MN and PQ are two uniform rods having a mass of 3 kg per metre length and suspen

Question

MN and PQ are two uniform rods having a mass of 3 kg per metre length and suspended at their upper ends and connected by a spring at their lower ends as shown in fig. Q2. When hanging freely, they are vertical and there is no force in the spring. The spring has a stiffness of 2.8 kN/m.
The spring is now compressed slightly and released.
Find:
Find:
a) the equation of motion of MN and PQ,.
[ Answer: 0.614 d2?/dt2 = -2.8 x103 x (0.85? + f) x 0.85,
1.d2?/dt2 = -2.8 x103 x (0.85? + f) x 1]
b) The frequency of vibration

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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