C1. Configuration of a mechanical system - Revision history

Bros: /* C1.4 Orientation of a rigid body moving in space */

2024-11-11T14:43:49Z

C1.4 Orientation of a rigid body moving in space

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	[[File:C1-Ex3-eng.png\|thumb\|center\|500px\|link=]]		[[File:C1-Ex3-eng.png\|thumb\|center\|500px\|link=]]
	::Starting from that orientation, the configuration <math>\psi = \varphi = 0</math> , <math>\theta = \pi/2</math> for the A family and the configuration <math>\psi = \theta =</math> 0, <math>\varphi = \pi/2</math> for the B family correspond to the same ship orientation. Family A goes through a singularity because the 1st and the 3rd axes are parallel, whereas in family B the 1st and 3d axes are orthogonal, thus far from the singularity.		::Starting from that orientation, the configuration <math>\psi = \varphi = 0</math> , <math>\theta = \pi/2</math> for the A family and the configuration <math>\psi = \theta =</math> 0, <math>\varphi = \pi/2</math> for the B family correspond to the same ship orientation. Family A goes through a singularity because the 1st and the 3rd axes are parallel, whereas in family B the 1st and 3d axes are orthogonal, thus far from the singularity.
			</small>


			In ships, airplanes and vehicles in general, Euler rotations have names associated with the direction of the axes in the reference (or initial) configuration (where all angles are zero). The rotation whose axis is initially parallel to the longitudinal direction of the vehicle is called '''roll'''. The rotation whose axis initially is parallel to the transverse direction of the vehicle is called '''pitch'''. That whose the axis initially is parallel to the direction perpendicular to the two previous ones (which coincides with the vertical direction if the vehicle is parked on flat ground) is called '''yaw'''.

	~~::In ships~~, ~~airplanes and vehicles~~ in ~~general~~, ~~Euler rotations have~~ names ~~associated with the direction of the axes in the reference~~ (~~or initial~~) ~~configuration (where all angles~~ are ~~zero). The rotation~~ whose ~~axis is initially parallel~~ to the ~~longitudinal direction of~~ the vehicle ~~is called '''roll'''. The rotation whose axis initially is parallel to the~~ transverse ~~direction of the vehicle is called '''pitch'''. That whose the axis initially is parallel to the direction~~ perpendicular to ~~the~~ two ~~previous ones (which coincides with the vertical direction if the vehicle is parked on flat ground~~) ~~is called '''yaw'''~~.		It should be noted that, when the vehicle is in configurations other than the initial one, these names (roll, pitch and yaw) are associated with Euler rotations, whose axes no longer have to coincide with the three directions fixed to the vehicle (longitudinal, transverse and perpendicular to these two) mentioned above.

	<~~/small~~>		In '''Video C1.4''' two alternative options are shown: in the first part of the video, the first rotation is roll, the second is pitch, and the third is yaw; in the second part of the video, the first is yaw, the second is pitch, and the third is roll. In most of the literature, yaw is taken as the first Euler rotation, pitch as the second, and roll as the third rotation (as in the second part of Video C1.4 and as in the "B family" of <span style="text-decoration: underline;">[[C1. Configuration of a mechanical system#✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship\|'''example C1-4.3''']]</span>).
	</span>

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Bros: /* ✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship */

2024-11-11T14:41:47Z

✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship

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	</small>		</small>
	</span>).		</span>

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Bros: /* ✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship */

2024-11-11T14:41:27Z

✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship

← Older revision		Revision as of 14:41, 11 November 2024
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	</small>		</small>
			</span>).
	::It should be noted that, when the vehicle is in configurations other than the initial one, these names (roll, pitch and yaw) are associated with Euler rotations, whose axes no longer have to coincide with the three directions fixed to the vehicle (longitudinal, transverse and perpendicular to these two) mentioned above.
	In '''Video C1.4''' two alternative options are shown: in the first part of the video, the first rotation is roll, the second is pitch, and the third is yaw; in the second part of the video, the first is yaw, the second is pitch, and the third is roll. In most of the literature, yaw is taken as the first Euler rotation, pitch as the second, and roll as the third rotation (as in the second part of '''Video C1.4''' and as in the "B family" of <span style="text-decoration: underline;">[[C1. Configuration of a mechanical system#✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship\|'''example C1-4.3''']]</span>).

