Td corrigé Physique vibratoire / 2ème année - Exercices corriges pdf

Physique vibratoire / 2ème année - Exercices corriges

ENIBe 2ème Année / TD Physique Vibratoire / Valeurs moyennes et efficaces. Bachard .... 2.5.1) Simple alternance 2.5.2) Double alternance. 2.6) y = a t2 pour t  ...




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{\title Physique vibratoire / 2\'e8me ann\'e9e }{\subject TD1 : Valeurs moyennes et efficaces}{\author Bachard Eric}{\operator ericb}{\creatim\yr1996\mo8\dy29\hr15\min7}{\revtim\yr2002\mo2\dy25\hr17\min14}{\printim\yr1999\mo6\dy21\hr9\min34}{\version46}
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\par Id\'e9e : sur une p\'e9riode, on calcule la valeur moyenne continue qui repr\'e9sente la m\'eame surface que celle obtenue en faisant : \tab }{\pard\plain \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright \lang1036\cgrid
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\par
\par Quand la fonction devient trop \ldblquote \~compliqu\'e9e\~\rdblquote , on calcule : }{\pard\plain \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright \lang1036\cgrid {\object\objemb\objw1820\objh760{\*\objclass Equation.3}
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\par Elle est int\'e9ressante car l\rquote \'e9nergie mise en jeu dans le ph\'e9nom\'e8ne est proportionnelle au carr\'e9 de l\rquote amplitude de la fonction p\'e9riodique.
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\par {\pntext\pard\plain\f3\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-283\li283\nowidctlpar\widctlpar{\*\pn \pnlvlblt\ilvl0\ls1\pnrnot0\pnf3\pnstart1\pnindent283\pnhang{\pntxtb \'b7}}\ls1\adjustright {\b T}{oute fonction p\'e9riodique est la somme d
\rquote une composante continue -ou valeur moyenne (qui peut \'eatre nulle)- et d\rquote une fonction alternative.
\par {\pntext\pard\plain\f3\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-283\li283\nowidctlpar\widctlpar{\*\pn \pnlvlblt\ilvl0\ls1\pnrnot0\pnf3\pnstart1\pnindent283\pnhang{\pntxtb \'b7}}\ls1\adjustright {\b U}{ne fonction alternative n\rquote
est pas forc\'e9ment sinuso\'efdale.
\par {\pntext\pard\plain\f3\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-283\li283\nowidctlpar\widctlpar{\*\pn \pnlvlblt\ilvl0\ls1\pnrnot0\pnf3\pnstart1\pnindent283\pnhang{\pntxtb \'b7}}\ls1\adjustright {\b U}{ne fonction sinuso\'efdale est forc\'e9
ment alternative (sinon, on parle de fonction p\'e9riodique).
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\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\b Exercice 1}{
\par
\par On consid\'e8re le signal p\'e9riodique suivant (figure 1).
\par
\par 1.1) Quelle est la valeur moyenne de ce signal?\tab
\par
\par 1.2) Calculer la valeur efficace de V (t).
\par
\par }{\b
\par Exercice 2 : R\'e9sistance chauffante
\par }{
\par On rappelle qu\rquote en continu, la puissance dissip\'e9e par effet Joule par une r\'e9sistance est donn\'e9e par
\par P}{\sub J}{ = R i }{\super 2}{ .
\par
\par Une r\'e9sistance R de valeur 10 }{\f3 \'57}{ est aliment\'e9e par un signal p\'e9riodique, dont l\rquote allure est donn\'e9e \'e0 la figure ci-dessous :
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\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {
\par 2.1) U = 220 V. Donner la valeur de }{\f3 \'61}{ pour que la r\'e9sistance dissipe 100 W.
\par
\par 2.2) Quelle est alors la valeur moyenne < U > de u (t) ?
\par
\par
\par
\par }\pard \nowidctlpar\widctlpar\box\brdrs\brdrw15\brsp20 \pvpara\posy0\dxfrtext141\dfrmtxtx141\dfrmtxty0{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright \shading2000 {\b\fs32 Solutions
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {
\par
\par
\par }{\b Exercice 1 :}{
\par
\par 1.1) Valeur moyenne de ce signal.
