JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS Previous Question Paper (September - 2021 - R16)
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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.
Answer
Given: Mass of the gun barrel m = 600 kg, stiffness k = 294000 N/m, and recoil distance x = 1.3 m. The motion of the barrel during recoil is governed by the conservation of energy, where the initial kinetic energy is converted into potential energy in the spring.
Substituting the values: 600 * v2 = 294000 * (1.3)2. Solving for initial velocity v: v = sqrt((294000 * 1.69) / 600) = 28.77 m/s.
The critical damping coefficient cc is defined as:
Substituting the values: cc = 2 * sqrt(294000 * 600) = 2 * sqrt(176400000) = 2 * 13281.57 = 26563.14 Ns/m.
For the return motion, the system is critically damped. The displacement x(t) for a critically damped system starting from rest at the maximum displacement x0 is given by:
Where ωn = sqrt(k/m) = sqrt(294000/600) = 22.136 rad/s. We need to find t when x(t) = 0.05 m, with x0 = 1.3 m:
0.03846 = (1 + 22.136t)e-22.136t. Solving this transcendental equation numerically, let f(t) = (1 + 22.136t)e-22.136t - 0.03846. By iteration, t ≈ 0.235 seconds.
A circular cylinder of mass 4 kg and radius 150 mm is connected by a spring of stiffness 4000 N/m as shown in Figure 1. It is free to roll on horizontal rough surface without slipping, determine the natural frequency.
Answer
To determine the natural frequency of the rolling cylinder, we apply the energy method or Newton's second law for a system undergoing pure rolling without slipping. For a cylinder of mass m and radius r rolling without slipping, the kinetic energy T is the sum of translational and rotational kinetic energy:
Given the cylinder rolls without slipping, the relationship between translational velocity v and angular velocity ω is v = rω. The mass moment of inertia I for a solid cylinder about its central axis is (1/2)mr2. Substituting these into the kinetic energy equation:
The potential energy V stored in the spring is:
Using the conservation of energy, the total energy E = T + V is constant. Differentiating with respect to time:
Dividing by (dx/dt) gives the equation of motion:
The equivalent mass meq is (3/2)m. The natural frequency ωn is given by:
Substituting the given values: m = 4 kg, k = 4000 N/m:
The natural frequency in Hertz is:
Investigate the terms involved in the equations of motion of one degree of freedom system as given by:
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What is the Duhamel integral? What is its use? Explain.
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Find the response of an underdamped single-degree-of-freedom system to a unit impulse.
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Solve the problem shown in Figure 2, m1 = 10 kg, m2 = 15 kg and k = 320 N/m.
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Find the flexibility and stiffness influence coefficients of the system shown in figure 3. Also, derive the equations of motion of the system.
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The vibration of a cantilever are given. Calculate the frequency with following data for the cantilever using Rayleigh’s method.
E = 2.1 × 1011 N/m2, second moment of area about bending axis = 0.02 m4, mass = 6 × 104 kg and length = 30 m.
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Derive an equation for the principal modes of longitudinal vibration of a uniform bar having both ends free.
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What are the boundary conditions for the free vibrations of a string fixed at x = 0 and attached to a spring of stiffness k at x = L?
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