A steel disc 300 mm diameter of mass 29kg, is suspended from the end of a wire 2
ID: 1820051 • Letter: A
Question
A steel disc 300 mm diameter of mass 29kg, is suspended from the end of a wire 2.5 mm diameter and 1.5 m long which is clamped into a central hole in the disc. The upper end of the wire being rigidly supported. When the disc is set in torsional vibration, it is found to make 10 complete oscillations in 78.2 seconds.a) Find the modulus of rigidity of the wire, [Answer: 82.4 GN/m2]
b) Calculate the amplitude of the oscillation which may be allowed if the maximum permissible
intensity of shearing stress in the wire is 140 MN/m2. [Answer: 117o]
Explanation / Answer
Free vibration with damping Mass Spring Damper Model We now add a "viscous" damper to the model that outputs a force that is proportional to the velocity of the mass. The damping is called viscous because it models the effects of an object within a fluid. The proportionality constant c is called the damping coefficient and has units of Force over velocity (lbf s/ in or N s/m). By summing the forces on the mass we get the following ordinary differential equation: The solution to this equation depends on the amount of damping. If the damping is small enough the system will still vibrate, but eventually, over time, will stop vibrating. This case is called underdamping – this case is of most interest in vibration analysis. If we increase the damping just to the point where the system no longer oscillates we reach the point of critical damping (if the damping is increased past critical damping the system is called overdamped). The value that the damping coefficient needs to reach for critical damping in the mass spring damper model is: To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. The formula for the damping ratio (?) of the mass spring damper model is: For example, metal structures (e.g. airplane fuselage, engine crankshaft) will have damping factors less than 0.05 while automotive suspensions in the range of 0.2–0.3. The solution to the underdamped system for the mass spring damper model is the following:
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