Vibration of Beam with Added Mass

[Marcheline]

1. A rigid uniform slender beam weighing 8kg is hinged at A and is in dynamic motion. A spring of stiffness 4.5kN/m and a damper giving a viscous resistance of 35Ns/m is connected at B. The beam carries a flywheel of mass 28kg at a distance of 0.9m from A. The beam, AB is 1.2m long, Find the natural frequency of the system



2. $$\sum$$M=J$$\ddot{}\theta$$


3. I have no problems summing up all the moments. What I am unsure of is the moment of inertia, J in this case. For a beam pivoted at one end, J=(1/3)mL^2. How to I account for the added mass of the flywheel? Do i treat it as a concentrated mass and just add the J of the flywheen to the beam's moment of inertia?

-kL^2 (Theta) - cL^2 (Theta/dot) = {(1/3)ML^2 + m[(3/4)L]^2} (Theta/dotdot)

When I do this and solve the equation harmonically, I get the natural frequency as 15.63 rad/s. Is this right? Or has the mass of the flywheel already been taken into account in the moment balance?

I tried to use Latex but failed miserably, as can be seen from the top.

Thank you.

 

[Valkarie]

Yes, your calculation is correct. You have correctly taken into account the mass of the flywheel and calculated the natural frequency of the system.

 

[Mene]

I would approach this problem by first understanding the physical properties and equations involved. The equation given in point 2 is the moment balance equation, where J represents the moment of inertia, \ddot{\theta} represents the angular acceleration, and \sumM represents the sum of all moments acting on the system. In this case, the sum of moments includes the spring force, the damping force, and the moment of the flywheel.

To answer the question about the moment of inertia, yes, you would need to account for the added mass of the flywheel. The moment of inertia for a concentrated mass is simply the mass multiplied by the square of its distance from the pivot point. Therefore, you would need to add the moment of inertia of the flywheel (m[(3/4)L]^2) to the moment of inertia of the beam (1/3)ML^2 in the moment balance equation.

Solving the equation harmonically will give you the natural frequency of the system, which in this case is 15.63 rad/s. This frequency represents the rate at which the system will naturally vibrate when disturbed from its equilibrium position. Therefore, it is important to take into account the added mass of the flywheel in order to accurately calculate this natural frequency.

Overall, it is important to carefully consider all physical properties and equations involved in a problem in order to accurately solve it and understand the behavior of the system.