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2003 Course Dynamics of Machinery

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Total No. of Questions : 12] P1310 [Total No. of Pages : 7 [3764]-132 B.E. (Mech. and Mech./SW) DYNAMICS OF MACHINERY (402042) (2003 Course) Time : 3 Hours] [Max. Marks :100 Instructions to candidates : 1) Answer 3 questions from Section I and 3 questions from Section II. 2) Answers to the two sections should be written in separate books. 3) Neat diagrams must be drawn wherever necessary. 4) Figures to the right indicate full marks. 5) Use of logarithmic tables, slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed. 6) Assume suitable data, if necessary. SECTION - I Unit - 1 Q1) a) b) A four wheeled trolley car has a total mass of 3000 kg. Each axle with its two wheels and gears has a total moment of inertia of 32 kg.m2. Each wheel is of 450 mm radius. The center distance between two wheels on a axle is 1.4 m. Each axle is driven by a motor with a speed ratio of 1:3. Each motor along with its gear has a moment of inertia of 16 kg m2 and rotating in the opposite direction to that of the axle. The centre of mass of the car is 1 m above the rails. Calculate the limiting speed of the car when it has to travel around a curve of 240 m radius without the wheels leaving the rails. [12] Explain the effect of gyroscopic couple on swinging table fan. [4] OR Q2) a) A uniform disc of diameter 125 mm and mass of 12 kg is mounted on one end on a frame such that the disc rotates at uniform speed of 15000 rpm with its axis horizontal. The other end of horizontal frame is pivoted to a vertical spindle which is free to turn in bearings. The frame is free to swing in vertical plane. The C.G. of the disc is at 450 mm from the pivot and C.G. of the frame of mass 5 kg is at a distance of 375 mm from the pivot. Find the velocity with which the system will precess about the spindle. Sketch the arrangement. [10] b) What do you understand by gyroscopic couple? Derive a relation for its magnitude. [6] P.T.O. Unit - 2 Q3) a) A shaft has three eccentrics each 7.5 cm diameter and 2.5 cm thick machined in one piece with the shaft. The central planes of the eccentrics are 6 cm apart. The distance of the centres from the axis of rotation are 1.2 cm, 1.8 cm and 1.2 cm and their angular positions are 120 apart. The metal weighs 0.007 kg/cm 3. Find the amount of the out of balance force and couple at 600 rpm. If the shaft is balanced by adding two weights of radius 7.5 cm and at a distance of 10 cm from the central plane of the middle eccentric, find the amounts of the weights and their angular positions. [12] b) What do you mean by balancing? Why it is necessary for high speed engines? [4] OR Q4) a) What are V-engines? How do they differ from the rest of the reciprocating engines. [4] b) Fig.1 shows the arrangement of the cranks in a 4 crank symmetrical engine, in which the masses of the reciprocating parts at cranks 1 and 4 are each equal to m1 and at cranks 2 and 3 are each equal to m2. Show that, the arrangement is balanced for primary forces and couples and for secondary forces provided that, m1 cos , x1 tan = = m2 cos x2 tan and cos cos = [3764]-132 1 . 2 [12] 2 Unit - 3 Q5) a) A combination of seven number of identical springs, each having a stiffness of K , supports a mass m as shown in Fig. 2. If the mass is slightly displaced from the mean position and released, find the natural frequency of the oscillations. [8] b) Discuss about Coulomb Damping. [6] c) A vibrating system is defined by the following parameters: [4] m = 3kg, k = 100 N/m, c = 3 N s/m. Determine: i) the damping factor. ii) the natural frequency of damped vibrations. OR Q6) a) b) [3764]-132 What is logarithmic decrement? Derive the relation for the same. [6] Show that for finding natural frequency of a spring mass system, the mass of the spring is taken into account by adding one-third (1/3) its mass to the main system. [6] 3 c) A rod of circular cross section and of uniform diameter 100 mm is fixed at one end. A disc of mass 500 kg is attached at the other end. Radius of gyration of the disc is 250 mm. Length of rod is 100 cm. E = 3 x 10 5 MN/m2 a nd G = 8 x 10 4 M N/m 2. Find the frequencies of natural longitudinal and torsional vibrations in Hz. [6] SECTION - II Unit - 4 Q7) a) b) A single cylinder engine of total mass 20 kg is to be mounted on an elastic support which permits vibratory movement in vertical direction only. Elastic support consist of 3 springs of stiffness K N/m each. If unit operates at 580 rpm. What should be the value of spring constant K , if only 10% of the shaking force of unit is to be transmitted to the support structure? [8] Derive graphically the expressions for i) amplitude of steady state vibrations and ii) phase angle for a spring mass - damper system subjected to an external periodic force f0 sin t . [8] OR Q8) a) A vehicle has a mass of 1000 kg. The suspension system has a spring constant 500 kN/m and damping ratio of 0.5. If the vehicle has a speed of 80 km/hr. Determine the displacement amplitude of the vehicle. The road surface varies sinusoidally with an amplitude of 10 cm and a wavelength of 5m. [6] b) Investigate the terms involved in the equation of motion of one degree of freedom system as given by 5 + 3 x + 12 x = 10 sin t . [10] x [3764]-132 4 Unit - 5 Q9) a) What do you understand by a semi-definite or degenerate system? Give two examples of systems that are degenerate? [8] b) A system is shown in Fig. 3. Find the equation of motions of masses for the condition if m = m1 = m2, if both the masses are displaced in the downward direction and released k1 = k3 = k. [8] OR [3764]-132 5 Q10)a) Determine the frequency of torsional vibrations of the disc shown in Fig. 4 if both the ends of the shaft are fixed and diameter of the shaft is 40 mm. The disc has a mass of 96 kg and a radius of gyration of 0.4 m. Take modulus of rigidity for the shaft material as 85 GN/m2 l1 = 1m and l2 = 0.8 m. [6] b) A reciprocating IC engine is coupled to a centrifugal pump through a pair of gears. The shaft from the flywheel of the engine to the gear wheel has a 48 mm diameter and is 800 mm long. The shaft from the pinion to the pump has 32 mm diameter and is 280 mm long. Pump speed is four times the engine speed. Moments of inertia of flywheel, gear-wheel, pinion and pump impeller are 1000 kgm 2, 14 kg.m 2, 5 kg.m 2 a nd 18 kg.m 2 r espectively. Find the natural frequency of the torsional oscillations of the system. G = 80 GN/m2. [10] Unit - 6 Q11)a) Explain with neat diagrams any two of the following: i) Single Reed frequency meter. ii) Piezoelectric Accelerometer. iii) Vibration model of seismic pickup. [3764]-132 6 [10] b) A rotor has a mass of 12 kg and is mounted midway on a 24 mm diameter horizontal shaft supported at the ends by two bearings. The bearings are 1m apart. The shaft rotates at 2400 rpm. If the center of mass of the rotor is 0.11 mm away from the geometric centre of the rotor due of a certain manufacturing defect, find the amplitude of steady state vibrations and the dynamic force transmitted to the bearing E = 200 GN/m2. [8] OR Q12)a) A seismic instrument having vibrating mass 5 kg is supported by a spring having stiffness 1000 N/m with negligible damping. A pen is attached to the mass which draws a curve on paper which is mounted on a rotating drum. The instrument is placed on a vibrating surface vibrating at frequency 5 rad/sec having amplitude of 10 mm. Find out amplitude of curve traced on paper if the vibrations are sinusoidal. b) Explain the term whirling speed of a shaft. Derive an expression for the same. [10] Y [3764]-132 [8] 7

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