{"id":996,"date":"2010-10-10T16:41:40","date_gmt":"2010-10-10T14:41:40","guid":{"rendered":"https:\/\/medianbox.fr\/wordpress\/?p=996"},"modified":"2014-08-13T19:07:11","modified_gmt":"2014-08-13T17:07:11","slug":"cyclisme-lenergie-cinetique-a-velo","status":"publish","type":"post","link":"https:\/\/medianbox.fr\/wordpress\/?p=996","title":{"rendered":"MeDiaN@Sports &#8211; \u00ab L\u2019\u00c9nergie cin\u00e9tique \u00e0 v\u00e9lo \u00bb"},"content":{"rendered":"<h3>Pr\u00e9sentation g\u00e9n\u00e9rale<\/h3>\n<p>Dans cet article, je me suis amus\u00e9 \u00e0 d\u00e9terminer l&rsquo;\u00e9nergie cin\u00e9tique d&rsquo;un v\u00e9lo lorsque celui-ci suit une trajectoire rectiligne. L&rsquo;id\u00e9e \u00e9tait de v\u00e9rifier que le changement des roues de mon Gitane pour des Mavic Ksyrium Elite \u00e9tait bien la bonne solution pour am\u00e9liorer son comportement.<\/p>\n<h3>Mod\u00e9lisation du probl\u00e8me<\/h3>\n<h4>Syst\u00e8me isol\u00e9<\/h4>\n<p>On choisit d&rsquo;isoler les \u00e9l\u00e9ments les plus significatifs du v\u00e9lo, c&rsquo;est \u00e0 dire les composants tournant et\/ou ayant une masse importante : cycliste, cadre avec les \u00e9l\u00e9ments qui lui sont li\u00e9s et en fixant la fourche, roues (avant et arri\u00e8re), p\u00e9dalier.<\/p>\n<h4>Sch\u00e9matisation<\/h4>\n<p><a href=\"https:\/\/medianbox.fr\/wordpress\/?attachment_id=1172\" rel=\"attachment wp-att-1172\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1172\" title=\"modelisation_E1\" src=\"https:\/\/medianbox.fr\/wordpress\/wp-content\/uploads\/2010\/10\/modelisation_E1.jpg\" alt=\"\" width=\"575\" height=\"392\" srcset=\"https:\/\/medianbox.fr\/wordpress\/wp-content\/uploads\/2010\/10\/modelisation_E1.jpg 575w, https:\/\/medianbox.fr\/wordpress\/wp-content\/uploads\/2010\/10\/modelisation_E1-300x204.jpg 300w\" sizes=\"auto, (max-width: 575px) 100vw, 575px\" \/><\/a><\/p>\n<h4 style=\"padding-left: 30px;\">Description<\/h4>\n<p style=\"padding-left: 60px;\">Le v\u00e9lo est sch\u00e9matis\u00e9 par le syst\u00e8me de centre d&rsquo;inertie G tel que :<\/p>\n<p style=\"padding-left: 90px;\">&#8211; Cadre, \u00e9l\u00e9ments fixes forme un solide S de masse M et dont le centre de gravit\u00e9\u00a0 G&rsquo; est positionn\u00e9 en [b, c, 0] dans le rep\u00e8re R.<\/p>\n<p style=\"padding-left: 90px;\">&#8211; Les roues avant <img src='https:\/\/s0.wp.com\/latex.php?latex=S+_%7B1%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S _{1} ' title='S _{1} ' class='latex' \/> et arri\u00e8re <img src='https:\/\/s0.wp.com\/latex.php?latex=S+_%7B2%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S _{2} ' title='S _{2} ' class='latex' \/> sont des cercles de rayon a et de masses <img src='https:\/\/s0.wp.com\/latex.php?latex=M+_%7B1%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='M _{1} ' title='M _{1} ' class='latex' \/> et<img src='https:\/\/s0.wp.com\/latex.php?latex=M+_%7B2%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='M _{2} ' title='M _{2} ' class='latex' \/> uniform\u00e9ment r\u00e9partie sur la circonf\u00e9rence. Les frottements sont n\u00e9glig\u00e9s. <img src='https:\/\/s0.wp.com\/latex.php?latex=O+_%7B1%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='O _{1} ' title='O _{1} ' class='latex' \/> en\u00a0 [d,e,0] et <img src='https:\/\/s0.wp.com\/latex.php?latex=O+_%7B2%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='O _{2} ' title='O _{2} ' class='latex' \/> en [f,e,0].