Mechanics - User contributions [en]

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[[File:Segwaywiki.png|right|top|200px|link=]]
This website is specially built to complement the learning of the '''Mechanics''' course in the bachelor's degrees of '''[https://etseib.upc.edu/en Barcelona School of Industrial Engineering (ETSEIB)]''' of the '''[https://www.upc.edu/en Polytechnic University of Catalonia (UPC) · BarcelonaTech]'''.

This aims to be an '''accessible''' and '''interactive''' '''open tool'''. It's development started on 2022 and it gathers more than 50 years of teaching experience. It's content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography.

The contentent is focused on '''general space movement of rigid bodies and muli-body systems''', but '''particles''' are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics.


'''About the status of the site''':

This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊.

The best experience will be through a computer or a tablet 📵.

The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯.


© Universitat Politècnica de Catalunya. [[Mechanics:Copyrights |All rights reserved]]
__NOTOC__
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===[[Introduction]]===
:: [[Introduction#I.1 What is mechanics?|I.1 What is mechanics?]]
:: [[Introduction#I.2 Models for material objects|I.2 Models for material objects]]
:: [[Introduction#I.3 Limitations of Newtonian mechanics|I.3 Limitations of Newtonian mechanics]]
:: [[Introduction#I.4 Reference frame|I.4 Reference frame]]

===[[Vector calculus]]===
::[[Vector calculus#V.1 Geometric representation of a vector|V.1 Geometric representation of a vector]]
::[[Vector calculus#V.2 Operations between vectors with geometric representation|V.2 Operations between vectors with geometric representation]]
::::[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]
::::[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]
::[[Vector calculus#V.3 Analytical representation of a vector|V.3 Analytical representation of a vector]]
::[[Vector calculus#V.4 Operations between vectors with analytical representation|V.4 Operations between vectors with analytical representation]]
::::[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]
::::[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]

==KINEMATICS==
===[[C1. Configuration of a mechanical system]]===
:: [[C1. Configuration of a mechanical system#C1.1 Position of a particle|C1.1 Position of a particle]]
:: [[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|C1.2 Configuration of a rigid body]]
:: [[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|C1.3 Orientation of a rigid body with planar motion]]
:: [[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|C1.4 Orientation of a rigid body moving in space]]
::::[[C1. Configuration of a mechanical system#Rotations about fixed axes|Rotations about fixed axes]]
::::[[C1. Configuration of a mechanical system#Euler rotations|Euler rotations]]
:: [[C1. Configuration of a mechanical system#C1.5 Independent coordinates|C1.5 Independent coordinates]]

===[[C2. Movement of a mechanical system]]===
:: [[C2. Movement of a mechanical system#C2.1 Velocity of a particle|C2.1 Velocity of a particle]]
:: [[C2. Movement of a mechanical system#C2.2 Acceleration of a particle|C2.2 Acceleration of a particle]]
:: [[C2. Movement of a mechanical system#C2.3 Intrinsic components of the acceleration|C2.3 Intrinsic components of the acceleration]]
:: [[C2. Movement of a mechanical system#C2.4 Angular velocity of a rigid body|C2.4 Angular velocity of a rigid body]]
::::[[C2. Movement of a mechanical system#Simple rotation|Simple rotation]]
::::[[C2. Movement of a mechanical system#Rotation in space|Rotation in space (Rotacions d'Euler)]]
:: [[C2. Movement of a mechanical system#C2.5 Angular acceleration of a rigid body|C2.5 Angular acceleration of a rigid body]]
:: [[C2. Movement of a mechanical system#C2.6 Particle kinematics versus rigid body kinematicsrígid|C2.6 Particle kinematics versus rigid body kinematics]]
:: [[C2. Movement of a mechanical system#C2.7 Degrees of freedom of a mechanical system|C2.7 Degrees of freedom of a mechanical system]]
:: [[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|C2.8 Usual constraints in mechanical systems]]

===[[C3. Composition of movements]]===
:: [[C3. Composition of movements#C3.1 Composition of velocities|C3.1 Composition of velocities]]
:: [[C3. Composition of movements#C3.2 Composition of accelerations|C3.2 Composition of accelerations]]
:: [[C3. Composition of movements#C3.3 Composition versus time derivative|C3.3 Composition versus time derivative]]

