--- title: "Torque capacity of a cone clutch" subject: "Machine Design II" chapter: 7 key: "semester_7/machine_design_ii:cone-clutch-torque-axial-force" source: https://www.bctengineer.com/q/semester_7/machine_design_ii/cone-clutch-torque-axial-force site: BCT Engineering --- # Torque capacity of a cone clutch Machine Design II · Chapter 7: Clutches and brakes - Asked in: year not given, 6 marks - Video: [Torque transmitting capacity of cone clutch](https://www.youtube.com/watch?v=WWEBt_DFCfA) — Mechanical Engineering Online Classes - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/machine_design_ii/cone-clutch-torque-axial-force - Whole chapter: https://www.bctengineer.com/md/semester_7/machine_design_ii/chapter-7 ## Question Derive the expression for the torque transmitted by a cone clutch of semi-cone angle alpha, outer radius R and inner radius r, on the basis of uniform wear. Obtain the axial force needed for engagement. State the advantages of a cone clutch over a plate clutch and the condition on the semi-cone angle to avoid sticking. ## Short answer Torque (uniform wear): T = μ F_a R_m / sin α, with R_m = (R+r)/2. Axial force for engagement: F_e = F_n(sin α + μ cos α). A cone clutch gives torque amplification due to wedge action; to avoid sticking, tan α > μ must hold. ## Answer Notation: $\alpha$ = semi-cone angle, $R$, $r$ = outer and inner radii of the cone face, $\mu$ = friction coefficient, $p_n$ = normal pressure, $F_n$ = total normal force on the cone surface, $F_a$ = axial force. ```text axial force Fa | ______v_____ / \ alpha \ R \ slant r \____\_______/ ``` #### Torque (uniform wear) Take an elementary ring of radius $x$ and slant width $dx/\sin\alpha$. Area $= 2\pi x\,dx/\sin\alpha$. Normal force $dF_n = p_n\,2\pi x\,dx/\sin\alpha$. Friction torque: $$ dT = \mu\,dF_n\,x = \frac{2\pi\mu p_n}{\sin\alpha}x^2dx $$ For uniform wear $p_n x = C$: $$ F_n = \frac{2\pi C}{\sin\alpha}\int_r^R dx = \frac{2\pi C (R-r)}{\sin\alpha} $$ $$ T = \frac{2\pi\mu C}{\sin\alpha}\int_r^R x\,dx = \frac{\pi\mu C (R^2-r^2)}{\sin\alpha} $$ Eliminating $C$: $$ T = \mu F_n R_m, \qquad R_m = \frac{R+r}{2} $$ The axial force transmitted by the cone face in steady running is $F_a = F_n\sin\alpha$, so $T = \dfrac{\mu F_a R_m}{\sin\alpha}$. (For uniform pressure, $R_m = \dfrac{2}{3}\dfrac{R^3-r^3}{R^2-r^2}$ and $F_n = \pi p_n(R^2-r^2)/\sin\alpha$.) #### Axial force for engagement While the clutch is engaging, friction acts along the cone surface, opposing the motion of the cone into the cup. Resolving along the axis: $$ F_e = F_n(\sin\alpha + \mu\cos\alpha) $$ Hence the spring force needed to engage the clutch is $F_e = F_n(\sin\alpha+\mu\cos\alpha)$; once it is engaged and running, $F_a = F_n\sin\alpha$ is enough to hold it. #### Advantages over plate clutch - The wedge action gives a normal force $F_n = F_a/\sin\alpha$ much larger than the axial force, so for the same $F_a$ a cone clutch transmits more torque (about $1/\sin\alpha$ times). - Smaller diameter for the same torque. - No need for many plates. Disadvantages: more expensive to make, needs close alignment, tends to stick. #### Condition to avoid sticking The cone must release easily. On disengagement the axial component of the normal force, $F_n\sin\alpha$, must overcome the friction along the slant, $\mu F_n\cos\alpha$, so $$ \tan\alpha > \mu, \qquad \alpha > \tan^{-1}\mu $$ (the semi-cone angle must be greater than the friction angle). The usual value $\alpha = 12.5^\circ$ ($\tan 12.5^\circ = 0.22$) suits linings with $\mu$ up to about 0.2, such as cork or leather. Linings with higher friction need a larger angle, up to about 20 degrees (for $\mu = 0.35$, $\alpha > 19.3^\circ$). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For uniform wear, the cone clutch torque can be written as A. T = mu F_n R_m, with R_m = (R + r)/2 B. T = mu F_a R_m, with R_m = (R + r)/2 C. T = mu F_n (R - r)/2 D. T = mu F_n R_m sin(alpha) Answer: A. T = mu F_n R_m, and since F_a = F_n sin(alpha), T = mu F_a R_m / sin(alpha). (Key point: Cone clutch torque) 2. The axial force needed to engage the cone is A. F_e = F_n sin(alpha) B. F_e = F_n cos(alpha) C. F_e = F_n (sin(alpha) + mu cos(alpha)) D. F_e = F_n (sin(alpha) - mu cos(alpha)) Answer: C. During engagement friction opposes motion into the seat, adding mu cos(alpha) to the sin(alpha) term. (Key point: Engagement force) 3. Why does a cone clutch transmit more torque than a plate clutch for the same axial force? A. It has more friction surfaces B. Wedge action makes normal force F_n = F_a/sin(alpha) C. Its mean radius is always larger D. It uses higher pressure lining Answer: B. Wedge action multiplies the normal force by about 1/sin(alpha). (Key point: Wedge advantage) 4. To avoid sticking, the semi-cone angle must satisfy A. alpha < tan^-1(mu) B. alpha > tan^-1(mu) C. alpha = 45 degrees D. alpha < 5 degrees Answer: B. The semi-cone angle must be greater than the friction angle so the cone releases easily. (Key point: Sticking condition) ## Related in Machine Design II - [Torque capacity of a plate clutch](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Aplate-clutch-uniform-pressure-wear) — Machine Design II, chapter 7 - [Number of plates in a multiplate clutch](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Amultiplate-clutch-power-numerical) — Machine Design II, chapter 7 ## Related in other subjects - [Actual grade on inner and outer lanes](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-64p-grade-inner-outer) — Transportation Engineering I, chapter 3 - [Jet on hinged plate, force and angle](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Ajet-on-hinged-plate-force-angle) — Fluid Mechanics, chapter 7 - [Inner/outer orbital complexes; VBT of [Fe(CN)₆]³⁻](https://www.bctengineer.com/question.md?key=semester_2%2Fengineering_chemistry%3Ainner-outer-vbt-fecn6) — Engineering Chemistry, chapter 6 - [Right circular cone with vertex (1,1,1), semi-vertical 30°](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cone-vertex-1) — Mathematics, chapter 3, asked 3×