JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV Year MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS Previous Question Paper (July/August - 2022 - R16)
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How can you find the natural frequency of a system by measuring its static deflection?
Answer
For a simple spring-mass system, the natural frequency can be determined by measuring the static deflection δst caused by the weight of the mass. The equilibrium condition is given by mg = kδst, where m is the mass, g is the acceleration due to gravity, and k is the spring stiffness. The natural frequency ωn in rad/s is defined as the square root of k/m. Substituting k/m = g/δst, the natural frequency is:
In terms of frequency in Hertz (fn), this becomes:
This method is highly effective for systems where the mass is supported by a spring, as it eliminates the need to know the mass or stiffness values individually, provided the static deflection can be accurately measured.
The maximum permissible recoil distance of a gun is specified as 0.5 m. If the initial recoil velocity is to be between 8 m/s and 10 m/s, find the mass of the gun and the spring stiffness of the recoil mechanism. Assume that a critically damped dashpot is used in the recoil mechanism and the mass of the gun has to be at least 500 kg.
Answer
For a critically damped system, the displacement x(t) is given by x(t) = (x0 + (v0 + ωnx0)t)e-ωnt. To find the maximum recoil distance xmax, we set the velocity to zero: dx/dt = 0, which occurs at t = 1/ωn. Substituting this into the displacement equation yields xmax = (v0 / ωn)e-1.
Given xmax = 0.5 m and v0 ranges from 8 m/s to 10 m/s, we solve for ωn = v0 / (xmax * e). For v0 = 8 m/s, ωn = 8 / (0.5 * 2.718) ≈ 5.88 rad/s. For v0 = 10 m/s, ωn = 10 / (0.5 * 2.718) ≈ 7.36 rad/s.
Using the minimum mass m = 500 kg, we calculate the required stiffness k using ωn = √(k/m), so k = m * ωn2. For ωn = 5.88 rad/s, k = 500 * (5.88)2 ≈ 17,285 N/m. For ωn = 7.36 rad/s, k = 500 * (7.36)2 ≈ 27,085 N/m. Thus, the gun mass is 500 kg and the spring stiffness must be adjustable or selected within the range of approximately 17.3 kN/m to 27.1 kN/m to satisfy the recoil constraints.
A cylinder of mass M radius ‘r’ rolls without slipping on a cylindrical surface of radius ‘R’. Find the natural frequency for small oscillation about the lowest point.
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A torsional pendulum has a natural frequency of 200 cycles/min when vibrating in a vacuum. The mass moment of inertia of the disc is it is then immersed in oil and its natural frequency is found to be 180 cycles/min. Determine the damping constant. If the disc, when placed in oil, is given an initial displacement of 2, find its displacement at the end of the first cycle.
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What is the frequency of the response of a viscously damped system when the external force is F0 sin ωt? Is this response harmonic?
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A spring-mass system with m = 10 kg and k = 5000 N/m is subjected to a harmonic force of amplitude 250 N and frequency ω. If the maximum amplitude of the mass is observed to be 100 mm, find the value of ω.
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How can we generate the frequency transfer function from the general transfer function?
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A spring-mass system is set to vibrate from zero initial conditions under a harmonic force. The response is found to exhibit the phenomenon of beats with the period of beating equal to 0.5 s and the period of oscillation equal to 0.05 s. Find the natural frequency of the system and the frequency of the harmonic force.
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How do you compute the frequency of the first harmonic of a periodic force?
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Find the displacement of a damped single-degree-of-freedom system under the forcing function F(t) = F0 e-αt where α is a constant.
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State the interpolation models that can be used for approximating an arbitrary forcing function.
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An automobile, having a mass of 1000 kg, runs over a road bump of the shape shown in Figure 1. The speed of the automobile is 50 km/hr. If the undamped natural period of vibration in the vertical direction is 1.0 sec, find the response of the car by assuming it as a single degree-of-freedom undamped system vibrating in the vertical direction.
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Set up the differential equations of motion for the double pendulum shown in Figure 2, using the coordinates x1 and x2 and assuming small amplitudes. Find the natural frequencies, the ratios of amplitudes, and the locations of nodes for the two modes of vibration when m1 = m2 = m, l1 = l2 = l.
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A stretched cable of length 2 m has a fundamental frequency of 3000 Hz. Find the frequency of the third mode. How are the fundamental and third mode frequencies changed if the tension is increased by 20 percent?
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