[{"data":1,"prerenderedAt":37},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Ftransmission-rotational-motion":3},{"topic":4,"trail":18,"children":32,"tasks":33,"alternates":34},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},219,"Transmission of Rotational Motion","en","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Ftransmission-rotational-motion","Transmission of Rotational Motion in Kinematics","Kinematic relations for gears, belts and friction drives: angular velocities, radii and transmission ratio.","This topic covers transmission of rotational motion between bodies without slipping and the relationships among angular velocities, radii and transmission ratio.","\u003Cp>\u003Cstrong>Rotational-motion transmissions\u003C\u002Fstrong> transfer motion from an input member to an output member. In kinematics, the main objective is to relate angular velocities, rotational frequencies, and geometric parameters of the members.\u003C\u002Fp>\u003Ch2>No-slip condition\u003C\u002Fh2>\u003Cp>For two wheels in contact or connected by a transmission without slipping, the magnitudes of the corresponding contact-point velocities are equal:\u003C\u002Fp>\u003Cp>$$\\omega_1 r_1=\\omega_2 r_2.$$\u003C\u002Fp>\u003Cp>Thus the transmission ratio by magnitude may be written:\u003C\u002Fp>\u003Cp>$$i=\\frac{\\omega_1}{\\omega_2}=\\frac{r_2}{r_1}.$$\u003C\u002Fp>\u003Cp>For gears, pitch-circle radii are proportional to tooth numbers, so $i=z_2\u002Fz_1$ using the same convention for input and output members.\u003C\u002Fp>\u003Cp>Equality of tangential velocities at contact is the basis for analyzing simple friction, belt, and gear transmissions under the ideal no-slip assumption.\u003C\u002Fp>\u003Ch2>Friction wheels\u003C\u002Fh2>\u003Cp>For two externally contacting wheels without slip, $|\\omega_1|r_1=|\\omega_2|r_2$. They rotate in opposite directions. With internal contact, their rotation directions are the same.\u003C\u002Fp>\u003Ch2>Belt drives\u003C\u002Fh2>\u003Cp>For an open belt drive without slip, belt speed is the same at both pulleys, so $\\omega_1r_1=\\omega_2r_2$ in magnitude. An open belt gives the pulleys the same rotation direction; a crossed belt gives opposite directions.\u003C\u002Fp>\u003Ch2>Gear drives\u003C\u002Fh2>\u003Cp>For an external gear pair, $|\\omega_1|\u002F|\\omega_2|=z_2\u002Fz_1$, where $z_1$ and $z_2$ are tooth numbers. External meshing reverses rotation direction, while internal meshing preserves it.\u003C\u002Fp>\u003Ch2>Transmission ratio\u003C\u002Fh2>\u003Cp>A transmission ratio must always be interpreted with its definition. Here $i=\\omega_1\u002F\\omega_2$, where member 1 is the input and member 2 is the output. If only the magnitude is required, rotation direction is handled separately.\u003C\u002Fp>\u003Ch2>Multistage transmissions\u003C\u002Fh2>\u003Cp>For successive stages, the overall transmission ratio is the product of the individual stage ratios. Output rotation direction follows from the number of external gear meshes or crossed-belt stages.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>An input gear has $z_1=20$ teeth and rotates at $\\omega_1=60$ rad\u002Fs. The output gear has $z_2=60$. Then $|\\omega_2|=60\\cdot20\u002F60=20$ rad\u002Fs, with opposite direction for external meshing.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>reversing the radius ratio $r_2\u002Fr_1$;\u003C\u002Fli>\u003Cli>defining $i$ without stating which member is input;\u003C\u002Fli>\u003Cli>ignoring rotation direction in external gear meshing;\u003C\u002Fli>\u003Cli>using equal contact speeds when the problem explicitly includes slip;\u003C\u002Fli>\u003Cli>considering only one pair in a multistage transmission.\u003C\u002Fli>\u003C\u002Ful>",[14],{"id":15,"code":16,"type":17,"locale":7},183,"kinematics-rotation-transmission-ratio","formula",[19,23,27,31],{"id":20,"name":21,"path":22},81,"Theoretical Mechanics","theoretical-mechanics",{"id":24,"name":25,"path":26},137,"Kinematics","theoretical-mechanics\u002Fkinematics",{"id":28,"name":29,"path":30},285,"Basic Motions of a Rigid Body","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body",{"id":5,"name":6,"path":8},[],[],{"en":35,"uk":36},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Ftransmission-rotational-motion","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fnaiprostishi-rukhy-tverdoho-tila\u002Fperedachi-obertalnoho-rukhu",1787712536426]