ME-503-B Dynamics of Machine - May 2024 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. 10 Marks
   aIn a four bar chain ABCD, link AD is fixed and the crank AB rotates at $10$ radians per second clockwise. Lengths of the links are $AB = 60\ mm$; $BC = CD = 70\ mm$; $DA = 120\ mm$. When angle $DAB = 60^{\circ}$ and both B and C lie on the same side of AD, find

   1. i)Angular velocities (magnitude and direction) of BC and CD
   2. ii)Angular acceleration of BC and CD.
   How to answer

   • Draw configuration of four-bar ABCD with given dimensions and crank angle
   • Draw velocity polygon / use relative velocity method to get velocities of BC and CD
   • Determine magnitude and direction (clockwise/anticlockwise) of angular velocities
   • Draw acceleration polygon / use relative acceleration (centripetal + tangential) method
   • Calculate angular accelerations of BC and CD with direction
2. 4 Marks
   bDerive the Tractive couple expression, for an uncoupled two-cylinder locomotive engine.
   How to answer

   • Define tractive effort variation for two cylinders with 90 degree crank
   • Write expressions for crank efforts neglecting/considering obliquity
   • Combine to get resultant tractive couple / turning moment
   • Show fluctuation as function of crank angle
   • State condition for uncoupled engine
3. 10 Marks
   aThe turning moment diagram for a multicylinder engine has been drawn to a scale of $1\ mm = 4500\ N$-$m$ vertically and $1\ mm = 2.4^{\circ}$ horizontally. The intercepted areas between output torque curve and mean resistance line taken in order from one end are $342, 23, 245, 303, 115, 232, 227, 164\ mm^{2}$, when the engine is running at $150\ r.p.m.$ If the mass of the flywheel is $1000\ kg$ and the total fluctuation of speed does not exceed $3\%$ of the mean speed, find the minimum value of the radius of gyration.
   How to answer

   • Convert scale to turning moment and crank angle per mm2
   • Calculate fluctuation of energy from intercepted areas
   • Find mean speed and permissible fluctuation of speed
   • Use $\Delta E = I \omega^2 C_s$ to solve for mass moment of inertia
   • Calculate minimum radius of gyration $k$ from $I = m k^2$
4. 4 Marks
   bDefine the terms 'coefficient of fluctuation of energy' and 'coefficient of fluctuation of speed', in the case of flywheels.
   How to answer

   • Define coefficient of fluctuation of energy as ratio of max fluctuation to work done per cycle
   • Give formula $C_e = \Delta E / W$
   • Define coefficient of fluctuation of speed as $(N1-N2)/N$
   • Explain their role in flywheel design
5. 14 Marks
   A Proell governor has arms of $300\ mm$ length. The upper arms are hinged on the axis of rotation, whereas the lower arms are pivoted at a distance of $35\ mm$ from the axis of rotation. The extension of lower arms to which the balls are attached are $100\ mm$ long. The mass of each ball is $8\ kg$ and the mass on the sleeve is $60\ kg$. At the minimum radius of rotation of $200\ mm$, the extensions are parallel to the governor axis. Determine the equilibrium speed of the governor for the given configuration. What will be the equilibrium speed for the maximum radius of $250\ mm$?
   How to answer

   • Write Proell governor equilibrium equation with arm geometry
   • Calculate equilibrium speed at minimum radius 200 mm with extensions parallel
   • Repeat calculation for maximum radius 250 mm
   • State effect of sleeve mass and ball mass
   • Give speeds in r.p.m.
6. 14 Marks
   A three cylinder radial engine driven by a common crank has the cylinders spaced at $120^{\circ}$. The stroke is $125\ mm$, length of the connecting rod $225\ mm$ and the mass of the reciprocating parts per cylinder $2\ kg$. Calculate the primary and secondary forces at crank shaft speed of $1200\ r.p.m.$
   How to answer

   • Calculate crank radius from stroke and angular speed from r.p.m.
   • Write primary force expression for 3-cylinder radial at 120 degrees
   • Show vector sum for primary balance
   • Write secondary force expression with connecting rod ratio
   • Compute numerical values of primary and secondary forces
7. 14 Marks
   A shaft has a number of a collars integral with it. The external diameter of the collars is $400\ mm$ and the shaft diameter is $250\ mm$. If the intensity of pressure is $0.35\ N/mm^{2}$ (uniform) and the coefficient of friction is $0.05$, estimate:

   1. i)Power absorbed when the shaft runs at 105 r.p.m. carrying a load of 150 kN
   2. ii)Number of collars required.
   How to answer

   • Calculate mean radius for uniform pressure theory
   • Compute frictional torque and power $P = T\omega$
   • Relate axial load to pressure and collar area
   • Determine number of collars for 150 kN load
   • State assumptions of uniform pressure
8. 14 Marks
   A single dry plate clutch transmits $7.5\ kW$ at $900\ r.p.m.$ The axial pressure is limited to $0.07\ N/mm^{2}$. If the coefficient of friction is $0.25$, find

   1. i)Mean radius and face width of the friction lining assuming the ratio of the mean radius to the face width as 4.
   2. ii)Outer and inner radii of the clutch plate.
   How to answer

   • Find torque from power $T = 60P/2\pi N$
   • Use uniform pressure theory for single plate with $R_m/w = 4$
   • Solve for mean radius and face width
   • Calculate outer and inner radii from $R_m$ and width
   • Check axial pressure limit
9. 14 Marks
   A simple band brake is operated by a lever of length $500\ mm$. The brake drum has a diameter of $500\ mm$ 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 $100\ mm$ from the fulcrum. If the effort applied to the end of the lever is $2\ kN$ and the coefficient of friction is $0.25$, find the maximum braking torque on the drum.
   How to answer

   • Draw band brake with lever, drum and angle of contact $5/8$ circumference
   • Apply belt friction formula $T1/T2 = e^{\mu\theta}$
   • Take moments about fulcrum for tight and loose sides
   • Determine braking torque for both directions and select maximum
   • Compute numerical value of maximum braking torque
10. 7 Marks
    aDefine and explain the following terms relating to governors:

    1. i)Stability
    2. ii)Sensitiveness
    3. iii)Isochronism
    4. iv)Hunting
    How to answer

    • Define stability, sensitiveness, isochronism and hunting
    • Explain each term with speed-load behaviour
    • Give formulas for sensitiveness where applicable
    • Distinguish between stable and hunting operation
11. 7 Marks
    bWhat is the stability of a governor? Sketch the controlling force versus radius diagrams for a stable, unstable, and isochronous governor. Derive the conditions for stability.
    How to answer

    • Define stability of governor in terms of equilibrium speed vs radius
    • Sketch controlling force vs radius for stable, unstable and isochronous cases
    • Derive condition $dF/dr > m\omega^2$ for stability
    • Explain isochronous as limiting case
    • State practical implications