===[[C4. Rigid body kinematics]]===
:: [[C4. Rigid body kinematics#C4.1 Velocity distribution|C4.1 Velocity distribution]]
:: [[C4. Rigid body kinematics#C4.2 Accelerations distribution|C4.2 Accelerations distribution]]
:: [[C4. Rigid body kinematics#Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)|Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)]]
::[[C4. Rigid body kinematics#C4.4 Fixed axode and moving axode|C4.4 Fixed axode and moving axode]]

===[[C5. Rigid body kinematics: planar motion]]===
:: [[C5. Rigid body kinematics: planar motion#C5.1 Instantaneous Center of Rotation (ICR)|C5.1 Instantaneous Center of Rotation (ICR)]]
:: [[C5. Rigid body kinematics: planar motion#C5.2 Examples|C5.2 Examples]]
:: [[C5. Rigid body kinematics: planar motion#C5.3 Introduction to vehicle kinematics|C5.3 Introduction to vehicle kinematics]]
:: [[C5. Rigid body kinematics: planar motion#C5.E General examples|C5.E General examples]]

==DYNAMICS==

===[[D7. Examples of 3D dynamics#|D7. Examples of 3D dynamics]]===
::[[D7. Examples of 3D dynamics#D7.1 Analysis of the equations of motion|D7.1 Analysis of the equations of motion]]
::[[D7. Examples of 3D dynamics#D7.2 General examples|D7.2 General examples]]

==ENERGETICS==
::''UNDER CONSTRUCTION''

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==Authors==

<center><gallery>
File:Autors Ana.png|alt=Ana Barjau Condomines|Ana Barjau Condomines|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1001000
File:Autors Ernest4.jpg|alt=Ernest Bosch Soldevila|Ernest Bosch Soldevila|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1099864
</gallery></center>

::'''Ilustrations:'''
<center><gallery>
File:Autors_Joaquim.png|alt=Joaquim Agulló i Batlle|Joquim Agulló i Batlle|link=https://www.agullobatlle.cat/
</gallery></center>

::'''Editing and interactive animations:'''
<center><gallery>
File:Autors_Arnau.png|alt=Arnau Marzábal Gatell|Arnau Marzábal Gatell|link=https://www.linkedin.com/in/arnau-marzabal/
File:Autors_Berta.png|alt=Berta Ros Blanco|Berta Ros Blanco|link=https://www.linkedin.com/in/berta-ros/
</gallery></center>

::'''Collaborators:'''
<center><gallery>
File:Autors_Daniel.png|alt=Daniel Clos Costa|Daniel Clos Costa|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1002252
File:Autors_Rosa.png|alt=Rosa Pàmies Vilà|Rosa Pàmies Vilà|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1066910
File:Autors_Albert.png|alt=Albert Peiret Giménez|Albert Peiret Giménez|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1115007
File:Autors_Javier.png|alt=Javier Sistiaga Vidal-Ribas|Javier Sistiaga Vidal-Ribas|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1114855
</gallery></center>

<center>
[https://www.youtube.com/channel/UCqWvnHTViRPI1wHlUQXqH-Q '''Mechanics Lab'''] ''' - ''' [https://etseib.upc.edu/en '''Barcelona School of Industrial Engineering (ETSEIB)''']

[https://em.upc.edu/en '''Mechanical Engineering Department'''] ''' - ''' [https://www.upc.edu/en '''Polytechic University of Catalonia (UPC) · BarcelonaTech''']
</center>

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==Bibliographic references==
Batlle, J. A., Barjau, A. (2020) “'''Rigid Body Kinematics'''” Cambridge Univerity Press. ISBN: 978-1-108-47907-3

Batlle, J. A., Barjau, A. (2022) “'''Rigid Body Dynamics'''” Cambridge Univerity Press. ISBN: 978-1-108-84213-6

Agulló, J. (2002) “'''Mecànica de la partícula i del sòlid rígid'''" Publicacions OK Punt. ISBN: 84-920850-6-1 (''Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent '''web de l'autor'''])

Agulló, J. (2000) “'''Mecánica de la partícula i del sólido rígido'''" Publicacions OK Punt. ISBN: 84-920850-5-3 (''Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent '''web de l'autor'''])

