the occurring loops are 
-either analytically solved, if this is possible, or are treated by table 
-look-up where the tables are constructed in a pre-processing phase. Without 
-these features the required real-time capability would be difficult to 
+It is well known that the non-linear
+algebraic equations of most mechanical loops in technical devices can be
+solved analytically. It is, however, difficult to perform this fully
+automatically and therefore none of the commercial, general purpose multi-body
+programs, such as MSC ADAMS, LMS DADS, SIMPACK, have this feature.
+These programs solve loop structures with pure numerical methods. Multi-body
+programs that are designed for real-time simulation of the dynamics of
+specific vehicles, such as ve-DYNA, usually contain manual implementations
+of a particular multi-body system (the vehicle) where the occurring loops are
+either analytically solved, if this is possible, or are treated by table
+look-up where the tables are constructed in a pre-processing phase. Without
+these features the required real-time capability would be difficult to
 achieve.
 </p>
@@ -499 +499 @@
 Prof. Hiller and his group in Duisburg, Germany,
 have developed systematic methods to handle mechanical
-loops analytically. The \"characteristic pair of joints\" method 
-basically cuts a loop at two joints and uses geometric 
-invariants to reduce the number of algebraic 
-equations, often down to one equation that can be solved analytically. Also 
-several multi-body codes have been developed that are based on this method, 
-e.g., MOBILE. Besides the very desired feature to solve non-linear 
-algebraic equations analytically, i.e., efficiently and in a robust way, there 
-are several drawbacks: It is difficult to apply this method automatically. 
-Even if this would be possible in a good way, there is always the problem that 
-it cannot be guaranteed that the statically selected states lead to no 
-singularity during simulation. Therefore, the \"characteristic pair of joints\" 
+loops analytically. The \"characteristic pair of joints\" method
+basically cuts a loop at two joints and uses geometric
+invariants to reduce the number of algebraic
+equations, often down to one equation that can be solved analytically. Also
+several multi-body codes have been developed that are based on this method,
+e.g., MOBILE. Besides the very desired feature to solve non-linear
+algebraic equations analytically, i.e., efficiently and in a robust way, there
+are several drawbacks: It is difficult to apply this method automatically.
+Even if this would be possible in a good way, there is always the problem that
+it cannot be guaranteed that the statically selected states lead to no
+singularity during simulation. Therefore, the \"characteristic pair of joints\"
 method is usually manually applied which requires know-how and experience.
 </p>
 <p>
-In the MultiBody library the \"characteristic pair of 
-joints\" method is supported in a restricted form such that it can be applied 
-also by non-specialists. The idea is to provide aggregations of joints in package 
+In the MultiBody library the \"characteristic pair of
+joints\" method is supported in a restricted form such that it can be applied
+also by non-specialists. The idea is to provide aggregations of joints in package
 <a href=\"Modelica://Modelica.Mechanics.MultiBody.Joints.Assemblies\">
-MultiBody.Joints.Assemblies</a>. 
-as one object that either have <b>6</b> degrees of freedom or 
-<b>3</b> degrees of freedom (for usage in planar loops). 
-</p>
-<p>
-As an example, a variant of the four bar mechanism is given in 
+MultiBody.Joints.Assemblies</a>.
+as one object that either have <b>6</b> degrees of freedom or
+<b>3</b> degrees of freedom (for usage in planar loops).
+</p>
+<p>
+As an example, a variant of the four bar mechanism is given in
 the figure below.
 </p>
@@ -531 +531 @@
 Here, the mechanism is modeled with one revolute
 joint, two spherical joints and one prismatic joint.
-In the figure below, the two spherical joints 
-and the prismatic joint are collected together in an assembly object 
+In the figure below, the two spherical joints
+and the prismatic joint are collected together in an assembly object
 called \"jointSSP\" from
 <a href=\"Modelica://Modelica.Mechanics.MultiBody.Joints.Assemblies.JointSSP\">
-MultiBody.Joints.Assemblies.JointSSP</a>. 
+MultiBody.Joints.Assemblies.JointSSP</a>.