<\/p>\n<p style=\"padding-left: 90px;\">-Le p\u00e9dalier <img src='https:\/\/s0.wp.com\/latex.php?latex=S+_%7BP%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S _{P} ' title='S _{P} ' class='latex' \/> de masse <img src='https:\/\/s0.wp.com\/latex.php?latex=m+_%7BP%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='m _{P} ' title='m _{P} ' class='latex' \/> munie de ses p\u00e9dales <img src='https:\/\/s0.wp.com\/latex.php?latex=m_%7Bp%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='m_{p} ' title='m_{p} ' class='latex' \/> de masse <img src='https:\/\/s0.wp.com\/latex.php?latex=S+_%7Bp%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S _{p} ' title='S _{p} ' class='latex' \/>. <img src='https:\/\/s0.wp.com\/latex.php?latex=O_%7BP%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='O_{P} ' title='O_{P} ' class='latex' \/> en [g,h,0]<\/p>\n<p style=\"padding-left: 90px;\">-Le cycliste <img src='https:\/\/s0.wp.com\/latex.php?latex=S+_%7BCycliste%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S _{Cycliste} ' title='S _{Cycliste} ' class='latex' \/> de masse <img src='https:\/\/s0.wp.com\/latex.php?latex=M+_%7BCycliste%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='M _{Cycliste} ' title='M _{Cycliste} ' class='latex' \/>. <img src='https:\/\/s0.wp.com\/latex.php?latex=G_%7BCycliste%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='G_{Cycliste} ' title='G_{Cycliste} ' class='latex' \/> en [i,j,0]<\/p>\n<h4 style=\"padding-left: 30px;\">Bilan des param\u00e8tres<\/h4>\n<p style=\"padding-left: 60px;\">4 param\u00e8tres :<\/p>\n<p style=\"padding-left: 90px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta+_%7B1%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\theta _{1}' title='\\theta _{1}' class='latex' \/> : Angle de rotation de la roue avant;<\/p>\n<p style=\"padding-left: 90px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta+_%7B2%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\theta _{2}' title='\\theta _{2}' class='latex' \/> : Angle de rotation de la roue arri\u00e8re;<\/p>\n<p style=\"padding-left: 90px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta+_%7Bp%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\theta _{p} ' title='\\theta _{p} ' class='latex' \/> : Angle de rotation du p\u00e9dalier;<\/p>\n<p style=\"padding-left: 90px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cxi+%3A&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\xi :' title='\\xi :' class='latex' \/> Abscisse de G dans <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cleft+%28+O_%7B0%7D%2C%5Cvec%7Bx_%7B0%7D%7D%2C%5Cvec%7By_%7B0%7D%7D+%5Cright+%29&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\left ( O_{0},\\vec{x_{0}},\\vec{y_{0}} \\right )' title='\\left ( O_{0},\\vec{x_{0}},\\vec{y_{0}} \\right )' class='latex' \/><\/p>\n<h3>R\u00e9solution<\/h3>\n<h4 style=\"padding-left: 30px;\">D\u00e9termination des param\u00e8tres ind\u00e9pendants<\/h4>\n<p style=\"padding-left: 60px;\">3 \u00c9quations de liaison (roulement sans glissement et transmission par cha\u00eene)<\/p>\n<p style=\"padding-left: 90px;\">&#8211; Roue avant : <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7BV_%7B%5Cleft+%28+I%5Cin+S_%7B1%7D%2FR_%7B0%7D+%5Cright+%29%7D%7D%3D%5Cvec%7B0%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{V_{\\left ( I\\in S_{1}\/R_{0} \\right )}}=\\vec{0}' title='\\vec{V_{\\left ( I\\in S_{1}\/R_{0} \\right )}}=\\vec{0}' class='latex' \/> &lt;=&gt; <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7BV_%7B%5Cleft+%28O%7B_%7B1%7D%2FR_%7B0%7D%7D%5Cright%29%7D%7D%2B%5COmega+_%7BS_%7B1%7D%2FR_%7B0%7D%7D%5Cwedge+%5Cvec%7BO_%7B1%7DI_%7B1%7D%7D%3D%5Cvec%7B0%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{V_{\\left (O{_{1}\/R_{0}}\\right)}}+\\Omega _{S_{1}\/R_{0}}\\wedge \\vec{O_{1}I_{1}}=\\vec{0}' title='\\vec{V_{\\left (O{_{1}\/R_{0}}\\right)}}+\\Omega _{S_{1}\/R_{0}}\\wedge \\vec{O_{1}I_{1}}=\\vec{0}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7BV_%7B%5Cleft+%28+I%5Cin+S_%7B1%7D%2FR_%7B0%7D+%5Cright+%29%7D%7D%3D%5Cleft+%28+%5Cdot%7B%5Cxi%7D%2Ba%5Cdot%7B%5Ctheta+_%7B1%7D%7D+%5Cright+%29.%5Cvec%7Bx_%7B0%7D%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{V_{\\left ( I\\in S_{1}\/R_{0} \\right )}}=\\left ( \\dot{\\xi}+a\\dot{\\theta _{1}} \\right ).\\vec{x_{0}}' title='\\vec{V_{\\left ( I\\in S_{1}\/R_{0} \\right )}}=\\left ( \\dot{\\xi}+a\\dot{\\theta _{1}} \\right ).\\vec{x_{0}}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">&#8211; Roue arri\u00e8re : <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7BV_%7B%5Cleft+%28+I%5Cin+S_%7B2%7D%2FR_%7B0%7D+%5Cright+%29%7D%7D%3D%5Cvec%7B0%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{V_{\\left ( I\\in S_{2}\/R_{0} \\right )}}=\\vec{0}' title='\\vec{V_{\\left ( I\\in S_{2}\/R_{0} \\right )}}=\\vec{0}' class='latex' \/> &lt;=&gt; <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7BV_%7B%5Cleft+%28O%7B_%7B2%7D%2FR_%7B0%7D%7D%5Cright%29%7D%7D%2B%5COmega+_%7BS_%7B2%7D%2FR_%7B0%7D%7D%5Cwedge+%5Cvec%7BO_%7B2%7DI_%7B2%7D%7D%3D%5Cvec%7B0%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{V_{\\left (O{_{2}\/R_{0}}\\right)}}+\\Omega _{S_{2}\/R_{0}}\\wedge \\vec{O_{2}I_{2}}=\\vec{0}' title='\\vec{V_{\\left (O{_{2}\/R_{0}}\\right)}}+\\Omega _{S_{2}\/R_{0}}\\wedge \\vec{O_{2}I_{2}}=\\vec{0}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7BV_%7B%5Cleft+%28+I%5Cin+S_%7B2%7D%2FR_%7B0%7D+%5Cright+%29%7D%7D%3D%5Cleft+%28+%5Cdot%7B%5Cxi%7D%2Ba%5Cdot%7B%5Ctheta+_%7B2%7D%7D+%5Cright+%29.%5Cvec%7Bx_%7B0%7D%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{V_{\\left ( I\\in S_{2}\/R_{0} \\right )}}=\\left ( \\dot{\\xi}+a\\dot{\\theta _{2}} \\right ).\\vec{x_{0}}' title='\\vec{V_{\\left ( I\\in S_{2}\/R_{0} \\right )}}=\\left ( \\dot{\\xi}+a\\dot{\\theta _{2}} \\right ).