JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS Previous Question Paper (February/March - 2022 - R16)
Answer any five questions. All questions carry equal marks.
Main
A mass m is attached to a cord that is under a tension T, as shown in figure 1. Assuming that T remains unchanged when the mass is displaced normal to the cord.
Write the differential equation of motion for small transverse vibrations.
Answer
Consider a mass m located at the midpoint of a cord of length L under constant tension T. When the mass is displaced by a small distance x in the transverse direction, the length of each segment becomes approximately L/2. The restoring force acting on the mass is the sum of the transverse components of the tension from both sides of the cord.
For a small displacement x, the angle θ made by the cord with the horizontal is tan(θ) ≈ sin(θ) = x / (L/2) = 2x/L. The total restoring force F is:
Applying Newton's Second Law, m(d2x/dt2) = F, the differential equation of motion is:
Find the natural frequency of vibration.
Answer
The differential equation derived is in the standard form for a simple harmonic oscillator, m(d2x/dt2) + keqx = 0, where the equivalent stiffness keq is 4T/L.
The natural frequency in radians per second ωn is given by:
The natural frequency in Hertz fn is:
A gun barrel of mass 600 kg has a recoil spring of stiffness 294000 N/m. If the barrel recoils 1.3 m on firing, determine the initial velocity of the barrel, the critical damping coefficient of the dashpot which is engaged at the end of the recoil stroke and the time required for the barrel to return to a position 5 cm from the initial position.
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A machine operates at 100 rad/s and has a rotating component of mass 5 kg whose center of mass is 3 cm from the axis of rotation. What is the amplitude of the harmonic excitation experienced by the machine?
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A 40 kg pump is to be placed at the mid-span of a 2.5-m long steel (E = 200 GPa) beam. The pump is to operate at 3000 rpm. For what values of the cross-sectional moment of inertia will the oscillations of the pump be within 3 Hz of resonance?
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The peak amplitude of a single-degree-of-freedom system, under a harmonic excitation, is observed to be 5 mm. If the undamped natural frequency of the system is 5 Hz, and the static deflection of the mass under the maximum force is 2.5 mm. Estimate the damping ratio of the system, and find the frequencies corresponding to the amplitudes at half power.
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Use the Duhamel’s integral method to derive expressions for the response of an undamped system subjected to the forcing function shown in figure 2.
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During operation, a 100 kg reciprocating machine is subject to a force F(t) = 200 sin 60t N. The machine is mounted on springs of an equivalent stiffness of 4.3 × 106 N/m. What is the machine’s steady-state amplitude?
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Derive the equation of motion of a system subjected to base excitation.
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Find the steady-state response of a viscously damped system to the forcing function obtained by F(t) and F0, as shown in figure 3.
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The mass and stiffness matrices and the mode shapes of a two-degree-of-freedom system are given. If the first natural frequency is given by ω1 = 1.7, determine the stiffness coefficients k12 and k22 and the second natural frequency of the system.
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A cable of length l and mass ρ per unit length is stretched under a tension P. One end of the cable is fixed and the other end is connected to a pin, which can move in a frictionless slot. Find the natural frequencies of vibration of the cable.
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