Vibration of Beam with Added Mass

In summary: The added mass of the flywheel can be treated as a concentrated mass and added to the moment of inertia of the beam. In summary, the natural frequency of the system with a rigid uniform slender beam, hinged at A and carrying a flywheel of mass 28kg, connected to a spring and damper, is 15.63 rad/s. The added mass of the flywheel is accounted for in the moment balance calculation.
  • #1
Marcheline
5
0
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. [tex]\sum[/tex]M=J[tex]\ddot{}\theta[/tex]


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.
 
Physics news on Phys.org
  • #2
Yes, your calculation is correct. You have correctly taken into account the mass of the flywheel and calculated the natural frequency of the system.
 
  • #3


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.
 

FAQ: Vibration of Beam with Added Mass

1. What is added mass in the context of beam vibration?

Added mass is the increase in effective mass of a vibrating beam due to the presence of a fluid or other external object attached to the beam. This added mass can significantly affect the natural frequency and mode shapes of the beam's vibration.

2. How does added mass impact the vibration of a beam?

The added mass increases the overall mass of the beam, leading to a decrease in the natural frequency of vibration. It also changes the distribution of mass along the beam, causing a shift in the mode shapes of vibration.

3. What factors affect the added mass of a beam?

The added mass of a beam is influenced by several factors, including the shape and size of the external object attached to the beam, the material properties of the beam and the external object, and the density and viscosity of the fluid surrounding the beam.

4. How can added mass be accounted for in beam vibration analysis?

Added mass can be accounted for by incorporating it into the equation of motion for the vibrating beam, using a modified mass matrix. Alternatively, the added mass can be included in the boundary conditions of the beam's vibration problem.

5. Can added mass have a beneficial effect on beam vibration?

In certain cases, added mass can have a beneficial effect on beam vibration. For example, it can be used to increase the damping of the beam, leading to reduced vibration amplitudes. It can also be utilized to shift the natural frequency of the beam to a desired value.

Similar threads

Replies
4
Views
2K
Replies
1
Views
1K
Replies
9
Views
2K
Replies
1
Views
1K
Replies
1
Views
3K
Replies
1
Views
2K
Back
Top