\par
\par Il appara\'eet clairement que le signal p\'e9riodique donn\'e9 \'e0 \'e9tudier est un signal triangulaire sym\'e9trique
\par -2V / +2V , d\'e9cal\'e9 de +1V.
\par On sait (cf cours) qu\rquote un signal p\'e9riodique est la somme d\rquote un signal alternatif (de valeur moyenne nulle) et d\rquote une composante continue, \'e9gale \'e0 la valeur moyenne.
\par
\par Ici, le signal triangulaire est la partie alternative du signal et la composante continue, \'e9gale \'e0 +1V est la valeur moyenne.
\par
\par Dit autrement : si on d\'e9cale V(t) de 1 V vers le bas, on retrouve un signal de valeur moyenne nulle.
\par }\pard \nowidctlpar\widctlpar\box\brdrs\brdrw15\brsp20 \pvpara\phpg\posx3652\posy239\dxfrtext141\dfrmtxtx141\dfrmtxty0{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright \shading1000 {Conclusion : V}{\sub moyen}{ = + 1V
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {
\par
\par
\par }\pard \qr\nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {\b [1 pt]}{
\par }\pard \nowidctlpar\widctlpar{\*\pn \pnlvlcont\ilvl0\ls0\pnrnot0\pndec }\adjustright {
\par 1.2) Valeur efficace de V (t).
\par
\par V (t) est un signal du type : V (t) = f (t) + C }{\super ste}{
\par
\par avec :
\par
\par f (t) = signal triangulaire sym\'e9trique, de valeur moyenne nulle et C }{\super ste}{ = valeur moyenne = +1 V.
\par
\par Par d\'e9finition :
\par
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\par
\par Il ne reste donc qu\rquote une int\'e9grale \'e0 calculer (la premi\'e8re), soit :
\par
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\par }\pard \nowidctlpar\widctlpar\adjustright {f(t) est donn\'e9e \'e0 la figure 1 ci-contre. C\rquote est un signal triangulaire sym\'e9trique -2V/+2V, de p\'e9riode T
\par }\pard \qj\nowidctlpar\widctlpar\adjustright {
\par Sur la figure 1 bis, qui repr\'e9sente f}{\super 2}{ (t), on s\rquote aper\'e7oit que le fait d\rquote \'e9lever au carr\'e9 donne une p\'e9riode qui devient T / 2 et que l\rquote axe Oy est axe de sym\'e9trie (en effet, la surface de -T/4 \'e0
0 est la m\'eame que de 0 \'e0 T/4....m\'eame chose sur chaque intervalle de dur\'e9e T/4).
\par
\par
\par }\pard \nowidctlpar\widctlpar\pvpara\phpg\posx1151\posy52\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright {\pard\plain \nowidctlpar\widctlpar\pvpara\phpg\posx1151\posy52\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \lang1036\cgrid {\object\objemb
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\par }\pard \nowidctlpar\widctlpar\adjustright {
\par
\par
\par
\par }{\b Exercice 2 :}{
\par
\par 2.1) La puissance dissip\'e9e par effet Joule (par une r\'e9sistance) en continu vaut P}{\sub J}{ = R i}{\super 2}{.
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\par
\par 2.2) Calcul de la valeur moyenne de U, not\'e9e < U > :
\par
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\par
\par }\pard \nowidctlpar\widctlpar\box\brdrs\brdrw15\brsp20 \pvpara\posy0\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \shading2000 {Application Num\'e9rique :}{\b < U > = 6,82 V}{
\par }\pard \nowidctlpar\widctlpar\adjustright {\page
\par }\pard \nowidctlpar\widctlpar\box\brdrs\brdrw15\brsp20 \pvpara\phpg\posx1060\posy233\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \shading2000 {\b\fs32 Exercices avec r\'e9ponses
\par }\pard\plain \s1\sb240\nowidctlpar\widctlpar\outlinelevel0\adjustright \b\f1\ul\lang1036\cgrid {
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \lang1036\cgrid {
\par }\pard\plain \s1\sb240\nowidctlpar\widctlpar\outlinelevel0\adjustright \b\f1\ul\lang1036\cgrid {Exercice 1 : Valeur moyenne et valeur efficace
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \lang1036\cgrid {
\par Rappeler les d\'e9finitions de ymoy, yeff lorsque y = f (t) est une fonction p\'e9riodique, avec t appartenant \'e0 [t1, t1+ T], T \'e9tant la p\'e9riode de y.