 </p>
 <p align=\"center\">
@@ -541 +541 @@
 </p>
 <p>
-The JointSSP joint aggregation has a frame at the 
-left side of the left spherical joint (frame_a) and a frame at the right side 
-of the prismatic joint (frame_b). JointSSP, as all other objects from the 
+The JointSSP joint aggregation has a frame at the
+left side of the left spherical joint (frame_a) and a frame at the right side
+of the prismatic joint (frame_b). JointSSP, as all other objects from the
 Joints.Assemblies package, has the property, that the <b>generalized
-coordinates, and all other frames 
-defined in the assembly, can be calculated given the movement of frame_a and 
-of frame_b</b>. This is performed by <b>analytically</b> solving non-linear 
-systems of equations (details are given in section xxx). From a 
-structural point of view, the equations in an assembly object are written in 
+coordinates, and all other frames
+defined in the assembly, can be calculated given the movement of frame_a and
+of frame_b</b>. This is performed by <b>analytically</b> solving non-linear
+systems of equations (details are given in section xxx). From a
+structural point of view, the equations in an assembly object are written in
 the form
 </p>
@@ -558 +558 @@
 </p>
 <p>
-where<b> r</b><sup>a</sup>, <b>R</b><sup>a</sup>,<b> 
-r</b><sup>b</sup>, <b>R</b><sup>b</sup><b> </b>are the variables defining the 
-position and orientation of the frame_a and frame_b connector, 
-<b>q</b> are the generalized positional coordinates inside the 
+where<b> r</b><sup>a</sup>, <b>R</b><sup>a</sup>,<b>
+r</b><sup>b</sup>, <b>R</b><sup>b</sup><b> </b>are the variables defining the
+position and orientation of the frame_a and frame_b connector,
+<b>q</b> are the generalized positional coordinates inside the
 assembly, e.g., the angle of a revolute joint. Given angle
-<span style=\"font-family:Symbol\">j</span> of revolute joint j1 from the four 
-bar mechanism, frame_a and frame_b of the assembly object can be computed by a 
+<span style=\"font-family:Symbol\">j</span> of revolute joint j1 from the four
+bar mechanism, frame_a and frame_b of the assembly object can be computed by a
 forward recursion
 </p>
@@ -572 +572 @@
 </p>
 <p>
-Since this is a structural property, the 
+Since this is a structural property, the
 symbolic algorithms can automatically select <span style=\"font-family:Symbol\">
-j</span> and its derivative as states and then all positional variables can be 
+j</span> and its derivative as states and then all positional variables can be
 computed in a forwards sequence. It is now understandable that a Modelica
 translator, such as Dymola, can
-transform the equations of the four bar mechanism to a recursive sequence of 
-statements that has no non-linear algebraic loops anymore(remember, 
+transform the equations of the four bar mechanism to a recursive sequence of
+statements that has no non-linear algebraic loops anymore(remember,
 the previous \"straightforward\" solution with 6 revolute joints and 1
 prismatic joint has a nonlinear system of equations of order 5).
 </p>
 <p>
-The aggregated joint 
-objects consist of a combination of either a revolute or prismatic joint and 
-of a rod that has either two spherical joints at its two ends or a spherical 
-and a universal joint, respectively. For all combinations, analytic solutions 
-can be determined. For planar loops, combinations of 1, 2 or 3 revolute joints 
-with parallel axes and of 2 or 1 prismatic joint with axes that are orthogonal 
-to the revolute joints can be treated analytically. The currently supported 
-combinations are listed in the table below. 
-The missing combinations (such as JointSUP 
-or Joint RPP) will be added in one of the next releases. 
+The aggregated joint
+objects consist of a combination of either a revolute or prismatic joint and
+of a rod that has either two spherical joints at its two ends or a spherical
+and a universal joint, respectively. For all combinations, analytic solutions
+can be determined. For planar loops, combinations of 1, 2 or 3 revolute joints
+with parallel axes and of 2 or 1 prismatic joint with axes that are orthogonal
+to the revolute joints can be treated analytically. The currently supported
+combinations are listed in the table below.
+The missing combinations (such as JointSUP
+or Joint RPP) will be added in one of the next releases.
 </p>
 <div align=\"center\">
@@ -634 +634 @@
 <p>
 On first view this seems to be quite restrictive. However, mechanical devices are usually built up with rods connected by spherical joints on each end,
-and additionally with revolute and prismatic joints. 
-Therefore, the combinations of the above table occur frequently. 
-The universal joint is usually not present in actual devices but is used 
-(a) if two JointXXX components can be connected such that a revolute 
+and additionally with revolute and prismatic joints.
+Therefore, the combinations of the above table occur frequently.