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Bros: /* C1.4 Orientation of a rigid body moving in space */

2024-11-11T14:41:01Z

C1.4 Orientation of a rigid body moving in space

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	</small>		</small>

			::It should be noted that, when the vehicle is in configurations other than the initial one, these names (roll, pitch and yaw) are associated with Euler rotations, whose axes no longer have to coincide with the three directions fixed to the vehicle (longitudinal, transverse and perpendicular to these two) mentioned above.
			In '''Video C1.4''' two alternative options are shown: in the first part of the video, the first rotation is roll, the second is pitch, and the third is yaw; in the second part of the video, the first is yaw, the second is pitch, and the third is roll. In most of the literature, yaw is taken as the first Euler rotation, pitch as the second, and roll as the third rotation (as in the second part of '''Video C1.4''' and as in the "B family" of <span style="text-decoration: underline;">[[C1. Configuration of a mechanical system#✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship\|'''example C1-4.3''']]</span>).

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Bros: /* ✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship */

2024-11-11T14:40:17Z

✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship

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	[[File:C1-Ex3-eng.png\|thumb\|center\|500px\|link=]]		[[File:C1-Ex3-eng.png\|thumb\|center\|500px\|link=]]
	::Starting from that orientation, the configuration <math>\psi = \varphi = 0</math> , <math>\theta = \pi/2</math> for the A family and the configuration <math>\psi = \theta =</math> 0, <math>\varphi = \pi/2</math> for the B family correspond to the same ship orientation. Family A goes through a singularity because the 1st and the 3rd axes are parallel, whereas in family B the 1st and 3d axes are orthogonal, thus far from the singularity.		::Starting from that orientation, the configuration <math>\psi = \varphi = 0</math> , <math>\theta = \pi/2</math> for the A family and the configuration <math>\psi = \theta =</math> 0, <math>\varphi = \pi/2</math> for the B family correspond to the same ship orientation. Family A goes through a singularity because the 1st and the 3rd axes are parallel, whereas in family B the 1st and 3d axes are orthogonal, thus far from the singularity.
	::~~When orientating a ship (~~and vehicles in general~~) with Euler angles~~, ~~the~~ Euler rotations ~~also~~ have ~~specific~~ names:
	~~::<math>\psi</math>: yaw~~		::In ships, airplanes and vehicles in general, Euler rotations have names associated with the direction of the axes in the reference (or initial) configuration (where all angles are zero). The rotation whose axis is initially parallel to the longitudinal direction of the vehicle is called '''roll'''. The rotation whose axis initially is parallel to the transverse direction of the vehicle is called '''pitch'''. That whose the axis initially is parallel to the direction perpendicular to the two previous ones (which coincides with the vertical direction if the vehicle is parked on flat ground) is called '''yaw'''.
	~~::<math>\theta</math>:~~ roll
	~~::<math>\varphi</math>:~~ pitch

	</small>		</small>

Bros at 19:36, 9 November 2024

2024-11-09T19:36:05Z

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	\newcommand{\evec}{\overline{\textbf{e}}}		\newcommand{\evec}{\overline{\textbf{e}}}
	\newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}		\newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}
			\newcommand{\Bs}{\textrm{B}}
			\newcommand{\Cs}{\textrm{C}}
	\newcommand{\ds}{\textrm{d}}		\newcommand{\ds}{\textrm{d}}
	\newcommand{\ts}{\textrm{t}}		\newcommand{\ts}{\textrm{t}}
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	\newcommand{\Rs}{\textrm{R}}		\newcommand{\Rs}{\textrm{R}}
	\newcommand{\Ts}{\textrm{T}}		\newcommand{\Ts}{\textrm{T}}
	~~\newcommand{\Ts}{\textrm{T}}~~
	~~\newcommand{\Bs}{\textrm{B}}~~
	\newcommand{\es}{\textrm{e}}		\newcommand{\es}{\textrm{e}}
	\newcommand{\is}{\textrm{i}}		\newcommand{\is}{\textrm{i}}

Bros at 19:35, 9 November 2024

2024-11-09T19:35:30Z

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	'''Figure C1.8''' illustrates this for a triangular object undergoing three rotations of 90<math>\deg</math> about three directions fixed in a reference frame R.		'''Figure C1.8''' illustrates this for a triangular object undergoing three rotations of 90<math>\deg</math> about three directions fixed in a reference frame R.