\par Etablir les relations existant entre ymoy, yeff, Ymoy et Yeff dans les cas suivants :
\par
\par 1.1) Y = y + C\tab \tab ( C est une constante r\'e9elle).
\par
\par 1.2) Y = |y|\tab \tab (fonction redressement).
\par
\par 1.3) Y = f(t +}{\f3 \'20\'74}{)\tab (Changement d'origine).
\par
\par 1.4) Y = f(-t)\tab \tab (Sym\'e9trie de le courbe /axe)
\par
\par 1.5) Y = -f(-t)\tab \tab (Sym\'e9trie de la courbe /\'e0 un point).
\par
\par \tab \tab \tab \tab }{\ul \tab \tab \tab \tab \tab }{
\par }\pard\plain \s1\sb240\nowidctlpar\widctlpar\outlinelevel0\adjustright \b\f1\ul\lang1036\cgrid {Exercice 2 : valeur moyenne, valeur efficace
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \lang1036\cgrid {
\par En utilisant au mieux les r\'e9sultats de l'exercice 1, calculer la valeur moyenne, puis la valeur efficace
\par des fonctions suivantes :
\par
\par 2.1) Fonction "cr\'e9neaux"
\par
\par }\pard \nowidctlpar\widctlpar\pvpara\phpg\posx3743\posy25\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright {\pard\plain \nowidctlpar\widctlpar\pvpara\phpg\posx3743\posy25\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \lang1036\cgrid {\object\objemb
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\par 2.2) m\'eame question dans le cas particulier o\'f9 }{\f3 \'71}{ = T/2
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\par }\pard \qj\nowidctlpar\widctlpar\adjustright {
\par
\par 2.6) y = a t}{\up6 2}{ pour t }{\f3 \'ce}{[0 ; T ]
\par
\par 2.7) y = a t}{\up6 3}{ pour t }{\f3 \'ce}{[0 ; T ]
\par
\par }\pard \nowidctlpar\widctlpar\adjustright {
\par }\pard \nowidctlpar\widctlpar\box\brdrs\brdrw15\brsp20 \pvpara\posy0\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \shading2000 {\b\fs32 R\'e9ponses :}{ }{\b\fs32
\par }\pard \nowidctlpar\widctlpar\adjustright {
\par
\par
\par Exercice 1 :
\par
\par }{\fs20 1.1) \tab Y = y + C\tab
\par
\par \tab }{\fs20\ul Relations entre valeurs moyennes}{\fs20 : Y}{\fs20\sub moy}{\fs20 = y}{\fs20\sub moy}{\fs20 + C\tab \tab
\par
\par \tab }{\fs20\ul Entre}{\fs20 }{\fs20\ul valeurs efficaces}{\fs20 : }{\pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1036\cgrid {\object\objemb\objw3760\objh760\objscalex88\objscaley88{\*\objclass Equation.3}{\*\objdata 01050000020000000b000000
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\par Remarque : l\rquote \'e9nergie de la somme des deux vibrations est \'e9gale \'e0 la somme de l\rquote \'e9nergie de chacune des vibrations plus un terme \ldblquote \~crois\'e9\~\rdblquote mettant en jeu le produit des valeurs moyennes.
\par Ce produit pouvant d\rquote ailleurs \'eatre nul si l\rquote une ou les deux vibrations ont une valeur moyenne nulle.