+The universal joint is usually not present in actual devices but is used
+(a) if two JointXXX components can be connected such that a revolute
 and a universal joint together form a spherical joint
-and (b) if the orientation of the connecting rod between two 
-spherical joints is needed, e.g., since a body shall be attached. 
-In this case one of the spherical joints might be replaced by a 
-universal joint. This approximation is fine as long as the mass 
+and (b) if the orientation of the connecting rod between two
+spherical joints is needed, e.g., since a body shall be attached.
+In this case one of the spherical joints might be replaced by a
+universal joint. This approximation is fine as long as the mass
 and inertia of the rod is not significant.
 </p>
 <p>
-Let us discuss item (a) in more detail: The 
-MacPherson suspension in the next figure 
-is from the Modelica VehicleDynamics library. 
+Let us discuss item (a) in more detail: The
+MacPherson suspension in the next figure
+is from the Modelica VehicleDynamics library.
 <p align=\"center\">
 <IMG SRC=\"../Images/MultiBody/Tutorial/LoopStructures/MacPherson1.png\">
 </p>
-It has three frame 
-connectors. The lower left one (frame_C) is fixed in the vehicle chassis. The 
-upper left one (frame_S) is driven by the steering mechanism, i.e., the 
-movement of both frames are given. The frame connector on the right (frame_U) 
-drives the wheel. The three frames are connected by a mechanism consisting 
-essentially of two rods with spherical joints on both ends. These are built up 
+It has three frame
+connectors. The lower left one (frame_C) is fixed in the vehicle chassis. The
+upper left one (frame_S) is driven by the steering mechanism, i.e., the
+movement of both frames are given. The frame connector on the right (frame_U)
+drives the wheel. The three frames are connected by a mechanism consisting
+essentially of two rods with spherical joints on both ends. These are built up
 by a jointUPS and a jointSSR assembly.
-As can be seen, the universal joint from the jointUPS 
-assembly is connected to the revolute joint of the jointSSR assembly. 
-Therefore, we have 3 revolute joints connected together at one point and if 
-the axes of rotations are chosen appropriately, this describes a spherical 
-joint. In other words, the two connected assemblies define the desired two 
-rods with spherical joints on each ends. 
-</p>
-<p>
-The movement of the chassis, frame_C, is computed 
-somewhere else. When the generalized coordinates of revolute joint 
-\"innerJoint\" (lower left part in figure) are used as states, then frame_a and 
-frame_b of the jointUPS joint can be calculated. After the non-linear loop 
-with jointUPS is (analytically) solved, all frames on this assembly are 
-known, especially, 
-the one connected to frame_b of the jointSSR assembly. Since frame_b of 
-jointSSR is connected to frame_S which is computed from the steering 
-mechanism, again the two required frame movements of the jointSSR assembly are 
-calculated, meaning in turn that also all other frames on the jointSSR 
-assembly can be computed, especially, the one connected to frame_U that drives 
-the wheel. From this analysis it is clear that a tool is able to solve these 
+As can be seen, the universal joint from the jointUPS
+assembly is connected to the revolute joint of the jointSSR assembly.
+Therefore, we have 3 revolute joints connected together at one point and if
+the axes of rotations are chosen appropriately, this describes a spherical
+joint. In other words, the two connected assemblies define the desired two
+rods with spherical joints on each ends.
+</p>
+<p>
+The movement of the chassis, frame_C, is computed
+somewhere else. When the generalized coordinates of revolute joint
+\"innerJoint\" (lower left part in figure) are used as states, then frame_a and
+frame_b of the jointUPS joint can be calculated. After the non-linear loop
+with jointUPS is (analytically) solved, all frames on this assembly are
+known, especially,
+the one connected to frame_b of the jointSSR assembly. Since frame_b of
+jointSSR is connected to frame_S which is computed from the steering
+mechanism, again the two required frame movements of the jointSSR assembly are
+calculated, meaning in turn that also all other frames on the jointSSR
+assembly can be computed, especially, the one connected to frame_U that drives
+the wheel. From this analysis it is clear that a tool is able to solve these
 coupled loops analytically.
 </p>
 <p>
-Another example is the model of the V6 engine, 
+Another example is the model of the V6 engine,
 see next figure for an animation view and the original definition
 of one cylinder with elementary joints.