	[[File:C1-8-eng.png\|thumb\|center\|500px\|link]]		[[File:C1-8-eng-new.png\|thumb\|center\|500px\|link]]
	<small><center>'''Figure C1.8''' Rotations about directions fixed in R: sequential method of orientation</center></small>		<small><center>'''Figure C1.8''' Rotations about directions fixed in R: sequential method of orientation</center></small>

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	:The following Euler axes can be defined:		:The following Euler axes can be defined:
	::* 1st axis: vertical (direction of the gravitational field <math>\Downarrow g</math>)		::* 1st axis: vertical (direction of the gravitational field <math>\Downarrow g</math>)
	::* 3rd axis: perpendicular to side 3		::* 3rd axis: perpendicular to side 4
	::* 2nd axis: orthogonal to the other two (thus, intersection of the horizontal plane with that of side 3)		::* 2nd axis: orthogonal to the other two (thus, intersection of the horizontal plane with that of side 4)
	\|}		\|}

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	::[[File:C1-Ex6-eng.png\|thumb\|left\|250px\|link=]]		::[[File:C1-Ex6-eng.png\|thumb\|left\|250px\|link=]]
	\|		\|
	:In a simplified model of a vehicle as that in example 5.2, the variable inclination of the wheels on the horizontal ground can be neglected. The configuration of any wheel is univocally defined through the <math>(x,y)</math> coordinates of its center <math>\Cs</math>, the angle <math>\psi</math> orientating the plane that contains the wheel, and the angle <math>\varphi</math> about its symmetry axis. It is a rigid body with four independent coordinates.		:In a simplified model of a vehicle as that in example C1-5.2, the variable inclination of the wheels on the horizontal ground can be neglected. The configuration of any wheel is univocally defined through the <math>(x,y)</math> coordinates of its center <math>\Cs</math>, the angle <math>\psi</math> orientating the plane that contains the wheel, and the angle <math>\varphi</math> about its symmetry axis. It is a rigid body with four independent coordinates.
	\|}		\|}
	</small>		</small>

Bros: /* C1.1 Position of a particle */

2024-02-23T16:21:51Z

C1.1 Position of a particle

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	[[File:C1-2-new-eng.png\|thumb\|center\|550px\|link=]]		[[File:C1-2 new -eng.png\|thumb\|center\|550px\|link=]]
	<small><center>'''Figure C1.2''' Description of the position of a particle Q relative to a reference frame R through three coordinates</center></small>		<small><center>'''Figure C1.2''' Description of the position of a particle Q relative to a reference frame R through three coordinates</center></small>

Bros: /* C1.2 Configuration of a rigid body */

2024-02-23T16:19:28Z

C1.2 Configuration of a rigid body

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	\|+		\|+
	\|-		\|-
	\|[[File:C1-5-neut.png\|center\|180px\|link=]] \|\|[[File:C1-5-eng-2.png\|center\|~~550px~~\|link=]]		\|[[File:C1-5-neut.png\|center\|180px\|link=]] \|\|[[File:C1-5-eng-2.png\|center\|450px\|link=]]
	\|}		\|}
	</center>		</center>

Bros: /* C1.2 Configuration of a rigid body */

2024-02-23T16:19:15Z

C1.2 Configuration of a rigid body

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	\|+		\|+
	\|-		\|-
	\|[[File:C1-5-neut.png\|center\|180px\|link=]] \|\|[[File:C1-5-2.png\|center\|550px\|link=]]		\|[[File:C1-5-neut.png\|center\|180px\|link=]] \|\|[[File:C1-5-eng-2.png\|center\|550px\|link=]]
	\|}		\|}
	</center>		</center>