\par
\par 1.2) Y = |y|
\par
\par }{\fs20\ul Valeurs moyennes}{\fs20 : si on appelle }{\b\fs20 A}{\b\fs20\super +}{\fs20 l\rquote aire totale au dessus de z\'e9ro sur une p\'e9riode, et }{\b\fs20 A}{\b\fs20\super -}{\fs20 l\rquote aire totale au dessous de z\'e9ro sur une p\'e9
riode, alors on a :
\par
\par }{\pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1036\cgrid {\object\objemb\objw1600\objh660\objscalex89\objscaley88{\*\objclass Equation.3}{\*\objdata 01050000020000000b0000004571756174696f6e2e3300000000000000000000100000
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\par }{\fs20\ul Valeurs efficaces}{\fs20 : lorsqu\rquote on \'e9l\'e8ve au carr\'e9, les surfaces sont les m\'eames, donc }{\pard\plain \nowidctlpar\widctlpar\adjustright \fs20\lang1036\cgrid {\object\objemb\objw980\objh360\objscalex82\objscaley81
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\par
\par }{\fs20 1.3) Y = f(t +}{\f3\fs20 \'20\'74}{\fs20 )
\par La valeur moyenne et la valeur efficace sont inchang\'e9es par changement d\rquote origine des temps car les surfaces d\'e9crites sont les m\'eames.
\par \page 1.4) Y = f(-t)\tab
\par
\par Lorsque on change t en -t, on ne change pas la valeur moyenne, ni la valeur efficace, car la forme des courbes est inchang\'e9e. Cela revient \'e0 faire d\'e9rouler le temps \ldblquote \~\'e0 l\rquote envers\~\rdblquote .
\par
\par 1.5) Y = -f(-t)\tab
\par
\par La valeur moyenne change de signe, mais pas la valeur efficace.
\par }{
\par }{\b Exercice 2}{ :
\par }\pard \qj\nowidctlpar\widctlpar\adjustright {
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\par }\pard \qj\nowidctlpar\widctlpar\adjustright {\fs20
\par Pour la valeur efficace, on utilise les m\'eames r\'e9sultats de l\rquote exercice 1.1) , ce qui donne :
\par
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\par Conseil : ces valeurs sont \'e0 conna\'eetre
\par parfaitement.....
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\par
\par Cette fois ci, la d\'e9pendance est en T}{\fs20\super 3}{\fs20 .
\par
\par
\par
\par }\pard\plain \s1\sb240\nowidctlpar\widctlpar\brdrl\brdrs\brdrw15\brsp20 \outlinelevel0\adjustright \b\f1\ul\lang1036\cgrid {\sect }\sectd \sbknone\linex0\headery680\footery1418\colsx709\sectdefaultcl \pard\plain \pagebb\nowidctlpar\widctlpar\box
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\par }\pard \nowidctlpar\widctlpar\adjustright {\b\fs32
\par }{
\par }{\b
\par Exercice 1 : Chaleur molaire moyenne}{
\par
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\par
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\par Calculer la chaleur molaire moyenne entre 0\'b0C et 300\'b0C.
\par
\par }{\b\fs32
\par }{\b Exercice 2 : Fonction hyperbolique.}{
\par }\pard \qj\nowidctlpar\widctlpar\adjustright {
\par On donne ( pour u tendant vers 0) : }{\b ln (1 + u) ~ u - u}{\b\up6 2}{\b /2 + u}{\b\up6 3}{\b /3 + ..... }{
\par
\par 2.1) Calculer la valeur moyenne de la fonction y = k / t entre les deux instants t1 et t2}{\b .}{
\par 2.2) Exprimer cette valeur moyenne sous la forme d'un d\'e9veloppement limit\'e9 de trois termes en fonction de }{\f3 \'44}{t = t2 - t1 .
\par 2.3) Donner le d\'e9veloppement limit\'e9 \'e0 3 termes de la valeur y3 de y pour l'instant t3 = ( t2 + t1) / 2 en fonction de }{\f3 \'44}{t . En d\'e9duire l'erreur absolue que l'on commet en confondant la valeur moyenne de y entre t1 et t2 avec y3 .
\tab
\par }\pard \nowidctlpar\widctlpar\adjustright {
\par }{\b Exercice 3 : redressement polyphas\'e9}{
\par On consid\'e8re un syst\'e8me de n tensions sinuso\'efdales de p\'e9riode T, toutes d\'e9phas\'e9es de }{\pard\plain \nowidctlpar\widctlpar\adjustright \lang1036\cgrid {\object\objemb\objw440\objh639{\*\objclass Equation.3}{\*\objdata 0105000002000000
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\par En utilisant les redresseurs appropri\'e9s, ces n tensions sinuso\'efdales sont redress\'e9es.
