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JNTU B.Tech II DYNAMICS OF MACHINERY Previous Question Paper (August/ September - 2022 - R16)
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In a Four bar mechanism shown in figure, the torque T3 and T4 have a magnitude of 3000 Nm and 2000 Nm respectively. The link lengths are AD = 800 mm, AB = 300 mm, BC = 700 mm, CD = 400 mm. For the static equilibrium of the mechanism determine the required torque T2 on link AB.
Answer
For the static equilibrium of a four-bar mechanism, the principle of virtual work states that the sum of the virtual work done by all external torques and forces must be zero. Given the torques T2, T3, and T4 acting on links AB, BC, and CD respectively, the equilibrium condition is expressed as:
Where ω2, ω3, and ω4 are the angular velocities of links AB, BC, and CD. The relationship between the angular velocities is determined by the velocity ratios (ω3/ω2) and (ω4/ω2), which depend on the instantaneous configuration of the mechanism. Using the method of instantaneous centers of rotation:
- Identify the instantaneous center of rotation I13 for the coupler link BC relative to the fixed link AD.
- The velocity ratio ω3/ω2 is given by the ratio of the distances of the instantaneous centers from the pivot points: (ω3/ω2) = (I12I13 / I14I13).
- Similarly, determine the ratio ω4/ω2 using the instantaneous center I14 and the geometry of the four-bar linkage.
- Substitute the given torque magnitudes T3 = 3000 Nm and T4 = 2000 Nm into the equilibrium equation. Note that torques acting against the direction of rotation are taken as negative.
- Solve for T2: T2 = -(T3 * (ω3/ω2) + T4 * (ω4/ω2)).
By calculating the specific velocity ratios based on the provided link lengths (AD=800mm, AB=300mm, BC=700mm, CD=400mm) for the given mechanism position, the input torque T2 required to maintain static equilibrium is determined. Assuming the torques are applied to oppose the motion, T2 must balance the combined effect of the external torques T3 and T4 acting through their respective mechanical advantages.
A single cylinder, single acting, four stroke gas engine develops 20 kW at 300 r.p.m. The work done by the gases during the expansion stroke is three times the work done on the gases during the compression stroke, the work done during the suction and exhaust strokes being negligible. If the total fluctuation of speed is not to exceed ± 2 percent of the mean speed and the turning moment diagram during compression and expansion is assumed to be triangular in shape, find the moment of inertia of the flywheel.
Answer
Given: Power (P) = 20 kW = 20,000 W, Speed (N) = 300 r.p.m., Mean speed (ω) = 2πN/60 = 31.416 rad/s. Total fluctuation of speed is ±2%, so coefficient of fluctuation of speed (Cs) = (0.02 - (-0.02)) = 0.04.
Work done per cycle (W) = P / (N/2) = 20,000 / 150 = 133.33 Nm. Let We be work during expansion and Wc be work during compression. Given We = 3Wc. Since net work W = We - Wc, we have 2Wc = 133.33, so Wc = 66.67 Nm and We = 200 Nm.
For a triangular turning moment diagram, the base of the expansion stroke is π and the base of the compression stroke is π. The peak torque Tmax = 2We/π = 400/π = 127.32 Nm. The peak compression torque Tc = 2Wc/π = 133.33/π = 42.44 Nm.
The maximum fluctuation of energy (ΔE) occurs during the expansion stroke. The area of the turning moment diagram above the mean torque line (Tmean = W / 4π = 133.33 / 12.566 = 10.61 Nm) represents the energy fluctuation. The area of the triangle above the mean line is ΔE = 0.5 * (base) * (height). The base of the triangle above the mean line is π * (Tmax - Tmean) / Tmax = π * (127.32 - 10.61) / 127.32 = 2.906 radians.
ΔE = 0.5 * 2.906 * (127.32 - 10.61) = 169.5 Nm.
Using the relation ΔE = I * ω2 * Cs:
Solving for I:
A shaft carries four masses in parallel planes A, B, C and D in this order along its length. The masses at B and C are 18 kg and 12.5 kg respectively, and each has an eccentricity of 60 mm. The masses at A and D have an eccentricity of 80 mm. The angle between the masses at B and C is 100° and that between the masses at B and A is 190°, both being measured in the same direction. The axial distance between the planes A and B is 100 mm and that between B and C is 200 mm. If the shaft is in complete dynamic balance, determine:
The magnitude of the masses at A and D
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the distance between planes A and D
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the angular position of the mass at D
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A disc of mass 4 kg is mounted between bearings which may be assumed simply supports. The bearing span is 48 cm the steel shaft which is horizontal, is 9 mm in diameter. The C.G. of the disc is displaced 3 mm from the geometric centre. The damping at the centre of the disc-shaft is 49 N-sec/m. If the shaft rotates at 760 r.p.m. find the maximum dynamic force on the shaft also find the power required to drive the shaft at this speed.
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A ship propelled by a turbine rotor which has a mass of 5 tonnes and a speed of 2100 r.p.m. The rotor has a radius of gyration of 0.5 m and rotates in a clockwise direction when viewed from the stern. Find the gyroscopic effects in the following conditions:
The ship sails at a speed of 30 km/h and steers to the left in a curve having 60 m radius.
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The ship pitches 6° above and 6° below the horizontal position. The bow is descending with its maximum velocity. The motion due to pitching is simple harmonic and the periodic time is 20 seconds.
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The ship rolls and at a certain instant it has an angular velocity of 0.03 rad/s clockwise when viewed from stern.
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A car moving on a level road at a speed 50 km/h has a wheel base 2.8 metres, distance of C.G. from ground level 600 mm, and the distance of C.G. from rear wheels 1.2 metres. Find the distance travelled by the car before coming to rest when brakes are applied:
to the rear wheels
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to the front wheels
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to all the four wheels. The coefficient of friction between the tyres and the road may be taken as 0.6.
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A centrifugal clutch is to transmit 15 kW at 900 r.p.m. The shoes are four in number. The speed at which the engagement begins is 3/4th of the running speed. The inside radius of the pulley rim is 150 mm and the centre of gravity of the shoe lies at 120 mm from the centre of the spider. The shoes are lined with Ferrodo for which the coefficient of friction may be taken as 0.25. Determine:
Mass of the shoes
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Size of the shoes, if angle subtended by the shoes at the centre of the spider is 60° and the pressure exerted on the shoes is 0.1 N/mm².
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A cantilever shaft 50 mm diameter and 300 mm long has a disc of mass 100 kg at its free end. The Young's modulus for the shaft material is 200 GN/m². Determine the frequency of longitudinal and transverse vibrations of the shaft.
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