\\vec{x_{0}}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">&#8211; Transmission : Soit t le facteur de transmission : <img src='https:\/\/s0.wp.com\/latex.php?latex=t%3D%5Cfrac%7B%5Cdot%7BO_%7B2%7D%7D+%7D%7B%5Cdot%7BO_%7Bp%7D%7D%7D%3D%5Cfrac%7BNdents_%7Bplateau%7D+%7D%7BNdents_%7Bpignon%7D%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='t=\\frac{\\dot{O_{2}} }{\\dot{O_{p}}}=\\frac{Ndents_{plateau} }{Ndents_{pignon}}' title='t=\\frac{\\dot{O_{2}} }{\\dot{O_{p}}}=\\frac{Ndents_{plateau} }{Ndents_{pignon}}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">Comme elles ont le m\u00eame diam\u00e8tre,\u00a0 les 2 roues tournent \u00e0 la m\u00eame vitesse <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cleft+%28+%5Cdot%7BO_%7B2%7D%7D%3D%5Cdot%7BO_%7B1%7D%7D+%5Cright+%29&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\left ( \\dot{O_{2}}=\\dot{O_{1}} \\right )' title='\\left ( \\dot{O_{2}}=\\dot{O_{1}} \\right )' class='latex' \/>. En connaissant t, on peut donc tout exprimer en fonction de la vitesse de rotation du p\u00e9dalier:<\/p>\n<p style=\"padding-left: 90px; text-align: center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbf%7B%5Cdot%7B%5Ctheta+_%7B1%7D%7D%3D%5Cdot%7B%5Ctheta+_%7B2%7D%7D%3D%5Cfrac%7B-%5Cdot%7B%5Cxi+%7D%7D%7Ba%7D%3D%5Cdot%7B%5Ctheta_%7Bp%7D%7D.t%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\mathbf{\\dot{\\theta _{1}}=\\dot{\\theta _{2}}=\\frac{-\\dot{\\xi }}{a}=\\dot{\\theta_{p}}.t}' title='\\mathbf{\\dot{\\theta _{1}}=\\dot{\\theta _{2}}=\\frac{-\\dot{\\xi }}{a}=\\dot{\\theta_{p}}.t}' class='latex' \/> (1) : une relation liant tous les param\u00e8tres<\/p>\n<p style=\"padding-left: 60px; text-align: left;\">Relation en exprimant la vitesse de p\u00e9dalage sous la forme d&rsquo;une cadence :<\/p>\n<p style=\"padding-left: 60px; text-align: left;\">On exprime traditionnellement la cadence de p\u00e9dalage en tr\/min alors que la relation ci-dessus est en rad\/sec. Soit N la cadence de p\u00e9dalage en tr\/min, la conversion des unit\u00e9s donne :<\/p>\n<p style=\"padding-left: 60px; text-align: center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdot%7B%5Ctheta+_%7B1%7D%7D%3D%5Cdot%7B%5Ctheta+_%7B2%7D%7D%3D%5Cfrac%7B-%5Cdot%7B%5Cxi+%7D%7D%7Ba%7D%3D%5Cfrac%7B%5Cpi+.N.t%7D%7B30%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\dot{\\theta _{1}}=\\dot{\\theta _{2}}=\\frac{-\\dot{\\xi }}{a}=\\frac{\\pi .N.t}{30}' title='\\dot{\\theta _{1}}=\\dot{\\theta _{2}}=\\frac{-\\dot{\\xi }}{a}=\\frac{\\pi .N.t}{30}' class='latex' \/> (2)<\/p>\n<h4 style=\"padding-left: 30px; text-align: left;\">Calcul de l&rsquo;\u00e9nergie cin\u00e9tique du syst\u00e8me<\/h4>\n<p style=\"padding-left: 60px;\">Je vais maintenant pouvoir \u00e9crire l&rsquo;\u00e9quation de l&rsquo;\u00e9nergie cin\u00e9tique de mon v\u00e9lo ramen\u00e9e \u00e0 ma cadence de p\u00e9dalage uniquement. L&rsquo;\u00e9nergie cin\u00e9tique totale est la somme des \u00e9nergie propre \u00e0 chacun des sous-ensembles composant le syst\u00e8me.<\/p>\n<p style=\"padding-left: 60px; text-align: center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T+%3D+T%5Cleft+%28+S_%7B0%7D%2FR_%7B0%7D+%5Cright+%29%2B+T%5Cleft+%28+S_%7B1%7D%2FR_%7B0%7D+%5Cright+%29%2BT%5Cleft+%28+S_%7B2%7D%2FR_%7B0%7D+%5Cright+%29%2BT%5Cleft+%28+S_%7BP%7D%2FR_%7B0%7D+%2BT%5Cleft+%28+Cycliste%2FR_%7B0%7D+%5Cright+%29+%5Cright+%29&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='T = T\\left ( S_{0}\/R_{0} \\right )+ T\\left ( S_{1}\/R_{0} \\right )+T\\left ( S_{2}\/R_{0} \\right )+T\\left ( S_{P}\/R_{0} +T\\left ( Cycliste\/R_{0} \\right ) \\right )' title='T = T\\left ( S_{0}\/R_{0} \\right )+ T\\left ( S_{1}\/R_{0} \\right )+T\\left ( S_{2}\/R_{0} \\right )+T\\left ( S_{P}\/R_{0} +T\\left ( Cycliste\/R_{0} \\right ) \\right )' class='latex' \/><\/p>\n<p style=\"padding-left: 60px;\">Chaque \u00e9nergie peut donc se calculer de mani\u00e8re ind\u00e9pendante.