<center>
<gallery>
File:RBK portada.png|alt=Rigid body kinematics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-kinematics?format=HB&isbn=9781108479073
File:RBD portada.png|alt=Rigid body dynamics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&isbn=9781108842136
File:Llibre verd.png|alt=Mecànica de la partícula i del sòlid rígid|link=https://www.agullobatlle.cat/activitat-docent
File:Llibre vermell.jpg|alt=Mecánica de la partícula i del sólido rígido|link=https://www.agullobatlle.cat/activitat-docent
</gallery>
</center>

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Vectors

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Peiret:

This is a vector <math>\vec{\bf u}</math>.

The time derivative of a vector <math>\vec{\bf u}</math> with respect to a reference frame <math>R</math> is

<math>\left. \frac{d \vec{\bf u}}{dt} \right]_\text{R} = \left. \frac{d \vec{\bf u}}{dt} \right]_\text{REL} + \vec{\bf\Omega}^\text{REL}_\text{R} \times \vec{\bf u}</math>

Vectors

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Peiret:

This is a vector <math>\vec{\bf u}</math>.

The time derivative of a vector <math>\vec{\bf u} = u \vec{\bf e}</math>, where <math>\| \vec{\bf e} \| = 1</math>, is:

<math>\left. \frac{d \vec{\bf u}}{dt} \right]_R = \dot{u} \vec{\bf e} + \vec{\bf\Omega}^\vec{\bf e}_R \times \vec{\bf u}</math>

Vectors

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Peiret:

This is a vector <math>\vec{\bf u}</math>.

The time derivative of a vector <math>\vec{\bf u} = u \vec{\bf e}</math> is:

<math>\frac{d \vec{\bf u}}{dt} = \dot{u} \vec{\bf e} + \vec{\bf\Omega} \times \vec{\bf u}</math>

Vectors

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Peiret:

This is a vector <math>\vec{\bf u}</math>.

The time derivative of a vector <math>\vec{\bf u} = u \vec{\bf e}</math>is:

<math>\frac{d \vec{\bf u}}{dt} = \dot{u} \vec{\bf e} + \vec{\bf\Omega} \times \vec{\bf u}</math>

Vectors

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Peiret:

This is a vector <math>\vec{\bf u}_0</math>.

[math]E=mc^2[/math]

Content

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#REDIRECT [[Main Page]]

Main Page

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Welcome to the Wiki of Mechanics.

== Space and Time ==

: [[Frame of reference]]

: [[Vectors]]

== Point Kinematics ==

: [[Trajectory, velocity and acceleration]]

: [[Composition of movements]]

== Rigid Body Kinematics ==

: [[Rigid body orientation]]

: [[Angular velocity and acceleration]]



__NOTOC__

Main Page

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Pàgina Principal

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Benvingut!

Referència

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Això és una referència.

Main Page

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Main Page

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Welcome to the Wiki of Mechanics.

== Espacio y tiempo ==

: [[Frame of reference]]

: [[Vectors]]

== Point Kinematics ==

: [[Trajectory, velocity and acceleration]]

: [[Composition of movements]]

== Rigid Body Kinematics ==

: [[Rigid body orientation]]

: [[Angular velocity and acceleration]]



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Trajectory, velocity and acceleration

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pre: "<math>",
post: "</math>"
}
}
}
}
} );
} );
}

MediaWiki:Common.js

2022-02-16T13:35:47Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

// Check if we're editing a page.
if ( [ 'edit', 'submit' ].indexOf( mw.config.get( 'wgAction' ) ) !== -1 ) {
// Add a hook handler.
mw.hook( 'wikiEditor.toolbarReady' ).add( function ( $textarea ) {

// Add Math button
$( '#wpTextbox1' ).wikiEditor( 'addToToolbar', {
section: 'advanced',
group: 'format',
tools: {
"math": {
label: 'Math',
type: 'button',
icon: '<img src="https://latex.codecogs.com/png.image?\dpi{200}&space;\sqrt{x}&space;"/>',
//icon: '//upload.wikimedia.org/wikipedia/commons/2/2e/Button_math.png',
action: {
type: 'encapsulate',
options: {
pre: "<math>",
post: "</math>"
}
}
}
}
} );
} );
}

