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Questions:

Title: Revision on the single degree of freedom vibration

A steel ball of mass M is held at a distance H above a massless rigid tabletop supported by undamped springs whose equivalent stiffness is K. If the ball is left to fall from rest on the tabletop. Deduce an expression for the velocity at which the ball will bounce back off the tabletop. On its way back, will the mass bounce back to its initial height? If it will not, where will the difference in energy go? Discuss mathematically!

A model was designed to simulate the response of a human head to a frontal blow. The model has a mass of 4.7 kg horizontally-supported by a spring of a stiffness 2000 N/m. If a blow of 400 N was applied to the mass over a period of 0.003 seconds. Find maximum acceleration which the mass is subject to as a result of this blow. What's the maximum deflection in the spring?

A 700 kg (total mass) elevator cage travels at a speed of 1 m/s and encounters a resistance of 30 kN s/m in its vertical guides. The cage is suspended by a 10 mm steel cable wrapped around a driving drum. As the cage was travelling downwards, and when its point of suspension was 15 m below the drum centre line, the brake was applied suddenly to the drum. Find the frequency to which the passengers of the elevator would be subject. Calculate the maximum tensile stress developed in the cable upon braking due to both static and dynamic loadings.

A hoisting system consists of a light 2 m bar hanging vertically of a small servo-driven trolley. The bar is allowed to swing on its hinge which is controlled by a spring of stiffness 10 kNm/rad and a damper whose damping coefficient is 800 Nms/rad. The trolley travels on a track built 3 m about the floor level at the speed profile shown in the figure. The total mass of the basket and the slewing mechanism (which maintains its horizontal position) is equivalent to 70 kg acting at the lower end of the bar. The system is designed to handle a payload of 800 kg.

 Velocity (m/s) 1.4 m/s

A viscous damper consists of a piston in a cylinder containing oil of viscosity 0.38 NS/m as a damping fluid. The length of the piston is 100 mm and it has four 6 mm holes between its faces. If the damping constant is required to be 26 kN.s/m, find the piston diameter.

Hint: velocity of laminar flow in a duct is given by

Determine the moment of inertia of a two-blade aircraft propeller about its rotational axis from the observation of natural frequency of the free oscillation of the propeller when suspended from two light wires attached to the tips of the blades. The length of each wire is h and the diameter of the propeller is D and its weight is W. Repeat the problem for the case of three-blade propellers suspended by three wires. Does the result depend on the number of wires?

A vibration instrument made up of a light steel cantilever carrying a weight of 25 N located 430 mm from the fixed end. The cross section of the cantilever is 25 mm wide and 6 mm thick. The instrument also incorporates a light recording gear that gives a magnification of 10 times the movements of the weight relative to the frame. When the cantilever is caused to perform free vibration, the ratio of two successive amplitudes is found to be 1:0.8. The instrument was placed on a machine whose vibration, x(t), is given as follows: x(t) = 1.2 cos(100t)

Find the damping factor of the instrument and answer the following questions:

1. What was the amplitude of the recorded vibration?
2. If the information regarding the vibration of the machine (i.e. x(t)) is not known to the operators, what would they take the amplitude of the machine vibration to be?
3. What is the percentage error associated with their approximation? Comment!.

Take the mass effects of the gear and cantilever as equivalent to 3 N acting at the weight position and Young’s Modulus for steel to be 205 GPa.

The floor of a laboratory was found to vibrate in a simple harmonic pattern with a frequency that ranges from 15 to 60 Hz depending on the speed of some nearby machinery. It is required to install a sensitive instrument of mass 40 kg in the laboratory and isolate it in such a fashion that the vibration transmitted to the instrument does not increase beyond 10% of the floor vibration. The suspension system of the instrument has 3 similar springs arranged to carry equal loads, and its equivalent damping factor is 0.2. Find the maximum allowable value of the spring stiffness to be used in the system.

ENGIN2020 Vibration and Machine Dynamics

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