<\/p>\n<p style=\"padding-left: 90px;\">&#8211; <img src='https:\/\/s0.wp.com\/latex.php?latex=S_%7B0%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S_{0}' title='S_{0}' class='latex' \/> (S\/E cadre+\u00e9l\u00e9ments fixes) : Cas d&rsquo;un corps en translation.<\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T%5Cleft+%28+S_%7B0%7D%2FR_%7B0%7D+%5Cright%29+%3D%5Cfrac%7B1%7D%7B2%7D.M.%5Cleft+%5B+%5Cvec%7BV_%7B%5Cleft+%28+G_%7B0%7D%5Cin+S_%7B0%7D%2FR_%7B0%7D+%5Cright+%29%7D%7D+%5Cright+%5D%5E%7B2%7D+%2B0+%3D+%5Cfrac%7B1%7D%7B2%7D.M.+%5Cdot%7B%5Cxi%7D%5E%7B2%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='T\\left ( S_{0}\/R_{0} \\right) =\\frac{1}{2}.M.\\left [ \\vec{V_{\\left ( G_{0}\\in S_{0}\/R_{0} \\right )}} \\right ]^{2} +0 = \\frac{1}{2}.M. \\dot{\\xi}^{2} ' title='T\\left ( S_{0}\/R_{0} \\right) =\\frac{1}{2}.M.\\left [ \\vec{V_{\\left ( G_{0}\\in S_{0}\/R_{0} \\right )}} \\right ]^{2} +0 = \\frac{1}{2}.M. \\dot{\\xi}^{2} ' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">&#8211; <img src='https:\/\/s0.wp.com\/latex.php?latex=S_%7B1%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S_{1}' title='S_{1}' class='latex' \/> (S\/E roue avant) : Cas d&rsquo;un corps en translation et rotation simultan\u00e9e.<\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T%5Cleft+%28+S_%7B1%7D%2FR_%7B0%7D+%5Cright%29+%3D%5Cfrac%7B1%7D%7B2%7D.M_%7B1%7D.%5Cleft+%5B+%5Cvec%7BV_%7B%5Cleft+%28+O_%7B1%7D%5Cin+S_%7B1%7D%2FR_%7B0%7D+%5Cright+%29%7D%7D+%5Cright+%5D%5E%7B2%7D+%2B%5Cfrac%7B1%7D%7B2%7D.%5Cvec%7B%5COmega+%5Cleft+%28+S_%7B1%7D%2FR_%7B0%7D%5Cright+%29%7D.%5Cvec%7B%5Csigma+%5Cleft+%28+O_%7B1%7D%5Cin+S_%7B1%7D%2FR_%7B0%7D+%5Cright+%29%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='T\\left ( S_{1}\/R_{0} \\right) =\\frac{1}{2}.M_{1}.\\left [ \\vec{V_{\\left ( O_{1}\\in S_{1}\/R_{0} \\right )}} \\right ]^{2} +\\frac{1}{2}.\\vec{\\Omega \\left ( S_{1}\/R_{0}\\right )}.\\vec{\\sigma \\left ( O_{1}\\in S_{1}\/R_{0} \\right )}' title='T\\left ( S_{1}\/R_{0} \\right) =\\frac{1}{2}.M_{1}.\\left [ \\vec{V_{\\left ( O_{1}\\in S_{1}\/R_{0} \\right )}} \\right ]^{2} +\\frac{1}{2}.\\vec{\\Omega \\left ( S_{1}\/R_{0}\\right )}.\\vec{\\sigma \\left ( O_{1}\\in S_{1}\/R_{0} \\right )}' class='latex' \/> avec <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7B%5Csigma+%5Cleft+%28+O_%7B1%7D%5Cin+S_%7B1%7D%2FR_%7B0%7D+%5Cright%29%7D%3D%5Cbar%7BI_%7BO_%7B1%7D%7D%7D.%5Cvec%7B%5COmega+%5Cleft+%28+S_%7B1%7D%2FR_%7B0%7D%5Cright+%29%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\vec{\\sigma \\left ( O_{1}\\in S_{1}\/R_{0} \\right)}=\\bar{I_{O_{1}}}.\\vec{\\Omega \\left ( S_{1}\/R_{0}\\right )}' title='\\vec{\\sigma \\left ( O_{1}\\in S_{1}\/R_{0} \\right)}=\\bar{I_{O_{1}}}.