\par }\pard \qj\nowidctlpar\widctlpar\adjustright {La tension qui en r\'e9sulte est unidirectionnelle (en trait plein sur la figure 2).
\par Sur une p\'e9riode T, on compte n calottes compl\'e8tes et identiques, comme repr\'e9sent\'e9 sur la figure 2 ci-dessous :
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\par }\pard \nowidctlpar\widctlpar\adjustright {
\par
\par
\par
\par }\pard \qj\nowidctlpar\widctlpar\adjustright {La premi\'e8re de ces calottes sinuso\'efdales est sym\'e9trique par rapport \'e0 l\rquote axe des ordonn\'e9es, c\rquote est \'e0 dire entre les instants -T/2n et +T/2n
\par }\pard \nowidctlpar\widctlpar\adjustright {
\par
\par
\par
\par
\par
\par
\par 3.1) Calculer la valeur moyenne de cette tension unidirectionnelle, obtenue par redressement de n tensions sinuso\'efdales d\'e9phas\'e9es de 2}{\f3 \'70}{ / n.
\par 3.2) Calculer la valeur efficace de cette tension.
\par 3.3) Application num\'e9rique : V}{\sub max}{ = 300 V
\par
\par Calculer V}{\sub moy}{ et V}{\sub eff}{ pour n = 2 , n= 3 et n =6 et conclure.
\par
\par }\pard\plain \s1\sb240\nowidctlpar\widctlpar\outlinelevel0\adjustright \b\f1\ul\lang1036\cgrid {Exercice 4 : Principe d'un gradateur.
\par }\pard\plain \qj\nowidctlpar\widctlpar\adjustright \lang1036\cgrid {
\par On alimente un r\'e9cepteur, suppos\'e9 purement r\'e9sistif, \'e0 l'aide d'un interrupteur bidirectionnel command\'e9. Le dispositif de commande n'est pas \'e9tudi\'e9 dans cet exercice.
\par
\par }\pard \qj\nowidctlpar\widctlpar\pvpara\phpg\posx3167\posy19\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright {\pard\plain \qj\nowidctlpar\widctlpar\pvpara\phpg\posx3167\posy19\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \lang1036\cgrid {\object\objemb
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\par }\pard \qj\nowidctlpar\widctlpar\adjustright {
\par
\par
\par
\par
\par
\par
\par La tension d'alimentation est celle d\'e9livr\'e9e par le secteur (f=50 Hz), de valeur efficace 220 V.
\par Lorsque l'interrupteur est ouvert, le courant qui traverse le r\'e9cepteur, soit i, est nul.
\par Lorsque l'interrupteur est ferm\'e9, la chute de tension \'e0 ses bornes est nulle.
\par
\par }\pard \qj\nowidctlpar\widctlpar\pvpara\phpg\posx1583\posy1056\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright {\pard\plain \qj\nowidctlpar\widctlpar\pvpara\phpg\posx1583\posy1056\dxfrtext141\dfrmtxtx141\dfrmtxty0\adjustright \lang1036\cgrid {\object\objemb
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\par }\pard \qj\nowidctlpar\widctlpar\adjustright {Un exemple est donn\'e9, pour lequel on repr\'e9sente les impulsions de d\'e9blocage et les p\'e9riodes pendant lesquelles l'interrupteur command\'e9 est ferm\'e9 (valeur 1) ou ouvert (valeur 0).
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\par 4.1) Repr\'e9senter I aux bornes du r\'e9cepteur en fonction du temps pour t}{\dn6 1}{ quelconque, compris entre 0 et T/2.
\par
\par 4.2) Calculer le courant efficace I}{\dn6 EFF}{ traversant le r\'e9cepteur de m\'eame que U}{\dn6 EFF}{ en fonction d'un angle }{\f3 \'71}{, avec }{\f3 \'71}{ / 2}{\f3 \'70}{ = t}{\dn6 1}{ / T (T p\'e9riode de la tension d'alimentation).
\par Commenter le r\'e9sultat obtenu en fonction du param\'e8tre }{\f3 \'71}{.\tab \tab \tab
\par }}
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