JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV Year VIBRATION AND AERO-ELASTICITY Previous Question Paper (February/March - 2022 - R18)
Answer any Five Questions. All Questions Carry Equal Marks.
Main
Two helical springs, one made of steel and the other made of aluminum, have identical values of wire diameter d and mean coil diameter D.
If the number of turns in the steel spring is 10, determine the number of turns required in the aluminum spring whose weight will be same as that of the steel spring.
Answer
The weight of a helical spring is given by the product of the volume of the wire and the density of the material, W = ρ × V. The volume of the wire is V = (π × d2 / 4) × (π × D × n), where d is the wire diameter, D is the mean coil diameter, and n is the number of turns. Since the wire diameter d and mean coil diameter D are identical for both springs, the volume is directly proportional to the number of turns n. Given the weights are equal, ρs × ns = ρa × na, where the subscripts s and a denote steel and aluminum respectively. Taking standard densities of ρs ≈ 7850 kg/m3 and ρa ≈ 2700 kg/m3, and ns = 10, the number of turns for the aluminum spring is na = (ρs / ρa) × ns = (7850 / 2700) × 10 ≈ 29.07 turns.
Find the spring constants of the two springs.
Answer
The spring constant k of a helical spring is defined by the formula:
where G is the shear modulus of the material. For steel, Gs ≈ 80 GPa, and for aluminum, Ga ≈ 26 GPa. The ratio of the spring constants is:
Substituting the values Gs = 80, Ga = 26, ns = 10, and na = 29.07:
Thus, the steel spring is approximately 8.945 times stiffer than the aluminum spring for the given dimensions.
An electronic instrument, weighing 200 N, is supported on a rubber mounting whose force deflection relationship is given by
where the force (F) and the deflection (x) are in N and mm, respectively. Determine the following:
Equivalent linear spring constant of the mounting at its static equilibrium position.
Answer
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Deflection of the mounting corresponding to the equivalent linear spring constant.
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A spring-mass system has a natural frequency of 10 Hz. When the spring constant is reduced by 800 N/m, the frequency is altered by 45 percent. Find the mass and spring constant of the original system.
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A simple pendulum is set into oscillation from its rest position by giving it an angular velocity of 1 rad/s. It is found to oscillate with an amplitude of 0.5 rad. Find the natural frequency and length of the pendulum.
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A simple pendulum is found to vibrate at a frequency of 0.5 Hz in a vacuum and 0.45 Hz in a viscous fluid medium. Find the damping constant, assuming the mass of the bob of the pendulum is 1 kg.
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Derive an expression for the time at which the response of a critically damped system will attain its maximum value. Also find the expression for the maximum response.
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Solve the problem for a multi-degree of freedom system with m1 = 10 kg, m2 = 15 kg and k = 320 N/m.
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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
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Using Rayleigh's method, determine the first natural frequency of vibration of a system with springs k1 = k, k2 = 2k, k3 = 3k, k4 = k and masses m1 = m, m2 = 2m, m3 = 3m.
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Obtain the expression for the divergence speed of a two-dimensional wing.
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