\\vec{\\Omega \\left ( S_{1}\/R_{0}\\right )}' class='latex' \/><\/p>\n<p style=\"padding-left: 120px;\">or la matrice d&rsquo;inertie pour un anneau tournant autour de l&rsquo;axe orthogonal \u00e0 son plan et passant par son centre est <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbar%7BI_%7BO_%7B1%7D%7D%7D+%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7BmR%7D%7B2%7D%5E%7B2%7D+%26+0+%260+%5C%5C0+%26%5Cfrac%7BmR%7D%7B2%7D%5E%7B2%7D%260%5C%5C0%260%26mR%5E%7B2%7D%5Cend%7Bpmatrix%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='\\bar{I_{O_{1}}} =\\begin{pmatrix}\\frac{mR}{2}^{2} &amp; 0 &amp;0 \\\\0 &amp;\\frac{mR}{2}^{2}&amp;0\\\\0&amp;0&amp;mR^{2}\\end{pmatrix}' title='\\bar{I_{O_{1}}} =\\begin{pmatrix}\\frac{mR}{2}^{2} &amp; 0 &amp;0 \\\\0 &amp;\\frac{mR}{2}^{2}&amp;0\\\\0&amp;0&amp;mR^{2}\\end{pmatrix}' class='latex' \/><\/p>\n<p style=\"padding-left: 120px;\">donc <img src='https:\/\/s0.wp.com\/latex.php?latex=+T%5Cleft+%28+S_%7B1%7D%2FR_%7B0%7D%5Cright%29%3D+%5Cfrac%7B1%7D%7B2%7D+M_%7B1%7D.+%5Cdot%7B%5Cxi%7D%5E%7B2%7D%2B%5Cfrac%7B1%7D%7B2%7D.+M_%7B1%7D.a%5E%7B2%7D.%5Cdot%7B%5Ctheta+_%7B1%7D%5E%7B2%7D%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt=' T\\left ( S_{1}\/R_{0}\\right)= \\frac{1}{2} M_{1}. \\dot{\\xi}^{2}+\\frac{1}{2}. M_{1}.a^{2}.\\dot{\\theta _{1}^{2}}' title=' T\\left ( S_{1}\/R_{0}\\right)= \\frac{1}{2} M_{1}. \\dot{\\xi}^{2}+\\frac{1}{2}. M_{1}.a^{2}.\\dot{\\theta _{1}^{2}}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">&#8211; <img src='https:\/\/s0.wp.com\/latex.php?latex=S_%7B2%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S_{2}' title='S_{2}' class='latex' \/> (S\/E roue arri\u00e8re) : Idem.<\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=+T%5Cleft+%28+S_%7B2%7D%2FR_%7B0%7D%5Cright%29%3D+%5Cfrac%7B1%7D%7B2%7D+M_%7B2%7D.+%5Cdot%7B%5Cxi%7D%5E%7B2%7D%2B%5Cfrac%7B1%7D%7B2%7D.+M_%7B2%7D.a%5E%7B2%7D.%5Cdot%7B%5Ctheta+_%7B2%7D%5E%7B2%7D%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt=' T\\left ( S_{2}\/R_{0}\\right)= \\frac{1}{2} M_{2}. \\dot{\\xi}^{2}+\\frac{1}{2}. M_{2}.a^{2}.\\dot{\\theta _{2}^{2}}' title=' T\\left ( S_{2}\/R_{0}\\right)= \\frac{1}{2} M_{2}. \\dot{\\xi}^{2}+\\frac{1}{2}. M_{2}.a^{2}.\\dot{\\theta _{2}^{2}}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">&#8211; <img src='https:\/\/s0.wp.com\/latex.php?latex=S_%7B2%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S_{2}' title='S_{2}' class='latex' \/> (S\/E P\u00e9dalier + P\u00e9dales) :<\/p>\n<p style=\"padding-left: 120px;\">Le rayon de giration du sous ensemble est approxim\u00e9 \u00e0 13 cm, avec <img src='https:\/\/s0.wp.com\/latex.php?latex=m_%7BP%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='m_{P}' title='m_{P}' class='latex' \/> la masse du p\u00e9dalier et <img src='https:\/\/s0.wp.com\/latex.php?latex=m_%7Bp%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='m_{p}' title='m_{p}' class='latex' \/> la masse d&rsquo;une p\u00e9dale. Comme pour les roues, on a :<\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=+T%5Cleft+%28+S_%7Bp%7D%2FR_%7B0%7D%5Cright%29%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%28+m_%7BP%7D%2B2.m_%7Bp%7D%5Cright%29+.+%5Cdot%7B%5Cxi%7D%5E%7B2%7D%2B%5Cfrac%7B1%7D%7B2%7D.+%5Cleft+%28+m_%7BP%7D%2B2.m_%7Bp%7D%5Cright%29.0.13%5E%7B2%7D.%5Cdot%7B%5Ctheta+_%7Bp%7D%5E%7B2%7D%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt=' T\\left ( S_{p}\/R_{0}\\right)= \\frac{1}{2} \\left ( m_{P}+2.m_{p}\\right) . \\dot{\\xi}^{2}+\\frac{1}{2}. \\left ( m_{P}+2.m_{p}\\right).0.13^{2}.