@@ -692 +692 @@
 <IMG SRC=\"../Images/MultiBody/Tutorial/LoopStructures/EngineV6_2.png\">
 </p>
- 
+
 <p>
 It is sufficient to rewrite the basic cylinder model
-by replacing the joints with a JointRRP object that has two 
+by replacing the joints with a JointRRP object that has two
 revolute and one prismatic joint, see next figure.
 </p>
@@ -702 +702 @@
 </p>
 <p>
-Since 6 
-cylinders are connected together, 6 coupled loops with 6 JointRRP objects are 
-present. This model is available as 
+Since 6
+cylinders are connected together, 6 coupled loops with 6 JointRRP objects are
+present. This model is available as
 <a href=\"Modelica://Modelica.Mechanics.MultiBody.Examples.Loops.EngineV6_analytic\">
-MultiBody.Examples.Loops.Engine­V6_analytic</a>.
+MultiBody.Examples.Loops.EngineV6_analytic</a>.
 </p>
 <p>
@@ -716 +716 @@
 </p>
 <p>
-It can be seen that the revolute joint 
-of the crank shaft (joint \"bearing\" in left part of figure) might be 
-selected as degree of freedom. Then the 4 connector frames of all cylinders 
-can be computed. As a result the computations of the cylinders are decoupled 
+It can be seen that the revolute joint
+of the crank shaft (joint \"bearing\" in left part of figure) might be
+selected as degree of freedom. Then the 4 connector frames of all cylinders
+can be computed. As a result the computations of the cylinders are decoupled
 from each other. Within one cylinder
-the position of frame_a and frame_b of the jointRRP assembly 
-can be computed and therefore the generalized coordinates of the two revolute 
-and the prismatic joint in the jointRRP object can be determined. From this 
+the position of frame_a and frame_b of the jointRRP assembly
+can be computed and therefore the generalized coordinates of the two revolute
+and the prismatic joint in the jointRRP object can be determined. From this
 analysis it is not surprising that a Modelica translator, such as Dymola,
-is able to transform the DAE 
-equations into a sequential evaluation without any non-linear loop. 
+is able to transform the DAE
+equations into a sequential evaluation without any non-linear loop.
 Compare this nice result with the model using only elementary joints
-that leads to a DAE with 6 algebraic loops and 5 non-linear equations per 
+that leads to a DAE with 6 algebraic loops and 5 non-linear equations per
 loop. Additionally, a linear system of equations of order 43 is present.
-The simulation time is about 5 times faster with the analytic loop handling. 
+The simulation time is about 5 times faster with the analytic loop handling.
 </p>
 </HTML>
@@ -772 +772 @@
 whether automatic conversion shall take place or not.
 Problems are not to be expected. Still one should first make
-a copy of such a model as backup before the conversion 
+a copy of such a model as backup before the conversion
 is performed.
 </p>
@@ -787 +787 @@
 be adapted manually. Still, this script is useful, since
 many class names, parameters and modifiers are
-automatically converted. 
+automatically converted.
 </p>
 <p>
@@ -804 +804 @@
 that are programmed with <b>equations</b> are only partly converted:
 The Frame connectors will be converted to the \"new\"
-Frame connectors of the MultiBody library, but the equations 
-that reference variables of the Frame connectors will 
-<b>not</b> be converted. For a manual conversion, the following 
+Frame connectors of the MultiBody library, but the equations
+that reference variables of the Frame connectors will
+<b>not</b> be converted. For a manual conversion, the following
 table might be helpful showing how the <b>variables</b> of the
-\"old\" and the \"new\" <b>Frame connectors</b> are 
+\"old\" and the \"new\" <b>Frame connectors</b> are
 related to each other (resolve2 and angularVelocity2 are
 functions from library Modelica.Mechanics.MultiBody.Frames):
@@ -862 +862 @@
 (now Modelica.Mechanics.MultiBody.Frames.TransformationMatrices.Orientation).
 In 1.0 the orientation object is a record holding the
-transformation matrix from frame 1 to frame 2 and the angular 
+transformation matrix from frame 1 to frame 2 and the angular
 velocity of the transformation matrix resolved in frame 2.
 The reason is that this allows to compute the angular velocity
@@ -892 +892 @@
 <dt>Otter M., Elmqvist H., and Mattsson S.E.:</dt>
 <dd> <b>The New Modelica MultiBody Library</b>.
-     Modelica 2003 Conference, Link&ouml;ping, Sweden,