MediaWiki:Common.js

2022-02-16T13:33:01Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

// Check if we're editing a page.
if ( [ 'edit', 'submit' ].indexOf( mw.config.get( 'wgAction' ) ) !== -1 ) {
// Add a hook handler.
mw.hook( 'wikiEditor.toolbarReady' ).add( function ( $textarea ) {

// Add Math button
$( '#wpTextbox1' ).wikiEditor( 'addToToolbar', {
section: 'advanced',
group: 'format',
tools: {
"math": {
label: 'Math',
type: 'button',
icon: 'https://latex.codecogs.com/png.image?\dpi{200}&space;\sqrt{x}&space;',
//icon: '//upload.wikimedia.org/wikipedia/commons/2/2e/Button_math.png',
action: {
type: 'encapsulate',
options: {
pre: "<math>",
post: "</math>"
}
}
}
}
} );
} );
}

MediaWiki:Common.js

2022-02-16T13:27:48Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

// Check if we're editing a page.
if ( [ 'edit', 'submit' ].indexOf( mw.config.get( 'wgAction' ) ) !== -1 ) {
// Add a hook handler.
mw.hook( 'wikiEditor.toolbarReady' ).add( function ( $textarea ) {

// Add Math button
$( '#wpTextbox1' ).wikiEditor( 'addToToolbar', {
section: 'advanced',
group: 'format',
tools: {
"math": {
label: 'Math',
type: 'button',
icon: '//upload.wikimedia.org/wikipedia/commons/2/2e/Button_math.png',
action: {
type: 'encapsulate',
options: {
pre: "<math>",
post: "</math>"
}
}
}
}
} );
} );
}

MediaWiki:Common.js

2022-02-16T13:25:16Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

// Check if we're editing a page.
if ( [ 'edit', 'submit' ].indexOf( mw.config.get( 'wgAction' ) ) !== -1 ) {
// Add a hook handler.
mw.hook( 'wikiEditor.toolbarReady' ).add( function ( $textarea ) {

// Add Math button
$textarea.wikiEditor( 'addToToolbar', {
section: 'advanced',
group: 'format',
tools: {
"math": {
label: 'Math',
type: 'button',
icon: '//upload.wikimedia.org/wikipedia/commons/2/2e/Button_math.png',
action: {
type: 'encapsulate',
options: {
pre: "<math>",
post: "</math>"
}
}
}
}
} );
} );
}

Main Page

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Peiret:

Main Page

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Peiret:

Welcome to the Wiki of Newtonian Mechanics. Each section covers a topic and contains several pages.

== Space and Time ==

: [[Frame of reference]]

: [[Vectors]]

== Point Kinematics ==

: [[Trajectory, velocity and acceleration]]

: [[Composition of movements]]

== Rigid Body Kinematics ==

: [[Rigid body orientation]]

: [[Angular velocity and acceleration]]



__NOTOC__

Vectors

2022-02-16T11:18:51Z

Peiret:

This is a vector <math>\vec{\bf u}_0</math>.

Vectors

2022-02-16T11:17:41Z

Peiret:

This is a vector <math>\vec{\bf u}</math>.

Vectors

2022-02-16T11:14:47Z

Peiret:

This is a vector $\vec{\bf u}$.

MediaWiki:Common.js

2022-02-16T10:57:44Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

MediaWiki:Common.js

2022-02-16T10:54:12Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

/* add extra time to load math equations on mobile screens */
function waitForMathJax($content) {
if (typeof MathJax === 'undefined') {
setTimeout(function () { waitForMathJax($content); }, 4000);
} else {
MathJax.Hub.Queue(["Typeset", MathJax.Hub, $content[0]]).execute();
}
}

mw.hook('wikipage.content').add(waitForMathJax);

MediaWiki:Common.js

2022-02-16T10:53:23Z

Peiret:

/* Any JavaScript here will be loaded for all users on every page load. */

/* add extra time to load math equations on mobile screens */
function waitForMathJax($content) {
if (typeof MathJax === 'undefined') {
setTimeout(function () { waitForMathJax($content); }, 2000);
} else {
MathJax.Hub.Queue(["Typeset", MathJax.Hub, $content[0]]).execute();
}
}

mw.hook('wikipage.content').add(waitForMathJax);