\\dot{\\theta _{p}^{2}}' title=' T\\left ( S_{p}\/R_{0}\\right)= \\frac{1}{2} \\left ( m_{P}+2.m_{p}\\right) . \\dot{\\xi}^{2}+\\frac{1}{2}. \\left ( m_{P}+2.m_{p}\\right).0.13^{2}.\\dot{\\theta _{p}^{2}}' class='latex' \/><\/p>\n<p style=\"padding-left: 90px;\">&#8211; <img src='https:\/\/s0.wp.com\/latex.php?latex=S_%7BCycliste%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='S_{Cycliste}' title='S_{Cycliste}' class='latex' \/> (S\/E Cycliste) :<\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T%5Cleft+%28+S_%7BCycliste%7D%2FR_%7B0%7D+%5Cright%29+%3D%5Cfrac%7B1%7D%7B2%7D.M_%7BCycliste%7D.+%5Cdot%7B%5Cxi%7D%5E%7B2%7D+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='T\\left ( S_{Cycliste}\/R_{0} \\right) =\\frac{1}{2}.M_{Cycliste}. \\dot{\\xi}^{2} ' title='T\\left ( S_{Cycliste}\/R_{0} \\right) =\\frac{1}{2}.M_{Cycliste}. \\dot{\\xi}^{2} ' class='latex' \/><\/p>\n<p style=\"padding-left: 60px;\">L&rsquo;\u00e9nergie cin\u00e9tique totale du syst\u00e8me est donc la suivante :<\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T+%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%28%5Cleft+%28+M%2BM_%7B1%7D%2BM_%7B2%7D%2Bm_%7BP%7D%2B2.m_%7Bp%7D%2BM_%7BCycliste%7D+%5Cright%29+.+%5Cdot%7B%5Cxi%7D%5E%7B2%7D+%5Cright%29&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='T = \\frac{1}{2} \\left (\\left ( M+M_{1}+M_{2}+m_{P}+2.m_{p}+M_{Cycliste} \\right) . \\dot{\\xi}^{2} \\right)' title='T = \\frac{1}{2} \\left (\\left ( M+M_{1}+M_{2}+m_{P}+2.m_{p}+M_{Cycliste} \\right) . \\dot{\\xi}^{2} \\right)' class='latex' \/><\/p>\n<p style=\"padding-left: 120px;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%2B%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%28+a%5E%7B2%7D+%5Cleft+%28+M_%7B1%7D%5Cdot%7B%5Ctheta+_%7B1%7D%5E%7B2%7D%7D%2BM_%7B2%7D%5Cdot%7B%5Ctheta+_%7B2%7D%5E%7B2%7D%7D+%5Cright%29+%2B.0%2C13%5E%7B2%7D+%5Cleft+%28+m_%7BP%7D%2B2.m_%7Bp%7D%5Cright%29+.%5Cdot%7B%5Ctheta+_%7Bp%7D%5E%7B2%7D%7D+%5Cright%29+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='+\\frac{1}{2} \\left ( a^{2} \\left ( M_{1}\\dot{\\theta _{1}^{2}}+M_{2}\\dot{\\theta _{2}^{2}} \\right) +.0,13^{2} \\left ( m_{P}+2.m_{p}\\right) .\\dot{\\theta _{p}^{2}} \\right) ' title='+\\frac{1}{2} \\left ( a^{2} \\left ( M_{1}\\dot{\\theta _{1}^{2}}+M_{2}\\dot{\\theta _{2}^{2}} \\right) +.0,13^{2} \\left ( m_{P}+2.m_{p}\\right) .\\dot{\\theta _{p}^{2}} \\right) ' class='latex' \/><\/p>\n<p style=\"padding-left: 60px;\">Les relations (1) et (2) permettent de simplifier cette expression en ramenant l&rsquo;\u00e9nergie cin\u00e9tique \u00e0 la rotation du p\u00e9dalier. L&rsquo;\u00e9nergie cin\u00e9tique totale peut ainsi \u00eatre \u00e9crite sous la forme :<\/p>\n<p style=\"padding-left: 90px; text-align: center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=T+%3D+%5Cfrac%7B1%7D%7B2%7D+.+K+.+%5Cdot%7B%5Ctheta+_%7Bp%7D%5E%7B2%7D%7D%3D%5Cfrac%7B1%7D%7B2%7D+.+K+%5Cleft+%28%5Cfrac%7B%5Cpi+.+N%7D%7B30%7D+%5Cright%29%5E%7B2%7D&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='T = \\frac{1}{2} . K . \\dot{\\theta _{p}^{2}}=\\frac{1}{2} . K \\left (\\frac{\\pi . N}{30} \\right)^{2}' title='T = \\frac{1}{2} . K . \\dot{\\theta _{p}^{2}}=\\frac{1}{2} . K \\left (\\frac{\\pi . N}{30} \\right)^{2}' class='latex' \/> avec inertie \u00e9quivalente ramen\u00e9e au p\u00e9dalier <img src='https:\/\/s0.wp.com\/latex.php?latex=K%3Da%5E%7B2%7D.t%5E%7B2%7D+%5Cleft+%28+M%2B2.