ME-403 Theory of Machines - Jun 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 4 Marks
   aWhat is redundant degree of freedom of a mechanism?
   How to answer

   • Define redundant / passive degree of freedom as extra motion not affecting input-output relation
   • Example e.g. roller of cam-follower having self rotation
   • Distinguish from effective DOF
2. 4 Marks
   bWhat is the degree of freedom of a mechanism? How is it determined?
   How to answer

   • Define DOF as number of independent inputs to define configuration
   • State Kutzbach/Gruebler formula $F=3(l-1)-2j-h$
   • Explain counting links, lower and higher pairs
3. 3 Marks
   cWhat is Kutzback's criterion for degree of freedom of plane mechanisms? In what way is Gruebler's criterion different from it?
   How to answer

   • State Kutzbach criterion $F=3(l-1)-2j-h-F_r$
   • State Gruebler criterion for single DOF planar with all lower pairs $3l-2j-4=0$
   • Explain difference: Gruebler is special case with $h=0, F=1$
4. 3 Marks
   dDefine Grashof's law. State how is it helpful in classifying the four-link mechanisms into different types.
   How to answer

   • Define Grashof's law $s+l \le p+q$ for continuous rotation
   • State conditions for crank-rocker, double-crank, double-rocker
   • Explain classification procedure with shortest link fixed/adjacent
5. 14 Marks
   In a mechanism shown in Fig.1, the angular velocity of the crank OA is $15 \text{ rad/s}$ and the slider at E is constrained to move at $2.5 \text{ m/s}$. The motion of both the sliders is vertical and the link BC is horizontal in the position shown. Determine the

   1. i)Rubbing velocity at B if the pin diameter is 15 mm.
   2. ii)Velocity of slider D.

   

   How to answer

   • Draw configuration and velocity diagram / instantaneous centre method
   • Compute velocity of A and relate to B-C link and sliders D, E
   • Compute relative angular velocity at B for rubbing velocity $=r\cdot\omega_{rel}$ and velocity of D
6. 7 Marks
   aExplain the D'Alembert's principle in brief.
   How to answer

   • State D'Alembert's principle: inertia force reverses effective motion
   • Write $F - m a = 0$ for dynamic equilibrium
   • Mention application to mechanism analysis
7. 7 Marks
   bAn engine shaft running at $120 \text{ r.p.m.}$ is required to drive a machine shaft by means of a belt. The pulley on the engine shaft is of $2 \text{ m}$ diameter and that of the machine shaft is $1 \text{ m}$ diameter. If the belt thickness is $5 \text{ mm}$; determine the speed of the machine shaft, when

   1. i)There is no slip;
   2. ii)There is a slip of 3%.
   How to answer

   • Use belt speed ratio with thickness: $N_2/N_1 = (D_1+t)/(D_2+t)$
   • Compute speed with no slip
   • Reduce by 3% slip for second case
8. 7 Marks
   aExplain with help of suitable sketches the inversions of double slider crank chain mechanism.
   How to answer

   • Define double slider crank chain
   • Sketch three inversions: Scotch yoke, Oldham coupling, elliptical trammel
   • Explain fixing of each link and motion obtained
9. 7 Marks
   bFig:2, shows an epicyclic gear in which arm A is fixed to the shaft S, B is free to rotate on S and F is separately driven. A receives $7.5 \text{ kW}$ at $200 \text{ rpm}$ and F is driven in the same direction at $100 \text{ rpm}$. Determine the speed of B and the torque on its shaft.

   

   How to answer

   • Tabulate teeth: F=120T, B=80T, C=20T and arm A
   • Apply tabular method for epicyclic gear with arm fixed then add arm speed
   • Solve for speed of B and torque from power $P=2\pi N T/60$
10. 7 Marks
    aState and prove law of gearing. What is meant by conjugate gears? How is it satisfied by involute profile gear teeth?
    How to answer

    • State law of gearing: common normal passes through pitch point
    • Prove with angular velocity ratio constant
    • Define conjugate action and show involute satisfies it
11. 7 Marks
    bAn engine running at $150 \text{ rpm}$ drives a line shaft by means of a belt. The engine pulley is $750 \text{ mm}$ diameter and the pulley on the line shaft is $450 \text{ mm}$. A $900 \text{ mm}$ diameter pulley on the line shaft drives a $150 \text{ mm}$ diameter pulley keyed to a dynamo shaft. Find the speed of the dynamo shaft when there is no slip.
    How to answer

    • Compute line shaft speed from engine pulley ratio
    • Compute dynamo speed from second stage ratio $N_4 = N_3 D_3/D_4$
    • Combine for overall velocity ratio
12. 7 Marks
    aEstablish a formula for the frictional torque transmitted by a cone clutch.
    How to answer

    • Consider frustum element and uniform pressure/wear assumption
    • Integrate frictional moment $dT = \mu p 2\pi r^2 dr /\sin\alpha$
    • Obtain $T = (2/3)\mu W (R^3-r^3)/(R^2-r^2)/\sin\alpha$
13. 7 Marks
    bDefine the following terms as applied to cam with a neat sketch:

    1. i)Base circle
    2. ii)Pitch circle
    3. iii)Pressure angle
    4. iv)Stroke of the follower
    How to answer

    • Draw cam with base, pitch circles, pressure angle and stroke marked
    • Define each term with reference to diagram
    • Distinguish base vs prime vs pitch circle
14. 14 Marks
    A cam rotating clockwise at a uniform speed of $100 \text{ r.p.m.}$ is required to give motion to knife-edge follower as below:

    1. a)Follower to move outwards through 25 mm during 120° of cam rotation,
    2. b)Follower to dwell for the next 60° of cam rotation.
    3. c)Follower to return to its starting position during next 90° of cam rotation.
    4. d)Follower to dwell for the rest of the cam rotation.
    How to answer

    • Note data: $N=100$ rpm, $S=25$ mm, angles $120^\circ/60^\circ/90^\circ$
    • Use uniform acceleration-retardation formulas $v_{max}=2S\omega/\theta$, $a=4S\omega^2/\theta^2$
    • Compute values for outstroke and return stroke separately
15. 7 Marks
    aA $20^{\circ}$ involute pinion with $20$ teeth drives a gear having $60$ teeth. Module is $8 \text{ mm}$ and addendum of each gear is $10 \text{ mm}$.

    1. i)State whether interference occurs or not. Give reasons.
    2. ii)Find the length of path of approach and arc of approach if pinion is the driver.
    How to answer

    • Compute pitch radii, addendum radii and path of approach/recess
    • Check interference condition $Addendum > \sqrt{R^2 - R^2\cos^2\phi} - R\sin\phi$
    • Compute arc of approach $=$ path$/\cos\phi$
16. 7 Marks
    bA simple band brake is operated by a lever of length $500 \text{ mm}$. The brake drum has a diameter of $500 \text{ mm}$ and the brake band embraces $5/8$ of the circumference. One end of the band is attached to the fulcrum of the lever while the other end is attached to a pin on the lever $100 \text{ mm}$ from the fulcrum. If the effort applied to the end of the lever is $2 \text{ kN}$ and the coefficient of friction is $0.25$, find the maximum braking torque on the drum.
    How to answer

    • Draw lever-drum with tensions $T_1,T_2$ and $T_1/T_2=e^{\mu\theta}$
    • Take moment about fulcrum to find tight-side tension from effort $2$ kN
    • Compute braking torque $(T_1-T_2)r$ for both rotation directions and take maximum