[{"data":1,"prerenderedAt":35},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fshear-and-torsion\u002Fpower-transmission-by-shafts":3},{"topic":4,"trail":20,"children":30,"tasks":31,"alternates":32},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},118,"Power Transmission by Shafts","en","strength-of-materials\u002Fshear-and-torsion\u002Fpower-transmission-by-shafts","Power Transmission by Shafts — Torque, Power and Speed","Relationship between shaft power, torque, and rotational speed using P = Tω and T ≈ 9550P\u002Fn, followed by torsional strength and stiffness checks.","This topic explains the relationship between transmitted mechanical power, shaft torque, and rotational speed. It covers P = Tω and T ≈ 9550P\u002Fn, consistent units, conversion from drive parameters to design torque, and the subsequent shaft checks for torsional shear stress and angle of twist.","\u003Cp>In a mechanical drive, a shaft transmits energy between a motor, gearbox, coupling, and driven machine. For shaft strength calculations, power and rotational speed must first be converted into \u003Cstrong>torque\u003C\u002Fstrong>.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>Mechanical power in rotational motion is:\u003C\u002Fp>\u003Cp>$$P=T\\omega.$$\u003C\u002Fp>\u003Cp>If rotational speed is given in revolutions per minute:\u003C\u002Fp>\u003Cp>$$\\omega=\\frac{2\\pi n}{60},\\qquad T=\\frac{60P}{2\\pi n}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>P\u003C\u002Fstrong> — power, W;\u003C\u002Fli>\u003Cli>\u003Cstrong>T\u003C\u002Fstrong> — torque, N·m;\u003C\u002Fli>\u003Cli>\u003Cstrong>ω\u003C\u002Fstrong> — angular speed, rad\u002Fs;\u003C\u002Fli>\u003Cli>\u003Cstrong>n\u003C\u002Fstrong> — rotational speed, rpm.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>In the common engineering form, for P in kW and n in rpm: $T\\approx9550P\u002Fn$ N·m.\u003C\u002Fp>\u003C\u002Fdiv>\u003Ch2>Physical meaning\u003C\u002Fh2>\u003Cp>For the same transmitted power, reducing rotational speed increases torque. This is why an idealized reduction gearbox that lowers output speed increases the torque available at its output shaft.\u003C\u002Fp>\u003Ch2>From drive parameters to shaft calculation\u003C\u002Fh2>\u003Cp>After determining $T$, use it in the torsion relations. For a circular shaft, determine the polar section properties, maximum torsional shear stress, and angle of twist.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>For a straight solid or hollow circular shaft under elastic torsion:\u003C\u002Fp>\u003Cp>$$\\tau(\\rho)=\\frac{T\\rho}{J_p},\\qquad \\tau_{\\max}=\\frac{T}{W_p}.$$\u003C\u002Fp>\u003Cp>If $T$, $G$, and $J_p$ are constant over a segment, the angle of twist is:\u003C\u002Fp>\u003Cp>$$\\varphi=\\frac{TL}{GJ_p}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$T$\u003C\u002Fstrong> — torque;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\rho$\u003C\u002Fstrong> — radial distance from the shaft axis;\u003C\u002Fli>\u003Cli>\u003Cstrong>$J_p$\u003C\u002Fstrong> — polar second moment of area;\u003C\u002Fli>\u003Cli>\u003Cstrong>$W_p$\u003C\u002Fstrong> — polar section modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$G$\u003C\u002Fstrong> — shear modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$L$\u003C\u002Fstrong> — segment length;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\varphi$\u003C\u002Fstrong> — angle of twist, rad.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>For a stepped shaft, the total twist is the algebraic sum $\\varphi=\\sum_i T_iL_i\u002F(G_iJ_{p,i})$.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A shaft transmits $P=15\\ \\text{kW}$ at $n=1500\\ \\text{rpm}$. Then:\u003C\u002Fp>\u003Cp>$$T\\approx\\frac{9550\\cdot15}{1500}=95.5\\ \\text{N·m}.$$\u003C\u002Fp>\u003Cp>This is the basic torque for the subsequent shaft calculation. A real drive may additionally require allowances for transmission efficiency, load nonuniformity, starting conditions, and dynamic effects.\u003C\u002Fp>\u003Ch2>Units and a common error\u003C\u002Fh2>\u003Cp>In the formula containing the coefficient 9550, power $P$ must be entered in kW and rotational speed $n$ in rpm; torque $T$ is obtained in N·m. When using $P=T\\omega$, all quantities must be expressed in a consistent SI unit system.\u003C\u002Fp>",[14,17],{"id":5,"code":15,"type":16,"locale":7},"shaft-power-torque","formula",{"id":18,"code":19,"type":16,"locale":7},81,"circular-shaft-torsion",[21,25,29],{"id":22,"name":23,"path":24},45,"Strength of Materials","strength-of-materials",{"id":26,"name":27,"path":28},54,"Shear and Torsion","strength-of-materials\u002Fshear-and-torsion",{"id":5,"name":6,"path":8},[],[],{"en":33,"uk":34},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fshear-and-torsion\u002Fpower-transmission-by-shafts","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fzsuv-ta-kruchennya\u002Fperedacha-potuzhnosti-valamy",1787712539686]