+%5Cleft+%28M_%7B1%7D%2BM_%7B2%7D+%5Cright%29+%2B+m_%7BP%7D+%2B+2.+m_%7Bp%7D%2BM_%7BCycliste%7D+%5Cright%29%2B0%2C13%5E%7B2%7D+%5Cleft+%28m_%7BP%7D+%2B+2.+m_%7Bp%7D+%5Cright%29+&#038;bg=121315&#038;fg=ffffff&#038;s=0' alt='K=a^{2}.t^{2} \\left ( M+2. \\left (M_{1}+M_{2} \\right) + m_{P} + 2. m_{p}+M_{Cycliste} \\right)+0,13^{2} \\left (m_{P} + 2. m_{p} \\right) ' title='K=a^{2}.t^{2} \\left ( M+2. \\left (M_{1}+M_{2} \\right) + m_{P} + 2. m_{p}+M_{Cycliste} \\right)+0,13^{2} \\left (m_{P} + 2. m_{p} \\right) ' class='latex' \/><\/p>\n<p style=\"padding-left: 60px; text-align: left;\">L&rsquo;\u00e9nergie cin\u00e9tique d&rsquo;un cycliste d\u00e9pend donc principalement de sa masse, de celle de son v\u00e9lo et de sa vitesse (qui est elle m\u00eame fonction de la cadence de p\u00e9dalage (N) et du braquet utilis\u00e9 (t)). On remarque \u00e9galement que les roues comptent double dans ce calcul, et c&rsquo;est pourquoi il vaut mieux privil\u00e9gier un gain de poids sur les roues plut\u00f4t que sur un cadre pour upgrader son v\u00e9lo.<\/p>\n<p style=\"padding-left: 90px;\">\n","protected":false},"excerpt":{"rendered":"<p>Approximation du calcul de l&rsquo;\u00e9nergie cin\u00e9tique pour un v\u00e9lo.<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"ngg_post_thumbnail":0,"footnotes":""},"categories":[60],"tags":[],"class_list":["post-996","post","type-post","status-publish","format-standard","hentry","category-mediansports"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>MeDiaN@Sports - \u00ab L\u2019\u00c9nergie cin\u00e9tique \u00e0 v\u00e9lo \u00bb - MeDiaN@Tour<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/medianbox.fr\/wordpress\/?p=996\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"MeDiaN@Sports - \u00ab L\u2019\u00c9nergie cin\u00e9tique \u00e0 v\u00e9lo \u00bb - MeDiaN@Tour\" \/>\n<meta property=\"og:description\" content=\"Approximation du calcul de l&#039;\u00e9nergie cin\u00e9tique pour un v\u00e9lo.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/medianbox.fr\/wordpress\/?p=996\" \/>\n<meta property=\"og:site_name\" content=\"MeDiaN@Tour\" \/>\n<meta property=\"article:published_time\" content=\"2010-10-10T14:41:40+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2014-08-13T17:07:11+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/medianbox.fr\/wordpress\/wp-content\/uploads\/2010\/10\/modelisation_E1.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"575\" \/>\n\t<meta property=\"og:image:height\" content=\"392\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"MeDiaN\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"MeDiaN\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/medianbox.fr\\\/wordpress\\\/?p=996#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/medianbox.fr\\\/wordpress\\\/?p=996\"},\"author\":{\"name\":\"MeDiaN\",\"@id\":\"https:\\\/\\\/medianbox.fr\\\/wordpress\\\/#\\\/schema\\\/person\\\/e1cba8e776ddbc5e91ef69e134732c6d\"},\"headline\":\"MeDiaN@Sports &#8211; 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